Department of Mechanical Engineering

College of Science and Engineering

Wilkes University, Wilkes Barre, PA 18766

FLUID MECHANICS

LABORATORY MANUALSupplement

Dr. M. Ghamari

ME-323

Fall 2019

THE MANUAL BELONGS TO:

THIS MANUAL IS DESIGNED AS A REFERENCE

AND GUIDE FOR STUDENTS IN THE

FLUID MECHANICS LABORATORY - ME 323.

IT INCLUDES SOME THEORY AND INFORMATION ON

EXPERIMENTS TO BE PERFORMED IN THE LABORATORY.

2

# Name of Experiment Page

16. Pressure and Vacuum Measurements Using Manometer 2

17. Force and Moment on a Vertical Submerged Plane 8

18. Force of an Impinging Jet on Flat and Hemispherical Surfaces 16

19. Examining Flow Characteristics in Orifice 24

20. Examining Trajectory of Horizontal Jet Through Nozzle 39

EXPERIMENT # 1

PRESSURE AND VACCUM MEASUREMENT USING MANOMETER

OBJECTIVE

To measure pressure and vacuum using inclined and U-tube manometers and compare with

theoretical values.

APPARATUS

Pressure Measurement Bench

Figure 1. Pressure Measurement Bench Apparatus

INTRODUCTION Manometry is a fundamental method of measuring low pressure and, together with

Bourdon type gauges, is widely used in engineering. They are intrinsic parts of more complex

measuring instruments, such as pneumatic comparators and flow indicators, and hence, is

important to fully understand their operation.

The apparatus shown in Figure 1 consists of a steel framework/bench construction with

vertical and inclined manometers, and two Bourdon type gauges. One of the gauges measures

pressure, the other measures vacuum. The gauges and manometers each have pressure sockets for

BourdonGauge(notrequiredinthisexperiment)

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direct connection. ‘Tee’ connectors and tubing are available to allow many different connections.

A syringe is supplied as a means of creating an adjustable pressure and vacuum.

THEORY

Figure 2. Layout of Manometers

The manometer readings must be ‘adjusted’ by simple math and trigonometry to give actual

readings. They record a pressure difference and so the level of one tube must be subtracted from

the other to give actual pressure. One limb of the inclined manometer is set at an angle of 54° (see

Figure 2), so this needs additional calculation to give actual readings.

Pg = Pu = Pv Pu = P1 – P2

Pv = P1 – P2 sin 54 Where:

Pg = Gauge pressure (in mmH2O);

Pu = Vertical manometer pressure (in mmH2O)

Pv = Inclined manometer pressure (in mmH2O)

INSTALLATION AND PREPARATION To fill the vertical and inclined manometers of the main unit, insert the funnel into the short

filler tubes at the open end of each manometer tube and pour in water. If necessary, use the pipe

clamps to clamp the short pipes at the back of the apparatus, to help stop water leaking out of the

pressure sockets while you fill the tubes. Fill each manometer to roughly half way. Refit the pipe

clamps to the ends of the pipes when you have finished. The pipe clamps seal the open ends of the

filler tubes during experiments.

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Note: It may be necessary to gently tap the manometer tubes while you fill them, to allow trapped

air to escape.

Figure 3. Filling tubes and installing pipe clamps instruction

EXPERIMENTAL PROCEDURE The syringe has a nozzle that fits into any of the pressure sockets (see Figure 4) or ‘Tee’

pieces.

Figure 4. Fitting syringe nozzle into pressure socket

15

Note: Never apply pressure to the Vacuum Gauge. It will be damaged.

a) Create a blank results table, similar to Table 1.

b) For pressure tests, extend the syringe fully before you connect it.

c) For vacuum tests, press the syringe piston in fully before you connect it.

d) Use the ‘Tee’ pieces and spare pipes (supplied) to connect a pressure or vacuum gauge

and a manometer at the same time.

e) See Figure 5 for typical test connections.

f) Make sure all pipe connections are good and that the pipe clamps are shut.

g) Check that there are no small trapped pockets of air in the fluid of each manometer

(gently tap each tube until the air moves upwards and out of the fluid).

h) Slowly move the syringe in steps to create pressure gauge changes of 50 mm H2O. At

each step, record the change in both the gauge reading and the levels of the two

manometer tubes at the same time. Monitor the manometer levels so that water is not spilt

Note: the pressure in the left-hand tube of each manometer is termed P1, the right-hand

tube is termed P2.

i) From your manometer results, subtract the height of one tube from the other to give the

pressure difference Δh for each manometer as described in “Theory” section, and

calculating the adjusted reading for the inclined manometer.Table 1. Blank results table for pressure and vacuum measurement

16

Figure 5. Typical Pressure (left) and Vacuum (right) test connections

RESULT AND DISCUSSION Plot a graph of the manometer readings (adjusted readings for the inclined manometer)

against the pressure gauge readings for both inclined and U-tube manometers and in both pressure

and vacuum measurements (4 plots in total). The slope of the graphs will show the error on either

the Bourdon gauge or the manometer. Since manometers are fundamental and highly accurate

measuring instruments, it is expected to obtain a slope of 1 or a very close value as shown in the

typical plot of Figure 6. Find the slope for each plot and discuss the possible errors if your

manometer and gauge readings are different.

Figure 6. Typical Calibration Graph of the Vertical Manometer Reading Against the Pressure Gauge Reading

PressureTest VacuumTest

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EXPERIMENT # 2

FORCE AND MOMENT ON A VERTICAL SUBMERGED PLANE

OBJECTIVE

To find the relationship between water height and moment of force (fluid thrust) on a

vertical plane surface.

APPARATUS

Center of Pressure Apparatus

Figure 1. Main parts of Center of Pressure apparatus

INTRODUCTION

Water at rest (or relatively slow moving water) exerts a hydrostatic force (pressure) on the

walls of dams, large ducts, canals and against canal gates (in canal locks). This hydrostatic force

is not the same as that fluid force at the bottom of the dam or canal, as it is applied at a steeper

angle (usually right angle) and changes with the height of water (submersion of the plane).

Engineers need to know how to predict this force to help understand the strength and design of

structures needed to resist it.

The Centre of Pressure Apparatus (Figure 1) allows us to study the moment due to the

hydrostatic force on a fully or partially submerged vertical plane surface and compare it with

theory. It also allows us to study forces against an angled plane surface.

SetofWeights(10geach)

WeightHangers(10geach)

QuadrantTank

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17

THEORY

Figure 2. Schematics of fully and partially submerged vertical planes

As shown in Figure 2, the plane is fully submerged when the water level completely covers

the plane. With the quadrant tank, this is when the water level (h) is less than 100 mm for a vertical

plane. Also, the plane is partially submerged when the water only covers part of the surface, or in

terms of water level, when h is more than 100 mm for a vertical plane.

Figure 3. Hydrostatic pressure at depth

To find the hydrostatic force on the vertical wall, and consequently its resulting moment

about point O, first we need to refresh our mind about the pressure of water inside the quadrant

tank. Referring to Figure 3, the hydrostatic pressure exerted by a liquid of density r or specific

weight g, at depth h below the surface, is:

! = #$ℎ = &ℎ

where g for water is 9800 N/m3. This is the gauge pressure, due solely to the liquid column of

height h. To obtain the absolute pressure pa at depth h, we must add whatever pressure ps is applied

at the liquid's surface, giving:

FullySubmergedh<R1

PartiallySubmergedh>R1

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!' = !( + ! or !' = !( + &ℎ (1)

Figure 4. Schematic of a Center of Pressure apparatus for a general case

Now, and for a general case shown in Figure 4, consider an element at start depth y, width dy.

Force on element *+ = &(- cos 1 − ℎ)45-

and the Moment of Force (M) on element about point O is: *6 = &(-7 cos 1 − ℎ)4-5-

Where B is the breadth (width) of the plane surface and equal to 75 mm. Therefore, total moment

about O is:

8×:; = 6 = &4 (-< cos 1 − ℎ-)5- (2)

where W is the weigh and R3 = 200 mm.

Fully Submerged Plane (h < R1cosq):

When the plane is fully submerged, the water covers the full area of the plane and the

moment applied should be inversely proportional to the water height (M µ 1/h). In this case, the

limits of integral (2) are R1 = 100 mm and R2 = 200 mm:

6 = &4 -< cos 1 − ℎ- 5-=>

=?

=&4 cos 1

3:<; − :A

; −&42

:<< − :A

< ℎ(3)

Using constant values for g, B, R1, R2 and for a vertical plane (q = 0):

6 = 1.71675 − 11.03625ℎ (4)

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Note that Equation (3) is a first order polynomials of the form M = c + mh, so you can

expect that a chart of moment M against height h for both vertical and angled planes will produce

linear charts with the same gradient, but a different intercept (see Figure 5).

Figure 5. Typical M against h charts for the Fully Submerged Plane

Partially Submerged Plane (h > R1cosq):

When the plane is partially submerged, the water only covers part of the area of the plane

and the moment applied is not proportional to the water height. In this case, the lower limit of

integral (2) will not be R1. Instead:

6 = &4 -< cos 1 − ℎ- 5-=>

J KLM N

and hence,

6 =&4:<

; cos 13

−&4:<

<ℎ2

+&4ℎ; sec< 1

6(5)

Using constant values for g, B, R2 and for a vertical plane (q = 0):

6 = 1.962 − 14.715ℎ + 122.625ℎ; (6)

Note that Equation 5 is a third order polynomials of the form M = c + a1h + a2h3. Therefore,

vertical and angled walls will not produce straight lines, but just similar shape curves with different

intercepts (see Figure 6).

VerticalPlane

AngledPlane

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Figure 6. Typical M against h charts for the Partially Submerged Plane

INSTALLATION AND PREPARATION

• Carefully fit the quadrant tank to the pivot of the back panel as in Figure 1.

• Use the adjustable feet and spirit level to level the apparatus.

• Before you start your experiments, mix about 2 Liters of clean, cold water with a few drops

of the colored dye.

• This equipment uses non-toxic colored dye, which can stain skin and clothing. Take care

when using.

EXPERIMENTAL PROCEDURE a) Create a Blank Results Table similar to Table 1.

b) Hook one (empty) weight hanger to the support and add water to the trim tank until the

tank is level and the submerged plane is therefore vertical. The horizontal line on the

back of the tank should line up with the 0 mm line of the back panel. The empty weight

hanger is a trim weight of 10 g. You do not need to record the trim weight or the amount

of water in the trim tank, as they are simply to balance the empty tank at 0 degrees.

c) Add the second weight hanger with a 10 g weight, giving a total weight of 20 g (the

weight hanger is 10 g).

d) Pour your colored water into the quadrant tank until it returns to 0°. Note the weight and

the height reading of the water (h) with respect to 0 mm (expect around 170 to 180 mm).

VerticalPlane

AngledPlane

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e) Continue increasing the weight in 20 g increments, while adding water until the tank

becomes level. Stop when the water level (h) reaches 0 mm or you have used all the

weights. Record the weight and level at each increment.

Note: Each weight hanger only holds about 500 g, so you will need to add some of your

weights to the trim weight hanger.

Note: As the tank fills up, note the height at which the surface becomes fully submerged.

Figure 7. Setup for 0 degree (vertical plane)

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Table 1. Blank results table

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RESULTS AND DISCUSSION

For each weight in grams, convert the value into Newtons (the first line is done for you).

For each height in mm, convert the value into meters. Multiply the value in Newtons by the

moment arm length (R3) to find the Moment M for each line of results.

Use the fully submerged part of your results (h < R1) to create a chart of Moment M

(vertical axis) against height h (similar to Figure 8). You should note that your chart gives a straight

line (linear), confirming that M and h have an inversely proportional relationship when the plane

is fully submerged. Use Equation (4) to find the theoretical values of M for the fully submerged

part of your results and add them to your chart for comparison with actual results. What are the

coefficients of first order polynomial (slope and intercept) passing through experimental data?

Compare them with the coefficients in Equation (4).

Use the partially submerged part of your results (h > R1) to create a chart of Moment M

(vertical axis) against height h (similar to Figure 8). You should note that your chart gives a curved

line (non-linear), confirming that M and h are not proportional when the plane is partially

submerged. Use Equation (6) to find the theoretical values of M for the fully submerged part of

your results and add them to your chart for comparison with actual results. What are the

coefficients of third order polynomial passing through experimental data? How do the theoretical

and actual results compare?

Figure 8. For reference only: typical results for fully submerged (left) and partially submerged (right)

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EXPERIMENT # 3

FORCE OF AN IMPINGING JET ON FLAT AND HEMISOPHERICAL SURFACES

OBJECTIVE

To demonstrate that the force on a vane is proportional to the rate of delivery of momentum,

and to show that you can predict the force on a vane from a combination of its surface shape and

the properties of the jet directed at it.

APPARATUS

Impact of a Jet apparatus, Hydraulic Bench

Figure 1. Impact of Jet apparatus connected to a digital Hydraulic Bench

INTRODUCTION One way of producing mechanical work from fluid under pressure is to use the pressure to

accelerate the fluid to a high velocity in a jet. When directed on to the vanes of a turbine wheel,

the force of the jet rotates the turbine. The force generated is due to the momentum change or

‘impulse’ that takes place as the jet strikes the vanes. Water turbines working on this impulse

principle have been constructed with outputs of the order of 100,000 kW and with efficiencies

greater than 90%.

The Impact of a Jet apparatus, which fits onto a Hydraulic Bench, allows us experiment

with the force generated by a jet of water as it strikes a vane in the shape of a flat plate or

hemispherical cup, and to compare it with the momentum flow rate in the jet.

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DESCRIPTION OF APPARATUS Figure 1 shows the main parts of the equipment shown on the top of a Hydraulic Bench.

Figure 2 shows a more detailed drawing of the apparatus. The unit fits onto a Digital Hydraulic

Bench which supplies water and measures flow. The main part is a transparent cylindrical tank

held between a top and bottom plate by three threaded bars. The whole assembly sits on three

adjustable legs.

The water enters the tank through a vertical inlet pipe that ends in a tapered nozzle inside

the tank. This produces a jet of water which hits the vane in the form of a Flat Plate and

Hemispherical. The water leaves the tank through a drain pipe to allow you to direct it back into

the hydraulic bench.

Figures 1 and 2 show the vane (hemispherical cup or the flat plate) supported by a pivoted

beam, restrained by a light spring. The beam carries a jockey weight. Adjusting the jockey weight

sets the beam to a balanced position (as indicated by the tally) along with adjustment of the knurled

nut above the spring. Any force generated by impact of the jet on the vane may now be measured

by moving the jockey weight along the lever until the tally shows that the lever is back at its

original balanced position.

Figure 2. General layout of the main apparatus

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Figure 3. (left) Key dimensions of apparatus; (right) schematic of forces on the vane

THEORY Force on the Vane

The weight beam forms a lever, pivoted at one end, with the jet force upwards at a distance

of 150 mm from the pivot. The mass of the jockey weight and gravity are an opposing force

downwards. At initial balance, you cancel out the mass of the weigh beam itself using the balance

spring. This allows you to calculate the force using moments, so that:

!×0.15 = ()* (1)

or

! =()*

0.15(2)

As the mass M of the Jockey Weight is 0.6 kg, then

! = 4)* (3)

and as gravity g is constant, then

! = 39.24* (4)

Vane Theory

Consider a vane symmetrical about the X-axis as shown in Figure 4. A jet of fluid flowing

at the rate of 2 kg.s-1 along the X-axis with the velocity u0 m.s-1 strikes the vane and is deflected

by it through angle b, so that the fluid leaves the vane with the velocity u1 m.s-1 inclined at an angle

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b to the X-axis. Note that this ignores changes in elevation and in piezometric pressure in the jet

from striking the vane to leaving it.

Figure 4. Vertical jet of fluid striking a symmetrical vane

Momentum enters the system in the X direction at a rate of:

234(56.7

89) (5)

Momentum leaves the system in the same direction at the rate of:

23: cos > (56.7

89) (6)

The force on the vane in the X direction is equal to the rate of change of momentum change.

Therefore:

! = 2 34 − 3: cos > (@) (7)

Ideally, jets are of constant velocity, so that u0 = u1. Therefore:

! ≈ 234 1 − cos > (@) (8)

From this, and reference to schematics in Figure 5, you can calculate the theoretical force for each

vane as shown in Table 1.

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Figure 5. Force against flat and hemispherical vanes

Table 1. Theoretical force values for different vane shapes

Finding Velocity

Fluid mass flow is equal to the product of its density, the area of the flow and the velocity:

2 = BC3 (9)

From this, if you know the fluid (water) density, the cross-sectional area through which it passes

(the nozzle in this equipment) and the mass flow (from the hydraulic bench), then you may find

the velocity:

3 =7

DE (10)

Water density changes with temperature, but assuming that you use the equipment at normal room

temperature, you can use a value of 103 kg.m-3. Using this with the nominal value of A = 78.54mm2:

3 =7

:444×4.4444FGHI(11)

3 =7

:444×4.4FGHIJK3 = 12.76×2 (12)

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The velocity u0 of the jet as it is deflected by the vane is less than the velocity, u, at exit

from the nozzle because of the deceleration due to gravity. You can find the velocity u0 at the vane

using:

34N = 3N − 2)O (13)

Using nominal values (g = 9.81 m.s-2 and s = 35 mm):

34N = 3N − 2×9.81×0.035 (14)

therefore,

34N = 3N − 0.6867or34 = 3N − 0.6867 (15)

INSTALLATION AND PREPARATION

1. Use the retaining screw to fit your chosen vane (plate or hemispherical cup) and the cover

plate (see Figure 6).

2. Put the equipment on the top of a Hydraulic Bench.

3. Connect the supply pipe from the Hydraulic Bench to the inlet at the bottom of the tank.

4. Put the exit of the drain pipe tank directly over the hole in the middle of the Digital Bench.

5. Use the adjustable feet to level the tank.

6. Rest the Jockey weight on the zero mark of the weigh beam.

7. Use the adjusting nut of the balance spring to level the weigh beam. When level, the

grooves on the tally should be equally spaced on either side of the top of the tank. Adjust

the length of the tally suspension if necessary.

8. Use the hydraulic bench to create a jet of water, lifting the vane. Stop the water flow and

make sure that the weigh beam returns to level.

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Figure 6. Schematic of vane installation

Table 2. Blank results table

EXPERIMENTAL PROCEDURE a) Create a Blank Results Table similar to Table 2.

b) Make sure the weigh beam is at balance with the jockey weight at the zero position.

c) Start the hydraulic bench and set to maximum flow.

d) Move the jockey weight until the beam balances again. Note the distance y from the zero

position.

e) Record the flow rate using your hydraulic bench.

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f) Reduce the hydraulic bench flow in steps to give at least eight more readings of distance

y and flow in relatively equal increments.

g) Repeat for the other vanes (flat or hemispherical) that you need to test.

RESULTS AND DISCUSSION

Convert your flow rate into mass flow rate 2. Calculate the flow velocity u and use this to

calculate u0. Calculate the rate of delivery of momentum 234 and force F. Create a chart of force

F on the vane (vertical axis) against rate of delivery momentum234 (similar to Figure 8). Add to

the chart, the results for each of the plates (vanes) that you test and complete Table 3. Compare

your actual results with the theoretical results and identify any differences or causes of error

(explain what may have caused deviation from theoretical result).

Figure 7. For reference only: typical results for Force against Momentum

Table 3.Theroretical vs. Experimental gradients

Slope Flat Plate Hemispherical Cup Theoretical 1.0 2.0

Experimental

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EXPERIMENT # 4

EXAMINING FLOW CHARACTERISTICS IN ORIFICE

OBJECTIVE

- To find the coefficients of discharge, velocity and contraction at a fixed flow rate.

- To understand the relationship between head and flow for an orifice.

- To show how the coefficient of discharge varies with flow.

- To use the head and flow relationship to find the average coefficient of discharge for the

orifice.

APPARATUS Flow Through an Orifice apparatus, Hydraulic Bench

Figure 1. Flow Through an Orifice apparatus

INTRODUCTION When a fluid passes through a constriction, such as through a sharp-edged hole or over a

weir, the constriction contracts the fluid stream, so it is not uniform and parallel. This contraction

can continue some distance downstream of the constriction. The contraction reduces the overall

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flow, which means that you must understand and allow for it in any calculations you make for

flow through the nozzle or orifice.

The Flow Through an Orifice apparatus, which fits onto a Hydraulic Bench, allows us to

experiment with flow, coefficients and contraction of a vertical jet of water passing through

different nozzles and orifices, and compare our experimental observations with available

theoretical formulations.

DESCRIPTION OF APPARATUS Figure 1 shows the main parts of the equipment shown on the top of a Hydraulic Bench.

Figure 2 shows a more detailed drawing of the apparatus. The unit fits onto a Digital Hydraulic

Bench which supplies water and measures flow. The main part is a transparent cylindrical tank

held between a top and bottom plate by three threaded bars. The whole assembly sits on three

adjustable legs. The tank has a water inlet to the top, an overflow pipe that exits through the

bottom of the tank and an outlet hole in the middle of the bottom plate. The orifices fit into the

hole in the bottom of the tank.

The water enters the top of the tank through a vertical inlet pipe. The bottom of the inlet

pipe has a water settler which sits below the normal water level and helps stabilize the flow as it

enters the tank. The inlet pipe has an adjustable fitting so the user may adjust the height of the

inlet pipe for different experiments. An overflow pipe directs the surplus water back to drain. This

helps keep a constant level or ‘Head’ of water in the tank for some experiments. The water also

passes out through the orifice fitted in the hole in the bottom of the tank. The emerging ‘jet’ passes

over a Pitot tube assembly, then back to the hydraulic bench for flow measurement.

Two clear plastic tubes mount in front of a vertical scale, forming two manometers. The

manometer scale clips to one of the three bars that hold the top and bottom plate to the tank. One

manometer tube connects to a pressure tapping in the base of the tank to show the water level in

the tank with respect to the bottom of the tank. The other tube connects to the Pitot assembly to

show the total jet head (pressure in millimeters of water) measured at the Pitot tip.

The Pitot assembly includes a micrometer-style adjustment for precise control of Pitot

position. A 3 mm thick angled pointer allows you to accurately measure the width of the jet using

the micrometer (see Figure 3).

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Figure 2. General layout of the main apparatus

Figure 3. The pitot tube tip and pointer

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THEORY Nomenclature

Table 1. Nomenclature

Symbol Meaning Units

u Velocity of flow m.s-1

uc Velocity at the vena contracta m.s-1

u0 Velocity in the tank m.s-1

p Pressure Pa or N.m-2

D Orifice Diameter m

H Head m of water

Q Flow or discharge m3.s-1

Cu Coefficient of velocity -

Cc Coefficient of contraction -

Cd Coefficient of discharge -

A0 Orifice Area (at narrowest part) m2

AC Vena Contracta Area (of the fluid) m2

ρ Density of water kg.m-3

g Acceleration due to gravity m.s-2

z A distance or displacement m

Background

Figure 4 shows how two different nozzles and a sharp-edged orifice contract the fluid

flow. Drawing (a) shows fluid streaming through a smoothly contracting nozzle, making a

parallel jet. The overall increase in speed through the contraction reduces non-uniformity

in the approaching flow, so the fluid velocity should be uniform across the emerging jet.

The cross-sectional area of the jet is the same as that of the nozzle, so the flow rate is the

product of the jet velocity and the nozzle area.

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Figure 4. Schematic patterns of flow passing through nozzle and orifice

In drawings (b) and (c) however, the fluid does not emerge as a convergent stream, so

the cross- sectional area of the jet reduces to a contracted section or ‘vena contracta’. Over

this section, the streamlines are parallel and velocity is effectively uniform. The flow rate is

the product of the jet velocity and area of the contracted section.

Flow Theory

Figure 5 shows a theoretical cross-section of the tank, showing the main features of

flow through the orifice. The tank is large enough to assume there is no fluid flow except for

that part near to the orifice. In this part, the fluid accelerates towards the center of the orifice,

and the curvature of the streamlines (shown as MN) reduces the jet diameter and therefore area.

The reduction in area due to the curvature extends to (or the vena contracta starts at) a

downstream position at a distance of between half the orifice diameter (D/2) and a complete

diameter (D). Atmospheric pressure applies everywhere on the surface of the jet; but inside

the jet the pressure only falls to atmospheric when the acceleration is complete (when the

jet reaches the vena contracta).

Consider now the total head of the water at points M and N of a typical streamline, M

being in the surface and N being in the plane of the vena contracta. From Bernoulli, with

reference to a datum, the total head at M is:

!"#2% +

'"(% + )"

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Figure 5. Schematic of a theoretical flow through orifice and its main features

and at N is:

u+#2g +

p+ρg + z+

so that, if the energy were conserved, i.e. if there were no loss of total head:

!"#2% +

'"(% + )" = !2#

2% +'2(% + )2 (1)

As mentioned earlier, Pm and Pn are equal (both at atmospheric pressure) and um is negligibly

small according to our assumption. Also, from the diagram, you can see that,

)" − )2 = 45 (2)

so that, from Equations (1) and (2), the ideal velocity at ˆ is given by:

!2#2% = 45 (3)

This result applies to all points in the plane of the vena contracta, so changing the notation

to let u0 be the ideal velocity in the plane of the vena contracta, which would occur if there

was no energy loss.

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!5#2% = 45 (4)

Because of the energy loss, which in fact takes place as the water passes down the tank and

through the orifice, the actual velocity uc in the plane of the vena contracta is less than u0, and

may be calculated from the Pitot tube using equation 5:

!6#2% = 46 (5)

From these equations:

Energy loss = H0 – Hc

Coefficients

The coefficient of velocity (Cu) is the ratio of actual velocity uc and ideal velocity u0

through the orifice. From Equations (4) and (5), we obtain:

78 =!6!5= 46

45(6)

In a similar sense, the coefficient of contraction Cc is defined as the ratio of the cross-section of

the vena contracta Ac, to the cross-section of the orifice A0,

76 =9695

(7)

Finally, the coefficient of discharge Cd is the ratio of the actual discharge to that which would

take place if the jet discharged at the ideal velocity without any reduction of area. The actual

discharge Q is given by:

Q = !;9; (8)

and if the jet discharged at the ideal velocity u0 over the orifice area A0, the discharge Q0 would

be:

Q5 = !595 = 95 2%45 (9)

So, from the definition of the coefficient of discharge,

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7< ===5

= !;9;!595

(10)

or in terms of quantities measured experimentally,

7< ==

95 2%45(11)

From Equations (6), (7) and (10) it follows immediately:

7> = 7?7; (8)

Typical Values

Figure 6 shows the typical values of Cd for two different orifices. In general, the

smoother the entrance to the orifice or nozzle, the higher the value for coefficient of discharge.

Table 2 gives typical values of coefficients for a sharp-edged circular orifice. Table 2 also

gives typical values of coefficients for a sharp-edged circular orifice.

Figure 6. Typical Values for Two Different Circular Orifices

Table 2. Typical Coefficients for a Circular Sharp-edged orifice

Cu Cc Cd0.97to 0.99 0.61to 0.66 0.6 to 0.65

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Mean Coefficient of Discharge

A rearrangement of Equation 11 gives:

7< ==1

95 2%× =

45

Using the nominal value of 1.327 x 10-4 m2 for the standard orifice area (sharp-edged orifice),

this simplifies to:

7< = 1701.26× =45

(13)

or alternatively:

7< = 1701.26×F (14)

Where:

F = =45

From this, the slope of a chart of Q against H01/2 will give k, which you can use with

Equation 14 to find a mean value of Cd.

Figure 7. Determining mean discharge coefficient using measured flow and head data

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INSTALLATION AND PREPARATION

1. Put the equipment on the top of a Hydraulic Bench.

2. Connect the supply pipe from the Hydraulic Bench to the inlet at the top of the tank.

3. Put the exit of the tank directly over the open channel of the Volumetric Bench.

4. Use the adjustable feet to level the tank.

5. Use the pipes supplied to direct the tank overflow back through one of the spare holes in

the lid of the hydraulic bench, directly back to the sump tank of the hydraulic bench.

USEFUL NOTES BEFORE PERFORMING MEASUREMENTS

The theory shows the theoretical model of the tank with Heads (H0 and Hc) shown with

respect to the Vena Contracta. However, the vertical position of the Vena Contracta is very

difficult to measure, so for ease of use you may use the Head values shown on the manometer,

with respect to the upper surface of the orifice or the base of the tank (see Figure 8). The error

caused is less than 0.1% for most nozzles.

The exact vertical position of the Pitot pressure and jet width measurements in the

contracted section (vena contracta) is not important to within a few millimeters, as the velocity

and therefore pressure across and along this section should be equal, but as a general guide:

• The Pitot tip must be towards the center of the jet flow and in the contracted section.

• The Pitot tip must be at a distance greater than D from the upper edge of the orifice, or you

may be measuring pressure before the vena contracta.

• The Pitot tip must not be too far away from the orifice. You may be below the contracted

section and the flow may become turbulent (this will affect the width of the jet, giving bad

results).

• Measure the jet width of the vena contracta at the same height as the Pitot tip.

43

EXPERIMENTAL PROCEDURE-TASK 1 1. Create a blank results table similar to Table 3.

2. Start the pump of the hydraulic bench and adjust the flow so the level in the tank stays

just below the overflow pipe in the tank. Make sure that all of the flow coming from the

bench leaves the tank through the orifice (and not the overflow pipe).

3. Adjust the vertical inlet pipe upwards so that its outlet is just below the surface of the

water in the tank (to help reduce any disturbance near to the orifice).

4. Allow conditions to stabilize and use the hydraulic bench to measure the flow. Record the

Head inside the tank.

5. Adjust the Pitot so that its tip points directly upwards into the water flow and the tip is in

the middle of the contracted section of the jet.

6. Record the Pitot manometer reading. The Pitot reading (HC) will only be slightly less

than the Head reading (H0), so take extra care measuring the levels of the manometer.

7. Use the 3 mm thick pointer with the micrometer to measure the jet diameter at the same

position as you measured the pressure with the Pitot tip.Table 3. Blank result table for Task 1

Orifice:

OrificediameterD(m)

OrificeAreaA0(m2)

FlowrateQ(m3.s-1)

Head(mm)

HeadH0(m)

PitotReading(mm)

PitotReadingHC(m)

Jetdiameter(mm)

JetdiameterD(m)

Cd

Cu

Cc

CuxCc

44

RESULTS AND DISCUSSION-TASK 1 1. Use your readings with the equations (6, 7 and 11) in the theory section to calculate

a) coefficients of discharge,

b) coefficient of velocity and,

c) coefficient of contraction.

2. Check that your calculated coefficients agree with the values and calculations suggested in

the theory for a sharp-edged orifice. In case of using a different type of nozzle or orifice, use

the typical coefficients presented at the end of this instruction for comparison.

3. Verify the accuracy of Equation 12.

4. Explain the cause of any error or difference.

EXPERIMENTAL PROCEDURE-TASK 2 1. Create a blank results table similar to Table 4.

2. Start the pump of the hydraulic bench and adjust the flow so the level in the tank stays

just below the overflow pipe in the tank. Make sure that all of the flow coming from the

bench leaves the tank through the orifice (and not the overflow pipe).

3. Adjust the vertical inlet pipe upwards so that its outlet is just below the surface of the

water in the tank.

4. Allow conditions to stabilize and use the hydraulic bench to measure the flow. Record the

Head inside the tank.

5. Reduce the inlet flow (supply from the hydraulic bench) so that the head falls by about 30

mm. Readjust the vertical inlet pipe upwards so that its outlet is just below the surface of

the water in the tank.

6. Allow conditions to stabilize and use the hydraulic bench to measure the flow. Record the

Head inside the tank.

7. Repeat for decreasing values of head in about 30 mm steps, giving about eight results

over the height of the manometer scale, each time recording head and flow.

45

Table 4. Blank result table for Task 2

Orifice:

FlowQ

(m3.s-1)

HeadH0

(m)

H01/2

(m1/2) Cd

RESULTS AND DISCUSSION-TASK 2 1- Use equation 11 to find the coefficient of discharge for each line of your results.

2- Create a chart of coefficient (vertical axis) against flow to see how it varies with flow.

3- Create a chart of flow (vertical axis) against head to show the direct relationship.

4- Now find the square root (or H1/2) of each of your head readings to complete the table.

Add these values to your second chart (number 3 above) and note the linearity. You

should be able to extend the results down to the origin of your chart.

5- Find the slope of the linear results to find the constant for the orifice, then use this to find

the average coefficient of discharge for the orifice.

6- How does the mean discharge coefficient found above compare with the value found in

Task 1? What can you say about assuming a precise value for coefficient of discharge?

46

Figure 8. Typical coefficients for different types of orifice with circular cross section.

Cur

ved

Entra

nce

(Cyl

indr

ical

Thr

oat N

ozzl

e)C

urve

d En

tranc

e O

rific

eA

ngle

d En

tranc

e an

d Ex

it O

rific

eC

onve

rgin

g/D

iver

ging

N

ozzl

e

47

Figure 9. Typical coefficients for different types of orifice with non-circular cross sections.

48

EXPERIMENT # 5

EXAMINING TRAJECTOY OF A HORIZONTAL JET THROUGH NOZZLE

OBJECTIVE

- To understand the horizontal discharge characteristics of am orifice or nozzle.

APPARATUS Jet Trajectory apparatus, Hydraulic Bench

Figure 1. Jet Trajectory apparatus

DESCRIPTION OF APPARATUS Figure 1 shows the main parts of the equipment on top of a Hydraulic Bench. Figure 2

shows a more detailed drawing of the apparatus. The unit fits onto a Digital Hydraulic Bench

which supplies water and measures flow. The main part is a transparent cylindrical tank similar

to the one used in previous experiment (Examining Flow Characteristics in Orifice). Therefore,

only the new parts related to horizontal jet tests will be explained here.

For horizontal tests, the water also passes out through orifices fitted to the side of the tank

and just under a plotting board (see Figure 3 for dimensions). The board holds a set of thin

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adjustable-depth pins that allow the user to plot the curvature of the emerging horizontal jet. A

‘plug’ supplied with the equipment allows the user to block either the vertical or horizontal ports

when not in use.

Figure 2. General layout of the main apparatus

Figure 3. Location and dimensions of the side hole

50

Figure 4. Dimensions of different types of nozzles. Flow direction is from left to right.

A set of nozzles are available with the equipment. Simple screw fixings hold the nozzles

in position at the side of the tank for horizontal flow tests. Figure 4 shows the dimensions of the

nozzles.

Note: Each 'Nozzle' has a different 'orifice’ or ‘throat'. The whole unit is the Nozzle, the orifice or throat

is the internal constriction of the Nozzle. Technically, an orifice is a hole in a sheet of material or

side of a vessel, where the thickness of the sheet is less than the diameter of the orifice.

THEORY For nomenclature and background theory, refer to experiment: Examining Flow

Characteristics in Orifice

Figure 5. Schematic of a theoretical horizontal jet through nozzle and its main features

51

Trajectory from Horizontal

The jet emerges from the nozzle at a velocity uc. At any point P (see Figure 5) in its

trajectory it is subject to gravitational acceleration (g) in the vertical direction, and has a steady

velocity component in the horizontal direction. As gravity affects its path, the trajectory should

be parabolic. The head (H0) of water in the tank will affect the horizontal distance travelled and

therefore the curvature of the jet.

Consider a ‘packet’ of water that has travelled a distance x from the vena contracta to

point P in t seconds. It travels at velocity uc in the horizontal direction,

! = #$% (1)

and over the same time in the vertical direction:

& = (12)g%, (2)

Eliminating t from the two equations gives,

& = (12)g!,#$,

(3)

and

#$ =-!,2& (4)

From Equations 5 and 6 in previous experiment (Flow through Orifice):

#$, = 2-.$

#$ = /0#1 = /0 2-.1

and

/0 =!,4.1&

(5)

or

52

/0 =!&

14.1

(6)

In the absence of any specific nozzle, the theoretical trajectory of jet will be:

& = !,4.1

(7)

Where the head in the tank (H0) stays constant, you can find an average value for the velocity

coefficient from a chart of & and x (see Figure 6). The chart should give linear results. The

inverse value of the gradient (m) of this chart substituted back into Equation 6 should give this

average. So:

/0 =13

14.1

(8)

Note: Units of x, y and H0 must be identical (mm or meters)

Figure 6. Determining velocity coefficient using jet trajectory data

USEFUL NOTES BEFORE PERFORMING MEASUREMENTS For comparison with theory, the datum x must be the point at which the vena contracta

starts. Take care when judging this with the nozzles, as it may start just inside the nozzle. In

theory, the head at the vena contracta (Hc) is the value needed for calculations. However, due to

the nozzle being at the base of the tank (where the head is H0), you may use this value for the

calculations.

53

When using the depth pins, each pin will just touch the top of the water jet - at position A

in Figure 7. The theory relates to the position in the center of the jet at position B, so you must

add half the jet thickness to the readings to match the vertical (y) value in the theory.

Figure 7. Measuring jet trajectory

Table 1. Blank result table

Orifice/Nozzle:Head:

x(mm)Actualy(mm) y

Predictedy(mm)

EXPERIMENTAL PROCEDURE 1. Create a blank results table similar to Table 1.

2. Set up the tank on the hydraulic bench for horizontal flow tests.

3. Set the tank for a constant head.

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Note: The head for this experiment will be the head of water with respect to the center line of the nozzle. The manometer measures with respect to the bottom of the base plate - roughly 22 mm lower (see Figure 3).

4. Adjust the depth pins so they just touch the surface of the jet without disturbing it.

5. Mark the positions of the top of the pins onto the chart paper.

6. Slowly reduce (or increase) the head in the tank and note how it affects the jet. Repeat

steps 3 to 5 for two more heads and complete their result table

7. Switch off the water supply and remove the chart paper.

RESULTS AND DISCUSSION 1- Use equation 7 to calculate the predicted values for the y positions and add the results to

your results table. Find the square root of your actual y values to complete your results

table (for all of the three heads).

2- Create a chart of y position (vertical axis) against x position (horizontal axis) and add

your actual and predicted results for three head measurements.

Note: Make the origin (zero value) of the y axis at the top left of the chart so thecurves match those made by the jet (moving from top left to bottom right).

3- On your chart, create a second y axis of & and add your square root values from your

table. This should produce a straight line.

4- Find the inverse of the gradient of the straight line as in Equation 8 to find an average

value for the coefficient of velocity. Compare the value with the typical experimental

data shown in Table 2.Table 2. Typical experimental result for different types of nozzles

Nozzle Cd Cu Cc A 0.81 0.997 0.88 B 0.92 0.994 1.00 C 0.92 0.996 1.00 D 0.97 0.999 0.97

55