Subject: FLUID MECHANICS -II

Topic: Hydraulic Machines

Module 1 (Pumps)

Part - II

Institute: NIT Jamshedpur

Branch: Civil Engineering

Class & Sem: B.Tech (H), 4th Sem.

Teacher: Dr. Ch. Madhusudana Rao, Associate Professor

PRESSURE CHANGES IN A PUMP

Refer to the given figure which shows the installation of a centrifugal pumping set. Let S represent the surface of liquid in the sump. D represent the surface of the liquid in the discharge tank. A denotes inlet to pump. B denotes outlet of pump.

The generally accepted definitions of heads against which the pump has to work are: (i) Suction lift 𝑕𝑠, represents vertical distance between top surface of the liquid in the sump and the centre of pump impeller.

(ii) Discharge lift 𝑕𝑑 represents vertical distance between the centre of the pump impeller and the top of the surface of the liquid in the discharge tank. (iii) Total static or vertical lift represents the sum of suction and delivery lifts ; height (𝑕𝑠 + 𝑕𝑑 )

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(iv) Suction head 𝐻𝑠 of the pump:

𝐻𝑠=𝑕𝑖 + 𝑕𝑓𝑠 + 𝑕𝑠 + 𝑉𝑠

2

2𝑔

Where, 𝑕𝑖 = is the loss of head at inlet to suction pipe

Suffix s denotes suction pipe

Suffix d denotes delivery pipe

𝑕𝑓𝑠 = is the loss of head due to friction in suction pipe

𝑣𝑠 = 𝑓𝑙𝑜𝑤 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑡𝑕𝑒 𝑠𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑖𝑝𝑒

𝐻𝑠= 𝑝𝑠

𝑤 +

𝑉𝑠2

2𝑔

𝑝𝑠

𝑤 = pressure head in suction pipe

(v) Delivery head 𝐻𝑑 of the pump:

𝐻𝑑= 𝑕𝑑 + 𝑕𝑓𝑑 +𝑉𝑑

2

2𝑔

𝑕𝑓𝑑 = is the loss of head due to friction in delivery pipe

𝑣𝑑 = 𝑓𝑙𝑜𝑤 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑡𝑕𝑒 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑝𝑖𝑝𝑒

𝐻𝑑= 𝑝𝑑

𝑤 +

𝑉𝑑2

2𝑔

𝑝𝑠

𝑤 = pressure head in delivery pipe

(vi) Total external head H against which the pump has to work is given by;

H=(suction head)+(delivery head)-( velocity head in the suction pipe)

H=𝐻𝑠 + 𝐻𝑑 − 𝑉𝑠

2

2𝑔=

𝑝𝑠

𝑤 +

𝑉𝑠2

2𝑔+

𝑝𝑑

𝑤 +

𝑉𝑑2

2𝑔

=(𝑕𝑖+ 𝑕𝑓𝑠 + 𝑕𝑠) + 𝑕𝑑 + 𝑕𝑓𝑑 +𝑉𝑑

2

2𝑔

(vii) Manometric head 𝐻𝑚= 𝑝𝑑

𝑤 +

𝑉𝑑2

2𝑔+ 𝑕 −

𝑝𝑠

𝑤 +

𝑉𝑠2

2𝑔

Assume (𝑉𝑠 = 𝑉𝑑) and the gauges are at the same level (h=0)

𝐻𝑚 =𝑝𝑑

𝑤−

𝑝𝑠

𝑤

3

Manometric head = static head + head losses (friction and minor) in the suction and delivery pipe + velocity head in the delivery pipe

𝐻𝑚 = (𝑕𝑠 + 𝑕𝑑 ) + (𝑕𝑓𝑠 + 𝑕𝑓𝑑 ) + 𝑉𝑑

2

2𝑔= (𝑕𝑠 + 𝑕𝑓𝑠 ) + 𝑕𝑑 + 𝑕𝑓𝑑 +

𝑉𝑑2

2𝑔

= 𝐻𝑠 −𝑉𝑠

2

2𝑔+ 𝐻𝑑 = 𝐻𝑠 + 𝐻𝑑 −

𝑉𝑠2

2𝑔

(viii) Net positive suction head (NPSH) represents the combination of following heads:

NPSH = (absolute pressure at inlet to pump) – (vapour pressure of liquid being pumped)

+ (velocity head in suction pipe)

NPSH = 𝑝𝑠

𝑤−

𝑝𝑣

𝑤+

𝑉𝑠2

2𝑔=

𝑝𝑎

𝑤− 𝑕𝑠 − 𝑕𝑓𝑠 −

𝑉𝑠2

2𝑔−

𝑝𝑣

𝑤+

𝑉𝑠2

2𝑔

Where, 𝑝𝑎 denotes the atmospheric pressure on the surface of liquid in the section well. Henceforth:

NPSH = 𝑝𝑎

𝑤−

𝑝𝑣

𝑤− 𝑕𝑠 − 𝑕𝑓𝑠

Evidently , the NPSH represents suction head at the impeller eye. It represents the head required to make the liquid flow from the suction pipe to the impeller. For smooth and cavitation free operation of the pump, NPSH should have such a value that the flowing liquid does not boil under reduced pressure.

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VELOCITY VECTOR DIAGRAMS AND WORK DONE

The velocity vector diagrams at inlet and outlet of the impeller of a centrifugal pump are shown in Fig. Assumptions: • Infinite number of vanes : liquid flowing in the

passages follows the path outlined by the vanes. • No energy loss in the impeller due to fraction and

eddy formation • Even/uniform velocity distribution in the narrow

passages formed between two adjusting vanes. • Liquid enters the impellers eye in an axial/radial

direction, i.e., the whirl component𝑉𝑤1of the inlet absolute velocity 𝑉1 is Zero and the flow component 𝑉𝑓1 equals the absolute velocity itself.

• No loss due to shock at entry, i.e., inlet edge of the impeller blades is parallel to the relative velocity.

• Liquid enters the impeller eye in radial direction, α = angle made by the absolute velocity V1

β = angle made by the absolute velocity V2

𝑉𝑤1 = 0 𝑉𝑓1 = 𝑉1 ; α = 90ᵒ

Fig. velocity vector diagrams of a centrifugal impeller

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Work done by the impeller (or centrifugal pump) on liquid

Let, 𝐷1 − Diameter of the impeller at inlet 𝑅1 = 𝐷1

2

N − Speed of the impeller in R.P.M

ω − Angular velocity =2π𝑁

60 𝑅𝑎𝑑/𝑠𝑒𝑐

𝑢1 − Tangential velocity of the impeller at inlet

=π𝐷1𝑁

60=

2π𝑅1𝑁

60= ω𝑅1

Similarly at outlet 𝑢2 = ω𝑅2 While passing through the impeller, the velocity of whirl changes and there is a change of moment of momentum. Torque on the impeller = Rate of the changes of moment of momentum Moment of momentum at inlet = 0 Where; Vw1 = 0

Moment of momentum a outlet = 𝑊

𝑔(Vw1 𝑅2)

Therefore, Torque = 𝑊

𝑔(Vw2 𝑅2) (or) moment of momentum

Work done per second = Torque x Angular Velocity

= 𝑊

𝑔(Vw2 𝑅2) x ω

= 𝑊

𝑔Vw2 𝑢2 ( Where, 𝑢2 = ω 𝑅2) …………… ( 1 )

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Therefore, work done per second per unite weight of liquid: • if the flow is radial (or) energy given by the impeller /unit wt on liquid

=Vw2 𝑢2

𝑔 …….……….(1a)

(Note : reverse or radially inward flow reaction turbine in CP) • If the flow Is not radial:

WD/Sec = 𝑊

𝑔 (Vw2 𝑢2 - Vw1 𝑢1)

Or WD/sec/ unit weight or 𝐻𝑒 = 1

𝑔 (Vw2 𝑢2 - Vw1 𝑢1) …………………(2)

This is know as the Euler momentum, equation for centrifugal pumps

The term 1

𝑔 (Vw2 𝑢2 - Vw1 𝑢1) is referred as Euler Head 𝐻𝑒.

(Theoretical Head) Where W = Weight of liquid = ω x Q And discharge Q = π𝐷1𝐵1𝑥 𝑣𝑓1 =

π𝐷2𝐵2𝑥 𝑣𝑓2 = Volume of liquid.

From Eqn. ( 1 ) stipulation that for delivering liquid at high heads the 𝑢2 must be high ad vector 𝑉𝑤2 must be larger ( so as to provide adequate whirl to the liquid). The increase in 𝑉2 can be obtained by increasing the impeller diameter and speed of rotation. The Vw2 however can be augmented by providing number of valves of sustainable size and shape.

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Losses in centrifugal pump

When a centrifugal pump operates the various losses which occur are as follows:

1. Hydraulic losses :

(i) Hydraulic losses in the pump

a) Shock or eddy losses at the entrance to and exit from the impeller.

b) Losses due to friction in the impeller

c) Friction and eddy losses in the guide vanes/diffuser and casing

(ii) Other hydraulic losses

a) Friction and other minor losses in the suction pipe.

b) Friction and other minor losses in the delivery pipe.

2. Mechanical losses :

(i) Losses due to disc friction between the impeller and the liquid which fills the clearance spaces between the

impeller and casing.

(ii) Losses pertaining to friction of the main bearings and glands

3. Leakage loss : The loss of energy due to leakage of liquid is known as leakage loss.

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Efficiencies of a centrifugal pump In the case of centrifugal pump, the power is transmitted from the shaft of the electric motor to the shaft of the pump and then to the impeller, the power is given to the water. Thus power is decreasing from the shaft of the pump to the impeller and then to the water. a) Manometric efficiency , η𝑚𝑎𝑛 b) Mechanical efficiency , η𝑚 c) Overall efficiency , η𝑜 d) Volumetric efficiency (η𝑣) a) Manometric efficiency (): The ratio of manometric head to the head imparted by the impeller to the water

known as manometric efficiency

η𝑚𝑎𝑛 = Manometric head

Head imparted by impeller to water

= 𝐻𝑚

Vw2 𝑢2

𝑔

=

𝑔𝐻𝑚

Vw2 𝑢2

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b) Mechanical efficiency (η𝑚):

η𝑚 =Power at the impeller (smaller)

Power at the shaft (greater)

The power at the impeller = Work done by impeller per sec

75

= 𝑊

𝑔 *

Vw2 𝑢2

75

η𝑚 =

𝑊

𝑔

Vw2 𝑢275

𝑆.𝐻.𝑃

Where, S.H.P = Shaft horse power c) Overall efficiency (η𝑜):

Power output of the pump = 𝑊𝑒𝑖𝑔𝑕𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑙𝑖𝑓𝑡𝑒𝑑 ∗ 𝐻𝑚

75 =

𝑊 𝐻𝑚

75

Ratio of power output of the pump to the power input to the pump. power input of the pump = power supplied by the electric motor = SHP of the pump.

η𝑜 = W Hm

75

𝑆𝐻𝑃

(or) η𝑜 = 𝑚𝑚𝑎𝑛 * 𝑛𝑚

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d) Volumetric efficiency (η𝑣)

η𝑣 =Liquid discharged per second from the pump

Quantity of the 𝑙𝑖𝑞𝑢𝑖𝑑 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑡𝑕𝑟𝑜𝑢𝑔𝑕 𝑡𝑕𝑒 𝑖𝑚𝑝𝑒𝑙𝑙𝑒𝑟

η𝑣 =Q

Q + q

Q = actual liquid discharge at the pump outlet per second q = leakage of liquid per second from the impeller (through the clearances between the impeller and casing )

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Multi-stage centrifugal pumps

Two or more identical impellers mounted on the same shaft or on different shaft

Important functions: 1. To produce heads greater than that permissible with a single impeller, discharge remaining

constant (can be achieved by series arrangement, wherein the impellers are mounted on the same shaft and enclosed in the same casing).

2. To discharge a large quantity of liquid, head remaining same (can be achieved by parallel arrangement, wherein the impellers are mounted on separate shafts).

Pumps in series • If more no. of impellers are mounted on the

same shaft the pressure at the outlet will be increased further.

• In each stage, the manometric head imposed on the liquid is Hmano , then for n identical impellers the total head developed will be Hmano = Nh, However the discharge passing through each impeller is same.

Pumps in parallel • Two or more pumps are employed which are so

arranged that each of these pumps working separately.

• If Q is the discharge capacity of one pump and

there are n identical pumps arranged in parallel then total discharge will be Qtotal = nQ

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Specific Speed: The specific speed of a centrifugal pump is defined as the speed of a geometrically similar pump which would deliver unit quantity (one cubic meter of liquid per second) against a unit head (one meter). It is denoted by 𝑁𝑠 . The specific speed is a characteristic of pumps which can be used as a basic for comparing the performance of different pumps.

Specific speed

Where, N – Speed of rotation of impeller

Q – Discharge ( area x velocity of flow )

The ranges of specific speeds for different types of pumps are tabulated below:

3

4

s

mano

N QN

H

Type of pump

Slow speed radial flow

Medium speed

radial flow

High speed radial flow

Mixed flow (or screw

type)

Axial flow (or propeller type)

Specific speed 10 to 30 30 to 50 50 to 80 80 to 160 160 to 500

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Characteristics of Centrifugal Pumps The following four types of characteristic curves are usually prepared for Centrifugal pumps: 1. Main characteristic curves: The main characteristic curves are obtained when the pump run at constant speed and the discharge is varied over the desired range. Measurements are taken for manometric head ( ), and shaft power ( P ) for each discharge ( Q ) and calculations are made for the overall efficiency η𝑜. 2. Operating characteristic curves: When centrifugal pump operates at design speed, the maximum efficiency occurs. To obtain the operating characteristic curves the pump is run at design speed and the discharge is varied.

manoH

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3. Constant efficiency or Muschel curves The constant efficiency curves depicts the performance of a pump over its entire range of operations. For every given efficiency, the values of discharges are obtained and are projected on the head ( ) v/s discharge ( Q ) for that speed. Similarly, another value of efficiency and speed are obtained and projected. The points corresponding to one efficiency are joined and the curves are called constant efficiency or Muschel curves. 4. Constant head and constant discharge curves. The performance of variable speed pump for which the speed constantly varies can be determined by these curves. When the pump has a variable speed, the plots between Q and N, and and N may be obtained. In the first case is kept constant and in the second case Q is kept constant.

m a n oH

manoH

m a n oH

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Net Positive Suction Head (NPSH ) The net head that is required to make the liquid flow through the suction pipe from the sump to the impeller.

= Absolute Static pressure at pump inlet = Absolute atmospheric pressure = Vapour pressure of the liquid for a given temperature Let = vertical distance between the centre line of pump and the free liquid surface of the sump. = Velocity of the liquid in the suction pipe. = Loses in the suction pipe up to the pump in it. If > by an amount equal to that required by the liquid for the increase in velocity head when entering the impeller, denoted by , we can write …….. ( i ) ………(ii)

From (i) and (ii), we have

Where, =Total suction Head =

1P

aP

vP

sV

fsh

1P vP

svH

2

2s

fs s

Vh h

g

aS

PH

w vP

w

aa

PH

w v

v

PH

w

1 vSV

PPH

w w

1 aPP

w w

2

2

sfs s

Vh h

g

SH

sh

sv a s vH H H H

16

2

2

sfs s

Vh h

g

Cavitation in Centrifugal pumps:

Cavitation begins to appear in centrifugal pump when the pressure at the suction falls bellow the vapour pressure of the liquid. The intensity of cavitation increases with the decrease in the value of NPSH.

The cavitation in a pump can be noted by a sudden drop in efficiency, head and power requirement.

Thoma’s cavitation factor :

Where, = Atmospheric pressure expressed in meters of water head.

= Vapour pressure expressed in meters of water head.

=Total suction Head =

= Net positive suction head ( NPSH) or total suction head

= Manometric head

If - Cavitation will occur

- At which cavitation just begins

- function of specific speed, efficiency of pump, number of vanes.

aH

vH

sH

manoH

svH

2

2s

fs s

Vh h

g

c

c

a s v sv

mano mano mano

H H H H NPSH

H H H

c

43

0.1031000

sc

N

Value for

Harmful effects a) Pitting and erosion of surface b) Sudden drop in head, efficiency c) Noise and vibration

Factors which facilitates cavitation a) Restricted suction b) High specific speed c) High runner speed d) High temperature of the liquid

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Priming of centrifugal pumps:

• The operation of filling the suction pipe, casing of the pump and a portion of the delivery pipe completely

from outside source with the liquid to be raised, before starting the pump, to remove any air, gas or vapour

from these parts of the pump is called priming of a centrifugal pump.

• If a centrifugal pump is not primed before starting, air pockets inside the impeller may give rise to vortices

and cause discontinuity of flow.

• Small pumps are usually primed by pouring liquid in to the funnel provide for the purpose . The priming is

continued till all air from the suction pipe , impeller and casing has been removed.

• Large pumps are primed by evacuating the casing and the suction pipe by a vacuum pump or by a an ejector.

• A supply of liquid are provided in the suction pipe due to which automatic priming of the pump occurs. Such

pumps are known as ‘self priming pumps’.

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