Fluid mechanics and hydraulics lab manual -...

Dr. Khalil M. Alastal

Eng. Mohammed Y. Mousa

2015

Fluid Mechanics and Hydraulics Lab Manual

Table of contents :

Experiment (1): Hydrostatic force on a plane surface ................................................................. 1

Experiment (2): Metacentric height of floating bodies ................................................................ 7

Experiment (3): Impact of jet ....................................................................................................... 15

Experiment (4): Flow measurement ............................................................................................ 22

Experiment (5): Flow through small orifices .............................................................................. 32

Experiment (6): Flow over weirs ................................................................................................. 40

Experiment (7): Investigation of Bernoulli's theorem ............................................................... 46

Experiment (8): Minor losses ....................................................................................................... 52

Experiment (9): Centrifugal pump ............................................................................................... 62

Experiment (10): Series and parallel pumps .............................................................................. 72

Experiment (11): Cavitation demonstration ............................................................................... 80

Experiment (12): Major losses ..................................................................................................... 86

Experiment (13): Flow channel .................................................................................................... 92

References .................................................................................................................................. 104

Fluid mechanics and hydraulics lab manual Islamic University – Gaza (IUG)

Dr. Khalil M. Alastal Eng. Mohammed Y. Mousa 1

Experiment (1): Hydrostatic force on a plane surface

Introduction:

The study of pressure forces acting on plane submerged surfaces is a fundamental topic in the

subject of hydrostatic involving assessment of the value of the net thrust and the concept of center

of pressure, which are so important in the design of innumerable items of hydraulic equipment and

civil engineering projects.

Purpose:

To investigate the pressure acting on a submerged surface and to determine the position of the

center of pressure.

Apparatus:

1. Center of pressure apparatus (Figure 1).

Figure 1: Center of pressure apparatus



Figure 2: Schematic diagram of center of pressure apparatus

Theory:

Figure 3: Experimental set-up

Referring to figure (3) which shows the experimental set-up consider the forces which result in

turning moments of the beam and submerged part of the model about the knife edged fulcrum.

Liquid pressures on the curved surfaces act at right angles to the curved surfaces, and the design of

the model ensures that these forces pass through the line of action of the knife edges and therefore

do not exert any turning moment. The hydrostatic pressure on the vertical end surface exerts a

force 𝐹 at the center of pressure which is at depth 𝐻𝑃 below the surface. The resulting turning

moment about the knife edge from the hydrostatic forces is therefore given by:



𝐹 (𝑎 + 𝑑 – 𝑦 + 𝐻𝑝)

Which is resisted by the weight of the mass 𝑀 on the balance arm at distance 𝐿 from the knife edge:

𝑀𝑔𝐿

Now considering the cases of partial immersion and complete immersion separately:

(A) Partial immersion

Figure 4: Surface is partially submerged

When the vertical end face of the quadrant is only partially immersed, the geometric properties of

the wetted portion of the end face are:

Area 𝐴 = 𝑏𝑦

Depth of center 𝐻 = 𝑦/2

Second moment of area 𝐼𝑜 = 𝑏𝑦3/12

Depth of center of pressure

𝐻𝑃 = 𝐻 +𝐼𝑜

𝐴𝐻=

𝑦

2+

𝑏𝑦3/12

𝑏𝑦 𝑦/2=

2𝑦

3

The force acting on the submerged part of the end surface of the model is:

𝐹 = 𝜌𝑔𝐻𝐴

= 𝜌𝑔 𝑦/2 𝑏𝑦 = 1/2𝜌𝑔𝑏𝑦2

Taking moments about the knife edge:

𝑀𝑔𝐿 = 𝐹 (𝑎 + 𝑑– 𝑦 + 𝐻𝑃)



Substituting for 𝐻𝑃 and rearranging the above equation shows that the force acting on the wetted

end surface can be calculated from the experimental results of 𝑀 and 𝑦:

𝐹 =𝑀𝑔𝐿

𝑎 + 𝑑 − 𝑦 +2𝑦3

=𝑀𝑔𝐿

𝑎 + 𝑑 −𝑦3

Which can then be compared with the theoretical result:

𝐹 = 1/2𝜌𝑔𝑏𝑦2

(B) Complete Immersion

Figure 5: Surface is fully submerged

When the end surface is fully immersed, the properties of the submerged end face are:

Area 𝐴 = 𝑏𝑑

Depth of center of area 𝐻 = 𝑦 − 𝑑/2

Second moment of area 𝐼𝑜 = 𝑏𝑑3/12


𝐻𝑃 = 𝐻 +𝐼𝑜

𝐴𝐻= 𝑦 −

𝑑

2+

𝑏𝑑3/12

𝑏𝑑𝐻= 𝑦 −

𝑑

2+

𝑑2

12𝐻

The force acting on the end surface is:

𝐹 = 𝜌𝑔𝐻𝐴

= 𝜌𝑔 (𝑦 −𝑑

2) 𝑏𝑑

Taking moments about the knife edge:



𝑀𝑔𝐿 = 𝐹 (𝑎 + 𝑑– 𝑦 + 𝐻𝑃)

Substituting for 𝐻𝑃 and rearranging the above equation shows that the force acting on the wetted

end surface can be calculated from the experimental results of 𝑀 and 𝑦:

𝐹 =𝑀𝑔𝐿

𝑎 + 𝑑 − 𝑦 + 𝑦 −𝑑2

+𝑑2

12𝐻

=𝑀𝑔𝐿

𝑎 +𝑑2

+𝑑2

12𝐻

Which can be compared with the theoretical result calculated from:

𝐹 = 𝜌𝑔 (𝑦 −𝑑

2) 𝑏𝑑

Equipment preparation:

Position the apparatus on the work surface of the hydraulic bench and adjust the feet to level the

base. Attach a length of hose to the drain cock and direct the other end of the hose into the overflow

pipe of the volumetric measuring tank. If the quadrant is not assembled to the balance arm then

locate the quadrant on the two dowel pins and fasten it to the balance arm by the central screw.

Procedures:

1. If necessary measure the dimensions 𝑎, 𝑏 and 𝑑 of the quadrant, and the distance between the

pivot and the weight hanger 𝐿. Lightly apply wetting agent to reduce surface tension effects.

2. Insert the quadrant into the tank locating the balance arm on the knife edges. Adjust the

counter-balance weight until the balance arm is horizontal, as indicated on the datum level

indicator.

3. Add all the weights supplied to the weight carrier. Fill the tank with water until the balance

beam tips lifting the weights then drain out a small quantity of water to bring the balance arm

horizontal, do not level the balance arm by adjustment of the counter balance weight or the

datum setting of the balance arm will be lost. Record the water level shown on the scale. Fine

adjustment of the water level may be achieved by over-filling and slowly draining, using the

drain cock.

4. Remove one or more weights from the weight carrier and level the balance arm by draining out

more of the water. When the arm is level record the depth of immersion shown on the scale on

the quadrant.



5. Repeat reading for reducing masses on the weight carrier.

Data & Results:

𝐿 = 275 mm, 𝑎 = 100 mm, 𝑑 = 100 mm, 𝑏 = 75 mm

(A) Partial Immersion

Trials 1 2 3

Total weight on arm 𝑴 (grams)

Depth of water 𝒚 (mm)

Force on end surface (experimental)

𝑭 =𝑴𝒈𝑳

𝒂 + 𝒅 −𝒚𝟑

(𝐍)

Force on end surface (theoretical)

𝑭 = 𝟏/𝟐𝝆𝒈𝒃𝒚𝟐 (𝐍)


𝑯𝑷 =𝟐𝒚

𝟑 (mm)

(B) Complete Immersion

Trials 1 2 3

Total weight on arm 𝑴 (grams)

Depth of water 𝒚 (mm)

𝑭 =𝑴𝒈𝑳

𝒂 +𝒅𝟐

+𝒅𝟐

𝟏𝟐�̅�

(𝐍)

Force on end surface (experimental)

𝑭 = 𝝆𝒈 (𝒚 −𝒅

𝟐) 𝒃𝒅 (𝐍)

Force on end surface (theoretical)

𝑯𝑷 = 𝒚 −𝒅

𝟐+

𝒅𝟐

𝟏𝟐�̅� (𝐦𝐦)




Experiment (2): Metacentric height of floating bodies

Introduction:

The Stability of any vessel which is to float on water, such as a pontoon or ship, is of paramount

importance. The theory behind the ability of this vessel to remain upright must be clearly

understood at the design stage. Archimedes’ principle states that the buoyant force has a magnitude

equal to the weight of the fluid displaced by the body and is directed vertically upward. Buoyant

force is a force that results from a floating or submerged body in a fluid which results from different

pressures on the top and bottom of the object and acts through the centroid of the displaced

volume.

Apparatus:

1. Flat bottomed pontoon (Figure 1).

2. Hydraulic bench.

Figure 1: Flat bottomed pontoon

Equipment set up:

The flat bottomed pontoon is constructed from non-ferrous materials and has a detachable bridge

piece and loading system. Provision is made to alter the keel weight and the mast weight so

obtaining a variety of loading conditions. For off balance loadings, the degree of list can be directly

measured using the plumb-bob line attached to the mast and swinging over a scale mounted on the



bridge piece. The floatation experiments can be carried out using the measuring tank of the

hydraulics bench.

Floatation characteristics of flat bottomed pontoon.

Depth 𝐻 = 170mm.

Length 𝐿 = 380mm.

Width 𝐷 = 250mm.

Distance from pontoon center line to added weight 𝑥 = 123mm.

Center of gravity of vessel with mast 𝑂𝐺𝑣𝑚 = 125mm approximately from outer surface of vessel

base.

Weight of vessel with mast 𝑊𝑣𝑚 = 3000g.

Height of mast loading position above water surface of vessel base = 790mm.

Theory:

Consider a ship or pontoon floating as shown in figure 2. The center of gravity of the body is at 𝐺

and the center of buoyancy is at 𝐵. For equilibrium, the weight of the floating body is equal to the

weight of the liquid it displaces and the center of gravity of the body and the centroid of the

displaced liquid are in the same vertical line. The centroid of the displaced liquid is called the

"center of buoyancy". Let the body now be heeled through an angle 𝜃 as shown in a subsequent

figure, 𝐵1 will be the position of the center of buoyancy after heeling. A vertical line through 𝐵1 will

intersect the center line of the body at 𝑀 and this point is known as the metacenter of the body

when an angle 𝜃 is diminishingly small. The distance 𝐺𝑀 is known as the metacentric height. The

force due to buoyancy acts vertically up through 𝐵1 and is equal to 𝑊. The weight of the body acts

downwards through 𝐺.

Figure 2: Illustrative figure of flat bottomed pontoon



Figure 3: Centers of buoyancy of floating and submerged objects

Stability of submerged objects:

Stable equilibrium: if when displaced, it returns to equilibrium position.

If the center of gravity is below the center of buoyancy, a righting moment will produced and the

body will tend to return to its equilibrium position (Stable).

Unstable equilibrium: if when displaced it returns to a new equilibrium position.

If the center of gravity is above the center of buoyancy, an overturning moment is produced and the

body is unstable.

Note : As the body is totally submerged, the shape of displaced fluid is not altered when the body is

tilted and so the center of buoyancy unchanged relative to the body.

Figure 4: Stability of submerged objects

Stability of floating objects:

Metacenter point 𝑀: the point about which the body starts oscillating.

Metacentric height 𝐺𝑀: is the distance between the center of gravity of floating body and the

metacenter.

If 𝑀 lies above 𝐺 a righting moment is produced, equilibrium is stable and 𝐺𝑀 is regarded as

positive.

If 𝑀 lies below 𝐺 an overturning moment is produced, equilibrium is unstable and 𝐺𝑀 is regarded

as negative.



If 𝑀 coincides with 𝐺, the body is in neutral equilibrium.

Figure 5: Stability of floating objects

Determination of Metacentric height

1- Practically

𝑊𝐺𝑀𝑠𝑖𝑛(𝜃) = 𝑃𝑥

𝐺𝑀 =𝑃𝑥

𝑊𝑠𝑖𝑛(𝜃)

Where 𝑥 = distance from pontoon centerline to added weight.

𝑊 = weight of the vessel including 𝑃.

2- Theoretically

𝐺𝑀 = 𝐵𝑀 + 𝑂𝐵 − 𝑂𝐺

𝐵𝑀 =𝐼

𝑉

𝑂𝐵 = 0.5𝑉

𝐿 × 𝐷

Where 𝑉 = volume of displaced liquid

𝐼 =𝐿𝐷3

12

'



Exercise A

Purpose:

To determine the floatation characteristics for unloaded and for loaded pontoon.

Procedures:

1. Assemble the pontoon by positioning the bridge piece and mast i.e. locate the mast in the hole

provided in the base of the vessel and clamp the bridge piece fixing screws into the locating

holes in the sides of the vessel.

The 'plumb-bob' is attached to the mounting dowel located on the mast and is allowed to swing

clear of and below the scale provided

2. Weigh the pontoon and determine the height of its center of gravity up the line of the mast by

balancing the mast on a suitable knife edge support and measuring the distance from knife

edge to outside base of pontoon.

3. Fill the hydraulic bench measuring tank, or other suitable vessel, with water and float the

pontoon in it. Trim the balance of the pontoon by applying one of the small weights provided to

the bridge piece at the required position so that the vessel floats without any list, this condition

being indicated by the plumb-bob resting on the zero mark.

4. Apply a weight of 50g on the bridge piece loading pin then measure and record the angle of list

and value of applied weight.

5. Take readings of list angle and applied weights (100, 150 & 200g). Repeat the procedure for

lists in the opposite direction i.e. apply the weights to the opposite side of the bridge piece.

6. Calculate GM practically. Draw a relationship between θ (x-axis) and GM (y-axis), then obtain

GM when θ equals zero.

7. Calculate GM theoretically.

8. Repeat the above procedures for increasing ballast loading conditions i.e. 2000 and 4000g.



Data & Results:

Bilge Weight Off balance wt.

Mean

Defln.

Exp.

𝑮𝑴

𝑮𝑴 at

𝜽 = 𝟎 𝑩𝑴 𝑶𝑩

Theo.

𝑮𝑴

𝑾𝒃 (gm) 𝑷 (gm) 𝜽

(degree) (mm)

from

graph (mm) (mm) (mm)

0.00

50

100

150

200

2000.00

x1 = 30mm

50

100

150

200

4000.00

x1 = 37.5mm

100

150

200

250

Conclusion:

Comment on the closeness or otherwise of the practically acquired values of metacentric height

against theoretically derived results.



Exercise B

Purpose:

To determine the effect on floatation characteristics of altering the center of gravity of the pontoon,

with given total loading.

Procedures:

1. Replace the large bilge weights by 4×50g weights.

2. Apply a weight of 300gm on a height of 190 mm from the pontoon surface.

3. Using the method of exercise A, determine the metacentric height 𝐺𝑀 (using applied weights

40, 80 &120g).

4. Move one 50g bilge weight to the mast head and once again determine 𝐺𝑀.

5. Repeat 100, 150 and 200g moved from the bilge weight to the mast head. Measure the position

of the center of gravity from the base of the pontoon for each loading condition.

6. Determine the theoretical 𝐺𝑀 for each condition and also a height of a metacenter above water

level.

Note 1: 𝐵𝑀 & 𝑂𝐵 values are constants for all loading conditions, since the dimensions & the weight

of a pontoon do not alter.

Note 2: Once the center of gravity of the unloaded pontoon has been determined, then the center of

gravity for other loaded conditions can be evaluated by taking moments about the base of the

pontoon.

𝑂𝐺 =3000(125) + 𝑊𝑏(35) + 300(190) + 𝑊𝑚(790 +

𝐿2

)

3500

Where : 𝑊𝑏 + 𝑊𝑚 = 200𝑔

𝐿 = 10 mm when 𝑊𝑚 = 50𝑔






Data & Results:

Off balance wt. Mean

Defln. Exp. 𝑮𝑴

𝑮𝑴 at

𝜽 = 𝟎 𝑩𝑴 𝑶𝑮

Theo.

𝑮𝑴

𝑴 above

water

level

(mm) 𝑷 (gm)

𝜽

(degree) (mm)

from

graph (mm) (mm) (mm)

Mast Weight = 0.0

40

80

120

Mast Weight = 50.0

40

80

120

Mast weight = 100.0

20

40

80

Mast Weight = 150.0

10

20

40

Mast weight = 200.0

Unstable

Conclusion:

Show the variation of depth of submergence and position of metacentric height under different

loading conditions

Discuss what will happen if the ballast weights were added at the center of gravity so that the

resultant center of gravity was unchanged.



Experiment (3): Impact of jet

Introduction:

Impact of jets apparatus enables experiments to be carried out on the reaction force produced on

vanes when a jet of water impacts on to the vane. The study of these reaction forces is an essential

step in the subject of mechanics of fluids which can be applied to hydraulic machinery such as the

Pelton wheel and the impulse turbine.

Purpose:

To investigate the reaction force produced by the impact of a jet of water on to various target vanes.

Apparatus:

1. Impact of jet apparatus (Figure 1).

2. Hydraulic bench.

Figure 1: Impact of jet apparatus



Equipment set up:

Set up the apparatus on the top of a hydraulics bench with the left hand support feet of the impact

of jet apparatus located on the two left hand locating pegs of the hydraulics bench so that the

apparatus straddles the weir channel. Connect the feed tube from the hydraulics bench to the boss

on the rear of the base of the impact of jet apparatus. Fit the 5mm nozzle and the normal flat target.

Figure 2: Illustrative figure of impact of jet apparatus

Theory:

When a jet of water flowing with a steady velocity strikes a solid surface the water is deflected to

flow along the surface. If friction is neglected by assuming an inviscid fluid and it is also assumed

that there are no losses due to shocks, then the magnitude of the water velocity is unchanged. The

pressure exerted by the water on the solid surface will everywhere be at right angles to the surface.

Consider a jet of water which impacts on to a target surface causing the direction of the jet to be

changed through an angle 𝜃 as shown in figure 3 below. In the absence of friction the magnitude of

the velocity across the surface is equal to the incident velocity Vi. The impulse force exerted on the

target will be equal and opposite to the force which acts on the water to impart the change in

direction.



Applying Newton’s second law in the direction of the incident jet

𝐹𝑜𝑟𝑐𝑒 = 𝑀𝑎𝑠𝑠 × 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

= 𝑀𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 × 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝐹 = �̇� (𝑣2 − 𝑣1)

𝐹 = 𝜌 𝑄 (𝑣2 − 𝑣1)

This is the resultant force acting on the fluid in the direction of motion.

This force is made up of three components:

𝐹1 = 𝐹𝑅 = Force exerted in the given direction on the fluid by any solid body touching the

control volume.

𝐹2 = 𝐹𝐵 = Force exerted in the given direction on the fluid by body force (e.g. gravity).

𝐹3 = 𝐹𝑃 = Force exerted in the given direction on the fluid by fluid pressure outside the control

volume.

By Newton’s third law, the fluid will exert an equal and opposite reaction on its surroundings.

The force exerted by the fluid on the solid body touching the control volume is equal and opposite

to FR . So the reaction force R is given by:

𝑅 = −𝐹𝑅

𝑅 = �̇� (𝑣1 − 𝑣2)

𝑅 = �̇� (𝑣𝑖 − 𝑣𝑖𝑐𝑜𝑠𝜃)

𝑅 = 𝜌 𝑄 𝑣𝑖 (1 − 𝑐𝑜𝑠𝜃)

𝑅

𝜌 𝑄 𝑣𝑖

= (1 − 𝑐𝑜𝑠𝜃)

Figure 3: Impact of a jet



Application to impact of jet apparatus

In each case it is assumed that there is no splashing or rebound of the water from the surface so

that the exist angle is parallel to the exit angle of the target.

The jet velocity can be calculated from the measured flow rate and the nozzle exit area.

𝑣𝑛 =𝑄

𝐴

However, as the nozzle is below the target the impact velocity will be less than the nozzle velocity

due to interchanges between potential energy and kinetic energy so that :

𝑣𝑖2 = 𝑣𝑛

2 − 2𝑔ℎ

where ℎ is the height of target above the nozzle exit.

1. Impact on normal plane target

For the normal plane target 𝜃 = 90°

Therefore 𝑐𝑜𝑠𝜃 = 0

𝑅

𝜌 𝑄 𝑣𝑖= (1 − 𝑐𝑜𝑠𝜃) = 1

2. Impact on conical target

The cone semi-angle 𝜃 = 45°

Therefore 𝑐𝑜𝑠𝜃 = 0.7071

𝑅

𝜌 𝑄 𝑣𝑖= (1 − 𝑐𝑜𝑠𝜃) = 0.2929

3. Impact on semi-spherical target

The target exit angle 𝜃 = 135°

Therefore 𝑐𝑜𝑠𝜃 = −0.7071

𝑅

𝜌 𝑄 𝑣𝑖= (1 − 𝑐𝑜𝑠𝜃) = 1.7071



Figure 4: Interchangeable target vanes

Procedures:

1. Position the weight carrier on the weight platform and add weights until the top of the target is

clear of the stop and the weight platform is floating in mid position. Move the pointer so that it

is aligned with the weight platform. Record the value of weights on the weight carrier.

2. Start the pump and establish the water flow by steadily opening the bench regulating valve until

it is fully open.

3. The vane will now be deflected by the impact of the jet. Place additional weights onto the weight

carrier until the weight platform is again floating in mid position. Measure the flow rate and

record the result on the test sheet, together with the corresponding value of weight on the tray.

Observe the form of the deflected jet and note its shape.

4. Reduce the weight on the weight carrier in steps and maintain balance of the weight platform by

regulating the flow rate in about three steps, each time recording the value of the flow rate and

weights on the weight carrier.

5. Close the control valve and switch off the pump. Allow the apparatus to drain.

6. Replace the 5mm nozzle with the 8mm diameter nozzle and repeat the tests.

7. Replace the normal vane with the 45° conical vane and repeat the test with both the 5mm and

8mm nozzles.

8. Replace the 45° conical vane with the hemispherical vane and repeat the tests with both the

5mm and 8mm nozzles.



Results:

1. Record the results on a copy of the results sheet provided.

2. Calculate for each result the flow rate and the nozzle exit velocity. Correct the nozzle velocity

for the height of the target above the nozzle to obtain the impact velocity.

3. Calculate the impact momentum and plot graphs of impact force 𝑅 against impact momentum

𝜌𝑄𝑣𝑖 and determine the slope of the graphs for each target. Compare with the theoretical

values of 1, 0.2929 and 1.7071 for the normal plane target, conical target and hemispherical

target respectively.

Target

Vanes

(degrees)

Nozzle

Dia. 𝑫

(mm)

Height of target

above nozzle 𝒉

(mm)

Additional Weights

𝒎

(g)

Volume of water

collected 𝑽

(Liter)

Time 𝒕

(sec)

°

5

30

90

30

𝜽=

30

Fla

t

8

30

30

30

°

5

25

45

25

𝜽=

25

Co

nic

al

8

25

25

25

°

5

30

13

5

30

𝜽=

30

Sem

i-

sph

eric

al

8

30

30

30



Target

Vanes

(degrees)

Nozzle

Dia. 𝑫

(mm)

𝑸 (m3/s) 𝑽𝒏 (m/s) 𝑽𝒊 (m/s) 𝑹 (N) 𝝆𝑸𝑽𝒊 (N) Slope 𝟏 − 𝐜𝐨𝐬 𝜽

°

5

1

90

𝜽=

Fla

t

8

°

5

0.2929

45

𝜽=

Co

nic

al

8

°

5

1.7071

13

5

𝜽=

Sem

i-

sph

eric

al

8



Experiment (4): Flow measurement

Introduction:

The flow measuring apparatus is used to familiarize the students with typical methods of flow

measurement of an incompressible fluid and at the same time demonstrate applications of the

Bernoulli's equation.

The flow is determined using a sudden enlargement, venturi meter, orifice plate, elbow and a

rotameter. The pressure drop associated with each meter is measured directly from the

manometers.

Purpose:

To investigate the flow rate using particular flow measuring apparatus.

Apparatus:

1. Water flow measuring apparatus (Figure 1).

2. Hydraulic bench.

Figure 1: Water flow measuring apparatus



Figure 2: Schematic diagram of water flow measuring apparatus

Water flow measuring apparatus is designed as a free-standing apparatus for use on the hydraulics

bench, although it could be used in conjunction with a low pressure water supply controlled by a

valve and a discharge to drain. Water enters the apparatus through the lower left-hand end and

flows horizontally through a sudden enlargement into a transparent venturi meter, and into an

orifice plate, a 90° elbow changes the flow direction to vertical and connects to a variable area flow

meter, a second bend passes the flow into a discharge pipe which incorporates an atmospheric

break.

The static head at various points in the flow path may be measured on a manometer panel. The

water flow through the apparatus is controlled by the delivery valve of the hydraulics bench and

the flow rate may be confirmed by using the volumetric measuring tank of the hydraulics bench.



Theory:

(1) Pressure drop at sudden enlargement

Figure 3: Schematic diagram of sudden enlargement

The test section consists of a 10mm diameter bore with a sudden enlargement to 20mm diameter.

Two manometer are provided.

Consider a sudden enlargement in pipe flow area from area A1 to area A2.

Applying Newton’s second law, the net force acting on the fluid equals the rate of increase of

momentum.

𝑃1𝐴1 + 𝑃′(𝐴2 − 𝐴1) − 𝑃2𝐴2 = 𝜌𝑄(𝑉2 − 𝑉1)

Where 𝑃′ is the mean pressure of the eddying fluid over the annular area of the expansion. It is

known from experimental evidence that 𝑃´ = 𝑃1, since the jet issuing from the smaller pipe is

essentially parallel.

(𝑃1 − 𝑃2)𝐴2 = 𝜌𝑄(𝑉2 − 𝑉1)

(𝑃1 − 𝑃2) = 𝜌𝑉2(𝑉2 − 𝑉1)

From Bernoulli equation:

𝑃1

𝛾+

𝑉12

2𝑔+ 𝑍1 =

𝑃2

𝛾+

𝑉22

2𝑔+ 𝑍2 + ℎ𝑒

Since the flow direction is horizontal 𝑍1 = 𝑍2.

ℎ𝑒 =𝑃1 − 𝑃2

𝛾+

𝑉12 − 𝑉2

2

2𝑔

And substituting 𝑃1 − 𝑃2 from Newton’s second law:

ℎ𝑒 =𝑉2(𝑉2 − 𝑉1)

𝑔+

𝑉12 − 𝑉2

2

2𝑔=

(𝑉1 − 𝑉2)2

2𝑔

Using Bernoulli equation:



𝑃1

𝛾+

𝑉12

2𝑔+ 𝑍1 =

𝑃2

𝛾+

𝑉22

2𝑔+ 𝑍2 + ℎ𝑒

𝑉12

2𝑔−

𝑉22

2𝑔− ℎ𝑒 = (

𝑃2

𝛾+ 𝑍2) − (

𝑃1

𝛾+ 𝑍1)

𝑉12

2𝑔−

𝑉22

2𝑔−

(𝑉1 − 𝑉2)2

2𝑔= ℎ2 − ℎ1

But by continuity equation 𝐴1𝑉1 = 𝐴2𝑉2 then:

𝑉12

2𝑔−

𝑉12 (

𝐴1𝐴2

)2

2𝑔−

(𝑉1 − 𝑉1𝐴1𝐴2

)2

2𝑔= ℎ2 − ℎ1

𝑉12

2𝑔−

𝑉12

2𝑔(

𝐴1

𝐴2)

2

−𝑉1

2 (1 − 𝐴1𝐴2

)2

2𝑔= ℎ2 − ℎ1

𝑉12

2𝑔[1 − (

𝐴1

𝐴2)

2

− (1 − 𝐴1

𝐴2)

2

] = ℎ2 − ℎ1

𝑉12

2𝑔[1 − (

𝐴1

𝐴2

)2

− (1 − 2𝐴1

𝐴2

+ (𝐴1

𝐴2

)2

)] = ℎ2 − ℎ1

𝑉12

2𝑔[1 − (

𝐴1

𝐴2)

2

− 1 + 2𝐴1

𝐴2− (

𝐴1

𝐴2)

2

] = ℎ2 − ℎ1

𝑉12

2𝑔[2

𝐴1

𝐴2− 2 (

𝐴1

𝐴2)

2

] = ℎ2 − ℎ1

𝑉1 =√

𝑔(ℎ2 − ℎ1)

𝐴1𝐴2

− (𝐴1𝐴2

)2

𝑄𝑡ℎ𝑒𝑜. = 𝐴1𝑉1

𝑄𝑎𝑐𝑡. = 𝐶𝑑𝑄𝑡ℎ𝑒𝑜.

Where 𝐶𝑑 is the coefficient of discharge.



(2) The venturi meter

Figure 4: Schematic diagram of venture meter

The venturi is manufactured from transparent acrylic materials and follows the classic 21°-10°

convergent-divergent design which forms the basis of most engineering standards for venturi flow

meters.

From consideration of continuity between the mouth of the venturi of area A1 and the throat of area

A2:

𝑄 = 𝐴1𝑉1 = 𝐴2𝑉2

And on introducing the diameter ratio 𝛽 = 𝐷2 𝐷1⁄ then:

𝐴2

𝐴1= 𝛽2 =

𝑉1

𝑉2

Applying Bernoulli’s theorem to the venturi meter between section 1 and section 2, neglecting

losses and assuming the venturi is installed horizontally:

𝑃1

𝛾+

𝑉12

2𝑔=

𝑃2

𝛾+

𝑉22

2𝑔

Rearranging

𝑃1 − 𝑃2

𝛾= 𝐻 =

𝑉22 − 𝑉1

2

2𝑔

And solving for 𝑉2:

𝑉2 =√

2(𝑃1 − 𝑃2)

𝜌 (1 −𝑉1

2

𝑉22)

= √2𝑔𝐻

1 − 𝛽4

The volumetric flow rate is then given by:



𝑄𝑡ℎ𝑒𝑜. = 𝐴2𝑉2 = 𝐴2√2𝑔𝐻

1 − 𝛽4

The actual discharge will be less than this due to losses causing the velocity through the throat to be

less than that predicted by Bernoulli’s theorem, therefore it is necessary to introduce an

experimentally determined coefficient of discharge 𝐶𝑑 . The actual discharge will then be given by:

𝑄𝑎𝑐𝑡. = 𝐶𝑑𝐴2√2𝑔𝐻

1 − 𝛽4

The coefficient of discharge varies with both the Reynolds number and area ratio. Typical values for

a machined venturi meter are between 0.975 and 0.995.

(3) Orifice plate

Figure 5: Schematic diagram of orifice plate

The orifice flow meter consists of a 20mm bore tube with an orifice of 12mm. The downstream

bore of the orifice is chamfered at 40° to provide an effective orifice plate thickness of 0.35mm.

Manometer tappings are positioned 20mm before the orifice and 10mm after the orifice plate.

Due to the sharpness of the contraction in flow area at the orifice plate, a vena contracta is formed

downstream of the throat in which the area of the vena contracta is less than that of the orifice.

Applying the continuity equation between the upstream conditions of area A1 and the vena

contracta of area Ac:

𝑄 = 𝐴1𝑉1 = 𝐴𝑐𝑉𝑐

Where suffix 𝑐 denotes the vena contracta.

Applying Bernoulli’s equation, neglecting losses and assuming a horizontal installation:

𝑃1

𝛾+

𝑉12

2𝑔=

𝑃𝑐

𝛾+

𝑉𝑐2

2𝑔



Rearranging

𝑃1 − 𝑃𝑐

𝛾= 𝐻 =

𝑉𝑐2 − 𝑉1

2

2𝑔

And solving for 𝑉𝑐:

𝑉𝑐 =√

2(𝑃1 − 𝑃𝑐)

𝜌 (1 −𝑉1

2

𝑉𝑐2)

= √2𝑔𝐻

1 − 𝛽4

The volumetric flow rate is then given by:

𝑄𝑡ℎ𝑒𝑜. = 𝐴𝑐𝑉𝑐 = 𝐴𝑐√2𝑔𝐻

1 − 𝛽4

The flow area at the vena contracta is not known and therefore a coefficient of contraction may be

introduced so that

𝐶𝑐 = 𝐴𝑐 𝐴2⁄

The coefficient of contraction will be included in the coefficient of discharge and the equations

rewritten in terms of the orifice area 𝐴2 with any uncertainties and errors eliminated by the

experimental determination of the coefficient of discharge. The volumetric flow rate is then given

by:

𝑄𝑎𝑐𝑡. = 𝐶𝑑𝐴2√

2(𝑃1 − 𝑃2)

𝜌 (1 −𝑉1

2

𝑉22)

= 𝐶𝑑𝐴2√2𝑔𝐻

1 − 𝛽4

The position of the manometer tappings has a small effect on the values of the discharge

coefficients which also vary with area ratio, with pipe size and with Reynolds number. The

variations of 𝐶𝑑 with Reynolds number is tabulated below for orifice plates with 𝛽 = 0.5.

Reynolds

Number 2 x 104 3 x 104 5 x 104 7 x 104 1 x 105 3 x 105 1 x 106 1 x 107

D & D/2

taps 0.6127 0.6102 0.6079 0.6068 0.6060 0.6043 0.6036 0.6032



(4) Flow Round Bends and Elbows

A 90º elbow is placed immediately after the orifice plate and before the rotameter. There is a

constant bore diameter throughout the bend with manometer tappings positioned normal to the

plane of the bend before and after the elbow.

Whenever the direction of the flow is changed at a bend or elbow, the velocity distribution across

the pipe is disturbed. A centrifugal effect causes the maximum velocity to occur towards the outside

of the bend or elbow whilst at the inside of the bend or elbow, the flow is slowed or even reversed

in direction if the flow separates from the wall and a vena-contracta formed. A secondary flow is set

up at right angles to the pipe cross section which increases the velocity gradient and hence the

shear stress of the wall.

The loss of head is related to the velocity head by defining a bend loss coefficient 𝐾𝑏 so that

ℎ𝑏 = 𝐾𝑏

𝑉2

2𝑔

Values of 𝐾𝑏 are related to the pipe friction factors 𝑓 or 𝑓′ by a constant which is dependent on the

ratio of the bend radius to the pipe diameter R/D. This constant may also be treated as an

equivalent length of straight pipe expressed as diameters by using the Darcy Weisbach equation :

ℎ𝑏 = 𝐾𝑏

𝑉2

2𝑔=

𝑓𝐿𝑒

𝐷

𝑉2

2𝑔

𝐻𝑒𝑛𝑐𝑒 𝐾𝑏 =𝑓𝐿𝑒

𝐷

𝑉 = √2𝑔ℎ𝑏

𝐾𝑏

𝑄𝑡ℎ𝑒𝑜. = 𝐴𝑉 = 𝐴√2𝑔ℎ𝑏

𝐾𝑏

𝑄𝑎𝑐𝑡. = 𝐶𝑑𝑄𝑡ℎ𝑒𝑜.

For single 90° bends and elbows, the bend resistance coefficient 𝐾𝑏 and the equivalent length 𝐿𝑒 are

typically:



R/D Elbows 1 1.5 2 4 6 8 10 12 14

𝐿𝑒 30D 20D 14D 12D 14D 17D 24D 30D 34D 38D

𝐾𝑏 30 𝑓 20 𝑓 14 𝑓 12 𝑓 14 𝑓 17 𝑓 24 𝑓 30 𝑓 34 𝑓 38 𝑓

(5) Rotameter

The rotameter utilises a transparent tube and a stainless steel float providing a visual indication of

the flow rate by measuring the position of the float relative to the position of the tube by using the

integral scale, which is calibrated from 1.5 to 10 litres/minute.

𝑄𝑡ℎ𝑒𝑜. = 𝑄𝑟𝑜𝑡.

𝑄𝑎𝑐𝑡. = 𝐾𝑄𝑟𝑜𝑡.

Procedures:

1. Position the water flow measuring apparatus on the horizontal operating surface of the

hydraulics bench using the locating pegs on the top surface of the bench. Connect the delivery

hose from the bench to the inlet connection of the water flow measuring apparatus. Insert the

overflow hose from the inlet tank into the overflow pipe of the volumetric measuring tank.

2. Prepare the instruments such that the water passes sudden enlargement, then venturi meter,

orifice plate, elbow, and finally rotameter .

3. With the flow regulating valve of the bench closed, switch on the bench pump and allow water

to be pumped into the apparatus by controlling the opening of the flow regulating valve until

water just begins to flow into the equipment and just overflows through the air vent above the

rotameter. Ensure that there are no air bubbles trapped in the manometer tubes, if necessary

open the supply valve until water spills out of the top of the manometer tubes so that the water

flushes out all air bubbles.

4. Adjust the supply valve to obtain 6 or 7 readings with the height of water in the left hand

manometer tube increasing in increments of approximately 50mm. The maximum flow which

can be achieved for the experiment is when the height of water in the left hand manometer tube



reaches the top of the manometer scale. At each steady state condition record the heights on

each manometer tube and the flow shown on the rotameter. Also measure the flow using the

volumetric flow tank of the hydraulics bench with a stop watch.

Data & Results:

Volume flow (Liters)

Time (min)

Head at tapping 1 (cm)








Rotameter flow rate



Experiment (5): Flow through small orifices

Introduction:

An orifice is an opening in the side or base of tank or reservoir through which fluid is discharge in

the form of a jet. The discharge will depend up on the head of the fluid (H) above the level of the

orifice. The term small orifice means that the diameter of the orifice is small compared with the

head producing flow.

The analysis of the quantity of water which can be discharged through an orifice is arrived at in a

simple, straightforward manner by the application of Bernoulli's equation. However, experimental

tests typically produce a result which is only some 65% of the solution indicated by the simple

analysis. The study of water flow through an orifice is therefore a classic topic to illustrate the need

for a semi-empirical approach which is so often required in Mechanics of Fluids.

Exercise A: Flow through a small orifice

Purpose:

To investigate the discharge characteristics of circular orifices subjected to a constant head.

Apparatus:

1. Constant head inlet tank (Figure 1).

2. Circular orifices with different diameters.

3. Hydraulic bench.



Figure 1: Constant head inlet tank with circular orifice

Equipment set up:

1. If the hook gauge and scale are to be used to measure the trajectory of horizontal jets then place

the two positioning rails on the worktop of the hydraulics bench engaging them onto the

locating pegs. Ensure that the engraved rail is placed closest to the front of the hydraulics bench

with the engraved side uppermost.

2. Position the constant head inlet tank onto the worktop of the hydraulics bench (over the hook

gauge positioning rails, if fitted) at the left hand side engaging two of the feet of the inlet tank

onto the locating pegs. If the orifice is to be fitted into the side of the inlet tank then it should be

moved to the left so that the right hand support feet engage with the locating pegs.

3. Remove the hexagonal (37mm across flats) bush and adaptor from the side of the inlet tank. Fit

the required orifice into the screwed hole in the side and plug the unused hole using the

blanking plug provided.

4. Connect the hydraulics bench flexible delivery tube to the connection provided on the rear of

the inlet tank base. Insert the flexible overflow take off pipe, which is connected to the boss on

the front of the inlet tank, into the overflow pipe of the volumetric measuring tank.

5. Remove or refit the overflow extention tube (screwed) in the inlet head tank to obtain a

nominal head of 250mm or 500mm above the side orifice.



A. Setting the overflow: Switch on the pump and control the flow rate by either adjusting the

hydraulics bench delivery valve or by adjusting the pump speed. The flow should be adjusted

carefully to produce a small but constant overflow and then fine adjusted to give 250 or 500mm

head as required.

B. Flow measurement: The discharge from the orifice may be measured using the volumetric

measuring tank and taking the time required to collect a quantity of water. The quantity should

be chosen so that the time to collect the quantity is at least 120 seconds to obtain a sufficiently

accurate result.

C. Measurement of jet trajectory: Use the hook gauge to measure the trajectory of the jet.

D. Measurement of head: The scale attached to the side of the inlet tank has its zero level with the

center line of the side outlet boss.

Theory:

Consider a small orifice in the side of a vessel with the head of water above the orifice kept

constant.

Figure 2: Discharge through an orifice

Applying Bernoulli's theorem between the surface of the water 1 and the orifice O yields

𝑍1 +𝑃1

𝜌𝑔+

𝑉12

2𝑔= 𝑍𝑜 +

𝑃𝑜

𝜌𝑔+

𝑉𝑜2

2𝑔

However 𝑃1 = 𝑃𝑜 = atmospheric pressure

𝑉1 = 0 𝑎𝑛𝑑 𝑍1 − 𝑍𝑜 = 𝐻𝑜

hence substituting these into Bernoulli's equation gives



𝐻𝑜 =𝑉𝑜

2

2𝑔

In other words, the theoretical velocity of the water passing through the orifice is given by

𝑉𝑜 = √2𝑔𝐻𝑜

and hence the quantity of water being discharged through the orifice is given by

𝑄 = 𝐴𝑉𝑜 = 𝐴√2𝑔𝐻𝑜

However in practice the discharge is always less than this theoretical amount due to the viscosity of

the fluid, to surface tension and due to resistance of the air. The disparity between the theoretical

discharge velocity and the actual discharge velocity is allowed for by introducing a factor

𝐶𝑣 known as the coefficien of velocity so that

𝑉𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐶𝑣√2𝑔𝐻𝑜

If the discharge from a sharp edged orifice is examined closely, it will be observed that the

minimum diameter of the jet of water discharging from the orifice is smaller than the orifice

diameter. The plane at which this occurs is known as the vena contracta, which is the plane where

stream lines first become parallel. Applying the discharge equation at the vena contracta

𝑄 = 𝐴𝑎𝑐𝑡𝑢𝑎𝑙𝐶𝑣√2𝑔𝐻𝑜

which can be written as

𝑄 = 𝐶𝑐𝐴𝐶𝑣√2𝑔𝐻𝑜

Where: 𝐶𝑐 = Coefficient of contraction.

or more simply as

𝑄 = 𝐶𝑑𝐴√2𝑔𝐻𝑜

Where: 𝐶𝑑 = 𝐶𝑐𝐶𝑣 = Coefficient of discharge .

Typical values of Cd range from 0·6 to 0·65, i.e. the actual flow through a sharp edged orifice is

approximately 60% of the theoretical value. The value of the coefficient of discharge may be

determined by measuring the quantity of water discharged over a period of time whilst the head is

maintained at a constant level.



Procedures:

1. Fit the 5mm diameter orifice into the side of the inlet head tank. Remove the overflow extension

pipe. Start the pump and set up an inlet head of 25cm. Measure the flow rate using the

volumetric measuring tank.

2. Replace the overflow extension pipe and set up an inlet head of 50cm. Measure the flow rate.

3. Repeat the procedure using the 8mm orifice.

Data & Results:

1. Record the results on a copy of the result sheet for discharge characteristics.

2. For each result calculate the flowrate

3. Plot a graph of square root of the head against the flow rate for each orifice diameter, the

results should lie on a straight line passing through the origin to confirm that:

𝑄 𝛼 √𝐻

Measure the slope of each graph and calculate the coefficient of discharge for each orifice from

𝐶𝑑 =𝑆𝑙𝑜𝑝𝑒

𝐴√2𝑔

D (mm) 5 8

H (cm) 50 25 50 25

√𝐇 (m)

V (L)

T (sec)

𝐐𝐚𝐜𝐭𝐮𝐚𝐥 (m3/s)



Exercise B: Trajectory of horizontal jet

Purpose:

To investigate the trajectory of a horizontal jet issuing from an orifice and hence determine the

coefficient of velocity for the orifice.

Apparatus:

1. Constant head inlet tank (Figure 1).

2. Circular orifices with different diameters.

3. Hook gauge and scale.

4. Hydraulic bench.

Theory:

Consider the trajectory of a jet formed by the discharge of water through an orifice mounted in the

side of a tank. The jet will be subjected to a downward acceleration of g due to gravity.

Figure 3: Trajectory of horizontal jet

Taking the origin of co-ordinates at the vena-contracta and applying the laws of motion in the

horizontal and vertical planes then ignoring any effect of air resistance on the jet.

In the horizontal direction𝑥 = 𝑉𝑡

In the vertical direction 𝑦 = 1 2⁄ 𝑔𝑡2

Solving simultaneously by eliminating t

𝑉 =𝑥

𝑡 𝑏𝑢𝑡 𝑡 = √

2𝑦

𝑔



ℎ𝑒𝑛𝑐𝑒 𝑉 = √𝑔𝑥2

2𝑦

𝑏𝑢𝑡 𝑉 = 𝐶𝑣√2𝑔𝐻𝑜

𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐶𝑣 = √𝑥2

4𝑦𝐻𝑜

Procedures:

1. Fit the 5mm diameter orifice into the side of the inlet head tank. Remove the overflow

extension pipe. Start the pump and set up an inlet head of 25cm.

2. Measure the trajectory of the jet using the hook gauge. Record the horizontal and vertical

distances.

3. Replace the overflow extension tube and establish an inlet head of 500mm. Measure the

trajectory of the jet

4. Repeat the experiment using the 8mm diameter orifice.

Data & Results:

1. Draw a graph of y against x to represent the trajectory.

2. Draw a graph of x against √𝑦 and draw the best straight line through the points to represent

the results. Measure the slope of the line and hence calculate the coefficient of velocity from:

𝐶𝑣 =𝑆𝑙𝑜𝑝𝑒

2√𝐻



D (mm) 5 8

H (cm) 25 50 25 50

x (cm) Vertical distance below orifice center line y (cm)

0

5

10

15

20

25

30

35

40

45

50

Slope of graph



Experiment (6): Flow over weirs

Introduction:

In open channel hydraulics, weirs are commonly used to either regulate or to measure the

volumetric flow rate, they are of particular use in large scale situations such as irrigation schemes,

canals and rivers. For small scale applications, weirs are often referred to as notches and invariably

are sharp edged and manufactured from thin plate material.

Purpose:

To investigate the discharge-head characteristics of a rectangular and triangular weirs.

Apparatus:

1. Rectangular and triangular notches (Figure 2).

2. Hydraulic bench.

3. Basket of glass spheres.

Figure 1: Flow over weirs apparatus



Figure 2: Rectangular and triangular notches

Equipment set up:

1. Place the flow stilling basket of glass spheres into the left end of the weir channel and attach

the hose from the bench regulating valve to the inlet connection into the stilling basket.

2. Remove the five thumb nuts which hold the standard weir in place, remove the standard weir

and replace it with the specific weir plate which is to be tested first. Ensure that the square

edge of the weir faces upstream.

A) Flow measurement : The discharge from the weir may be measured using either the rotameter

(if fitted) or by using the volumetric measuring tank and taking the time required to collect a

quantity of water. The quantity should be chosen so that the time to collect the water is at least

120 seconds to obtain a sufficiently accurate result.

B) Measuring the weir datum : Fill the weir channel with water up to the weir plate crest level

and use the hook gauge to measure the level of the water. This will be the zero or datum level

for the weir.

C) Measuring the head : The surface of the water as it approaches the weir will fall, this is

particularly noticeable at high rates of discharge caused by high heads. To obtain an accurate

measure of the undisturbed water level above the crest of the weir it is necessary to place the

hook gauge at a distance at least three times the head.



Figure 3: Rectangular and triangular weirs

Theory:

1. Flow through a rectangular notch

A rectangular notch in a thin square edged weir plate installed in a weir channel as shown in figure

4.

Figure 4: Rectangular notch

Consider the flow in an element of height 𝛿ℎ at a depth ℎ below the surface. Assuming that the flow

is everywhere normal to the plane of the weir and that the free surface remains horizontal up to the

plane of the weir, then

Velocity through element 𝑉 = √2𝑔ℎ

Theoretical discharge through element 𝛿𝑄 = 𝐵 √2𝑔ℎ 𝛿ℎ

Integrating to obtain the total discharge between ℎ = 0 and ℎ = 𝐻

𝑄𝑡ℎ𝑒𝑜. = 𝐵 √2𝑔 ∫ ℎ1/2𝐻

0

𝑑ℎ



Total theoretical discharge 𝑄𝑡ℎ𝑒𝑜. =2

3 𝐵 √2𝑔 𝐻3/2

In practice the flow through the notch will not be parallel and therefore will not be normal to the

plane of the weir. The free surface is not horizontal and viscosity and surface tension will have an

effect. There will be a considerable change in the shape of the nappe as it passes through the notch

with curvature of the stream lines in both vertical and horizontal planes as indicated in figure 5, in

particular the width of the nappe is reduced by the contractions at each end.

Figure 5: Shape of a nappe

The discharge from a rectangular notch will be considerably less, approximately 60% of the

theoretical analysis due to these curvature effects. A coefficient of discharge 𝐶𝑑 is therefore

introduced so that:

𝑄𝑎𝑐𝑡. = 𝑄𝑡ℎ𝑒𝑜.

𝑄𝑎𝑐𝑡. = 𝐶𝑑 2

3 𝐵 √2𝑔 𝐻3/2

ln(𝑄𝑎𝑐𝑡.) = ln (𝐶𝑑 2

3 𝐵 √2𝑔) +

3

2 ln(𝐻)

𝑦 = 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 +3

2 𝑥

𝐶𝑑 =𝑒𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡

23

𝐵√2𝑔

2. Flow through a triangular notch

A sharp edged triangular notch with an included angle of 𝜃 is shown in figure 6.



Figure 6: Triangular notch

Again consider an element of height 𝛿ℎ at a depth ℎ

Breadth of element 𝐵 = 2(𝐻 − ℎ) 𝑡𝑎𝑛𝜃

2

Hence area of element 𝐴 = 2(𝐻 − ℎ) 𝑡𝑎𝑛𝜃

2 𝛿ℎ

Velocity through element 𝑉 = √2𝑔ℎ

Discharge through element 𝛿𝑄 = 2(𝐻 − ℎ) 𝑡𝑎𝑛𝜃

2 √2𝑔ℎ 𝛿ℎ

Integrating to obtain the total discharge between ℎ = 0 and ℎ = 𝐻

𝑄𝑡ℎ𝑒𝑜. = 2𝑡𝑎𝑛𝜃

2 √2𝑔 ∫ [𝐻ℎ1/2 − ℎ3/2]

𝐻

0

𝑑ℎ

𝑄𝑡ℎ𝑒𝑜. =8

15 𝑡𝑎𝑛

𝜃

2 √2𝑔 𝐻5/2

Again, a coefficient of discharge 𝐶𝑑 has to be introduced.

𝑄𝑎𝑐𝑡. = 𝐶𝑑 8

15 𝑡𝑎𝑛

𝜃

2√2𝑔 𝐻5/2

The triangular notch has advantages over the rectangular notch since the shape of the nappe does

not change with head so that the coefficient of discharge does not vary so much. A triangular notch

can also accommodate a wide range of flow rates.

ln(𝑄𝑎𝑐𝑡.) = ln (𝐶𝑑 8

15 𝑡𝑎𝑛

𝜃

2√2𝑔) +

5

2 ln(𝐻)

𝑦 = 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 +5

2 𝑥

𝐶𝑑 =𝑒𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡

815 √2𝑔 𝑡𝑎𝑛

𝜃2



Procedures:

1. Start the pump and slowly open the bench regulating valve until the water level reaches the

crest of the weir and measure the water level to determine the datum level.

2. Adjust the bench regulating valve to give the first required head level of approximately 10mm.

Measure the flow rate using the volumetric tank or the rotameter. Observe the shape of the

nappe.

3. Increase the flow by opening the bench regulating valve to set up heads above the datum level

in steps of approximately 10mm until the regulating valve is fully open. At each condition

measure the flow rate and observe the shape of the nappe.

4. Close the regulating valve, stop the pump and then replace the weir with the next weir to be

tested. Repeat the test procedure.

Data & Results:

1. Record the results on a copy of the results sheet. Record any observations of the shape and type

of nappe paying particular attention to whether the nappe was clinging or sprung clear, and of

the end contraction and general change in shape.

2. Plot a graph of ln(𝑄𝑎𝑐𝑡.) against ln(𝐻) for each weir. Measure the slopes and the intercepts.

Calculate the coefficient of discharge from the intercept and confirm that the slope is

approximately 1.5 for the rectangular notch and 2.5 for the triangular notch.

Trials 1 2 3

𝑯 (mm)

𝑽 (L)

𝑻 (sec)

𝑸𝒂𝒄𝒕.(m3/s)

𝐥𝐧(𝑸𝒂𝒄𝒕.)

𝐥𝐧(𝑯)



Experiment (7): Investigation of Bernoulli's theorem

Introduction:

The flow of a fluid has to conform with a number of scientific principles in particular the

conservation of mass and the conservation of energy. The first of these when applied to a liquid

flowing through a conduit requires that for steady flow the velocity will be inversely proportional

to the flow area. The second requires that if the velocity increases then the pressure must decrease.

Bernoulli's apparatus demonstrates both of these principles and can also be used to examine the

onset of turbulence in an accelerating fluid stream.

Both Bernoulli's equation and the continuity equation are essential analytical tools required for the

analysis of most problems in the subject of mechanics of fluids.

Purpose:

To verify Bernoulli's equation by demonstrating the relationship between pressure head and

kinetic head.

Apparatus:

1. Bernoulli's apparatus (Figure 1).

2. Hydraulic bench.

Figure 1: Bernoulli's apparatus



Bernoulli's apparatus consists essentially of a two dimensional rectangular section convergent

divergent duct designed to fit between constant head inlet tank and variable head outlet tank. An

eleven tube static pressure manometer bank is attached to the convergent divergent duct. The

differential head across the test section can be varied from zero up to a maximum of 450mm. The

test section, which is manufactured from acrylic sheet, is illustrated in figure below

Figure 2: Test section of Bernoulli's apparatus

The convergent divergent duct is symmetrical about the center line with a flat horizontal upper

surface into which the eleven static pressure tappings are drilled. The lower surface is at an angle of

4° 29'. The width of the channel is 6·35 mm. The height of the channel at entry and exit is 19·525

mm and the height at the throat is 6·35 mm. The static tappings are at a pitch of 25 mm distributed

about the centre and therefore about the throat. The flow area at each tapping is tabulated below

the dimensions which are shown in figure 3.



Figure 3: Duct dimensions

Tapping number

1 2 3 4 5 6 7 8 9 10 11

Flow area

(mm2) 102.56 90.11 77.66 65.22 52.77 40.32 52.77 65.22 77.66 90.11 102.56

Equipment set up:

Position the inlet head tank and the variable head outlet tank on the mounting studs provided on

the hydraulic bench working surface and connect the Bernoulli apparatus between them using the

union connections. Connect the bench feed hose to the inlet head tank and attach an overflow hose

to the overflow outlet of the inlet head tank.

Prepare the equipment to the following specification :

Inlet : Constant head inlet tank with overflow extension fitted.

Test section : Bernoulli's apparatus.

Exit : Variable head outlet tank.

Manometer: Insert a sheet of graph paper 440mm high by 325mm wide behind the manometer

tubes to provide an easy method of obtaining a record of the results.



Theory:

Bernoulli's theorem

Bernoulli's equation is applicable to the steady flow of an incompressible and inviscid fluid.

Bernoulli's equation shows that the sum of the three quantities :

𝑃

𝜌𝑔= 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 ℎ𝑒𝑎𝑑

𝑉2

2𝑔= 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ℎ𝑒𝑎𝑑

𝑍 = 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 ℎ𝑒𝑎𝑑

are constant. Therefore the three terms must be interchangeable so that, for example, if in a

horizontal system the velocity head is increased then the pressure head must decrease

𝑃

𝜌𝑔+

𝑉2

2𝑔+ 𝑍 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Loss of head due to friction

If the fluid is not inviscid then there will be a small loss of head due to friction within the fluid and

between the fluid and the walls of the passage. Bernoulli's equation can then be modified by the

inclusion of the frictional head loss 𝐻𝑓

𝑃1

𝜌𝑔+

𝑉12

2𝑔+ 𝑍1 =

𝑃2

𝜌𝑔+

𝑉22

2𝑔+ 𝑍2 + 𝐻𝑓

Where Bernoulli's equation has been written in the integrated form and has been applied between

the upstream section 1 and the downstream section 2.

Since the passage is horizontal 𝑍1 = 𝑍2. At two positions of equal area the two velocities will be

equal thus the equation reduces to

𝐻𝑓 =𝑃1

𝜌𝑔−

𝑃2

𝜌𝑔

Most of the pressure loss in the converging part of the duct is recovered in the diverging part of the

duct. The degree of pressure recovery is given by :

𝑅 =𝐻𝑜𝑢𝑡𝑙𝑒𝑡 − 𝐻𝑡ℎ𝑟𝑜𝑎𝑡

𝐻𝑖𝑛𝑙𝑒𝑡 − 𝐻𝑡ℎ𝑟𝑜𝑎𝑡



The continuity equation

The continuity equation is a statement of the conservation of mass. Consider the steady flow of a

fluid through a streamtube of varying cross sectional area as shown in figure 4. For steady flow the

mass of fluid entering the streamtube at section 1 must equal the mass of fluid leaving the

streamtube at section 2. The mass flow rate of fluid at any section along the streamtube must be

constant so that :

�̇� = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

�̇� = 𝜌𝐴𝑉 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

For an incompressible fluid the density is constant and the continuity equation can be written as :

𝐴𝑉 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

For an incompressible fluid flowing in a converging duct it follows that as the area reduces then the

velocity must increase, whilst in a diverging duct as the area increases then the velocity must

decrease. Applying Bernoulli's equation if the velocity increases then the pressure must decrease

whilst as the velocity decreases the pressure must increase.

Figure 4: Element in a streamtube

Procedures:

1. Start the pump and initiate a flow of water through the test section. Regulate the flow to the

inlet head tank so that there is a small but steady overflow from inlet tank. Adjust the swivel

tube of the outlet tank to obtain a differential head of 50mm.



2. Measure the height of the water level in each manometer tube by marking the paper positioned

behind the tubes and record on the test sheet. Measure the time taken to fill the bench

measuring tank from zero to 10 liters and record.

3. Increase the differential head between the inlet and outlet head tanks by 5O mm increments,

until the water level in the centre manometer tubes drops off the scale. For each condition,

record the heights of liquid in the manometer tubes by once again marking the paper positioned

behind the tubes and measure the flow rate.

Data & Results:

1. Record the results on a copy of the result sheet provided.

2. Calculate the flow rate for each set of results.

3. For each set of results calculate at the cross-section adjacent to each manometer tube and the

flow velocity.

4. Plot a graph of head against distance and also 𝐻 + 𝑉2

2𝑔 against distance.

Quantity of water collected (liters)

Time to collect water (Sec)

Volumetric flow rate 𝑸 (m3/s)

Tapping number

1 2 3 4 5 6 7 8 9 10 11

Flow area

(mm2) 102.56 90.11 77.66 65.22 52.77 40.32 52.77 65.22 77.66 90.11 102.56

Static head 𝑯

Velocity (m/s)

Total head



Experiment (8): Minor losses

Introduction:

Minor (secondary) head losses occur at any location in a pipe system where streamlines are not

straight, such as at pipe junctions, bends, valves, contractions, expansions, and reservoir inlets and

outlets. The specific hydraulic model that we are concerned with for this experiment is the energy

losses in bends and fittings. A full description of the apparatus is given later in these texts.

Purpose:

To determine the loss factors for flow through a range of pipe fittings including bends, a

contraction, an enlargement and a gate valve.

Apparatus:

1. Energy losses in bends and fittings apparatus (Figure 1).

2. Hydraulic bench.

3. Clamps for pressure tapping connection tubes.

Figure 1: Energy losses in bends and fittings apparatus



The following dimensions from the equipment are used in the appropriate calculations. If required

these values may be checked as part of the experimental procedure and replaced with your own

measurements.

Internal diameter of pipework d = 0.0183 m

Internal diameter of pipework at enlargement outlet and contraction inlet d = 0.024 m

Figure 2: Schematic diagram of energy losses in bends and fittings apparatus

The accessory is designed to be positioned on the side channels of the hydraulics bench top

channel.

The following fittings are connected in a series configuration to allow direct comparison:

Long bend.

Area enlargement.

Area contraction.

Elbow bend.

Short bend.

Valve fitting.

Mitre bend.

Flow rate through the circuit is controlled by a flow control valve.



Pressure tappings in the circuit are connected to a twelve bank manometer, which incorporates an

air inlet/outlet valve in the top manifold. An air bleed screw facilitates connection to a hand pump.

This enables the levels in the manometer bank to be adjusted to a convenient level to suit the

system static pressure.

A clamp which closes off the tappings to the mitre bend is introduced when experiments on the

valve fitting are required. A differential pressure gauge gives a direct reading of losses through the

gate valve.

Commissioning:

1. Locate the apparatus over the moulded channel in the top of the bench and ensure that the base

plate is horizontal. Adjust the feet on the base plate if necessary.

2. Connect the flexible inlet tube at the left hand end to the quick release fitting in the bed of the

channel.

3. Place the free end of the flexible outlet tube in the volumetric tank of the bench.

4. Fully open the gate valve and the outlet flow control valve at the right hand end of the

apparatus.

5. Close the bench flow control valve then start the service pump.

6. Gradually open the bench flow control valve and allow the pipework to fill with water until all

air has been expelled from the pipework.

7. In order to bleed air from the pressure tapping points and the manometers, close both the

bench flow control valve and the outlet flow control valve and open the air bleed screw. Remove

the cap from the adjacent air inlet/outlet connection. Connect a length of small bore tubing from

the air valve to the volumetric tank. Now, open the bench flow control valve and allow flow

through the manometers to purge all air from them; then, tighten the air bleed screw and partly

open both the bench valve and the outlet flow control valve. Next, open the air bleed screw

slightly to allow air to enter the top of the manometers, re-tighten the screw when the

manometer levels reach a mid height.

8. Gradually increase the volume flowrate until the pattern just fills the range of the manometer

(adjust the bench flow control valve and the outlet flow control valve in combination to

maintain all of the readings within the range of the manometer). If the pattern is too low on the



manometer, open the bench flow control valve to increase the static pressure. If the pattern is

too high open the outlet flow control valve on the apparatus to reduce the static pressure.

9. These levels can be adjusted further by using the air bleed screw and the hand pump supplied.

The air bleed screw controls the air flow through the air valve, so when using the hand pump,

the bleed screw must be open. To retain the hand pump pressure in the system, the screw must

be closed after pumping.

10. If the levels in the manometer are too high then the hand pump can be used to pressurise the

top manifold. All levels will decrease simultaneously but retain the appropriate differentials.

11. If the levels are too low then the hand pump should be disconnected and the air bleed screw

opened briefly to reduce the pressure in the top manifold. Alternatively the outlet flow control

valve can be closed to raise the static pressure in the system which will raise all levels

simultaneously.

12. If the level in any manometer tube is allowed to drop too low then air will enter the bottom

manifold. If the level in any manometer tube is too high then water will enter the top manifold

and flow into adjacent tubes.

Note: If the static pressure in the system is excessive, Eg. with the bench flow control valve fully

open and the outlet flow control valve almost closed, it will not be possible to use the hand

pump to lower the levels in the manometer tubes. The valves should be adjusted to provide

the required flowrate at a lower static pressure.

13. In operation the pressure drop across each fitting is compared with the volume flowrate which

is measured using the volumetric measuring tank and a stopwatch.

14. To check the operation of the differential pressure gauge associated with the gate valve, close

off the flexible connecting tubes to the mitre bend pressure tappings using the clamps supplied

before closing the gate valve (to prevent air being drawn into the system).

15. Open the bench flow control valve and the outlet flow control valve. As the gate valve is closed

the differential pressure across the valve will be displayed on the gauge.

16. Close the bench flow control valve then switch off the service pump, the Energy Losses in Bends

apparatus is ready for use.



Equipment set up:

1. Set up the losses apparatus on the hydraulic bench so that its base is horizontal (this is

necessary for accurate height measurements from the manometers). Connect the test rig inlet to

the bench flow supply and run the outlet extension tube to the volumetric tank and secure it in

place.

2. Open the bench valve, the gate valve and the flow control valve and start the pump to fill the test

rig with water. In order to bleed air from pressure tapping points and the manometers, close

both the bench valve and the test rig flow control valve and open the air bleed screw and

remove the cap from the adjacent air valve. Connect a length of small bore tubing from the air

valve to the volumetric tank. Now, open the bench valve and allow flow through the

manometers to purge all air from them; then, tighten the air bleed screw and partly open both

the bench valve and the test rig flow control valve. Next, open the air bleed screw slightly to

allow air to enter the top of the manometers, re-tighten the screw when the manometer levels

reach a convenient height.

3. Check that all manometer levels are on scale at the maximum volume flow rate required

(approximately 17 liters/minute). These levels can be adjusted further by using the air bleed

screw and the hand pump supplied. The air bleed screw controls the air flow through the air

valve, so when using the hand pump, the bleed screw must be open. To retain the hand pump

pressure in the system, the screw must be closed after pumping.

Theory:

The energy loss which occurs in a pipe fitting (so-called secondary loss) is commonly expressed in

terms of a head loss (h, metres) in the form:

∆ℎ =𝐾𝑣2

2𝑔

Where K = the loss coefficient and v = mean velocity of flow into the fitting.

Because of the complexity of flow in many fittings, K is usually determined by experiment. For the

pipe fitting experiment, the head loss is calculated from two manometer readings, taken before and

after each fitting, and K is then determined as :



𝐾 =∆ℎ

𝑣2/2𝑔

Due to the change in pipe cross-sectional area through the enlargement and contraction, the system

experiences an additional change in static pressure. This change can be calculated as :

𝑣12

2𝑔−

𝑣22

2𝑔

To eliminate the effects of this area change on the measured head losses, this value should be added

to the head loss readings for the enlargement and the contraction. Note that (ℎ1 − ℎ2) will be

negative for the enlargement and (𝑣1

2

2𝑔−

𝑣22

2𝑔 ) will be negative for the contraction.

For the gate valve experiment, pressure difference before and after the gate is measured directly

using a pressure gauge. This can then be converted to an equivalent head loss using the equation :

1 bar = 10.2 m water

The loss coefficient may then be calculated as above for the gate valve.

Procedures:

It is not possible to make measurements on all fittings simultaneously and, therefore, it is necessary

to run two separate tests.

Exercise A measures losses across all pipe fittings except the gate valve, which should be

kept fully open.

1. Adjust the flow from the bench control valve and, at a given flow rate, take height readings from

all of the manometers after the levels have steadied.

2. In order to determine the volume flow rate, you should carry out a timed volume collection

using the volumetric tank. This is achieved by closing the ball valve and measuring (with a

stopwatch) time taken to accumulate a known volume of fluid in the tank, which is read from

the sight glass. You should collect fluid for at least one minute to minimise timing errors.

3. Repeat this procedure to give a total of at least five sets of measurements over a flow range from

approximately 8 - 17 litres per minute.

Exercise B measures losses across the gate valve only.

1. Clamp off the connecting tubes to the mitre bend pressure tappings (to prevent air being drawn

into the system).



2. Start with the gate valve closed and open fully both the bench valve and the test rig flow control

valve. Now open the gate valve by approximately 50% of one turn (after taking up any

backlash). For each of at least 5 flow rates, measure pressure drop across the valve from the

pressure gauge; adjust the flow rate by use of the test rig flow control valve. Once

measurements have started, do not adjust the gate valve.

3. Determine the volume flow rate by timed collection.

4. Repeat this procedure for the gate valve opened by approximately 70% of one turn and then

approximately 80% of one turn.

Data & Results:

1. Record the results on a copy of the result sheet provided.

2. For exercise A, plot graphs of head loss (∆ℎ) against dynamic head, and K against volume flow

rate Q.

3. For exercise B, plot graphs of equivalent head loss (∆ℎ) against dynamic head, and K against Q.

Table 1 : Raw data for all fittings except gate valve

Case No. I II III IV V

Volume (L)

Time (sec)

Pie

zom

ete

r R

ea

din

gs

(mm

)

Enlargement 1

2

Contraction 3

4

Long Bend 5

6

Short Bend 7

8

Elbow 9

10

Mitre Bend 11

12



Table 2 : Raw data for gate valve


50

% O

pe

ned

Volume (L)

Time (sec)

Gauge

Reading

(bar)

Red

(upstream)

Black

(downstream)

70

% O

pe

ned

Volume (L)

Time (sec)

Gauge

Reading

(bar)

Red

(upstream)

Black

(downstream)

80

% O

pe

ned

Volume (L)

Time (sec)

Gauge

Reading

(bar)

Red

(upstream)

Black

(downstream)



Table 4 : Loss coefficient for all fittings except gate valve


Q (m3/sec)

V (m/s)

V2/2g (m)

Lo

ss C

oe

ffic

ien

ts

Enlargement

Contraction

Long Bend

Short Bend

Elbow

Mitre Bend

Table 3 : Minor head losses of all fittings except gate valve


Q (m3/sec)

V (m/s)

V2/2g (m)

Min

or

He

ad

Lo

sse

s (m

)

Enlargement

Δh

Δh +V12/2g -

V22/2g

Contraction

Δh

Δh +V12/2g -

V22/2g

Long Bend

Short Bend

Elbow

Mitre Bend



Table 5 : Equivalent minor head loss and Loss coefficient for gate valve


50

% O

pe

ned

Q (m3/sec)

V (m/sec)

V2/2g (m)

Minor Head

Loss (m)

Loss

Coefficient

70

% O

pe

ned

Q (m3/sec)

V (m/sec)

V2/2g (m)

Minor Head

Loss (m)

Loss

Coefficient

80

% O

pe

ned

Q (m3/sec)

V (m/sec)

V2/2g (m)

Minor Head

Loss (m)

Loss

Coefficient



Experiment (9): Centrifugal pump

Introduction:

Pumps fall into two main categories: positive displacement pumps and rotodynamic pumps. In a

positive displacement pump, a fixed volume of fluid is forced from one chamber into another. The

centrifugal pump is, by contrast, a rotodynamic machine. Rotodynamic (or simply dynamic)

pumps impart momentum to a fluid, which then causes the fluid to move into the delivery

chamber or outlet. Turbines and centrifugal pumps all fall into this category.

Centrifugal pumps are widely used in industrial and domestic situations. Due to the characteristics

of this type of pump, the most suitable applications are those where the process liquid is free of

debris, where a relatively small head change is required, and where a single operating capacity or a

narrow range of capacities is required. The general design is usually simple with few mechanical

parts to fail, however, and it is possible to operate a centrifugal pump outside ideal parameters

while maintaining good reliability.

The centrifugal pump converts energy supplied from a motor or turbine, first into kinetic energy

and then into potential energy.

The motor driving the impeller imparts angular velocity to the impeller. The impeller vanes then

transfer this kinetic energy to the fluid passing into the center of the impeller by spinning the fluid,

which travels outwards along the vanes to the impeller casing at increasing flow rate.

This kinetic energy is then converted into potential energy (in the form of an increase in head) by

the impeller casing (a volute or a circular casing fitted with diffuser vanes) which provides a

resistance to the flow created by the impeller, and hence decelerates the fluid. The fluid decelerates

again in the outlet pipe. As the mass flow rate remains constant, this decrease in velocity produces a

corresponding increase in pressure as described by Bernoulli ’s equation.

Exercise A

Purpose:

To create head, power and efficiency characteristic curves for a centrifugal pump.



Apparatus:

1. Centrifugal pump demonstration unit (Figure 1).

2. Interface device.

3. PC with a suitable software installed.

Figure 1: Centrifugal pump demonstration unit

Figure 2: Interface of one of the suitable softwares

Theory:

The operating characteristics of a centrifugal pump may be described or illustrated by using graphs

of pump performance. The three most commonly used graphical representations of pump

performance are:



Change in total head produced by the pump, Ht

Power input to the pump, Pm

Pump efficiency, E

The change in total head produced as a result of the work done by pump can be calculated as:

Ht = Change in pressure head + change in velocity head + change in elevation = Hs + Hv + He

Where:

Hs = Change in pressure head = (𝑃𝑜𝑢𝑡−𝑃𝑖𝑛)

𝜌𝑔

Where Pin is the fluid pressure at inlet in Pa and Pout is the fluid pressure at outlet in Pa.

Hv = Change in velocity head = (𝑉𝑜𝑢𝑡−𝑉𝑖𝑛)2

2𝑔

Where Vin is the fluid velocity at inlet in m/s and Vout is the fluid velocity at outlet in m/s.

He = Change in elevation.

The vertical distance between inlet and outlet, which is O.075m for the available pump.

The mechanical power input to the pump may be calculated as:

Pm = rotational force x angular distance = 2.π.n.t/60

where n is the rotational speed of pump in revolutions per minute and t is the shaft torque in N.m .

The efficiency of the pump may be calculated as :

𝐸 = 100 ×𝑃ℎ

𝑃𝑚

Where Q is the volume flow rate in m3/s, and Pm is the mechanical power absorbed by pump:

Each of these parameters is measured at constant pump speed, and is plotted against the volume

flow rate Q through the pump. An example of this type of graphical representation of pump

performance is given in Figure 2.



Figure 3: Characteristic curves for a centrifugal pump

Examining Figure 3, the general performance of the pump can be determined.

The Ht-Q curve shows the relationship between head and flow rate. The head decreases as flow rate

increases. This type of curve is referred to as a rising characteristic curve. A stable head-capacity

characteristic curve is one in which there is only one possible flow rate for a given head, as in the

example here.

The Pm-Q curve shows the relationship between the power input to the pump and the change in

flow rate through the pump. Outside the optimum operating range of the pump this curve flattens,

so that a large change in pump power produces only a small change in flow velocity.

The E-Q curve shows the pump capacity at which the pump operates most efficiently. In the

example here, the optimum operating capacity is 0.7 dm3/s, which would give a head of 0.95m.

When selecting a pump for an application where the typical operating capacity is known, a pump

should be selected so that its optimum efficiency is at or very near that capacity.

Equipment set up:

If the equipment is not yet ready for use, proceed as follows:

1. Ensure the drain valve is fully closed.

2. If necessary, fill the reservoir to within 20cm of the top rim.

3. Ensure the inlet valve and gate valve are both fully open.

4. Ensure the equipment is connected to the interface device and the interface device is connected

to a suitable PC. The red and green indicator lights on the interface device should both be

illuminated.



5. Ensure the interface device is connected to an appropriate mains supply, and switch on the

supply.

6. Run the software. Check that 'IFD: OK' is displayed in the bottom right corner of the screen and

that there are values displayed in all the sensor display boxes on the mimic diagram.

Procedures:

1. Switch on the interface device, then switch on the pump within the software using the pump

on/standby button.

2. Using the software, set the speed to 80%. The interface will increase the pump speed until it

reaches the required setting. Allow water to circulate until all air has been flushed from the

system. Slightly closing and opening the inlet valve and gate valve a few times will help in

priming the system and eliminating any bubbles caught within the valve mechanism. Leave the

inlet valve fully open.

3. In the results table, rename the spreadsheet (Select Format > Rename Sheet) to 80%.

4. Close the gate valve to give a flow rate Q of 0. (Note that the pump will not run well with the

gate valve closed or nearly closed, as the back pressure produced is outside normal operating

parameters. The pump should begin to run more smoothly as the experiment progresses).

5. Select the (Go) icon to record the sensor readings and pump settings on the results table of the

software.

6. Open the gate valve to allow a low flow rate. Allow sufficient time for the sensor readings to

stabilise then select the (Go) icon to record the next set of data.

7. Increase the flow in small increments, allowing the sensor readings to stabilise then recording

the sensor and pump data each time.

8. Using the arrow buttons on the software display, reduce the pump speed to 0%. Select "Save" or

"Save as…" from the "file" menu and save the results with a suitable file name.

9. Switch off the pump within the software using the power on/standby button, then switch off

the interface device and close the software.



Results:

Using the graph facility, plot a graph of head against flow rate. On the secondary axis plot a graph of

mechanical power and efficiency against flow rate.

Alternatively, the results sheet may be exported to an alternative spreadsheet program (or results

may be manually plotted on graph paper) to produce a chart.

Conclusion:

Examine and describe the shapes of the graphs obtained, relating this to the changing performance

of the pump as the flow rate changes. Locate the point of maximum efficiency and the flow rate at

which it occurs.

Exercise B

Purpose:

To obtain a head - flow curve for the piping system through which the fluid is to be pumped. To

determine the operating point of the pump .

Apparatus:

1. Centrifugal pump demonstration unit (Figure 1).


3. PC with a suitable software installed.

4. Tape measure.

Theory:

System analysis for a pumping installation is used to select the most suitable pumping units and to

define their operating points. System analysis involves calculating a head - flow curve for the

pumping system (valves, pipes, fittings, etc.) and using this curve in conjunction with the

performance curves of the available pumps to select the most appropriate pump(s) for use within

the system.



The system curve is a graphic representation of the flow rate in the system with respect to system

head. It represents the relationship between flow rate and hydraulic losses in a system. Such losses

are due to the system design (e.g. bends and fittings, surface roughness) and operating conditions

(e.g. temperature).

Assuming that:

Flow velocity is proportional to volume flow rate.

Losses in the system are proportional to the square of the flow velocity.

It follows that system head loss must be proportional to the square of the volume flow rate, and the

system head-flow graph will therefore be parabolic in shape.

A predicted system head-flow curve may be calculated using standard coefficients for the system

design and a measurement of the system head at zero flow. The simplest method of calculation is

Hazen-Williams equation for major pipe losses. This uses a coefficient based on the pipe material,

along with values for the pipe length and diameter and the flow rate within the system. This is not

the most accurate method and is only valid for water flowing at ordinary temperatures (approx. 5

to 40°C), but it is sufficient for many practical purposes. Accuracy may be improved by adding a

second equation for calculating the minor losses due to pipe fittings. The resulting calculation is as

follows:

h = total head loss in system = hf + hm

where :

hf = major losses in pipe = 𝐿 [𝑉

0.85𝐶(

4

𝑑)

0.63]

1/0.54

hm = minor losses in pipe = 𝑘𝑉2

2𝑔

Where L is the total pipe length, V is the flow velocity, d is the pipe diameter, C is a coefficient

obtained from standard values (acrylic pipe = 140) and k is a coefficient obtained from standard

values, as follows:

Pipe entrance 0.5 (reservoir to pipe)

Pipe exit 1 (pipe to reservoir)

90° Bend 0.3

45° Bend 0.4

Ball Valve Negligible when fully open



Gate Valve 2.1 (half open)

As noted previously, pump characteristic curves illustrate the relationship between head,

discharge, efficiency and power over a wide range of possible operating conditions, but they do not

indicate at which point on the curves the pump will operate. The operating point (or duty point) is

found by plotting the pump head-discharge curve with the system head-flow curve. The

intersection of the two curves gives the duty point for the pump in that system, as illustrated in

figure 4 below.

It will be seen that the optimum operating condition is achieved if this operating point coincides

with the maximum point in the efficiency-discharge curve of the pump.

Figure 4: Definition sketch for determination of pump operating point

Equipment set up:







illuminated.


supply.





Procedures:

1. Measure the pipe length of the system, not including the path through the pump. Keep the

measurement as close to the centerline of the pipework as possible. Enter the result in meters

on the mimic diagram screen in the box for Pipe Length.

2. Add up the coefficient values for all the pipe fittings in the system. Do not include the entry and

exit into the pump but do include the pipes entering and exiting the reservoir, all bends. valves

and flow meter. Assume the pressure sensors have no effect on the coefficient. Enter the total

on the mimic diagram screen of the software in the box for coefficient k.

3. Switch on the interface device, then switch on the pump within the software. In the software,

set the pump to 100%.

4. Allow water to circulate until all air has been flushed from the system.


software.

6. Set the pump to 90%, and select the (Go) icon again.

7. Repeat while reducing the pump speed in 10% steps, recording a data sample at each step, with

a final set of data taken at 0%.

8. Select the (New) icon to create a new results sheet.

9. Set the pump to 70% (the design speed of the pump).

10. Select the (Go) icon to record the sensor readings and pump settings on the new results table in

the software.

11. Close the gate valve to give a small but noticeable reduction in flow rate. Allow a few moments

for the system to settle then select the (Go) icon again.

12. Repeat while closing the gate valve in small increments, recording the data at each step, until

the valve is fully closed.

13. Set the pump to 0%, then select 'Save' or 'Save As .. .' from the 'File' menu and save the results

with a suitable file name (e.g. the date and the exercise).

14. Switch off the pump within the software, then switch off the interface device.



Results:

On a base of flow rate, on one y-axis plot the system head from the first set of data and the total

head from the second set of data. On the second y-axis plot the pump efficiency from the second set

of data.

Mark the point on the graph at which the system head curve and pump curve (total head curve)

intersect to obtain the duty point of the pump.

Conclusion:

Compare the graph obtained with the example given.

Compare the point of intersection of the system head curve and pump head curve with the curve

for pump efficiency.

The k value for the gate valve was greatly simplified for this experiment. A more accurate value varies

depending on whether the valve is fully open or partially open (0.26 for 1/4 closed, 2.1 for 1/2 open, 17

for 3/4 closed). Discuss the effect on the results obtained on having used a single average value for

the gate valve.



Experiment (10): Series and parallel pumps

Introduction:

Centrifugal pumps are often used together to enhance either the flow rate or the delivery pressure

beyond that available from the single pump. For some piping system designs, it may be desirable to

consider a multiple pump system to meet the design requirements. Two typical options include

parallel and series configurations of pumps which require a specific performance criteria. In serial

operation the heads of the pumps are added and in parallel operation the flow rates (capacities) of

the pumps are added.

The experimental unit provides the determination of the characteristic behavior for single

operation and interaction of two pumps. The apparatus consists of a tank and pipework which

delivers water to and from two identical centrifugal pumps. The unit is fitted with electronic

sensors which measure the process variables. Signals from these sensors are sent to a computer via

an interface device, and the unit is supplied with data logging software as standard.

Purpose:

To investigate the result on discharge and total head of operating pumps in series and in parallel.

Apparatus:

1. Series and parallel pumps demonstration unit (Figure 1).


3. PC with suitable software installed.



Figure 1: Series and parallel pump demonstration unit

Figure 2: Interface of one of the suitable softwares

Setting the flow path

The system may be configured to drive flow using single,

series or parallel pumps. The system valves are as shown:



Valves should be set to configure the system as follows. The software should also be set to the

corresponding flow path to ensure that the correct calculations are performed.

Single Pump:

Series Pumps:

Parallel Pumps:

The two pumps are motor-driven centrifugal pumps. On pump 1, the speed of the motor is

adjustable to give a range of 0 to 100%, allowing operation as a single pump for pump performance

analysis. Pump 2 is an identical model but is run at its design speed, which is equivalent to a setting

of 80% on the variable-speed pump for a 50Hz electrical supply, or 100% for a 60 Hz supply.



Exercise A (Series pumps)

Theory:

A single pump may be insufficient to produce the performance required. Combining two pumps

increases the pumping capacity of the system. Two pumps may be connected in series, so that water

passes first through one pump and then through the second. When two pumps operate in series, the

flow rate is the same as for a single pump but the total head is increased. The combined pump head-

capacity curve is found by adding the heads of the single pump curves at the same capacity.

Figure 3: Pump curve for two pumps in series

Equipment Set Up:




3. Check that both pumps are fitted with similar impellers (the impellers may be viewed through

the clear cover plate of each pump).


5. Set the 3-way valve for flow in series.



illuminated.




supply. Switch on the interface device.



Procedures:

1. Both pumps must be used at the same setting in this experiment to ensure identical

performance. As the speed of pump 2 is fixed at its design operational point, pump 1 should be

set to match - select 80% for a 50Hz electrical supply, or 100% for 60 Hz.


3. If results are already available for a single pump across its full flow range, load those results into

the software now and jump to the section of this exercise using two pumps. If results are not

available then proceed as follows:

Single pump performance:

a. Close pump 2 outlet valve and open pump 1 outlet valve.

b. In the software, on the mimic diagram, set the 'Mode' to 'Single' by selecting the appropriate

radio button.

c. Rename the results sheet to 'Single'.

d. Select the (Go) icon to record the sensor readings and pump settings on the results table of the

software.

e. Close the gate valve to reduce the flow by a small amount. Select the (Go) icon again.

f. Continue to close the gate valve to give incremental changes in flow rate, recording the sensor

data each time.

g. After taking the final set of data, fully open the gate valve.

Series pump performance:

4. Create a new results sheet using the (New) icon. Rename this new results sheet to 'Series'. In the

software, on the mimic diagram, set the 'Mode' to 'Series' by selecting the appropriate radio

button.

5. Open pump 2 outlet valve, close pump 1 outlet valve and wait for any air to circulate out of the

system.




software.

7. Close the gate valve to reduce the flow by a small increment. Select the (Go) icon again.

8. Continue to close the gate valve to give incremental changes in flow rate, recording the sensor

data each time.

9. After taking the final set of data, fully open the gate valve again.

Pumps in parallel exercise may be performed immediately after this experiment without closing the

software; otherwise, save the results and ensure they are available for exercise B when required. (It

may also be advisable to save the results from this exercise before starting exercise B even if

continuing straight on, to ensure that the data is not lost in the event of a computer failure. The

results sheet may be overwritten with the combined results once exercise B has been completed).

Results:

On a base of flow rate, plot a graph of total head gain for the single pump and for two pumps

connected in series. Calculate the difference between the total head gain for single and series

pumps.

Conclusion:

Does the total head gain for the two pumps in series match the theoretical prediction of twice the

head gain for a single pump (assuming the two pumps used gave identical performance)?

Exercise B (Parallel pumps)

Theory:

A single pump may be insufficient to produce the performance required. Combining two pumps

increases the pumping capacity of the system. Two pumps may be connected in parallel, so that half

the flow passes through one of the pumps and the other half through the second pump. When two

pumps operate in parallel, the total head increase remains unchanged but the flow rate is increased.

The head-capacity curve is found by adding the capacities of the single pump curves at the same



head.

Figure 4: Pump curve for two pumps in parallel

Equipment Set Up:




3. Check that both pumps are fitted with similar impellers (the impellers may be viewed through

the clear cover plate of each pump).


5. Set the 3-way valve for flow in parallel.

6. Fully open the pump 1 outlet valve and pump 2 outlet valve. Opening both valves fully ensures

that the outlet pressure on both pumps is equal.



illuminated.


supply. Switch on the interface device.



10. In the software, on the mimic diagram, set the "Mode" to "parallel" by selecting the appropriate

radio button.



Procedures:

1. Both pumps must be used at the same setting in this experiment, to ensure identical

performance. As the speed of pump 2 is fixed at its design operational point, pump 1 should be

set to match - select 80% for a 50Hz electrical supply, or 100% for 60 Hz.


3. Exercise A should be performed before this experiment, and the results loaded into the software

if the software is not still open from that exercise. If the software is still open from exercise A,

then create a new results sheet by selecting the (New) icon. Rename the current (blank) results

sheet to 'Parallel'.


software.

5. Close the gate valve to reduce the flow by a small increment. Select the (Go) icon again.

6. Continue to close the gate valve to give incremental changes in flow rate, recording the sensor

data each time.

7. After taking the final set of data, fully open the gate valve. Set Pump 1 to 0% and switch off both

pumps.

Results:

On a base of flow rate, plot a graph of total head gain for the single pump and for two pumps

connected in parallel. Calculate the difference between the capacity for single and parallel pumps.

Conclusion:

Does the total head gain for the two pumps in parallel match the theoretical prediction of twice the

capacity of a single pump (assuming the two pumps used gave identical performance)?

Compare the graphs for pumps in series and pumps in parallel, and describe the similarities and

differences.



Experiment (11): Cavitation demonstration

Introduction:

Cavitation is demonstrated by forcing water through a contraction so that the static pressure of

the water reduces. When the static pressure is reduced, any dissolved air in the water is released

as bubbles. When the static pressure is reduced to the vapour pressure of the water, violent

cavitation (vaporisation of the water) occurs. By restricting the flow downstream of the test

section, the static pressure in the test section is increased. When the static pressure is maintained

above the vapour pressure, increased flowrate is possible through the test section without

cavitation occurring.

Purpose:

To demonstrate the appearance and sound of cavitation in a hydraulic system.

To demonstrate the conditions for cavitation to occur (liquid at its vapour pressure).

To show how cavitation can be prevented by raising the static pressure of a liquid above its vapour

pressure.

Apparatus:

1. Cavitation demonstration apparatus.

The following dimensions from the equipment are applicable:

Upstream diameter = 16mm.

Contraction included angle = 20°.

Contraction length = 33mm.

Throat diameter = 4.5mm.

Throat length = 20mm.

Expansion included angle = 12°.

Expansion length = 55mm.

Downstream diameter = 16mm.



2. Hydraulics Bench to supply water to the cavitation demonstration apparatus (the flow of water

can be measured by timed volume collection).

3. A 0 - 50°C thermometer to determine the temperature of the water.

4. A stopwatch to time the accumulation of water in the volumetric tank.

Figure 1 : Cavitation demonstration apparatus

1. Bourdon gauge for the pressure at the upstream (P1), Range

0-2 Bar (gauge).

2. Bourdon gauge for the pressure at the throat (P2), Range 0-

1 Bar vacuum.

3. Bourdon gauge for the pressure at the downstream (P3),

Range 0-1 Bar (gauge).

4. Support plate.

5. A quarter-turn ball valve (ball perforated to prevent

excessive back pressure in the test section when the valve is

fully closed).

6. Right-hand / outlet end.

7. The test section (circular venture-shaped).

8. Left-hand / inlet end.

9. Diaphragm valve.

Figure 2 : Front view of cavitation demonstration apparatus

Theory:

In accordance with Bernoulli’s equation, the pressure at the throat of the venturi-shaped

test section falls as the velocity of the water increases. However, the pressure can only fall as far as

the vapour pressure of the water at which point the water starts to vaporise - cavitation occurs. Any

further increase in velocity cannot reduce the pressure below the vapour pressure, so the water

vaporises faster - stronger cavitation occurs and Bernoulli’s equation is not obeyed.



Equipment Set Up:

Locate the cavitation demonstration apparatus on top of the hydraulics bench.

Connect the flexible tube at the left hand end of the cavitation demonstration apparatus to the

water outlet on the hydraulic bench (it will be necessary to remove the yellow quick release

connector before screwing the fitting onto the outlet). To aid assembly, the flexible tube can be

disconnected from the cavitation demonstration apparatus by unscrewing the union on the valve.

Ensure that the union is tightened (hand tight only) following reassembly.

Locate the flexible tube at the right hand end of the cavitation demonstration apparatus inside the

volumetric tank of the hydraulic bench with the end inside the stilling baffle to minimize

disturbances in the volumetric tank.

Note: that when operating the cavitation demonstration apparatus near or at the vapour pressure

of the liquid, the vacuum gauge will be slow to respond. This is because the liquid inside the gauge

turns to water vapour when operating at the vapour pressure and this process will not be

instantaneous. The effect is more noticeable when the pressure is raised and cavitation stops –

there will be a delay before the reading changes on the vacuum gauge after cavitation ceases visibly

and audibly in the test section.

Procedures:

1. Open the ball valve (right hand end) fully then close the inlet diaphragm valve (left hand end)

fully.

2. Close the flow control valve on hydraulic bench. Switch on the hydraulic bench, then slowly

open the flow control valve on hydraulic bench until it is fully open.

3. Slowly open the inlet diaphragm valve at the left hand end of cavitation apparatus and allow

water to flow through the cavitation apparatus until the clear acrylic test section and flexible

connecting tubes are full of water with no air entrained.

4. Continue to open the inlet diaphragm valve slowly until fully open to obtain maximum flow

through the system. Note the milky formation at the throat indicating the presence of cavitation.

Also note the loud audible crackling sound accompanying the cavitation.



Observe that the visible cavitation occurs in the expansion of the test section and not in the throat

where the pressure is at its lowest (with the exception of the pressure tapping hole in the throat

that causes a local disturbance to the flow).

5. If a thermometer is available measure and record the temperature of the water.

6. Close the inlet diaphragm valve until water flows slowly through the equipment with no

cavitation in the test section (typically 0.1 Bar on the upstream gauge P1) ensuring that the test

section remains full of water.

7. Record the following parameters:

Upstream water pressure P1 Bar.

Pressure at the throat P2 Bar (Vacuum).

Downstream water pressure P3 Bar.

8. Determine the flowrate by timing the collection of a known volume of water.

9. Gradually open the inlet diaphragm valve to increase the upstream pressure in small steps

(typically 0.1 Bar increments on the upstream gauge p1). At each setting repeat steps (7) and

(8) and note the presence of any tiny bubbles in the water. At each setting wait for the vacuum

gauge to settle before recording the pressure at the throat (there will be a long delay before the

reading changes on the gauge when near to or at cavitation because water inside the gauge is

converting to vapour).

Observe the change in appearance and change in sound when the pressure at the throat reaches the

vapour pressure of the water (air bubbles released from the water at higher static pressure make a

softer noise that is not true cavitation).

Also observe that the pressure at the throat does not continue to fall below the vapour pressure of

the water as the flow of water is increased.

10. Continue opening the inlet diaphragm valve in steps and recording / observing the

characteristics of the water until the maximum flow of water is achieved with the valve fully

open.

11. Gradually close the inlet diaphragm valve and observe that the Cavitation ceases as the pressure

rises above the vapour pressure of the water (again there will be a long delay before the reading

on the pressure gauge starts to fall because vapour inside the gauge is converting back to

water).



12. Close the inlet diaphragm valve until water flows slowly through the equipment with no

cavitation in the test section (typically 0.1 Bar on the upstream gauge P1) ensuring that the test

section remains full of water.

13. Close the outlet ball valve fully (the valve is perforated to allow water to flow when fully closed).

14. Repeat steps 6 – 10 with the outlet restricted.

15. Repeat step 14 with different settings of the outlet ball valve (partially closed).

16. Close the flow control valve on the hydraulic bench, then switch off the pump.

Results:

For each set of readings calculate the volume flowrate, then plot the graph of P2 against volume

flowrate Q for each set of results.

If the temperature of the water is known, determine the vapour pressure of the water using the

table below. From your results determine the minimum static pressure achieved at the throat of the

test section and confirm that this agrees with the vapour pressure of the water.

Analysis of the graphs should show that increasing flowrate (increasing velocity) through the test

section results in decreasing static pressure (as predicted by the Bernoulli equation) until the water

reaches its vapour pressure.

Results obtained with the downstream ball valve throttled should show that the onset of cavitation

is delayed due to the higher static pressure in the system (higher flowrate is possible before

cavitation occurs).

Conclusion:

The graphs should clearly show how the pressure at the throat falls as the flow / velocity of the

water is increased as predicted by the Bernoulli equation. It should also show that the pressure

reaches a minimum value that cannot be exceeded despite increasing water velocity.

Consider the effect of cavitation if allowed to occur in a hydraulic system.

The exercise shows that the cavitation can be prevented by increasing the static pressure of the

fluid. However, this technique can only be applied to delay the effect (a slight increase in flowrate

without cavitation occurring) and is not efficient as additional energy / a larger pump is required to

overcome the additional losses in the system. Cavitation is therefore best avoided by careful system



design to eliminate any high velocities, low pressures or high temperatures that could lead to

cavitation.

Table 1: Vapour pressure of water at different temperature

Temp. °C

Vapour pressure

KN/m2

Vapour pressure Bar (abs.)

Temp. °C

Vapour pressure

KN/m2

Vapour pressure Bar (abs.)

4 0.8130 0.0081 28 3.7814 0.0378

5 0.8720 0.0087 29 4.0074 0.0401

6 0.9348 0.0093 30 4.2451 0.0425

7 1.0015 0.0100 31 4.4949 0.0449

8 1.0724 0.0107 32 4.7574 0.0476

9 1.1477 0.0115 33 5.0332 0.0503

10 1.2276 0.0123 34 5.3326 0.0533

11 1.3123 0.0131 35 5.6264 0.0563

12 1.4002 0.0140 36 5.9451 0.0595

13 1.4974 0.0150 37 6.2793 0.0628

14 1.5983 0.0160 38 6.6296 0.0663

15 1.7051 0.0171 39 6.9967 0.0700

16 1.8180 0.0182 40 7.3812 0.0738

17 1.9375 0.0194 42 8.2053 0.0821

18 2.0639 0.0206 44 9.1075 0.0911

19 2.1974 0.0220 46 10.094 0.1009

20 2.3384 0.0234 48 11.171 0.1117

21 2.4872 0.0249 50 12.345 0.1235

22 2.6443 0.0264 52 13.623 0.1362

23 2.8099 0.0281 54 15.013 0.1501

24 2.9846 0.0298 56 16.522 0.1652

25 3.1686 0.0317 58 18.160 0.1816

26 3.3625 0.0336 60 19.933 0.1993

27 3.5666 0.0357



Experiment (12): Major losses

Introduction:

When a fluid is flowing through a pipe, it experiences some resistance due to which some of energy

(head) of fluid is lost. Energy loss through friction in the length of pipeline is commonly termed the

major loss (hf) which is the loss of head due to pipe friction and to viscous dissipation in flowing

water.

The resistance to flow in a pipe is a function of the pipe length, pipe diameter, mean velocity,

properties of the fluid and roughness of the pipe (if the flow is turbulent), but it is independent of

pressure under which the water flows.

Friction head losses in straight pipes of different sizes can be investigated over a range of Reynolds'

numbers from 103 to nearly 105, thereby covering the laminar, transitional, and turbulent flow

regimes in smooth pipes. A further test pipe is artificially roughened and, at the higher Reynolds'

numbers, shows a clear departure from typical smooth bore pipe characteristics.

Exercise A (Fluid friction in a smooth bore pipe)

Purpose:

To determine the relationship between head loss due to fluid friction and velocity for flow of water

through smooth bore pipes and to confirm the head loss predicted by a pipe friction equation.

Apparatus:

1. Fluid friction apparatus.

2. Hydraulics bench to supply water to the fluid friction apparatus (the flow of water can be

measured by timed volume collection).



Figure 1: Fluid friction apparatus

Theory:

Professor Osborne Reynolds demonstrated that two types of flow may exist in a pipe.

1. Laminar flow at low velocities where h α u

2. Turbulent flow at higher velocities where h α un

Where h is the head less due to friction and u is the fluid velocity. These two types of flow are

separated by a transition phase where no definite relationship between h and u exists.

Graphs of h versus u and log(h) versus log(u) show these zones.

Figure 2: Fluid friction apparatus

Furthermore, for a circular pipe flowing full, the head loss due to friction may be calculated from

the formula:

ℎ =𝑓𝐿𝑢2

2𝑔𝑑



where L is the length of the pipe between tappings, d is the internal diameter of the pipe, u is the

mean velocity of water through the pipe in m/s, g is the acceleration due to gravity in m/s2 and f is

pipe friction coefficient.

The Reynolds' number, Re, can be found using the following equation:

𝑅𝑒 =𝜌𝑣𝑑

𝜇

Where 𝜇 is the molecular viscosity (1.15 x 10-3 Ns/m2 at 15°C) and 𝜌is the density (999 kg/m3at

15°C).

Having established the value of Reynolds' number for flow in the pipe, the value of f may be

determined using a Moody diagram as shown below.

Figure 3: Moody diagram

Equipment Set Up:

Additional equipment required: Stop watch, Internal Vernier calliper.

Arrange the valves on the equipment to allow flow through only the test pipe under observation.

If using the data logging accessory, ensure that the console is powered and connected to the PC via

the USB connection. Load the software and choose a suitable exercise.



Procedures:

1. Prime the pipe network with water. Open and close the appropriate valves to obtain flow of

water through the required test pipe.

2. Take readings at several different flow rates, altering the flow using the control valve on the

hydraulics bench (ten readings is sufficient to produce a good head-flow curve).

3. Measure flow rates using the volumetric tank (if using a software, flow rate is measured

directly). For small flow rates use the measuring cylinder. Measure head loss between the

tappings using the portable pressure meter or pressurised water manometer as appropriate.

4. Obtain readings on all four smooth test pipes.

5. Measure the internal diameter of each test pipe sample using a Vernier calliper.

Data & Results:

All readings should be tabulated as follows:

Volume 𝑽

(liters)

Time 𝑻

(Sec)

Flow rate 𝑸

(m3/s)

Pipe diam.

𝒅 (m)

Velocity 𝒖

(m/s)

Reynolds No 𝑹𝒆

𝒇

Calculated head loss

𝒉𝒄 (m H2O)

Measured head loss

𝒉 (mH2O)

𝜌𝑢𝑑

𝜇

From Moody

Diagram

𝑓𝐿𝑢2

2𝑔𝑑 (hC-hD)

1. Plot a graph of h versus u for each size of pipe. Identify the laminar, transition and turbulent

zones on the graphs.

2. Confirm that the graph is a straight line for the zone of laminar flow h α u .

3. Plot a graph of log h versus log u for each size of pipe. Confirm that the graph is a straight line

for the zone of turbulent flow h α un. Determine the slope of the straight line to find n.

4. Estimate the value of Reynolds number (Re = 𝜌𝑢𝑑/𝜇) at the start and finish of the transition

phase. These two values of Re are called the upper and lower critical velocities.

5. Compare the values of head loss determined by calculation with those measured using the

manometer.

6. Confirm that the head loss can be predicted using the pipe friction equation provided the

velocity of the fluid and the pipe dimensions are known.

It is assumed that the molecular viscosity 𝜇 is 1.15 X 10-3 Ns/m2 at 15°C and the density 𝜌 is 999



kg/m3 at 15°C.

Exercise B (Fluid friction in a roughened pipe)

Purpose:

To determine the relationship between fluid friction coefficient and Reynolds' number for flow of

water through a pipe having a roughened bore.

Apparatus:

1. Fluid friction apparatus.

2. Hydraulics bench to supply water to the fluid friction apparatus (the flow of water can be


Theory:

The head loss due to friction in a pipe is given by:

ℎ =𝑓𝐿𝑢2

2𝑔𝑑

Where L is the length of the pipe between tappings, d is the internal diameter of the pipe, u is the

mean velocity of water through the pipe in m/s, g is the acceleration due to gravity in m/s 2 and 𝑓 is

pipe friction coefficient.

The Reynolds' number, Re, can be found using the following equation:

𝑅𝑒 =𝜌𝑣𝑑

𝜇

Where 𝜇 is the molecular viscosity (1.15 x 10-3 Ns/m2 at 15°C) and 𝜌 is the density (999 kg/m3 at

15°C).

Having established the value of Reynolds' number for flow in the pipe, the value of 𝑓 may be

determined using a Moody diagram.

Equipment Set Up:

Additional equipment required: Stop watch, Internal Vernier calliper.

Open and close the ball valves as required to obtain flow through only the roughened pipe.



If using the Data Logging accessory, ensure that the console is powered and connected to the PC via

the USB connection. Load the software and choose a suitable exercise.

Procedures:

1. Prime the pipe network with water. Open and close the appropriate valves to obtain flow of

water through the roughened pipe.

2. Take readings at several different flow rates, altering the flow using the control valve on the

hydraulics bench.

3. Measure flow rates using the volumetric tank (if using the software, flow rate is measured

directly). For small flow rates use the measuring cylinder.

4. Measure head loss between the tappings using the hand-held meter, sensors or manometer as

appropriate.

5. Estimate the nominal internal diameter of the test pipe sample using a Vernier calliper (not

supplied). Estimate the roughness factor e/d.

Data & Results:

All readings should be tabulated as follows:

Volume 𝑽

(liters)

Time 𝑻

(Sec)

Flow rate 𝑸

(m3/s)

Pipe diam.

𝒅 (m)

Velocity 𝒖

(m/s)

Reynolds No 𝑹𝒆

Measured head loss

𝒉 (mH2O)

Friction

coefficient

𝒇

𝜌𝑢𝑑

𝜇 (ℎ𝐶 − ℎ𝐷)

2𝑔𝑑ℎ

𝑙𝑢2

1. Plot a graph of pipe friction coefficient versus Reynolds' number (log scale).

2. Note the difference from the smooth pipe curve on the Moody diagram when the flow is

turbulent.



Experiment (13): Flow channel

Introduction:

An open channel is a duct in which the liquid flows with a free surface exposed to atmospheric

pressure. Along the length of the duct, the pressure at the surface is therefore constant and the flow

can not be generated by external pressures but only by differences in potential energy due to the

slope of the surface.

The flow channel is one of the most important tools available for the teaching of hydraulic

principles. The flow channel has been designed to allow students a wide range of experiments on

water flow in an open channel under different flow conditions and analyze the effects of test models

of various shapes on water flow. It also allows the verification of the Chezy equation and Mannings

friction factor. In addition studies of 'specific energy-depth' relationships, the effect of various

weirs and flumes, hydraulic jump and the determination of hydraulic mean depth can also be

carried out.

Flow channel: Flow channel is designed to allow a series of experiments on water flow through a rectangular

channel to be conducted. The channel is of rectangular cross section 175mm high x 55mm wide and

2500mm long. The flow channel incorporates a specially designed entry section which incorporates

a stilling pond, filled with glass spheres, to provide smooth non turbulent flow conditions at entry

to the channel. At the discharge end of the channel an adjustable undershot sluice gate is provided

which can be used to control the exit flow.

The channel is supported on a steel framework which incorporates a variable height support at the

right hand end allowing the slope of the channel to be varied. A measuring point is provided

together with a clock distance gauge and the calibration is such that 1 revolution of the clock dial is

equivalent to a slope of 1 :1500.



Figure 1: Flow channel apparatus

Exercise A (Flow in open channels)

Purpose:

To investigate the flow of water through a rectangular open channel.

Apparatus: 1. Flow channel.

2. Hydraulics bench to supply water to the flow channel apparatus (the flow of water can be


Theory:

Consider an open channel of uniform width B and with a flat but sloping bed as illustrated below, in

which a liquid flows from left to right.

Figure 2: Rectangular open channel



At plane X let the

Height of the channel bed above datum = 𝑍

Depth of liquid in the channel = 𝐷

Width of the channel = 𝐵

Wetted perimeter = 𝑃 = 𝐵 + 2𝐷

Mean velocity of the liquid = 𝑉

The hydraulic mean radius 𝑅ℎ is defined as:

𝑅ℎ = 𝐴

𝑃=

𝐵. 𝐷

𝐵 + 2𝐷

Applying Bernoulli's equation to the liquid at plane X then the total energy head above the datum is:

𝐻 = 𝑍 + 𝐷 +𝑉2

2𝑔

It is often advantageous to use the channel bed as the datum. The total energy head above the

channel bed is known as the specific energy, 𝐸 is:

𝐸 = 𝐷 +𝑉2

2𝑔

Rearranging to obtain the mean velocity:

𝑉 = √2𝑔(𝐸 − 𝐷)

Depending on the slope of the channel, the depth of the liquid along the channel may be constant or

it may either decrease or increase. Consideration of continuity of flow rate between two planes X1

and X2 requires that the flow rate 𝑄 is the same at each of the planes so that:

𝑄 = 𝐴1. 𝑉1 = 𝐴2. 𝑉2

and for a rectangular channel of uniform width 𝐵:

𝑞 =𝑄

𝐵= 𝑉1𝐷1 = 𝑉2𝐷2

For a uniform or steady flow in a constant width channel the depth of liquid will be constant along

the length of the channel 𝐷1=𝐷2 and therefore the slope of the surface 𝑆𝑆 must be parallel to the

slope of the bed 𝑆𝐵 so that 𝑆𝑆 = 𝑆𝐵.

If the velocity along the bed increases then the depth decreases in the direction of flow 𝐷1 > 𝐷2 and

the slope of the surface is greater than the slope of the channel bed 𝑆𝑆 > 𝑆𝐵 or if the velocity

decreases then the depth increases 𝐷1 < 𝐷2 and 𝑆𝑆< 𝑆𝐵.

Consider the case of constant flow, a mean velocity 𝑉 in a rectangular duct can be calculated using



Chezy formula:

𝑉 = 𝐶√𝑅ℎ𝑆𝐵

where 𝐶 is the Chezy coefficient for the channel.

The Manning formula is used exclusively for open channels and is usually used in large civil

engineering applications. Whilst the formula of Darcy-Weisbach and Colebrook-White applied to

channels are derived from flow in circular and non-circular pipes which are more suited to

mechanical engineering problems.

From the study of frictional forces in pipes, the friction factor is found to be dependent on the

Reynolds number and on the relative roughness of the pipe wall, however for very rough pipes the

friction factor becomes independent of Reynolds number and depends only on the roughness of the

pipe wall. For many open channels found in civil engineering problems the channel walls are very

rough and in studying these civil engineering channels Robert Manning found from experimental

work that the Chezy coefficient varied as the sixth root of the hydraulic mean radius and inversely

to the roughness of the channel.

𝐶 =𝑅ℎ

1/6

𝑛

Where 𝑛 is the Manning roughness factor.

The velocity in an open channel is given by the Manning formula:

𝑉 = 𝐶√𝑅ℎ𝑆𝐵 =𝑅ℎ

2/3𝑆𝐵

1/2

𝑛

Typical values of Manning's roughness factor are:

Irregular rock channels 0.035 to 0.045

Rough earth channels 0.025 to 0.040

Smooth earth channels 0.017 to 0.025

Rubble masonry 0.017 to 0.03

Clean smooth brick or wood channels 0.010 to 0.017

Smooth metal channels 0.008 to 0.010

General experimental procedures:

1. Position the flow channel to the left hand side of the hydraulics bench so that the discharge

from the flow channel will enter the weir channel of the hydraulics bench.

2. Adjust the feet of the flow channel support frame so that it does not rock.



3. Connect the delivery hose from the hydraulics bench to the inlet connection of the flow channel.

4. Lower the sluice gate at the discharge end of the tunnel to seal the exit from the tunnel.

5. Start the hydraulics bench pump and allow water to enter the channel until it is filled to a depth

of approximately 20mm.

6. Measure the distance of the water level from the top edge of the channel wall at each end and by

means of the slope adjusting knob, make the measurements equal.

7. Set the clock dial to zero and note the reading of the dial counter gauge.

8. Check that the depth of water in the channel is constant along the length of the channel. This is

the setting for zero slope.

Procedures:

1. Set up the hydraulics bench using the general experimental procedures.

2. Fully raise the sluice gate at the discharge end of the channel so that it will not restrict the flow.

3. Set the flow channel slope to a downwards gradient from left to right of 1·25 in 1500 i.e. 1¼

revolutions of the clock dial from the zero point.

4. Start the hydraulics bench pump and adjust the flowrate to approximately 1·5 liters/sec.

5. When the flow conditions have become stable measure the flow rate using the volumetric tank

of the hydraulics bench and measure the depth of water in the flow channel at 50cm from the

left hand end.

6. Keeping the flow rate constant flow repeat the above measurements for the following different

downward gradients.

Slope Revs. on dial from

'zero slope'

1.7/1500 17/10

1.8/1500 18/10

2.5/1500 21/2

3.0/1500 3

4.0/1500 4

5.5/1500 51/2



Results:

1. Record the results on a copy of the results sheet.

2. For each value of slope of the channel calculate:-

The water flow rate Q

The flow area from A = B. D

The mean velocity from V = Q/A

The hydraulic mean radius from Rh = A/(2D + B)

The slope of the channel bed 𝑆𝐵

The expression Rh2/3

S1/2

3. Plot a graph of the mean velocity V against Rh2/3

S1/2 and determine the Manning roughness value

from the slope of the graph.

Conclusion:

Comment on the value of Manning's roughness coefficient n and compare it with values quoted in

text books.

Exercise B (Flow under a sluice gate)

Purpose:

To investigate the flow of water under a sluice gate.

Apparatus:

1. Flow channel.





Theory:

By applying Bernoulli's equation to the flow in a channel it was shown that the specific energy

measured from the bed of the channel at any plane is given by:

𝐸 = 𝐷 +𝑉2

2𝑔= 𝐷 +

𝑄2

2𝑔𝐴2

the mean velocity is:

𝑉 = √2𝑔(𝐸 − 𝐷)

and the discharge is:

𝑄 = 𝐴√2𝑔(𝐸 − 𝐷)

Substituting 𝐵 · 𝐷 for 𝐴 in the specific energy equation and defining 𝑞 as the volume flow per unit of

channel width so that 𝑞 = 𝑄/𝐵

𝐸 = 𝐷 +𝑄2

2𝑔𝐴2= 𝐷 +

𝑞2

2𝑔𝐷2

Now differentiating with respect to depth and equating to zero to determine the conditions for the

minimum value of the specific energy 𝐸𝐶 :

𝐷𝐶 = √𝑞2

𝑔

3

𝐸𝐶 =3

2𝐷𝐶

The critical depth corresponding to minimum specific energy is:

𝐷𝐶 =2

3𝐸𝐶

and the velocity at this critical condition is:

𝑉𝐶 = √2𝑔(𝐸𝐶 − 𝐷𝐶) = √2𝑔(3

2𝐷𝐶 − 𝐷𝐶) = √𝑔. 𝐷𝐶

For a given value of discharge 𝑞 there will be two possible depths for a given value of specific

energy as shown in the graph below.



Figure 3: Specific energy-depth curve

For depths greater than the critical depth, the flow is said to be sub critical or tranquil and for

depths less than the critical depth, the flow is described as supercritical or shooting.

The flow under a sluice gate is dependent on the upstream head and the height under the sluice

gate.

Assuming tranquil conditions upstream, the flow under the sluice gate may be either tranquil or

supercritical, if it is supercritical then a downstream hydraulic jump can occur if the slope is either

insufficient to maintain the supercritical flow or if there is a downstream restriction.

Figure 4: Flow under a sluice gate

Procedures:

1. Set up the hydraulics bench using the general experimental procedures.

2. Fit the sluice gate in the channel at a distance of 50cm or more from the water flow entry.


a) Free discharge – Unrestricted downstream

b) Flooded discharge – Downstream restricted



4. Turn on the hydraulics bench and adjust the water flow to approximately 1·5 liter/second.

5. Check that stable conditions are achieved at the upstream measuring point (20cm upstream

from the sluice gate) and when stable flow conditions are established in the channel measure

the water depth:

20cm upstream of sluice gate.

10cm downstream of sluice gate.

20cm downstream of sluice gate.

6. Check the width of the flow channel at each of the three measuring points.

Results:

Record the results on a copy of the results sheet.

1. Calculate the water flow rate.

2. Calculate the specific energy for each of the three measurement points from:E = D +Q2

2gA2

3. Calculate the critical point using: DC = √q2

g

3 and EC =

3

2DC

4. Draw a graph of depth D against specific energy E for the three measured points and for the

calculated critical point. Draw a smooth curve through the four points.

5. Superimpose on to the graph a line from the origin at a slope of 2

3 to represent DC =

2

3EC

Conclusion:

Discuss the shape of the graph of depth against specific energy.



Exercise C (Demonstration of a hydraulic jump)

Purpose:

To investigate the phenomenon of a hydraulic jump.

Apparatus:

1. Flow channel.



Theory:

If the flow in a channel is supercritical and there is insufficient slope for the gravity forces to

overcome the frictional forces then the flow will suddenly change to a sub critical flow by means of

a hydraulic Jump which is illustrated in the figure below.

Figure 5: Hydraulic jump

The depth of water before the jump is less than the critical depth and the depth after the hydraulic

jump is greater than the critical depth. The specific energy before and after the hydraulic jump must

be higher than the critical energy value. The hydraulic jump is a highly irreversible process, there is

a loss in kinetic energy, and although there is a gain in potential energy, the irreversibility of the

process requires that the specific energy downstream of the hydraulic jump is less than the specific

energy upstream of the hydraulic jump. A hydraulic jump will occur in a supercritical flow if the

downstream water level is raised above the critical depth by an obstruction.

For continuity of flow through the hydraulic jump:

𝑄 = 𝐵𝐷1𝑉1 = 𝐵𝐷2𝑉2

𝑉2 = 𝑉1

𝐷1

𝐷2



In a hydraulic jump, the velocity changes from 𝑉1 to 𝑉2 and hence there is a change in momentum

through the jump. The force producing this change in momentum is due to the difference in

hydrostatic pressure resulting from the change of depth.

By equating the resultant hydrostatic force to the rate of change of momentum, the conjugate depth

𝐷2 can be calculated from the initial depth 𝐷1 from the following relationship:

𝐷2 = −𝐷1

2+ √

2𝑉12𝐷1

𝑔+

𝐷12

4

The loss of specific energy due to irreversibility in a hydraulic jump can be calculated as the

following:

𝐻𝐿 = 𝐸1 − 𝐸2 =(𝐷2 − 𝐷1)3

4𝐷1𝐷2

Procedures :

1. Set up the flow channel as for exercise B.


3. Turn on the hydraulics bench and adjust the water flow to approximately 1·5 liter/second.

4. Adjust the height of the sluice gate fitted to the discharge end of the flow channel until the

bottom of the sluice gate just touches the water surface.

5. A hydraulic jump will then form, make fine adjustments to the discharge sluice gate until a

stable stationary position of the hydraulic jump is obtained between the two sluice gates.

6. Measure the water depth each side of the hydraulic jump.

Results:

1. Record the results on a blank copy of the results sheet.

2. Calculate the water flow rate.

3. Calculate the specific energy for each of the three measurement points from: E = D +Q2

2gA2

4. Provided that the flow rate is unchanged from that for expercise B, superimpose on the graph

for exercise B the two points for the depth and specific energy before and after the hydraulic

jump.



Conclusion:

Discuss the shape of the water surface before and after the hydraulic jump.

104

References :

1. Fluid mechanics, by Douglas, J.F., Gasiorek, J.M., Swaffield, J.A., Jack, L.B., Fifth Edition, 2005.

2. Fundamentals of Hydraulic Engineering Systems, Houghtalen, R.J., Akan, A.O., Hwang, N.H.,

Fourth Edition, 2009.

3. Lecture Notes of Fluid Mechanics and Hydraulics, Alastal, K.M., 2014.

4. http://www.cussons.myzen.co.uk/SOFTWARE/FrontEnd.htm. Access date 01/06/2014.

5. http://www.discoverarmfield.com. Access date 01/06/2014.

http://www.cussons.myzen.co.uk/SOFTWARE/FrontEnd.htm

http://www.cussons.myzen.co.uk/SOFTWARE/FrontEnd.htm

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