chapter ii – 3 pumps for pipelinesme.metu.edu.tr/courses/me437/pl-chapter ii-3 pumps-web.pdf ·...

Chapter II – 3 Pumps for PipelinesII.3.1. Energy Relations for a Pumping SystemII.3.2. Momentum Relations in a Hydraulic MachineII.3.3. Dimensional Analysis and Similitude for a Hydraulic MachineII.3.4. Specific Speed and TypificationII.3.5. Performance Characteristics

II.3.5.a Operating PointII.3.5.b Flowrate Adjustment in a Pumping SystemII.3.5.c Similar Pumps with Speed ChangeII.3.5.d Similar Pumps with Diameter Change

II.3.6. CavitationII.4.1. System Characteristics and Operating PointII.4.2. Pump System Combinations

II.3.8.a Series Combination of PumpsII.3.8.b Parallel Combination of PumpsII.3.8.c Single Pump and 2 Parallel Systems

II.4.3. Selection of PumpsII.4.4. Operational Problems in Pumping Systems

II.3.1. Energy Relations for a Pumping System

DischargeReservoir

SuctionReservoir

Pd

Ps

Pdr

Psr

Hgd

zsr

zs zd

Hgp

Hgs

Hgt

Datum

Ps : Pump suction end PressurePd : Pump discharge end Press.Psr : Suction Reservoir Press.Pdr : Discharge Reservoir Press.Hgt : Total geometric headHgd : Discharge geometric headHgs : Suction geometric headHgp : Pump geometric head

Suction Side

Discharge side

Typical Pumping System

Extended Bernoulli equation written between the reservoir free surfaces;

2 2

12 2

Net et bt bt

et j btj

p V p VZ H h Zg g g gρ ρ=

+ + + − = + +∑Total head Loss in the System

j : Element with head Loss ( Pipe, elbow, valve, etc.)

For LargeReservoirs :

022

22

≈≈g

Vg

V btet

If reservoirs are open toatmosphere ;

gp

gp btet

ρρ=


etbtgt ZZH −= DischargeReservoir

SuctionReservoir

Pd

Ps

Pdr

Psr

Hgd

zsr

zs zd

Hgp

Hgs

Hgt

Datum

Ps : Pump suction end PressurePd : Pump discharge end Press.Psr : Suction Reservoir Press.Pdr : Discharge Reservoir Press.Hgt : Total geometric headHgd : Discharge geometric headHgs : Suction geometric headHgp : Pump geometric head

Suction Side

Discharge side

Thus the Pump Head; H

Typical Pumping System

Extended Bernoulli equation written between the reservoir free surfaces;

Total Geometrical Head Hgt

1

N

gt jj

gt t

H H h

H H h

=

= +

= +

∑

Pump Head “H” reguired in a systembetween two large reservoirs open to

atmosphere is equal to the sum of total geometric head “Hgt“ and total head

loss in the system “ht”


Total head loss in a system can be expressed as the sum of total head losses in the suction(s) and discharge(d) sides.

( ) ( )s d

1 1 1

N N N

j j jj j j

h h h= = =

= +∑ ∑ ∑

Extended Bernoulli equation written between the suction anddischarge sides of the pump gives the total pump head, H.

2 21 1 2 2

1 22 2s s d d

s dp V p VZ H Z

g g g gρ ρ+ + + = + +

2 22 2 1 1

2 12 2d d s s

d sp V p VH Z Z

g g g gρ ρ⎛ ⎞ ⎛ ⎞

= + + − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


For small pumps or when the suction and discharge side of a pump areat the same geometrical head and of equal pipe diameter; the pumphead can be mesured and the differential static pressure between them.

gppH

ρ12 −

=

The Power transformed to the fluid is the product of Specific Weight, ( γ ≡ ρg) ; volumetric flowrate, Q and total pump headH, and referred to as Hydraulic Power Ph.

hP gQHρ=

The effective power to operate the pump, Peff is referred to as ; brake horse power (bHP) or is Shaft Power. The shaft power can be expressed in terms of the shaft toque and speed as ;

shP Tω= ::

shaft ro ta tional speedT torqueω

As the shaft power is tranformed to hydraulic power, someportion of it is lost due to various reasons, resulting in a smallerhydraulic power. The loses expressed as the ratio of input tooutput powers is exsprssed as follows defining the pumpefficiency η.

TgQH

PP

m

H

ωρη ==

The losses in a pump are :• Losses in the mechanical transmission system losses (shaft, bearing, sealing

and windage). These are expressed in terms of the mechanical efficiency ( ηm )• Fluid losses inside the pump (impeller, diffuser,volute, casing) due to friction

turbulence and seperation. These are expressed in terms of the hydraulicefficiency (ηh).

• Leakage losses between the impeller and the casing. These are expressed in terms of the volumetric efficiency ( ηv ).

The pump designer aims to design a pump which has the highest efficiencyin a wide range of flowrates.


The pump efficiency (overall) is the product of these three efficiencies.

η = ηm ηv ηh


II.3.3 Dimensional Analysis and Similitude in Turbomachinery

Ω

• Turbomachinery performance is a function of many physicalparameters. Some of these parameters are geometric, some aredynamic and others are fluid property parameters.

• Full performance can hardly be analytically expressed through theseparameters, so an empirical approach is necessary to obtain thecharacteristics of a turbomachine.

• A “Dimensional Analysis” is very useful in reducing the number of parameters especially reducing the number of experiments and in compacting the results.

• Here the non-dimensional parameters (π terms) effective in performance are derived for a general turbomachine to express theperformance non-dimensionally; but more important to establishsimilarity of operation between various cases and benefit in applyingothers empirical results to our cases of interest.

II.3.3 Dimensional Analysis and Similitudein (Pumps) Turbomachinery

Q : Volumetric flowrate [ m3 / s ]gH : Energy per unit weight [ m2 / s2 ] D : Size of the turbomachine, Impeller diameter [ m ]Ω : Rotational speed [ rad / s ]P : Power of the turbomachine [ W ]ρ : Fluid density [ kg / m3 ]μ : Fluid viscosity [ Pa - s ]

Power of the turbomachine can be expressed as :

( )μρω ,,,,, DgHQP =

•The physical parameters of interest are given below

II.3.3 Dimensional Analysis and Similitude in Pumps

A dimensional analysis method such as Buckingham pi (π) teoremcan be used to obtain the non-dimensional parameters as:

3QQD

πω

= 2 2HgH

Dπ

ω= 3 5P

PD

πρω

= Re 2Dμπ

ρω=

3 5 3 2 2 2, ,P Q g HD D D D

μρ ω ω ω ρ ω

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Here the 7 physical parameters effective in turbomachinery performanceare expressed in terms of 4 π terms as non-dimensional performanceparameters.

53DP

ρω 3DQ

ω 22DgH

ω2Dρω

μ

Power parameter Flow parameter Load parameter Reynolds Number

In geometrically similar machines full similitude can be obtained if allπ terms are the same. This can be hardly satisfied, so the equality of Reynolds number is relaxed. The error made due to this is latercorrected by empirical correlations. Thus for similitude ;

3 5 3 2 2,P Q gHD D Dρω ω ω

⎛ ⎞= ⎜ ⎟⎝ ⎠

Thus, for the similitude of geometrically similar turbo-machinesit is enough to have the flow and load parameters to be the samein both machines.


The performance parameter efficiency as previously defined may be obtained by modifying the π terms.

3 2 2

3 5

QD

P

Q Hpump

P

gHgQHD

P PD

η

π π ρω ωπ ηπ

ρω

= = = =

3 5

3 2 2QD

t

Pturbine

Q H

PPD

gH gQHD

ηπ ρωπ η

π π ρω ω

= = = =

Pump Efficiency

Turbine Efficiency

For a pump (or turbine) opreating in similitude disregarding ReynoldsNumber effects the efficiencies are the same.


II.3.4. Specific Speed and Typification

When the performance of geometrically similar pumps (of differentsizes) are expressed in terms of similarity parameters, they collapseto a single curve. Any deviation from this is due to ; • Measurement errors• Scale factor effect• Reynolds number effect.

ExperimentalResults

Eliminating D between πQ ve πH a ND-parameter independent of size factor can be obtained.

( )

121 1

2 3 2

3 3 34 4 4

2 2

Qs

H

QQDN

gH gHD

π ωω

πω

⎛ ⎞⎜ ⎟⎝ ⎠= = =⎛ ⎞⎜ ⎟⎝ ⎠


For a given type (family) of turbomachinery the typifiying“Specific Speed NS” is defined, based on the H and Q values at

the best efficiency operating point (design point *).

Thus the Specific Speed typifying the pump Ns is defined basedon the H, Q and N values at the best efficiency (design) pointdesignated by ( )* is expressed as;

( ) ( )1/ 2* * 3

2 * 3/ 4

2 [ ] / 60 . [ / ]( [ / ]. [ ])s

N rpm Q m sN

g m s mπ

=

II.3.4. Specific Speed and TypificationSpecific speed is the type parameter corresponding to the required flowrate andHead to designate the shape of the pump giving the highest efficiency.

Radial Flow Mixed Flow Axial Flow

Radial Mixed Flow Axial

Pump typification according to NS

Radial Mixed Axial

Turbine Typification according to NSP

Impulse turbine

The Specific Speed is used in a dimensional form in practice .

For Pumps

For Turbines

Nsp is thePower Specific

Speed

Ω: rotational speed (rad/s)

( )

12

34

sQN

H

ω=

( )

12

1 13 52 2

5 5 1 54 4 2 4

2 2

Psp

H

PD PN

gH gHD

ρωπ ω

π ρω

⎛ ⎞⎜ ⎟⎝ ⎠= = =⎛ ⎞⎜ ⎟⎝ ⎠


Pump performanceCharacteristics are obtainedthrough extensive tests. Characteristics expressedinterms of non-dimensionalparameters help the user tocompare the types of pumpswithout including the effect of size factor. In the figure a comparison of 3 different pumps having specificspeeds ;NS =0.4 → Centrifugal,NS =3.0 → Mixed Flow,NS =5.8 → Axial Flow,are given for comparison.

Ns=0.4 Ns=3

Ns=5.8

II.3.5. Performance Characteristics

Unstable ¢

Stable ¢

H

Q

Figure : Stable and UnstablePump Characteristics

Figure : Surging Pumps withUnstable Characteristics


Q

Figure : Three DimensionalPump ¢ (H-Q-η )

H

Q

η

NPSH-Req.

Figure : Two Dimensional Pump ¢( H-Q-η & NPSH-Req. )


H

Q

Figure : Volumetric (Positive Displacement) Pump Characteristic

Figure : Turbine Characteristics


Operating Point (OP) is defined as the intersecting point of Pump Characteristicand System Characteristic

Figure : Pump Characteristic, System Characteristic& Operating Point

Operating Point

System ¢ ( H = Hgt + C Q2 )

Pump ¢Hgt

H

Q

CQ2

II.3.5. Performance CharacteristicsII.3.5.a Operating Point

Change of Operating Point due to Changes in System Characteristics in Rotodinamic Pumps

1- System ¢ Change due to Geometrical Head (Hgt) Variations

Pump ¢

Hgt2

Hgt1

21

Operating PointsH

Q

II.3.5.a Operating Point - Rotodynamic Pumps

1- System ¢ Change due to Geometrical Head (Hgt) Variations

Q

Hgt2

Hgt1

DHgt

H

η

H

2- System ¢ Change due to Valve Throttling (CQ2 Variations)

H

1

Q

23

II.3.5.a Operating Point - Rotodynamic Pumps

2- System ¢ Change due to Valve Throttling (CQ2 Variations)

Hgt

H

Valveclosing

Loss

h

H

Q

Change of Operating Point due to Changes in System Characteristics in Positive Displacement Pumps

Hgt2 Hgt1

H

Q

2

1

Pump Characteristc

OP

1. System ¢ Change due to Geometrical Head ( Hgt ) Variations

II.3.5.a Operating Point - Positive Displacement Pumps

H3

2

1

Q

Pump Characteristic

SystemCharacteristics

2. System ¢ Change due to Valve Throttling (CQ2 Variations)

II.3.5.a Operating Point - Positive Displacement Pumps

In practice the non-dimensional parameters are simplified to form simpler but dimensional performance parameters.

3QQ

N Dπ ′ =

2 2HH

N Dπ ′ =

Flow parameter

Load parameter

N : rpm

These can be used as similarity parameters.

II.3.3 Dimensional Analysis and Similitude in PumpsII.3.3.1 Simplified Non-Dimensional Groups

As the rotational speed of the pump N changes, the Pump Characteristicchanges. As a consequence the Opeating Point changes. The Pump ¢ at this new N can be found through similitude. İf the change in N is too large, a Reynolds Number correction may be necessary. For this case thesimplified similarity parameters for pumps are;

QQN

π ′ =2H

HN

π ′ = 3PPN

π ′ =

These dimensional π’ terms have to be the same for similaroperating points at two different N values.

1 2

1 2

Q QN N

= 1 22 21 2

H HN N

= 1 23 31 2

P PN N

=

II.3.3.1 Simplified Non-Dimensional GroupsII.3.3.1.a Speed Change in Pumps

If the pump characteristic at N1 is known, the characteristic at N2 could be foundusing the similarity parameters.

If N is eliminated between these 2 similarity relations, a relation between H and Q can be found. This is the “Similarity Curve for geometrically equivqlent (same) pumps operating at different speeds.

222

'

CQQHQ

H =⎟⎠⎞⎜

⎝⎛ ′

=π

π→

2QHC =

This relation gives similar operating points at different N values.

The efficiencies of similar operating points can be taken to be the same.


Pump ¢N=1500 rpm

Similar OP


Figure : Similar Operating Points in Pumps with Speed Changes.Similar Operating Points

Flowrate Adjustment by Speed Control in a Pumping System

OPo

OP2

H

(H1,Q1)(Ho,Qo)

OP1

(H2,Q2)

Ho

H2

Q2 Qo

N2

N1

Q

For Geometrically similar pumps the Pump impeller diameter, D can be changed, keeping the pump speed, N constant. If the geometric similarity is not disturbed by this diameter change, the similarityparameters (“affinity laws”) may be used to obtain the modified characteristics. Thus ;

3DQ

Q =″π 2DH

H =″π 5DP

P =″π

The π” terms given above are dimensional. For the similar operating points of twodifferent diameters they have to be the same.

232

131

DQ

DQ

=2

22

121

DH

DH

=2

52

151

DP

DP

=

If the characteristics of a pump of diameter D1 at speed N is known, the characteristics of another geometrically similar pump of diameter D2 operating at the same speed N can be found through the above relations.

II.3.3.1 Simplified Non-Dimensional GroupsII.3.3.1.a Diameter Change in Pumps

From these relations if “D” is eliminated, a relation between H andQ may be obtained.

3DQ

Q =″π2DH

H =″π

32

32

32 CQQH

Q

H =

⎟⎠⎞⎜

⎝⎛ ″

′′=

π

π

32

Q

HC =

This relation gives the locus of similar operating points forgeometrically similar pumps of different size ( D ), but operatingat the same rotational speed ( N ).


Similarity curveH3 = KQ2

Pump ¢ ( D )

Figure : Similar OP in Similar Pumps with Diameter Change


Paralel Pipeline Systems ( Single Pump – 2 Reservoirs )

Hgt1

Hgt2

1

2

Pump

Paralel Pipeline Systems ( Single Pump – 2 Reservoirs )

Q2 Q1 QA Q

OP

A

H2=Hgt2+K2Q2 H1=Hgt1+K1Q2

H

CombinedSystemCharacteristics

• Overloading Pumps• Pump Combinations

Series and Parallel Pumps• System Combinations• Flowrate Control• Cavitation

chapter ii – 3 pumps for pipelinesme.metu.edu.tr/courses/me437/pl-chapter ii-3 pumps-web.pdf ·...

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