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Applied Fluid Mechanics (2160602) 4. Turbo Machinery

Department of Mechanical Engineering Prepared By: Jigar J. Vaghela Darshan Institute of Engineering & Technology, Rajkot Page 4.1

4 TURBO MACHINERY

Course Contents

4.1 Introduction

4.2 Classification of Hydraulic

Turbines

4.3 Efficiencies & Head of

Hydraulic Turbines

4.4 Impulse Turbine (Pelton Wheel)

4.5 Reaction Turbine

4.6 Inward and Outward Flow

Reaction Turbine

4.7 Francis turbine

4.8 Axial Flow Reaction Turbine

4.9 Draft Tube Theory

4.10 Specific Speed

4.11 Unit Quantities

4.12 Performance Curves of

Hydraulic Turbines

4.13 Solved Numerical



4.1 Introduction

Hydraulic Machines

“It is defined as those machines which convert either hydraulic energy into

mechanical energy (i.e. turbines) or mechanical energy into hydraulic energy (i.e.

pumps).”

Turbines

“It is defined as the hydraulic machines which converts hydraulic energy into

mechanical energy.”

− This mechanical energy is used in running an electric generator which is directly

coupled to the shaft of the turbine. Thus the mechanical energy is converted into

electric energy.

− The electric power, which is obtained from the hydraulic energy is known as Hydro-

electric power.

4.2 Classification of Hydraulic Turbines

A. According to the type of energy at inlet or the action of the water on

the blade

I. Impulse turbine

− In an Impulse turbine, all the available energy of the water is converted into

kinetic energy or velocity head by passing it through a convergent nozzle

provided at the end of penstock.

− So at the inlet of the turbine, only kinetic energy is available.

− Here the pressure of water flowing over the turbine blades remains constant.

(i.e. atmospheric pressure)

− Examples: Pelton wheel, Turgo-impulse turbine, Girard turbine, Banki

turbine, Jonval turbine, etc.

II. Reaction turbine

− In a reaction turbine, at the entrance to the runner, only a part of the

available energy of water is converted into kinetic energy and a substantial

part remains in the form of pressure energy.

− So at the inlet of the turbine, water possesses kinetic energy as well as

pressure energy.

− As the water flows through the turbine blades, the change from pressure

energy to kinetic energy takes place gradually.

− For this gradual change of pressure, the runner must be completely enclosed

in an air-tight casing and the passage should be full of water.



− The difference of pressure between the inlet and outlet of the runner is

called reaction pressure, and hence these turbines are known as reaction

turbine.

− Examples: Francis turbine, Kaplan turbine, Propeller turbine, Thomson

turbine, Fourneyron turbine, etc.

B. According to the direction of flow through runner

I. Tangential flow turbine

− In tangential flow, the water strikes the runner in the direction of tangent to

the path of rotation of runner. OR The water strikes the vane/bucket along

the tangent of the runner.

− Example: Pelton wheel

II. Radial flow turbine

− In radial flow, water flows through the turbine along the direction normal to

the axis of rotation (i.e. radial direction).

− A radial flow turbine is further classified as inward or outward flow

depending upon whether the flow is inward from the periphery to the center

or outward from center to periphery.

− Example: Old Francis turbine

III. Axial flow turbine

− In an axial flow, water flows along the direction parallel to the axis of rotation

of the runner.

− Here water flows parallel to the turbine shaft.

− Examples: Kaplan turbine, Propeller turbine

IV. Mixed flow turbine

− In mixed flow, water enters the runner in the radial direction and leaves in

the direction parallel to the axis of rotation (i.e. axial direction).

− Example: Modern Francis turbine.

C. According to the head at the inlet of the turbine

I. High head turbine

− High head turbines which operates under high head (above 250 m) and

requires relatively less quantity of water.

− Example: Pelton wheel turbine


Department of Mechanical Engineering Prepared By: Jigar J. Vaghela Darshan Institute of Engineering & Technology, Rajkot .4

II. Medium head turbine

− Medium head turbines which operate under medium head (60 m to 250 m)

and require medium flow rate.

− Example: Modern Francis turbine

III. Low head turbine

− Low head turbines which operate under head up to 30 m and require very

large quantity of water.

− Example: Kaplan and Propeller turbine

D. According to the specific speed of the turbine

The specific speed of a turbine is the speed of a geometrically similar turbine that

would develop 1 KW power when working under a head of 1 m.

All geometrically similar turbines (irrespective of the sizes) will have the same

specific speeds when operating under the same head.

𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑠𝑝𝑒𝑒𝑑, 𝑁𝑠 =𝑁√𝑃

𝐻5 4⁄

Where,

𝑁 = Normal working speed

𝑃 = Power output of the turbine, and

𝐻 = Net or effective head in meter

Turbines with low specific speeds work under high head and low discharge

conditions, while high specific speed turbines work under low head and high

discharge conditions.

I. Low specific speed turbine

− For Pelton wheel turbine with single jet, 𝑁𝑠 = 8.5 𝑡𝑜 30

− For Pelton wheel turbine with double jet, 𝑁𝑠 = 40

II. Medium specific speed turbine

− For Francis turbine, 𝑁𝑠 = 50 𝑡𝑜 340

III. High specific speed turbine

− Kaplan and other Propeller turbine, 𝑁𝑠 = 255 𝑡𝑜 860



4.3 Efficiencies & Head of Hydraulic Turbines

1. Gross Head (𝑯𝒈)

− It is the difference between headrace level and tail race level when no water is

flowing.

− It is also known as total head of the turbine.

Fig. 4.1 Layout of hydroelectric power plant using an impulse turbine

2. Effective Head or Net Head (𝑯)

− Net head or effective head is the actual head available at the inlet of the turbine.

− When water is flowing from head race to the turbine, a loss of head due to friction

between water and penstock occurs.

− Though there are other losses also such as loss due to bend, pipe fittings, loss at

entrance of the penstock, etc. These all having small magnitude as compared to

head loss due to friction.

So,

𝐻 = 𝐻𝑔 − ℎ𝑓 − − − − − − − −(4.1)

Where,

𝐻 = Net head or Effective head

𝐻𝑔 = Gross head

ℎ𝑓 = Head loss due to friction between penstock and water and is given by,

ℎ𝑓 =4𝑓𝐿𝑉2

2𝑔𝐷− − − − − − − −(4.2)

𝑓 = Coefficient of friction of penstock depending on the type of material of penstock

𝐿 = Total length of penstock

𝑉 = Mean velocity of water through the penstock

𝐷 = Diameter of penstock and

𝑔 = Acceleration due to gravity



3. Hydraulic Efficiency (𝜼𝒉)

− It is the ratio of the power developed by the runner of a turbine to the power

supplied by the water at the inlet of a turbine.

− Since the power supplied is hydraulic, and the probable loss is between the striking

jet and vane it is rightly called hydraulic efficiency.

𝜂ℎ =𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑟𝑢𝑛𝑛𝑒𝑟

𝑃𝑜𝑤𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑤𝑎𝑡𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑙𝑒𝑡

𝜂ℎ =𝑅𝑢𝑛𝑛𝑒𝑟 𝑃𝑜𝑤𝑒𝑟

𝑊𝑎𝑡𝑒𝑟 𝑃𝑜𝑤𝑒𝑟=

𝑅. 𝑃.

𝑊. 𝑃.− − − − − − − −(4.3)

4. Mechanical Efficiency (𝜼𝒎)

− The power delivered by water to the runner of a turbine is transmitted to the shaft

of the turbine.

− It is the ratio of the power available at the shaft of the turbine to the power

developed by the runner of a turbine.

− This depends on the slips and other mechanical problems that will create a loss of

energy i.e. friction.

𝜂𝑚 =𝑃𝑜𝑤𝑒𝑟 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑠ℎ𝑎𝑓𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑟𝑏𝑖𝑛𝑒

𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑟𝑢𝑛𝑛𝑒𝑟 𝑜𝑓 𝑎 𝑡𝑢𝑟𝑏𝑖𝑛𝑒

𝜂𝑚 =𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟

𝑅𝑢𝑛𝑛𝑒𝑟 𝑃𝑜𝑤𝑒𝑟=

𝑆. 𝑃.

𝑅. 𝑃.− − − − − − − −(4.4)

5. Overall Efficiency (𝜼𝒐)

− It is the ratio of the power available at the shaft to the power supplied by the water

at the inlet of a turbine.

𝜂𝑜 =𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟

𝑊𝑎𝑡𝑒𝑟 𝑃𝑜𝑤𝑒𝑟

Runner Power

• 𝐹𝑜𝑟 𝑃𝑒𝑙𝑡𝑜𝑛 𝑤ℎ𝑒𝑒𝑙, 𝑅. 𝑃. = �̇�[𝑉𝑤1 ± 𝑉𝑤2] × 𝑢, Watt

• 𝐹𝑜𝑟 𝑅𝑎𝑑𝑖𝑎𝑙 𝑓𝑙𝑜𝑤 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑡𝑢𝑟𝑏𝑖𝑛𝑒, 𝑅. 𝑃. = �̇�[𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2], Watt

Water Power

𝑊. 𝑃. =𝑊 × 𝐻

1000, 𝐾𝑊

𝑊. 𝑃. =𝜌𝑔𝑄𝐻

1000, 𝐾𝑊

Where,

𝑊 = Weight of water striking the vanes per second = �̇�𝑔 = 𝜌𝑔𝑄

𝐻 = Net available head on the turbine



𝜂𝑜 =𝑆. 𝑃.

𝑊. 𝑃.×

𝑅. 𝑃.

𝑅. 𝑃.

𝜂𝑜 =𝑆. 𝑃.

𝑅. 𝑃.×

𝑅. 𝑃.

𝑊. 𝑃.

𝜂𝑜 = 𝜂𝑚 × 𝜂ℎ − − − − − − − −(4.5)

6. Volumetric Efficiency (𝜼𝒗)

− The volume of the water striking the runner of a turbine is slightly less than the

volume of the water supplied to the turbine.

− Some of the volume of the water is discharged to the tail race without striking the

runner of the turbine.

− Thus the ratio of the volume of the water actually striking the runner to the volume

of water supplied to the turbine is defined as volumetric efficiency.

𝜂𝑣 =𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑎𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑠𝑡𝑟𝑖𝑘𝑖𝑛𝑔 𝑡ℎ𝑒 𝑟𝑢𝑛𝑛𝑒𝑟

𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑡𝑢𝑟𝑏𝑖𝑛𝑒− − − − − − − −(4.6)

4.4 Pelton Wheel − A Pelton wheel turbine is:

Tangential Flow Turbine:

Water strikes the bucket/vane tangentially to the direction of the rotation.

Impulse Turbine:

At the inlet of the turbine, only kinetic energy is available. (Pressure will

remain constant at the inlet and outlet, i.e. Atmospheric pressure)

− Pelton wheel is generally used at a very high head and low discharge.

− Pelton wheel is named after an American engineer L. A. Pelton.

Components of Pelton Wheel

− The main components of Pelton wheel are:

1. Nozzle and Flow Regulating Arrangement (Spear)

2. Runner and Buckets

3. Casing and

4. Breaking Jet



Fig. 4.2 Components of a Pelton wheel

1. Nozzle and Flow Regulating Arrangement (Spear)

− Depending on load fluctuations, the speed of the turbine is to be kept constant by

controlling the quantity of water flowing through the nozzle.

− The amount of water striking the buckets of the runner is controlled by providing a

spear in the nozzle as shown in Fig. 4.3.

− The spear is a conical needle which is operated either by a hand wheel or

automatically by governor in an axial direction depending upon the size of the unit.

− Spear reciprocates in nozzle and hence changes the annular area through which

water can pass.

Fig. 4.3 Flow regulating Arrangement



− When the spear is pushed forward into the nozzle, the amount of water striking the

runner is reduced. On the other hand, if the spear is pushed back, the amount of

water striking the runner increases.

2. Runner and Buckets

− It consists of a circular disc, on the periphery of which a number of buckets evenly

spaced are fixed.

− The shape of the buckets is of a double hemispherical cup or bowl. Each bucket is

divided into two symmetrical parts by a dividing wall which is known as splitter.

− The jet of water strikes on the splitter. The splitter divides the jet into two equal

parts and the jet comes out at the outer edge of the bucket.

− The buckets are shaped in such a way that the jet gets deflected through 160° or

170°. Maximum work is obtained if the jet is deflected through 180° i.e. the bucket is

semicircular.

− If semicircular bucket is used, an outgoing jet may strikes to the next incoming

bucket and hence opposes the motion of the rotor. Hence the angle of jet deflection

is generally kept 160° to 170°.

− Material of buckets: Cast iron, Cast steel, Bronze or S.S., depending upon the head

at inlet.

− The inner surface of the bucket is highly polished to minimize the frictional losses.

− As the splitter has to bear total impact of jet, it must be made very strong. Therefore

it is not practical to have a sharp edge with a zero inlet angle at the center of the

bucket.

− Usually this angle is made 3 to 6 degrees even though, for practical purpose the inlet

vane angle is assumed to be zero.

3. Casing

− The function of the casing is to prevent the splashing of the water and to discharge

water to the tailrace.

− It also acts as a safe-guard against accidents.

− Material: Cast iron or fabricated steel plates.

− The casing of the Pelton wheel does not perform any hydraulic function.

Fig. 4.4 Bucket of Pelton Wheel



4. Breaking Jet

− When the nozzle is completely closed by moving the spear in the forward direction,

the amount of water striking the runner reduces to zero.

− But the runner due to inertia goes on revolving for a long time.

− To stop the runner in a shorter time, a small nozzle is provided which directs the jet

of water on the back of the vanes as shown in Fig. 4.2. This jet of water is called

breaking jet.

Working of Pelton Wheel

− Pelton wheel works on Impulse-Momentum principle. (i.e. 𝐹. 𝑑𝑡 = 𝑚. 𝑑𝑉)

− The water from the reservoir (or head race) is conveyed to turbine house through a

penstock, at the outlet of which a nozzle is fitted.

− When water flows through a penstock and comes out of nozzle, all pressure energy

and potential energy is converted into kinetic energy.

− Hence at the outlet of the nozzle, the water out in the form of jet (at atmospheric

pressure) and strikes the buckets/vanes of the runner.

− The impact of water on the bucket makes runner to rotate.

− Runner is mounted on the shaft and hence mechanical energy is available at the

shaft which is coupled with generator, which converts mechanical energy into

electrical energy and produce electricity.

− After performing work on the buckets water is discharged into the tail race.

Velocity Triangles, Work done and Efficiency of Pelton Wheel

− The jet of water from the nozzle strikes the bucket at the splitter, which splits up the

jet into two parts.

− These parts of the jet, glides over the inner surfaces and comes out at the outer edge

of the bucket.

− The splitter is the inlet tip and outer edge of the bucket is the outlet tip of the

bucket.



Fig. 4.5 Velocity diagram of Pelton wheel

Let,

𝑉1 = Absolute velocity of water at the inlet, and is given by,

𝑉1 = 𝐶𝑣√2𝑔𝐻

Where,

𝐻 = Net head acting on the Pelton wheel = 𝐻𝑔 − ℎ𝑓

𝐻𝑔 = Gross head

ℎ𝑓 = Head loss due to friction and is given by,

ℎ𝑓 =4𝑓𝐿𝑉2

2𝑔𝐷∗

𝐷∗ = Diameter of penstock

𝑉 = Velocity of water in the penstock

𝑓 = Co-efficient of friction of penstock depending on the type of material of

penstock

𝐿 = Total length of penstock

𝑉2 = Absolute velocity of water at the outlet

𝑉𝑟1 = Relative velocity of jet and vane at inlet

𝑉𝑟2 = Relative velocity of jet and vane at outlet

𝑢 = Peripheral velocity of a runner which has same value at inlet and outlet of the

runner at mean pitch (𝑢 = 𝑢1 = 𝑢2)

𝑢 =𝜋𝐷𝑁

60

𝐷 = Diameter of runner

𝑁 = Speed of the runner in RPM

𝑉𝑤1 = Velocity of whirl at inlet

𝑉𝑤2 = Velocity of whirl at outlet

𝛼 = Guide blade angle (𝐹𝑜𝑟 𝑝𝑒𝑙𝑡𝑜𝑛 𝑤ℎ𝑒𝑒𝑙, 𝛼 = 0)

𝜃 = Vane angle at the inlet (𝐹𝑜𝑟 𝑝𝑒𝑙𝑡𝑜𝑛 𝑤ℎ𝑒𝑒𝑙, 𝜃 = 0)



𝜑 = Vane angle at the outlet

− From inlet velocity triangle,

𝛼 = 0 𝑎𝑛𝑑 𝜃 = 0 (𝑤𝑎𝑡𝑒𝑟 𝑒𝑛𝑡𝑒𝑟𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙𝑙𝑦)

So,

𝑉1 = 𝑉𝑤1 𝑎𝑛𝑑

𝑉𝑟1 = 𝑉1 − 𝑢1

− From outlet velocity triangle,

𝑉𝑟1 = 𝑉𝑟2 (𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠𝑒𝑠 𝑎𝑟𝑒 𝑛𝑒𝑔𝑙𝑒𝑐𝑡𝑒𝑑 𝑜𝑟 𝑣𝑎𝑛𝑒 𝑖𝑠 𝑣𝑒𝑟𝑦 𝑠𝑚𝑜𝑜𝑡ℎ)

𝑉𝑤2 = 𝑉𝑟2 cos 𝜑 − 𝑢2

− Force exerted by the jet of water in the direction of motion is given by,

𝐹𝑥 = 𝜌𝑎𝑉1[𝑉𝑤1 + 𝑉𝑤2]

(𝑉1 is taken instead of 𝑉𝑟1 because of series of vanes)

Also for Pelton wheel 𝛽 is an acute angle i.e. 𝛽 < 90° ℎ𝑒𝑛𝑐𝑒 + 𝑣𝑒 sign should be

taken.

− Work done by the jet on runner per sec,

= 𝐹𝑥 × 𝑢

𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝜌𝑎𝑉1[𝑉𝑤1 + 𝑉𝑤2] × 𝑢,𝑁𝑚

𝑠𝑒𝑐− − − − − − − −(4.7)

− Hydraulic efficiency,

𝜂ℎ =𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑

𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑗𝑒𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑

𝜂ℎ =𝜌𝑎𝑉1[𝑉𝑤1 + 𝑉𝑤2] × 𝑢

1

2(𝜌𝑎𝑉1)𝑉1

2

𝜂ℎ =2𝑢[𝑉𝑤1 + 𝑉𝑤2]

𝑉12

Now substituting values of 𝑉𝑤1 & 𝑉𝑤2 in above equation, we get,

𝜂ℎ =2𝑢[𝑉1 + {𝑉𝑟1 cos 𝜑 − 𝑢2}]

𝑉12

But,

𝑉𝑟1 = 𝑉𝑟2 = 𝑉1 − 𝑢 (∵ 𝑢 = 𝑢1 = 𝑢2)

So,

𝜂ℎ =2𝑢[𝑉1 + (𝑉1 − 𝑢) cos 𝜑 − 𝑢]

𝑉12

𝜂ℎ =2𝑢 (𝑉1 − 𝑢)[1 + cos 𝜑]

𝑉12



Condition for Maximum efficiency,

𝑑(𝜂ℎ)

𝑑𝑢= 0

𝑑

𝑑𝑢[2𝑢 (𝑉1 − 𝑢)[1 + cos 𝜑]

𝑉12 ] = 0

[1 + cos 𝜑]

𝑉12 [

𝑑

𝑑𝑢(2𝑢𝑉1 − 2𝑢2)] = 0

2𝑉1 − 4𝑢 = 0

2𝑉1 = 4𝑢

𝒖 =𝑽𝟏

𝟐− − − − − − − −(4.8)

− Hydraulic efficiency of a Pelton wheel will be maximum when the velocity of wheel is

half the velocity of the jet of water at inlet.

− Maximum efficiency,

𝜂ℎ𝑚𝑎𝑥=

2 ×𝑉1

2× (𝑉1 −

𝑉1

2) [1 + cos 𝜑]

𝑉12

𝜂ℎ𝑚𝑎𝑥=

[1 + cos 𝜑]

2− − − − − − − −(4.9)

Design Aspects of Pelton Wheel For design aspect following points should be considered:

1. The velocity of jet (𝑽𝟏) at inlet of the turbine,

𝑉1 = 𝐶𝑣√2𝑔𝐻 − − − − − − − −(4.10)

Where, 𝐶𝑣 = 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ≅ 0.98 𝑡𝑜 0.99

2. The velocity of wheel (𝒖),

𝑢 = 𝜑√2𝑔𝐻 − − − − − − − −(4.11)

Where, 𝜑 = 𝑠𝑝𝑒𝑒𝑑 𝑟𝑎𝑡𝑖𝑜 ≅ 0.43 𝑡𝑜 0.48

3. The angle of deflection of the jet through bucket is taken at 165° (average of 160° to

170°), if no angle of deflection is given.

4. The mean diameter or pitch diameter (𝑫) of the Pelton wheel is given by,

𝑢 =𝜋𝐷𝑁

60

∴ 𝐷 =60 × 𝑢

𝜋𝑁



5. Jet ratio (𝒎): It is the ratio of pitch diameter (D) to diameter of jet (d).

𝑚 =𝐷

𝑑 (≅ 12 𝑖𝑛 𝑚𝑜𝑠𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒𝑠) − − − − − − − −(4.12)

6. No. of buckets (𝒁) on a runner is given by,

𝑍 = 15 +𝐷

2𝑑

∴ 𝑍 = 15 + 0.5𝑚 − − − − − − − −(4.13)

7. No. of jets: It is obtained by dividing the total rate of flow through the turbine (𝑄) by

the rate of flow of water through a single jet (q).

𝑁𝑜. 𝑜𝑓 𝑗𝑒𝑡𝑠 =𝑄

𝑞− − − − − − − −(4.14)

8. Working proportions for buckets:

1. Width of the bucket = 3𝑑 𝑡𝑜 5𝑑 ≅ 5𝑑

2. Depth of the bucket = 0.8𝑑 𝑡𝑜 1.2𝑑 ≅ 1.2𝑑

− Size of bucket means width and depth of the buckets.

Design of Pelton Wheel means……….

To determine,

a. Diameter of jet (𝑑)

b. Diameter of wheel (𝐷)

c. Size of the bucket (Width and Depth)

d. No. of buckets on the wheel (𝑍)



Fig. 4.6 A close-up view of a Pelton wheel showing the detailed design of the buckets; the electrical generator is on the right.

Fig. 4.7 A view from the bottom of an operating Pelton wheel illustrating the splitting and turning of the water jet in the bucket.

Fig. 4.8 The runner

of a Modern Francis

turbine. There are

17 runner blades of

outer diameter 20.3

ft. The turbine

rotates at 100 rpm

and produces 194

MW of power at a

volume flow rate of

375 m3/s from a

net head of 54.9 m.



4.5 Reaction Turbine − In reaction turbine, water at the inlet of the turbine possesses kinetic energy as well

as pressure energy.

− As water flows through runner, a part of pressure energy goes on changing into

kinetic energy.

− Thus the water through runner is under pressure and the runner is completely

enclosed in an air-tight casing.

− Casing and the runner is always full of water.

− Different types of reaction turbine are:

A. Inward radial flow reaction turbine (Water flows from outward to inward)

B. Outward radial flow reaction turbine (Water flows from inward to outward)

C. Mixed flow or Francis turbine (Water enters radially but leaves axially)

D. Axial flow turbine (Water enters and leaves axially)

I. Kaplan turbine:- Runner blades are adjustable

II. Propeller turbine:- Runner blades are fixed

Main Components of a Radial Flow Reaction Turbine

− There are many components used in radial flow reaction turbine but the main

components of radial flow reaction turbine are:

1. Casing

2. Guide Mechanism

3. Runner and

4. Draft tube

Fig. 4.9 Main components of radial flow reaction turbine

− Main parts of radial flow reaction turbine are shown in Fig. 4.9 and are discussed

below:



1. Casing

− In case of reaction turbine, casing and runner are always full of water.

− The cross-section area of this casing decreases uniformly along the circumference to

keep the fluid velocity constant in magnitude along its path towards the guide vane.

− This is so because the rate of flow along the fluid path in the volute decreases due to

continuous entry of the fluid to the runner through the openings of the guide vanes.

− Material: Concrete, Cast steel or Plate steel

2. Guide Mechanism or Guide Blades

− It is a stationary circular wheel. Guide vanes are fixed on guide mechanism between

two rings in form of wheel.

− The guide vanes allow the water to strike the vanes fixed on the runner without

shock at inlet.

− Material: Cast iron

− The quantity of water passing through the guide blades depends on the position of

the guide vanes.

3. Runner

− It is a circular wheel on which a series of radial curved vanes are fixed.

− Surface of the vanes are made very smooth.

− The radial curved vanes are so shaped that the water enters and leaves the runner

without shock.

− Material: Cast steel, Cast iron or Stainless steel.

− Runner is keyed to the shaft.

4. Draft Tube

− The pressure at the exit of the runner of a reaction turbine is generally less than

atmospheric pressure.

− Hence water at exit cannot be directly discharged to the tail race.

− A tube or pipe of gradually increasing area is used for discharging water from the exit

of the turbine to the tail race. This tube of increasing area is called draft tube.

4.6 Inward and Outward Radial Flow Reaction Turbine

Inward Radial Flow Reaction Turbine Outward Radial Flow Reaction Turbine

Water enters at the outer periphery, flows

inward and towards the center of the turbine

and discharges at the inner periphery.

Water enters at the inner periphery, flows

outward and discharges at the outer

periphery.

The outer diameter of the runner is inlet and The inner diameter of the runner is inlet



the inner diameter is the outlet.

∴ 𝐷1 > 𝐷2

And hence,

𝑢1 > 𝑢2

and the outer diameter is the outlet.

∴ 𝐷1 < 𝐷2

And hence,

𝑢1 < 𝑢2



(Work done and hydraulic efficiency are same for both inward and outward flow reaction

turbines)

Work done per sec,

𝑊𝐷 𝑠𝑒𝑐⁄ = 𝜌𝑎𝑉1[𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2] − − − − − − − −(4.15)

Work done per unit weight,

𝑊𝐷 𝑠𝑒𝑐⁄ 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 =1

𝑔[𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2] − − − − − − − −(4.16)

Hydraulic efficiency,



𝜂ℎ =�̇� (𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2)

𝜌𝑔𝑄𝐻=

𝜌𝑄 (𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2)

𝜌𝑔𝑄𝐻

𝜂ℎ =(𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2)

𝑔𝐻− − − − − − − −(4.17)

4.7 Francis turbine

− A Francis turbine is:

a. Mixed Flow Turbine:

Water enters radially and leaves axially to the direction of rotation of shaft.

b. Reaction Turbine:

At the inlet of the turbine both kinetic as well as pressure energy is available.

− It is generally operated under medium head and medium flow rate.

− It is designed by an American engineer J. B. Francis in 1849.

Components of Francis Turbine

− Different components of Francis turbine are:

A. Penstock

B. Spiral Casing

C. Guide Blades

D. Governing Mechanism

E. Runner

F. Draft Tube

A. Penstock

− Penstock is a large diameter conduit, which carries water from a dam or a reservoir

to the turbine house.

− Since Francis turbine requires large volume of water than Pelton wheel, size of the

penstock is bigger in the case of Francis turbine.

− Material: Generally steel is used.

B. Spiral Casing

− Water from the penstock enters into the spiral casing which completely surrounds

the runner.

− This casing is also known as scroll casing or volute.



− The cross-section area of this casing decreases uniformly along the circumference to

keep the fluid velocity constant in magnitude along its path towards the guide vane.

− This is so because the rate of flow along the fluid path in the volute decreases due to

continuous entry of the fluid to the runner through the openings of the guide vanes.

− Material:

▪ For low head: Concrete

casing with steel plate

lining

▪ For medium head:

Welded rolled steel

plate casing

▪ For high head: Cast

steel

C. Guide Blades

− A series of airfoil shaped vanes

called the guide vanes or

wicket gates, are mounted on

the casing.

− Guide vanes are fixed between

the two rings in form of a

wheel; however they can

swing about their own axis.

− The basic purpose of the guide

vanes is to convert a part of

pressure energy at its entrance

in to the kinetic energy and to

direct the water or fluid on to

the runner blades at an angle

appropriate to the design.

− The quantity of water passing through the guide vanes depends on the position of

the guide vanes, which can be controlled either by means of a hand wheel or

automatically by a governor.

− Material: Cast steel

D. Governing Mechanism

− Turbine must rotate at constant speed irrespective of the load variation on

generator.

− Governing mechanism keeps the speed of the turbine constant by controlling the

quantity of water to the turbine.

− Guide blades can move on its pivot centers and hence can change the area of flow.

Fig. 4.11 Components of Francis turbine



− Depending on load fluctuations, governing mechanism changes the position of guide

blades and hence the area of flow so that the turbine rotates with constant speed.

E. Runner

− It is the most important component of the Francis turbine.

− The runner of a Francis turbine consists of a series of curved vanes evenly arranged

around the circumference in the annular space between two plates.

− The runner vanes are so shaped that water enters the runner radially at the outer

periphery and leaves it axially at the inner periphery.

− Most of the portion of pressure energy is converted into kinetic energy as water

flows through the runner.

− The driving force on the runner is both due to impulse (deviation in the direction of

flow) and reaction (change in kinetic and pressure energy) effects.

− The number of runner blades are usually varies between 16 to 24.

− The runner is keyed to the shaft which is usually of forged steel.

− Material:

▪ Cast iron or Cast steel

▪ Sometimes Stainless steel or Bronze is used to avoid corrosion.

F. Draft Tube

− It is a pipe or passage of gradually increasing cross-sectional area towards its outlet

end. It connects the runner exit to the tail race.

− As the pressure of reaction turbine decreases continuously as water passes through

the guide vanes and the runner, it does below atmospheric pressure at the outlet of

the runner.

− Draft tube is used to discharge the water to the tail race by increasing pressure

above atmospheric.

− Draft tube must be submerged below the level of water in the tail race.

− Material: Steel plate

Working of a Francis Turbine

− Water through the penstock under pressure enters the spiral casing which

completely surrounds the runner.

− From casing water passes through a series of guide vanes, which directs the water to

the runner at a proper angle.

− The pressure energy of water reduces continuously as it passes over the guide vanes

and moving vanes.

− The difference in pressure at stationary guide vanes and moving runner is

responsible for the motion of the runner vanes.

− Finally water is discharged to the tail race through a draft tube.



Velocity Triangles, Work done and Efficiency of Francis Turbine

Fig. 4.12 Velocity Triangles for Francis turbine

− The velocity triangles at inlet and outlet of the Francis turbine are drawn as shown in

Fig. 4.12(a).

− General expression for work done by runner will be derived in the same manner as

in the case of series of radial curved vanes

𝑊𝐷 𝑠𝑒𝑐⁄ = �̇�(𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2)

𝑊𝐷 𝑠𝑒𝑐⁄ = 𝜌𝑎𝑉1(𝑉𝑤1𝑢1 ± 𝑉𝑤2𝑢2) − − − − − − − −(4.18)

𝑖𝑓 𝛽 < 90° → +𝑣𝑒 𝑠𝑖𝑔𝑛 𝑡𝑎𝑘𝑒𝑛

𝑖𝑓 𝛽 > 90° → −𝑣𝑒 𝑠𝑖𝑔𝑛 𝑡𝑎𝑘𝑒𝑛

− For maximum output, runner of the Francis turbine is so designed that there occurs

a radial discharge at the outlet tip of the blades.

− For radial discharge at the outlet, 𝛽 = 90° and 𝑉𝑤2 = 0, as shown in Fig. 4.12 (b).

∴ 𝑾𝑫 𝒔𝒆𝒄⁄ = �̇� (𝑽𝒘𝟏𝒖𝟏), 𝑵𝒎 𝒔𝒆𝒄⁄ − − − − − − − −(4.19)

Hydraulic Efficiency



𝜂ℎ =�̇� (𝑉𝑤1𝑢1)

𝜌𝑔𝑄𝐻=

𝜌𝑄 (𝑉𝑤1𝑢1)

𝜌𝑔𝑄𝐻

(b)



𝜼𝒉 =(𝑽𝒘𝟏𝒖𝟏)

𝒈𝑯− − − − − − − −(4.20)

Working Proportions for Francis Turbine

1. Flow Ratio (𝑲𝒇)

− Ratio of flow velocity at the inlet (𝑉𝑓1) to theoretical velocity (√2𝑔𝐻) is called flow

ratio. Its value lies between 0.15 to 0.30.

𝐾𝑓 =𝑉𝑓1

√2𝑔𝐻− − − − − − − −(4.21)

2. Speed Ratio (𝑲𝒖)

− Ratio of the peripheral velocity at the inlet (𝑢1) to theoretical velocity (√2𝑔𝐻) is

called speed ratio. Its value lies between 0.6 to 0.9.

𝐾𝑢 =𝑢1

√2𝑔𝐻− − − − − − − −(4.22)

3. Breadth Ratio (𝒏)

− Ratio of width of the runner (𝐵) to outside diameter of the runner (𝐷) is called

breadth ratio. Its value ranges from 0.1 to 0.4.

𝑛 =𝐵

𝐷− − − − − − − −(4.23)

Total Discharge through Francis Turbine

Let,

𝐷1 = Diameter of runner at inlet

𝐷2 = Diameter of runner at outlet

𝐵1 = Width of runner at inlet

𝐵2 = Width of runner at outlet

𝑉𝑓1 = Velocity of flow at inlet

𝑉𝑓2 = Velocity of flow at outlet

𝑛 = Number of vanes on runner

𝑡 = Thickness of each vane

𝑇ℎ𝑒𝑛, total discharge through the Francis turbine is given by,

𝑄 = 𝐴𝑟𝑒𝑎 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑜𝑤 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡

= 𝐴𝑟𝑒𝑎 𝑎𝑡 𝑜𝑢𝑡𝑙𝑒𝑡 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑜𝑤 𝑎𝑡 𝑜𝑢𝑡𝑙𝑒𝑡

∴ 𝑄 = 𝜋𝐷1𝐵1 × 𝑉𝑓1 = 𝜋𝐷2𝐵2 × 𝑉𝑓2 − − − − − − − −(4.24)

− If the thickness of the vanes are taken into consideration, then the area through

which flow takes place is given by, (𝜋𝐷1 − 𝑛𝑡)𝐵1

Hence,



𝑄 = (𝜋𝐷1 − 𝑛𝑡)𝐵1 × 𝑉𝑓1 = (𝜋𝐷2 − 𝑛𝑡)𝐵2 × 𝑉𝑓2 − − − − − − − −(4.25)

4.8 Axial Flow Reaction Turbine

− In an axial flow reaction turbine, the water flows parallel to the axis of the rotation

of the shaft.

− It is used under low head and high discharge conditions.

− For the axial flow reaction turbine the shaft of the turbine is vertical.

− The lower end of the shaft is made larger which is known as “Hub” or “Boss”.

− The vanes are fixed on the hub and hence hub acts as a runner for axial flow reaction

turbine.

Types of Axial Flow Reaction Turbine

1. Kaplan Turbine and

2. Propeller Turbine

− When the vanes are fixed to the hub and they are not adjustable, the turbine is

known as Propeller turbine.

− If the vanes on the hub are adjustable the turbine is known as a Kaplan turbine.

− The runner blades are adjusted automatically by servo-mechanism so that at all

loads the flow enters them without shock. This gives better part load efficiency for

Kaplan turbine.

− Components of Kaplan turbine and Propeller turbine are similar to that of the

Francis turbine, only the runner is different.

− Main parts of the Kaplan & Propeller turbine are:

A. Scroll casing

B. Guide vane mechanism

C. Hub with vanes or runner and

D. Draft tube

Key Point for Reaction Turbine

✓ 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔𝑡ℎ 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠 𝑯𝒆𝒂𝒅.

✓ 𝑯𝒆𝒂𝒅 𝒃𝒂𝒍𝒂𝒏𝒄𝒆:

𝐻𝑒𝑎𝑑 𝑢𝑡𝑖𝑙𝑖𝑧𝑒𝑑 = 𝐻𝑒𝑎𝑑 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑜𝑢𝑡𝑙𝑒𝑡

𝟏

𝒈[𝑽𝒘𝟏𝒖𝟏 ± 𝑽𝒘𝟐𝒖𝟐] = 𝑯 −

𝑽𝟐𝟐

𝟐𝒈− − − − − − − −(4.26)



Fig. 4.13 Components of Kaplan turbine

− The water from penstock enters the casing and then moves to the guide vanes. From

the guide vanes, the water turns through 90° and flows axially through the runner as

shown in Fig. 4.13.

Work done, Efficiency and Power Developed

− Expressions for work done, efficiency and power developed by Kaplan & Propeller

turbine are similar to that of Francis turbine.

Discharge through Runner of Kaplan & Propeller Turbine

− The discharge through the runner is obtained by,

𝑄 =𝜋

4(𝐷𝑜

2 − 𝐷𝑏2) × 𝑉𝑓1 − − − − − − − −(4.27)

Where,

𝐷𝑜 = Outer diameter of the runner

𝐷𝑏 = Diameter of the hub

𝑉𝑓1 = Velocity of flow at inlet

Working Proportions of Kaplan and Propeller Turbine

1. The peripheral velocity at inlet and outlet are equal,

∴ 𝑢1 = 𝑢2 =𝜋𝐷0𝑁

60− − − − − − − −(4.28)

2. Velocity of flow at inlet and outlet are equal,

∴ 𝑉𝑓1 = 𝑉𝑓2 = 𝐾𝑓√2𝑔𝐻 − − − − − − − −(4.29)

3. Area of flow at inlet and outlet are equal,

∴ 𝐴1 = 𝐴2 =𝜋

4(𝐷𝑜

2 − 𝐷𝑏2)



4.9 Draft Tube Theory

− Draft tube is an integral part of reaction turbine. It is an air tight diverging conduit

with cross-sectional area increasing along its length. One end of this diverging tube is

connected to runner exit and the other is located below the level of tail race.

− The function of the draft tube are:

• When water flows through the turbine it’s kinetic and pressure energy is

utilized to generate shaft power. Even though when water leaves the turbine

it possesses high kinetic energy and negative pressure head. If water is

discharged through a draft tube having gradually increasing cross sectional

area, the velocity is largely reduced at the outlet of the draft tube, and thus

resulting in a gain in kinetic head and also increases the negative pressure

head at the turbine exit so that net working head on the turbine increases. So

output of turbine and efficiency also increases.

• By providing a draft tube, a turbine can be installed above the tail race

without loss of any head. This helps to make inspection and maintenance of a

turbine easy.

− Different types of draft tubes used in reaction turbine are:

a) Straight divergent tube or Conical draft tube

b) Simple elbow tube

c) Moody spreading tube

d) Elbow tube with circular cross-section at inlet and rectangular at outlet

− Fig. 4.14 shows different types of draft tubes.

Fig. 4.14 Types of draft tubes



− Let,

𝐻𝑠 = Vertical height of draft

tube above the tail race

𝑦 = Distance of bottom of

draft tube from tail race

− Applying Bernoulli’s equation

to the inlet (section 2-2) and

outlet (section 3-3) of the draft

tube as shown in Fig. 4.15.

− Assuming section 3-3 as a datum line, we get,

𝑃2

𝜌𝑔+

𝑉22

2𝑔+ (𝐻𝑠 + 𝑦) =

𝑃3

𝜌𝑔+

𝑉32

2𝑔+ 0 + ℎ𝑓 − − − − − − − −(4.30)

Where,

ℎ𝑓 = Loss of energy between section 2-2 and 3-3.

But,

𝑃3

𝜌𝑔= 𝐴𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 ℎ𝑒𝑎𝑑 + 𝑦

∴𝑃3

𝜌𝑔=

𝑃𝑎

𝜌𝑔+ 𝑦

𝑆𝑜,

∴𝑃2

𝜌𝑔+

𝑉22

2𝑔+ (𝐻𝑠 + 𝑦) =

𝑃𝑎

𝜌𝑔+ 𝑦 +

𝑉32

2𝑔+ ℎ𝑓

∴𝑃2

𝜌𝑔+

𝑉22

2𝑔+ 𝐻𝑠 =

𝑃𝑎

𝜌𝑔+

𝑉32

2𝑔+ ℎ𝑓

∴𝑷𝟐

𝝆𝒈=

𝑷𝒂

𝝆𝒈− 𝑯𝒔 − (

𝑽𝟐𝟐

𝟐𝒈−

𝑽𝟑𝟐

𝟐𝒈− 𝒉𝒇) − − − − − − − −(4.31)

− In Equation 4.31, 𝑃2

𝜌𝑔 is less than atmospheric pressure.

Efficiency of Draft Tube (𝜼𝒅)

− It is defined as the ratio of actual conversion of kinetic head into pressure head in

the draft tube to the kinetic head at the inlet of the draft tube.

𝜂𝑑 =(

𝑉22

2𝑔−

𝑉32

2𝑔) − ℎ𝑓

𝑉22

2𝑔

− − − − − − − −(4.32)

Fig. 4.15 Draft tube theory



4.10 Specific Speed (𝑵𝒔)

− It is defined as the speed of a turbine which is identical in shape, geometrical

dimensions, blade angles, gate openings, etc. with the actual turbine but of such a

size that it will develop unit power when working under a unit head.

− The specific speed is used in comparing the different types of turbines as every type

of turbine has different specific speed.

In MKS system,

Unit power → 1 Horse power

Unit head → 1 meter

In SI system,

Unit power → 1 KW

Unit head → 1 meter

Derivation of Specific Speed for Hydraulic Turbine

− The overall efficiency of any turbine is given by,



𝑃𝜌𝑔𝑄𝐻

1000

∴ 𝑃 = 𝜂𝑜 ×𝜌𝑔𝑄𝐻

1000

∴ 𝑃 ∝ 𝑄𝐻 (𝑎𝑠 𝜌 𝑎𝑛𝑑 𝜂𝑜 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) − − − − − − − −(4.33)

− Let,

𝐷 = Diameter of actual turbine runner

𝐵 = Width of the actual turbine blade

𝑁 = Speed of actual turbine

𝑢 = Tangential velocity of the turbine wheel

𝑁𝑠 = Specific speed of the turbine

𝑉 = Absolute velocity of the jet of water

𝑢 ∝ 𝑉 𝑎𝑛𝑑 𝑉 ∝ √𝐻

∴ 𝑢 ∝ √𝐻 − − − − − − − −(4.34)

But,

𝑢 =𝜋𝐷𝑁

60

∴ 𝑢 ∝ 𝐷𝑁 − − − − − − − −(4.35)

From Equation 4.34 and 4.35 we have,

√𝐻 ∝ 𝐷𝑁

∴ 𝐷 ∝√𝐻

𝑁− − − − − − − −(4.36)



− The discharge through the turbine is given by,

𝑄 = 𝑎𝑟𝑒𝑎 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

But,

𝐴𝑟𝑒𝑎 ∝ 𝐵𝐷 ∝ 𝐷2 (∵ 𝐵 ∝ 𝐷)

And

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ∝ √𝐻

∴ 𝑄 ∝ 𝐷2√𝐻

∴ 𝑄 ∝ (√𝐻

𝑁)

2

× √𝐻 (𝐹𝑟𝑜𝑚 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 4.36)

∴ 𝑄 ∝𝐻

𝑁2× √𝐻

∴ 𝑄 ∝𝐻3 2⁄

𝑁2− − − − − − − −(4.37)

− Substituting the value of Q in Equation 4.33 we get,

𝑃 ∝𝐻3 2⁄

𝑁2× 𝐻

∴ 𝑃 ∝𝐻5 2⁄

𝑁2

∴ 𝑃 = 𝐾 ×𝐻5 2⁄

𝑁2− − − − − − − −(4.38)

Where, 𝐾 = Constant of proportionality

− If,

𝑃 = 1𝐾𝑊 and 𝐻 = 1𝑚, Then, 𝑁 = 𝑁𝑠

− Substituting these values in Equation (4.38) we get,

1 = 𝐾 ×15 2⁄

𝑁𝑠2

∴ 𝐾 = 𝑁𝑠2

− So,

𝑃 = 𝑁𝑠2 ×

𝐻5 2⁄

𝑁2

∴ 𝑁𝑠2 =

𝑃𝑁2

𝐻5 2⁄

∴ 𝑁𝑠 = √𝑃𝑁2

𝐻5 2⁄

∴ 𝑵𝒔 =𝑵√𝑷

𝑯𝟓 𝟒⁄− − − − − − − −(4.39)



Significance of Specific Speed

− Specific speed plays an important role for selecting the type of the turbine. Also the

performance of a turbine can be predicted by knowing the specific speed of the

turbine.

− The type of turbine for different specific speed are given in following table:

Sr. No. Specific Speed, 𝑵𝒔

Type of Turbine In MKS unit In SI unit

1 10 to 60 10 to 50 Pelton Wheel

2 60 to 300 51 to 225 Francis Turbine

3 300 to 1000 255 to 860 Kaplan turbine

4.11 Unit Quantities and Model Relationship − A turbine operates most efficiently at its design point, i.e., at a particular

combination of head, discharge, speed and power output. But in actual practice

hardly any turbine operates at its designed parameters.

− In order to predict the behavior of turbine operating at varying conditions of head,

discharge, speed and power output, the results expressed in terms of quantities

which may be obtained when the head on the turbine is reduced to unity (1m).

− The conditions of the turbine under unit head are such that the overall efficiency of

the turbine remains constant.

− Turbine can be compared with the help of the following common characteristics:

A. Unit Speed (𝑵𝒖)

It is defined as the speed of a turbine working under a unit head (1 m).

𝑢 ∝ 𝑉 𝑎𝑛𝑑 𝑉 ∝ √𝐻

∴ 𝑢 ∝ √𝐻

But,

𝑢 =𝜋𝐷𝑁

60

∴ 𝑢 ∝ 𝐷𝑁

For a given turbine, the diameter (𝐷) is constant.

∴ 𝑁 ∝ 𝑢

∴ 𝑁 ∝ √𝐻



∴ 𝑁 = 𝐾1√𝐻

From definition of unit speed, if 𝐻 = 1𝑚, 𝑁 = 𝑁𝑢

∴ 𝑁𝑢 = 𝐾1

Hence,

𝑵𝒖 =𝑵

√𝑯

B. Unit Discharge (𝑸𝒖)

It is defined as a discharge passing through a turbine, which is working under a unit

head (1m).

Total discharge, Q = Area of flow x Velocity of flow

But for a given turbine, area of flow is constant and,

𝑄 ∝ 𝑉𝑓 ∝ √𝐻

∴ 𝑄 = 𝐾2√𝐻

From definition of unit discharge, if 𝐻 = 1𝑚, 𝑄 = 𝑄𝑢

∴ 𝑄𝑢 = 𝐾2

Hence,

𝑸𝒖 =𝑸

√𝑯

C. Unit Power (𝑷𝒖)

It is defined as the power developed by a turbine, which is working under a unit head

(1m).

The overall efficiency,



𝑃𝜌𝑔𝑄𝐻

1000

∴ 𝑃 = 𝜂𝑜 ×𝜌𝑔𝑄𝐻

1000

∴ 𝑃 ∝ 𝑄𝐻 (𝑎𝑠 𝜌 𝑎𝑛𝑑 𝜂𝑜 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)

But,

𝑄 ∝ 𝑉𝑓 ∝ √𝐻

∴ 𝑃 ∝ √𝐻 × 𝐻

∴ 𝑃 ∝ 𝐻3 2⁄

∴ 𝑃 = 𝐾3𝐻3 2⁄

From definition of unit power, if 𝐻 = 1𝑚, 𝑄 = 𝑄𝑢

∴ 𝑃𝑢 = 𝐾3

Hence,

𝑷𝒖 =𝑷

𝑯𝟑 𝟐⁄



➢ Use of Unit Quantities (𝑵𝒖, 𝑸𝒖, 𝑷𝒖)

− If a turbine is working under different heads, the behavior of the turbine can be

easily known from the values of the unit quantities.

− Let,

𝐻1, 𝐻2 = Different heads under which a turbine works,

𝑁1, 𝑁2 = Corresponding speeds,

𝑄1, 𝑄2 = Corresponding discharge, and

𝑃1, 𝑃2 = Corresponding power developed by the turbine

From the definition of unit quantities, we get

𝑁𝑢 =𝑁1

√𝐻1

=𝑁2

√𝐻2

𝑄𝑢 =𝑄1

√𝐻1

=𝑄2

√𝐻2

𝑃𝑢 =𝑃1

𝐻13 2⁄

=𝑃2

𝐻23 2⁄

− Hence, if the speed, discharge and power developed by a turbine under a head are

known, then by using above relations the speed, discharge and power developed by

the same turbine under a different head can be obtained easily.

4.12 Performance (Characteristic) Curves of Hydraulic Turbines

− The turbines are generally designed to work at particular designed conditions. But

often the turbines are required to work at different conditions. Therefore it is

essential to determine the exact behavior of the turbines under the varying

conditions.

− “Characteristic curves of a hydraulic turbine are the curves, with the help of which

the exact behavior and performance of the turbine under different working

conditions can be known.”

− These curves are plotted from the results of the test performed on the actual turbine

or its model under different working conditions.

− The important parameters which are varied during a test on a turbine are:

(1) Speed (N), (2) Head (H), (3) Discharge (Q), (4) Power (P), (5) overall efficiency (ηo)

and (6) Gate opening (i.e. the percentage of the inlet passages provided for water to

enter the turbine)

− Out of these six parameters speed, head and discharge are independent parameters.

Different characteristic curves are obtained by keeping one independent parameter



constant and variation of any parameter with respect to remaining two independent

parameters.

− The following are the important characteristic curves for a hydraulic turbine:

1. Main Characteristic Curves or Constant Head Curves

2. Operating Characteristic Curves or Constant Speed Curves

3. Muschel Curves or Constant Efficiency Curves

1. Main Characteristic Curves or Constant Head Curves

− Main characteristic curves are obtained by maintaining a constant head and a

constant gate opening on the turbine.

− The speed of the turbine is varied by admitting different rates of flow by adjusting

the percentage of gate opening. The power (P) developed is measured mechanically.

From each test the unit power Pu, the unit speed Nu, the unit discharge Qu and the

overall efficiency ηo are determined. The characteristic curves drawn are:

a) Unit discharge vs unit speed

b) Unit power vs unit speed

c) Overall efficiency vs unit speed

Fig. 4.16 (a) Main Characteristic curves for a Pelton wheel

− For Pelton wheel since Qu depends only on the gate opening and independent of Nu,

Qu vs Nu plots are horizontal straight lines.

− However for low specific speed Francis turbines Qu vs Nu are drooping curves,

thereby indicating that as the speed increases the discharge through the turbine

decreases. This is so because in these turbines a centrifugal head is developed which

retards the flow. On the other hand for high specific speed Francis turbine as well as



Kaplan turbine, since the flow is axial there is no such centrifugal head developed

which may cause the retardation of flow.

Fig. 4.16 (b) Main Characteristic curves for a Reaction turbine

2. Operating Characteristic Curves or Constant Speed Curves

− Operating characteristic curves are plotted when the speed on the turbine is

constant. In case of turbines the head is generally constant. Hence the variation of

power and efficiency w.r.t. discharge Q is plotted.

− The power curve for turbines shall not pass through the origin because certain

amount of discharge is needed to produce power to overcome initial friction. Fig.

4.17 shows the variation of power and efficiency with respect to discharge.

Fig. 4.17 Operating characteristic curves



3. Muschel Curves or Constant Efficiency Curves

− These curves are obtained from the speed vs. efficiency and speed vs. discharge

curves (main characteristic curves) for different gate openings.

− For a given efficiency there are two values of speeds and two values of discharge for

a given gate opening, these can be plotted as shown in Fig. 4.18.

− The procedure is repeated for different gate openings and the curves Q vs. N are

plotted. The curves having the same efficiencies are joined. The curves having same

efficiency are called iso-efficiency curves. These curves are helpful in determining the

zone of constant efficiency and for predicting the performance of the turbine at

various efficiencies.

Fig. 4.18 Constant efficiency curve



3. A Francis Turbine is proposed to be installed at an available head of 60m and a

discharge of 40 m3/s. Determine the number of turbines and power available if the

specific speed is 210 and these are to run at 540 rpm with an overall efficiency of 85%.

Solution: Given Data:

Head, H1 = 60 m

Discharge, Q1 = 40 m3/sec

Specific Turbine = 210

Speed = N= 540 r.p.m.

Overall Efficiency = 85%

Find :(1) Power = P=?

(2)Number of Turbine =?



Overall Efficiency,𝑜

=P

ƍX gXQXH

1000

(𝟏) 𝑻𝒐𝒕𝒂𝒍 𝑷𝒐𝒘𝒆𝒓, 𝑃𝑡 = 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 (ƞ𝑜) 𝑋 ƍ𝑋 𝑔𝑋𝑄𝑋𝐻

1000

= 0.85 𝑋 1000𝑋9.81𝑋40𝑋 60

1000

= 20.0124 𝑘𝑊

𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑣𝑒𝑙𝑜𝑝 𝑏𝑦 𝑒𝑎𝑐ℎ 𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝑢𝑛𝑖𝑡𝑠,

𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑆𝑝𝑒𝑒𝑑, 𝑁𝑠 = 𝑁√𝑃

𝐻5/4

210 = 540𝑋 √𝑃

(60)5/4

√𝑃 = 64.9403

𝑃 = 4.217 𝑘𝑊

(𝟐) 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑻𝒖𝒓𝒃𝒊𝒏𝒆 = 20.0124

4.217

= 4.74

= 5 𝑇𝑢𝑟𝑏𝑖𝑛𝑒𝑠

4. A Francis turbine of 1 metre runner diameter working under a head of 4.5 metres at a

speed of 200 rpm develops 90 Kw when the rate of flow of water is 1.8 m3 /s. If the

head on the turbine is increased to 13.5 metres determine the new speed, discharge

and power.

Solution:

Given Data:

Head, H1= 4.5 m, Head, H2= 13.5 m

Discharge, Q1 = 1.8 m3/sec, Discharge, Q2 =?

Speed, N1= 200 r.p.m., Speed, N2=?

Power, P1= 90 Kw, Power, P2=?

𝑺𝒑𝒆𝒆𝒅:

𝑵𝟏

√𝑯𝟏

= 𝑵𝟐

√𝑯𝟐

𝑁2 = 200 𝑋 √13.5

√4.5

𝑁2 = 346.41 𝑟. 𝑝. 𝑚.



𝑷𝒐𝒘𝒆𝒓:

𝑷𝟏

𝑯𝟏

𝟑

𝟐

= 𝑷𝟐

𝑯𝟐

𝟑

𝟐

𝑃2 = 90 𝑋 (13.5)

3

2

(4.5)3

2

𝑃2 = 467.65 𝑘𝑊

𝑫𝒊𝒔𝒄𝒉𝒂𝒓𝒈𝒆:

𝑸𝟏

√𝑯𝟏

= 𝑸

𝟐

√𝑯𝟐

𝑄2 = 1.8 𝑋 √13.5

√4.5

𝑄2 = 3.117 𝑚3/𝑠𝑒𝑐



CENTRIFUGAL PUMPS

Course Contents

4.14 Introduction

4.15 Components and Working of a

Centrifugal Pump

4.16 Velocity Diagram and Work

Done of a Centrifugal Pump

4.17 Definitions of Heads &

Efficiencies of Pumps

4.18 Specific Speed

4.19 Minimum Starting Speed

4.20 Maximum Suction Lift

4.21 Net Positive Suction Head

4.22 Priming of Centrifugal Pump

4.23 Multi-stage Centrifugal Pump

4.24 Characteristic Curves of

Hydraulic Pumps

4.25 Cavitation of Pump & Turbine

4.26 Solved Numerical



4.14 Introduction

− “The hydraulic machines which convert the mechanical energy into hydraulic energy

are called pumps.”

− It increases pressure energy or kinetic energy or both by using mechanical energy.

The energy level of the fluid can be increased by either rotodynamic action or by

positive displacement of the fluid.

− “If the mechanical energy is converted into pressure energy or kinetic energy by

means of centrifugal force acting on the fluid, the hydraulic machine is called

Centrifugal pump.”

− They work on the same principle of a reaction turbine. The basic difference between

pump and a turbine is that in a turbine, flow takes place from the high pressure side

to low pressure side while in a pump flow takes place from low pressure side to high

pressure side.

Classification of Pumps on the Basis of Transfer of Mechanical Energy:

Applications of Hydraulic Pumps:

• Agriculture and irrigation work

• Municipal water works and drainage system

• Condensate, boiler feed, sump drain and such other services in a steam

power plant

• Hydraulic control system

• Oil pumping

Pumps

Rotodynamic

Centrifugal

Propeller

Turbine

Positive Displacement

Reciprocating

Piston

Plunger

Diaphram

Rotory

Gear

Vane

Lobe

Screw



• Transfer of material in industry.

4.15 Components and Working of a Centrifugal Pump

Components:

− Main parts of a centrifugal pump (refer Fig. 4.19) are:

1. Impeller

2. Casing

3. Suction pipe

4. Delivery pipe

Fig. 4.19 Main parts of a centrifugal pump

1. Impeller

− It is a wheel or rotor which is provided with a series of backward curves vanes or

blades. It is mounted on a shaft which is coupled to an external source of energy

(electric motor), which imparts required energy to the impeller.

− It gets mechanical energy and converts it to kinetic and pressure energy of the fluid.

− Liquid enters the impeller through an eye of the impeller, high energy liquid than

enters the pump casing.



2. Casing

− It is an air tight passage surrounding the impeller, designed in such a way that kinetic

energy of the water discharged at the outlet of the impeller is converted into

pressure energy before the water leaves the casing and enters the delivery pipe.

− Material of the casing is generally cast iron or cast steel.

− The efficiency of the pump depends on the type of casing used. The following three

types of casings are commonly used:

A. Volute Casing

B. Vortex Casing and

C. Casing with Guide Blades

A. Volute Casing

− It is of spiral type in which area of flow increases gradually. [𝐴(↑) → 𝑉(↓) →

𝑃(↑)]

− It is observed that in case of volute casing, large amount of kinetic energy is lost

due to eddy formation and hence lower overall efficiency.

− These pumps hence give comparatively low head.

B. Vortex Casing

− In this type of casing, a circular chamber is provided in between the casing and

the impeller, which is known as vortex or whirlpool chamber (refer Fig. 4.20 (a)).

− By introducing the circular chamber, the loss of energy due to the formation of

eddies is reduced to a considerable extent.

− Thus the efficiency of the pump is more than the efficiency when only volute

casing is provided.

Fig. 4.20 Types of casing



C. Casing with Guide Blades

− Most efficient casing.

− In this impeller is surrounded by a series of guide blades mounted on a ring

which is known as diffuser (refer Fig. 4.20 (b)).

− The guide vanes are designed in such a way that the water from the impeller

enters the guide vanes without shock which avoids hydraulic losses.

− Also the area of guide vanes increases, thus reducing velocity of flow and

consequently increases the pressure of water.

− Used for developing high heads and hence mostly used as vertical pumps and

very suitable for installations in deep wells, mines, etc.

− Casing is in most of the cases concentric with the impeller.

3. Suction Pipe

− It carries liquid from the sump to the pump.

− Its lower end is dipped into the sump and upper end is connected with the eye of the

pump (i.e. inlet of the pump).

− A strainer and foot-valve are connected with the lower end.

− Strainer keeps the debris away from entering into suction pipe and hence only clear

water enters the impeller.

− Foot-valve is a kind of non-return valve which does not allow the liquid to go back

into sump.

− Cavitation may be caused due to negative pressure at the suction of the pump and

hence losses in the inlet pipe must be minimized.

− To keep low velocity in suction pipe, normally diameter of the suction pipe is kept

more than that of the delivery pipe.

4. Delivery Pipe

− A pipe whose one end is connected to the outlet of the pump and other end delivers

the water at a required height is known as delivery pipe.

− The velocity of liquid in delivery pipe is kept slightly higher than that in suction pipe.

− A valve is provided just near the pump outlet to regulate the flow of liquid in the

delivery pipe.

Working:

− “A centrifugal pump works on a principle that when the liquid is rotated by an

external prime mover, it is thrown away from the axis of rotation and a centrifugal

head is imparted which makes it possible to raise to the higher elevation.”



− Before starting a centrifugal pump, liquid is filled in the suction pipe, impeller, casing

and a delivery pipe up to a delivery valve. This is known as priming. During priming

delivery valve is kept close.

− After priming, prime mover (electric motor) is started; delivery valve is still kept

closed.

− Energy given to the impeller by external source (i.e. prime mover) is transferred to

working fluid which increases the kinetic energy and pressure energy of the fluid.

− The rotation of the impeller causes strong suction at the eye of the pump.

− After the impeller attains its normal speed, the delivery valve is opened and liquid is

allowed to flow through the impeller vanes and it attains higher velocity at the outer

periphery.

− Liquid enters into casing, due to special design of casing the velocity of liquid

decreases and pressure energy hence increases.

− With high pressure energy and negligible kinetic energy liquid enters into delivery

pipe and is lifted to the required height.

− At that instant partial vacuum is created at the eye of pump due to centrifugal action

of impeller on liquid.

− This helps liquid to rush through the suction pipe towards the impeller eye, to take

place of liquid which has left the impeller vanes.

− When the pump is to be stopped the delivery valve should be first closed to stop the

back flow of liquid.

4.16 Velocity Diagram and Work Done of a Centrifugal Pump

− In case of the centrifugal pump,

work is done by the impeller on the

water. The expression for the work

done by the impeller on the liquid is

obtained by drawing velocity

triangles at the inlet and outlet of

the impeller in the same way as for

a turbine.

− Fig. 4.21 shows the vane of impeller

and velocity triangles at the inlet

and outlet of the impeller.

Fig. 4.21 Velocity triangles of

centrifugal Pump



− The water enters the impeller radially at inlet for the best efficiency of the pump,

which means the absolute velocity of water at inlet makes an angle of 90° with the

direction of motion of the impeller at inlet. Hence 𝛼 = 90° 𝑎𝑛𝑑 𝑉𝑤1 = 0.

Assumptions:

• No energy losses due to friction and eddy formation

• No loss due to shock at entry

• Liquid enters the impeller eye in radial direction

• Uniform velocity distribution in the passage between two adjacent vanes.

− Let,

𝑁 = Speed of the impeller in rpm.

𝐷1 = Diameter of impeller at the inlet

𝐷2 = Diameter of impeller at the outlet

𝑢1 = Tangential velocity of impeller at the inlet =𝜋𝐷1𝑁

60

𝑢2 = Tangential velocity of impeller at the outlet =𝜋𝐷2𝑁

60

𝑉1 = Absolute velocity of water at the inlet

𝑉𝑟1 = Relative velocity of water at the inlet

𝛼 = Angle made by absolute velocity at inlet with the direction of motion of vane

𝜃 = Angle made by relative velocity at inlet with the direction of motion of vane and

𝑉2, 𝑉𝑟2, 𝛽 𝑎𝑛𝑑 𝜑 are corresponding values at outlet.

− A centrifugal pump is the reverse of a radially inward flow reaction turbine. But in

case of a radially inward flow reaction turbine, the work done by the water on the

runner per sec per unit weight is given by,

=1

𝑔(𝑉𝑤1𝑢1 − 𝑉𝑤2𝑢2)

− Therefore, work done by the impeller on the water per sec per unit weight,

= −[𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑖𝑛 𝑐𝑎𝑠𝑒 𝑜𝑓 𝑡𝑢𝑟𝑏𝑖𝑛𝑒]

= −1


=1


=1

𝑔(𝑉𝑤2𝑢2) − − − − − − − −(4.40) (∵ 𝑉𝑤1 = 0)

− Work done by the impeller on water per sec,

= �̇�(𝑉𝑤2𝑢2)

= 𝜌𝑄(𝑉𝑤2𝑢2) − − − − − − − −(4.41)

− Discharge,

𝑄 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑓𝑙𝑜𝑤 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑜𝑤



∴ 𝑄 = 𝜋𝐷1𝐵1 × 𝑉𝑓1 = 𝜋𝐷2𝐵2 × 𝑉𝑓2 − − − − − − − −(4.42)

Where,

𝐵1 𝑎𝑛𝑑 𝐵2 are the width of the impeller at the inlet and outlet respectively.

− Equation (4.40) gives the head imparted to the water by the impeller or energy given

by impeller to water per sec per unit weight.

4.17 Definitions of Heads and Efficiencies of a Centrifugal Pump

Heads:

1. Suction Head or Suction Lift (𝒉𝒔)

− It is the vertical height of the center line of the pump shaft above the liquid surface

in the sump from which the liquid is being lifted. (refer Fig. 4.19)

2. Delivery Head (𝒉𝒅)

− The vertical distance between the center line of the pump shaft and the liquid

surface in the tank to which liquid is delivered. (refer Fig. 4.19)

3. Static Head or Static Lift (𝑯𝒔)

− As shown in Fig. 4.19 the static head is the vertical distance between the liquid

surface in the sump and the tank to which the liquid is delivered by the pump.

− Thus the static head may be expressed as,

𝐻𝑠 = ℎ𝑠 + ℎ𝑑 − − − − − − − −(4.43)

− Thus static head is the net total vertical height through which the liquid is lifted by

the pump.

4. Manometric Head (𝑯𝒎)

− It is defined as the head against which a centrifugal pump has to work.

Or

It is the total head that must be produced by the pump to satisfy the external

requirements.

− It is given by the following expressions:

a) If there are no losses in the impeller and casing of the pump, then the

manometric head will be equals to the energy given to the liquid by the

impeller.

∴ 𝐻𝑚 =1

𝑔(𝑉𝑤2𝑢2) − 𝑙𝑜𝑠𝑠 𝑜𝑓 ℎ𝑒𝑎𝑑 𝑖𝑛 𝑖𝑚𝑝𝑒𝑙𝑙𝑒𝑟 & 𝑐𝑎𝑠𝑖𝑛𝑔 − − − − − − − −(4.44)

∴ 𝐻𝑚 =1

𝑔(𝑉𝑤2𝑢2) (𝑖𝑓 𝑙𝑜𝑠𝑠𝑒𝑠 𝑎𝑟𝑒 𝑛𝑒𝑔𝑙𝑒𝑐𝑡𝑒𝑑)



b) Manometric head is the difference of total head at the outlet and total head

at the inlet of the pump.

∴ 𝐻𝑚 = (𝑃𝑜

𝜌𝑔+

𝑉𝑜2

2𝑔+ 𝑍𝑜) − (

𝑃𝑖

𝜌𝑔+

𝑉𝑖2

2𝑔+ 𝑍𝑖) − − − − − − − −(4.45)

c) Whole of the manometric head is not used to lift the liquid against the static

lift; a part of it is used to overcome the losses in the pipes and fittings and to

provide the kinetic energy at delivery outlet.

∴ Manometric head = static head + head losses in suction and delivery pipes +

velocity head in delivery pipe

∴ 𝐻𝑚 = (ℎ𝑠 + ℎ𝑑) + (ℎ𝑓𝑠 + ℎ𝑓𝑑) +𝑉𝑑

2

2𝑔− − − − − − − −(4.46)

Efficiencies:

− In case of a centrifugal pump, the power is transmitted from the shaft of the electric

motor to the shaft of the pump and then to the impeller. From the impeller, the

power is given to the water.

− The followings are the important efficiencies of a centrifugal pump:

1. Manometric Efficiency

2. Mechanical Efficiency and

3. Overall Efficiency

1. Manometric Efficiency (𝜼𝒎𝒂𝒏)

− It is defined as the ratio of the manometric head developed by the pump to the head

imparted by the impeller to the liquid.

∴ 𝜂𝑚𝑎𝑛 =𝑀𝑎𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 ℎ𝑒𝑎𝑑

𝐻𝑒𝑎𝑑 𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑 𝑏𝑦 𝑖𝑚𝑝𝑒𝑙𝑙𝑒𝑟 𝑡𝑜 𝑙𝑖𝑞𝑢𝑖𝑑

∴ 𝜂𝑚𝑎𝑛 =𝐻𝑚

(𝑉𝑤2𝑢2

𝑔)

=𝑔𝐻𝑚

𝑉𝑤2𝑢2− − − − − − − −(4.47)

− The power at the impeller of the pump is more than that the power given to the

liquid at outlet of the pump.

𝑷𝒐𝒘𝒆𝒓 𝒈𝒊𝒗𝒆𝒏 𝒕𝒐 𝒘𝒂𝒕𝒆𝒓 𝒂𝒕 𝒐𝒖𝒕𝒍𝒆𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒖𝒎𝒑 =𝑾𝑯𝒎

𝟏𝟎𝟎𝟎=

𝝆𝒈𝑸𝑯𝒎

𝟏𝟎𝟎𝟎 𝒌𝑾

𝑷𝒐𝒘𝒆𝒓 𝒂𝒕 𝒕𝒉𝒆 𝒊𝒎𝒑𝒆𝒍𝒍𝒆𝒓 =𝑾𝑫 𝒃𝒚 𝒕𝒉𝒆 𝒊𝒎𝒑𝒆𝒍𝒍𝒆𝒓 𝒑𝒆𝒓 𝒔𝒆𝒄

𝟏𝟎𝟎𝟎=

𝝆𝑸(𝑽𝒘𝟐𝒖𝟐)

𝟏𝟎𝟎𝟎 𝒌𝑾



2. Mechanical Efficiency (𝜼𝒎)

− It is defined as the ratio of the power actually delivered by the impeller to the power

at the shaft of the centrifugal pump.

∴ 𝜂𝑚 =𝑃𝑜𝑤𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑚𝑝𝑒𝑙𝑙𝑒𝑟

𝑃𝑜𝑤𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 𝑠ℎ𝑎𝑓𝑡

∴ 𝜂𝑚 =�̇�(𝑉𝑤2𝑢2) 1000⁄

𝑆. 𝑃. 𝑖𝑛 𝑘𝑊− − − − − − − −(4.48)

3. Overall Efficiency (𝜼𝒐)

− It is defined as the ratio of power output of the pump to the power input to the

pump.

𝑃𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑢𝑚𝑝 =𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑙𝑖𝑓𝑡𝑒𝑑 × 𝐻𝑚

1000=

𝑊𝐻𝑚

1000 𝑘𝑊

𝑃𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑢𝑚𝑝 = 𝑃𝑜𝑤𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑚𝑜𝑡𝑜𝑟

= 𝑆ℎ𝑎𝑓𝑡 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑢𝑚𝑝

∴ 𝜂𝑜 =(

𝑊𝐻𝑚

1000)

𝑆. 𝑃.− − − − − − − −(4.49)

∴ 𝜂𝑜 = 𝜂𝑚𝑎𝑛 × 𝜂𝑚

4.18 Specific Speed

− “The specific speed of a centrifugal pump is defined as the speed of a geometrically

similar pump which delivers unit quantity against a unit head.”

− It is used to compare the performance of different pumps.

− For a centrifugal pump,

𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒, 𝑄 = 𝐴𝑟𝑒𝑎 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑜𝑤

∴ 𝑄 = 𝜋𝐷𝐵 × 𝑉𝑓

∴ 𝑄 ∝ 𝐷𝐵𝑉𝑓 − − − − − − − −(4.50)

Where,

𝐷 = Diameter of the impeller of the pump

𝐵 = Width of the impeller

We know that,

𝐵 ∝ 𝐷

∴ 𝑄 ∝ 𝐷2𝑉𝑓 − − − − − − − −(4.51)

− Tangential velocity is given by,

𝑢 =𝜋𝐷𝑁

60∝ 𝐷𝑁 − − − − − − − −(4.52)



− Now tangential velocity (𝑢) and velocity of flow (𝑉𝑓) are related to the manometric

head (𝐻𝑚) as,

𝑢 ∝ 𝑉𝑓 ∝ √𝐻𝑚 − − − − − − − −(4.53)

− Substituting value of 𝑢 in equation (4.52), we get,

√𝐻𝑚 ∝ 𝐷𝑁

∴ 𝐷 ∝√𝐻𝑚

𝑁

− Substituting value of D in equation (4.51), we get,

𝑄 ∝𝐻𝑚

𝑁2𝑉𝑓

∴ 𝑄 ∝𝐻𝑚

𝑁2 √𝐻𝑚

∴ 𝑄 ∝𝐻𝑚

3/2

𝑁2

∴ 𝑄 = 𝐾𝐻𝑚

3/2

𝑁2− − − − − − − −(4.54)

Where, K = Constant of proportionality.

− By definition, if 𝐻 = 1𝑚 and 𝑄 = 1 𝑚3 𝑠𝑒𝑐⁄ , 𝑁 becomes 𝑁𝑠

Substituting these values in equation (4.54), we get,

1 = 𝐾 ×1

𝑁𝑠2

∴ 𝐾 = 𝑁𝑠2

− Substituting value of 𝐾 in equation (4.54), we get,

𝑄 = 𝑁𝑠2

𝐻𝑚3/2

𝑁2

∴ 𝑁𝑠2 =

𝑄𝑁2

𝐻𝑚3/2

∴ 𝑵𝒔 =𝑵√𝑸

𝑯𝒎𝟑/𝟒

− − − − − − − −(𝟒. 𝟓𝟓)

4.19 Priming of Centrifugal Pump − Before starting a centrifugal pump, the suction pipe, casing and portion of the

delivery pipe up to delivery valve is completely filled with water by external source

of water to remove the air from the suction pipe and casing. This is known as priming

of a pump.



− The work done by the impeller per unit weight of liquid per sec is known as the head

developed by an impeller.

− Head developed by the impeller is given by 𝑢2𝑉𝑤2

𝑔 meter. Since this equation is

independent of the density of the liquid, the head developed will be in terms of

meters of air when pump is running in the air.

− If the pump is primed with water, the head generated is same meter of water. But as

the density of air is very low, the generated head of air is negligible compared to

meter of water head. Hence the water may not be sucked from the pump. To avoid

this difficulty, priming is necessary.

4.20 Multi-stage Centrifugal Pump

− If a centrifugal pump consists of two or more impellers, the pump is called a multi-

stage centrifugal pump.

− The impellers may be mounted on the same shaft or different shaft.

− A multi-stage pump is having the two important functions:

I. To produce a high head and

II. To discharge a large quantity of water.

− When the pumps are connected in series discharge of the first pump enters the

second pump where the pressure is further increased. If two pumps are connected

final head would be

𝐻 = 𝐻1 + 𝐻2

Fig. 4.22 (a) Pump in series



− For high head, impellers are connected in series (on same shaft) as shown in Fig.4.22

(b).

Fig. 4.22 (b) Pump in series

− When a large quantity (high discharge) of liquid is required to be pumped against a

relatively small head, then it may not be possible for a single pump to deliver the

required discharge.

− In such cases two or more pumps are used which are so arranged that each of these

pumps working separately lifts the liquid from a common collecting pipe through

which it is carried to the required height. Since in this Case each of the pump delivers

the liquid against the same head, the arrangement is known as pumps in parallel.

𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑚𝑝𝑒𝑙𝑙𝑒𝑟𝑠

𝑄 = 𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 = 𝑛 × 𝐻𝑚

Fig. 4.23 (a) Pump in parallel



− For high discharge, the impellers are connected in parallel as shown in Fig. 4.23 (b).

Fig. 4.23 (b) Pumps in parallel

4.21 Characteristic Curves of Hydraulic Pumps

− Characteristic curves of centrifugal pumps are defined those curves which are

plotted from the results of a number of tests on the centrifugal pump. These curves

are necessary to predict the behavior and performance of the pump when the pump

is working under different flow rate, head and speed.

− The followings are the important characteristic curves for pumps:

1. Main Characteristic Curves,

2. Operating Characteristic Curves and

3. Constant Efficiency or Muschel Curves.

1. Main Characteristic Curves

− The main characteristic curves of a centrifugal pump consists of variation of head

(𝐻𝑚), power and discharge with respect to speed.

− For plotting curves of manometric head versus speed, discharge is kept constant. For

plotting curves of discharge versus speed, manometric head is kept constant and for

plotting curves of power versus speed, the manometric head and discharge are kept

constant.

− Fig. 4.24 shows main characteristic curves of a pump.

𝑛 = 𝑁𝑜. 𝑜𝑓 𝑝𝑢𝑚𝑝𝑠 𝑖𝑛 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

𝐻𝑒𝑎𝑑 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝𝑢𝑚𝑝𝑠

𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 = 𝑛 × 𝑄



Fig. 4.24 Main characteristic curves of a pump

2. Operating Characteristic Curves

− If the speed is kept constant, the variation of manometric head, power and efficiency

with respect to discharge gives the operating characteristics of the pump. Fig. 4.25

shows the operating characteristic curves of a pump.

Fig. 4.25 Operating characteristic curves of a pump

− The input power curve for pumps shall not pass through the origin. It will be slightly

away from the origin on the y-axis, as even at zero discharge some power is needed

to overcome mechanical losses.

− The head curve will have maximum value of head when discharge is zero.

− The output power curve will start from origin as at 𝑄 = 0, output power (𝜌𝑔𝑄𝐻)

will be zero.

− The efficiency curve will start from origin as at 𝑄 = 0, 𝜂 = 0.



3. Constant Efficiency or Muschel Curves

− For obtaining constant efficiency curves (iso-efficiency curves) for a pump, the head

versus discharge curves and efficiency versus discharge curves for different speeds

are used.

− Fig. 4.26 (a) shows the head versus discharge curves for different speeds. The

efficiency versus discharge curves for the different speeds are as shown in Fig. 4.9(b).

− By combining these curves(𝐻~𝑄 𝑐𝑢𝑟𝑣𝑒𝑠 𝑎𝑛𝑑 𝜂~𝑄 𝑐𝑢𝑟𝑣𝑒𝑠), constant efficiency

curves are obtained as shown in Fig. 4.26 (a).

− For obtaining constant efficiency curves, horizontal lines representing constant

efficiencies are drawn on the 𝜂~𝑄 curves.

− The points at which these lines cut the efficiency curves at various speeds, are

transferred to the corresponding 𝐻~𝑄 curves.

− The points having the same efficiency are then joined by smooth curves. These

smooth curves represents the iso-efficiency or constant efficiency curves.

Fig. 4.26 Constant efficiency curves of a pump

4.22 Cavitation of Pump & Turbine

− Cavitation is defined as the phenomenon of formation of vapor bubbles of a flowing

liquid in a region where the pressure of the liquid falls below its vapor pressure and

the sudden collapsing of these vapor bubbles in a region of higher pressure.



− When the vapor bubble collapse, a very high pressure is created. The metallic

surfaces, above which these vapor bubbles collapse, is subjected to these high

pressures, which cause pitting action on the surface. Thus cavities are formed on the

metallic surface and also considerable noise and vibrations are produced.

− Cavitation includes formation of vapor bubbles of the flowing liquid and collapsing of

the vapor bubbles.

Precaution against Cavitation: The following precautions should be taken against

Cavitation:

1) The pressure of the flowing liquid in any part of the hydraulic system should

not be allowed to fall below its vapor pressure.

2) The special materials or coatings such as aluminum-bronze and stainless

steel, which are cavitation resistant materials, should be used.

Effects of Cavitation: The following are the effects of cavitation:

1) The metallic surfaces are damaged and cavities are formed on the surfaces.

2) Due to sudden collapse of vapor bubble, considerable noise and vibrations

are produced.

3) The efficiency of a turbine decreases due to cavitation. Due to pitting action,

the surface of the turbine blades becomes rough and the force exerted by

water on the turbine blade decreases. Hence the work done by water or

output horse power becomes less and thus efficiency decreases.

Cavitation in Turbines:

− In turbines, only reaction turbines are subjected to cavitation.

− In reaction turbines, the cavitation may occur at the outlet of the runner or at the

inlet of the draft tube, where the pressure is considerably reduced (i.e. , which may

be below the vapor pressure of the liquid flowing through the turbine).

− Due to cavitation, the metal of the runner vanes and draft tube is gradually eaten

away, which results in lowering the efficiency of the turbine.

− Hence the cavitation in a reaction turbine can be noted by a sudden drop in

efficiency.

− In order to determine whether cavitation will occur in any portion of a reaction

turbine, the critical value of Thoma’s cavitation factor (𝜎) is calculated (Equation

4.56).

𝜎 =𝐻𝑏 − 𝐻𝑠

𝐻𝑛𝑒𝑡=

(𝐻𝑎𝑡𝑚 − 𝐻𝑣) − 𝐻𝑠

𝐻𝑛𝑒𝑡− − − − − − − −(4.56)



Cavitation in Centrifugal Pumps:

− In centrifugal pumps the cavitation may occur at the inlet of the impeller of the

pump, or at the suction side of the pumps, where the pressure is considerably

reduced.

− Hence if the pressure at the suction side of the pump drops below the vapor

pressure of the liquid then the cavitation may occur.

− The cavitation in a pump can be noted bay a sudden drop in efficiency and head.

− In order to determine whether cavitation will occur in any portion of the suction side

of the pump, the critical value of Thoma’s cavitation factor (𝜎) is calculated

(Equation 4.57).

𝜎 =𝐻𝑏 − 𝐻𝑠 − ℎ𝑓𝑠

𝐻𝑛𝑒𝑡=

(𝐻𝑎𝑡𝑚 − 𝐻𝑣) − 𝐻𝑠 − ℎ𝑓𝑠

𝐻𝑛𝑒𝑡− − − − − − − −(4.57)

− If the value of Thoma’s cavitation factor (𝜎) is greater than critical cavitation factor

(𝜎𝑐), the cavitation will not occur in that turbine or pump. The critical cavitation

factor (𝜎𝑐) may be obtained from tables or empirical relationships.

4.23 Ventilation System

− Ventilation is meant for supply of fresh air, and to replace the old hot Used up

(exhausted) air. The ventilation ensures the removal of bad effects of occupancy of

an enclosed space:

1) By providing necessary oxygen to remove oxygen deficit caused by

respiration;

2) By removing and diluting C02 in the air;

3) By lowering down the temperature by removing hot used up air and

replacing it by colder fresh air;

4) By reducing humidity

5) By reducing body odours.

Requirements of ventilation system

− A good ventilation system should generally fulfill the following requirements:

1) It should admit sufficient quantity of fresh air, and remove the requisite used

up or vitiated air.

2) Admitted air should be properly controlled with respect to its quantity as well

as velocity of movement.

3) The system should be capable of changing the old air thoroughly, without

leaving any stagnant pockets in the room.



4) Should avoid draughts, for which maximum permissible velocity of admitted

air should not exceed 15 m/min. i.e. 0.25 m/sec.

5) The system should admit clean and humid air.

6) The system should also be capable of controlling the temperature of

admitted air.

The ventilation system can be broadly divided in to two categories;

1) Natural ventilation:

− Natural ventilation is based upon providing suitable openings in a room, at lower

levels for admitting free atmospheric air, and also at upper levels for removing the

warmer and lighter used-up air. Doors and windows near the floor level, thus admit

fresh air and ventilators near the ceiling, take out the vitiated air from a room.

2) Artificial or mechanical ventilation:

− The artificial ventilation system can be broadly divided in to:

1) The extraction or vacuum system

2) The propulsion or plenum system

3) The air conditioning system

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