1. Chapter II – 4 .1 Pipe-Pump Systems for Pipelines
II Energy Relations for a Pumping System
II Momentum Relations in a Hydraulic Machine
II Dimensional Analysis and Similitude for a Hydraulic Machine
II Specific Speed and Typification
II Performance Characteristics
II.4.5.a Operating Point
II.4.5.b Flowrate Adjustment in a Pumping System
II.4.5.c Similar Pumps with Speed Change
II.4.5.d Similar Pumps with Diameter Change
II Cavitation
II System Characteristics and Operating Point
II Pump System Combinations
II.3.8.a Series Combination of Pumps
II.3.8.b Parallel Combination of Pumps
II.3.8.c Single Pump and 2 Parallel Systems
II Selection of Pumps
II.4.4. Operational Problems in Pumping Systems

2. II.4.1.1 Energy Relations for a Pumping System
Discharge Reservoir
Suction Reservoir Pd Ps
Pdr
Psr
Hgd
zsr zs zd
Hgp
Hgs
Hgt
Datum
Ps : Pump suction end Pressure
Pd : Pump discharge end Press.
Psr : Suction Reservoir Press.
Pdr : Discharge Reservoir Press.
Hgt : Total geometric head
Hgd : Discharge geometric head
Hgs : Suction geometric head
Hgp : Pump geometric head
Suction Side
Discharge side

3. II.4.1. Energy Relations for a Pumping System
Extended Bernoulli equation written between the reservoir free surfaces;
Discharge Reservoir
Suction Reservoir Pd Ps
Pdr
Psr
Hgd
zsr zs zd
Hgp
Hgs
Hgt
Datum
Suction Side
Discharge side
Total head Loss in the System, j : Element with head Loss ( Pipe, elbow, valve, etc.)
For Large Reservoirs :
If reservoirs are open to atmosphere ;
Typical Pumping System

4. II.4.1. Energy Relations for a Pumping System
Extended Bernoulli equation written between the reservoir free surfaces;
Total Geometrical Head Hgt
Thus the Pump Head; H
Pump Head “H” reguired in a system between two large reservoirs open to atmosphere is equal to the sum of total geometric head “Hgt“ and total head loss in the system “ht”

5. II.4.1. Energy Relations for a Pumping System
Total head loss in a system can be expressed as the sum of total head losses in the suction(s) and discharge(d) sides.
Extended Bernoulli equation written between the suction and discharge sides of the pump gives the total pump head, H.

6. II.4.1. Energy Relations for a Pumping System
For small pumps or when the suction and discharge side of a pump are at the same geometrical head and of equal pipe diameter; the pump head can be mesured and the differential static pressure between them is:
The Power transformed to the fluid is the product of
Specific Weight, ( g ≡ rg) ; volumetric flowrate, Q and total pump head H, and referred to as Hydraulic Power Ph.

7. II.4.1. Energy Relations for a Pumping System
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 as:
As the shaft power is tranformed to hydraulic power, some portion of it is lost due to various reasons. The loses expressed as the ratio of input to output powers and defines the pump efficiency η.

8. II.4.1. Energy Relations for a Pumping System
The losses in a pump are:
Losses in the mechanical transmission system (shaft, bearing, sealing and windage). Expressed in terms of the mechanical efficiency ( ηm )
Fluid losses inside the pump (impeller, diffuser, volute, casing) due to friction turbulence and seperation. Expressed in terms of the hydraulic efficiency (ηh).
Leakage losses between the impeller and the casing. These are expressed in terms of the volumetric efficiency ( ηv ).

9. II.4.1. Energy Relations for a Pumping System
The pump efficiency (overall) is the product of these three efficiencies.
The pump designer aims to design a pump which has the highest efficiency in a wide range of flowrates.
η = ηm ηv ηh

10. II Operating Point
Operating Point (OP) is defined as the intersecting point of Pump Characteristic and System Characteristic
Operating Point
System ¢ ( H = Hgt + C Q2 )
Pump ¢
Hgt H Q
CQ2
Figure : Pump Characteristic, System Characteristic
& Operating Point

11. II Operating Point
Change of Operating Point due to Changes in System Characteristics
 1.System ¢ Change due to Geometrical Head (Hgt) Variations
Hgt2
Hgt1 2 1
Operating Points H Q
Pump ¢ η

12. II Operating Point
2.System ¢ Change due to Valve Throttling (CQ2 Variations)
Operating Points H 1 Q 2 3

13. Example
What will be the flow rate through the following system and the required pump head (pump operating point)? 1 P 2
Pipes Ltot = 1000m; D=0,4 m
f = ; ke = 0.5 ; KELB = Kd =1
Z2 = 230 m
Z1 = 200 m

14. Pump Characteristics is given by :
Hp = 60 – 375Q2
To find the system characteristics, Extended Bernoulli equation between 1 & 2:

15. Thus the “System Characteristics” is found as:
hP= 30 +
* 39.3 = 30 m + 16 Q2 m.
Thus the “System Characteristics” is found as:
H= Q where the head is in m(meters) and the flow rate is in m3/s.
Solving the system and pump characteristics simultaneously, one gets the operating point of the pump (and the system).
Hp = 31.2 m ; Q = 0.28 m3/s

16. II.4.1.3. Dimensional Analysis and Similitude in Turbomachinery
Turbomachinery performance is a function of many physical parameters; which are geometric, dynamic or fluid property parameters.
Full performance can hardly be analytically expressed through these parameters, so an empirical approach is necessary to obtain the characteristics of a turbomachine.
A “Dimensional Analysis” is very useful in reducing the number of parameters and especially the number of experiments and thus the results.

17. II. 4.1.3 Dimensional Analysis and Similitude in Turbomachinery
The physical parameters of interest are given below
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 ]
 : Fluid density [ kg / m3 ]
 : Fluid viscosity [ Pa - s ]
Power of the turbomachine can be expressed as :

18. II. 4.1.3 Dimensional Analysis and Similitude in Turbomachinery
A dimensional analysis method such as Buckingham pi (π) teorem can be used to obtain the non-dimensional parameters as:
Here the 7 physical parameters effective in turbomachinery performance are expressed in terms of 4 p terms as non-dimensional performance parameters.
Power parameter
Flow parameter
Load parameter
Reynolds Number

19. II. 4.1.3 Dimensional Analysis and Similitude in Turbomachinery
In geometrically similar machines full similitude can be obtained if all p terms are the same. This can be hardly satisfied, so the equality of Reynolds number is relaxed. The error made due to this, is later corrected by empirical correlations. Thus for similitude ;
Thus, for the similitude of geometrically similar turbo-machines it is enough to have the flow and load parameters to be the same in both machines.

20. II. 4.1.3 Dimensional Analysis and Similitude in Turbomachinery
The performance parameter efficiency as previously defined may be obtained by modifying the p terms.
Pump Efficiency
Turbine Efficiency
For a pump (or turbine) opreating in similitude disregarding Reynolds Number effects, the efficiencies are the same.

21. II. 4.1.3 Dimensional Analysis and Similitude in Pumps Simplified Non-Dimensional Groups
In practice the non-dimensional parameters are simplified to form dimensional performance parameters.
Flow parameter
N : rpm
Load parameter
These can be used as similarity parameters.

22. Speed Change in Pumps
As the rotational speed of the pump (N) changes, Pump ¢ changes. As a consequence O.P. changes. Pump ¢ at this new N can be found through similitude. İf the change in N is too large, a Re correction may be necessary. For this case the simplified similarity parameters for pumps are;
These dimensional π’ terms have to be the same for similar operating points at two different N values.

23. Speed Change in Pumps
If the pump ¢ at N1 is known, ¢ at N2 could be found using 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 equivalent (same) pumps operating at different speeds.
This relation gives similar operating points at different N values.
The efficiencies of similar operating points can be taken to be the same.

24. Speed Change in Pumps
Pompa ¢
N=1500 rpm
Similar OP
Figure : Similar Operating Points in Pumps with Speed Changes.

25. System ¢( H = Hgt + CQ2) H H=kQ2 OP1 OPo (H1,Q1) Ho (Ho,Qo) H2 OP2
N2 (1500 rpm) N1 Q
H=kQ2
System ¢( H = Hgt + CQ2)

26. Diameter Change in Pumps
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 similarity parameters (“affinity laws”) may be used to obtain the modified characteristics.
Thus ;

27. Diameter Change in Pumps
The π” terms given above are dimensional. For the similar operating points of two different diameters they have to be the same.
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.

28. Diameter Change in Pumps
From these relations if “D” is eliminated, a relation between H and Q may be obtained.
This relation gives the locus of similar operating points for geometrically similar pumps of different size ( D ), but operating at the same rotational speed ( N ).

29. Diameter Change in Pumps
Pump ¢ ( D )
Similarity curve
H3 = KQ2
Figure : Similar OP in Similar Pumps with Diameter Change

30. Prototype: Model Find:
Example
A centrifugal pump is to be operated at 300 rpm to give 6.75 m3/s flowrate at 123 m head. A model will be designed for production, but the max. flowrate to be sustained in lab is m3/s and max. power consumption value is limited to 225kW. If model and prototype efficiencies are same and 80 %, find the operating speed of the model pump and Dprototip/Dmodel.
Prototype: Model Find:

31. For model:
Using load parameter:
 Equation A

32. Using flow parameter:
 Equation B
From A and B:
So Dprototip/Dmodel :

33. From A:
Model pump speed:

34. II.4.1.4. Specific Speed and Typification
When the performance of geometrically similar pumps (of different sizes) are expressed in terms of similarity parameters, they collapse to a single curve. Any deviation from this is due to ;
Measurement errors
Scale factor effect
Reynolds number effect.

35. II. 4.1.4. Specific Speed and Typification
Experimental Results
Eliminating D between πQ ve πH a Non Dimensional parameter independent of size factor can be obtained.

36. II. 4.1.4. Specific Speed and Typification
For a given type (family) of turbomachinery the typifiying “Specific Speed NS” is defined, based on H and Q values at the best efficiency operating point (design point).
Thus the Specific Speed typifying the pump Ns is defined based on the H, Q and N values at the best efficiency (design) point and expressed as;

37. II. 4.1.4. Specific Speed and Typification
Radial flow
Mixed flow
Axial flow
Radial
Mixed Flow
Axial
Specific speed is the type parameter corresponding to the required flowrate and Head to designate the shape of the pump giving the highest efficiency.
Pump typification according to NS

38. II. 4.1.4. Specific Speed and Typification
Radial
Mixed
Axial
Specific speed is the type parameter corresponding to the required flowrate and Head to designate the shape of the turbine giving the highest efficiency.
Francis
Axial flow
Impulse
turbine typification according to NS

39. Nsp is the Power Specific Speed
II Specific Speed and Typification
The Specific Speed is used in a dimensional form in practice .
For Pumps
Ω: rotational speed (rad/s)
For Turbines
Nsp is the Power Specific Speed

40. II.4.1.5. Performance Characteristics
Pump performance Characteristics are obtained through extensive tests. Characteristics expressed interms of non-dimensional parameters help the user to compare the types of pumps without including the effect of size factor.
In the figure a comparison of 3 different pumps having specific speeds ;
NS =0.4 → Centrifugal,
NS =3.0 → Mixed Flow,
NS =5.8 → Axial Flow,
are given for comparison.

41. II.4.1.5. Performance Characteristics
Ns=0.4
Ns=3
Ns=5.8

42. II.4.1.5. Performance CharacteristicsQ
Figure : Volumetric (Positive Displacement) Pump Characteristic
Figure : Turbine Characteristics

43. II.4.1.7. Caviation in Turbomachinary
Cavitation refers to the conditions at certain locations within the hydraulic machinery (pumps, turbines etc) where the local pressure drops to the vapor pressure of the liquid.
As a result of the cavitation, vapor filled cavities are formed.

44. II.4.1.7. Caviation in Turbomachinary
As the cavities transported through the machinery in forms of bubbles into higher regions of pressure, they collapse rapidly and generate extremely high localized pressures.

45. II.4.1.7. Caviation in Turbomachinary
Those bubbles that collapse close to the solid boundaries can weaken the solid surface, and fatigue can occur.

46. II.4.1.7. Caviation in Turbomachinary
This may demage the turbomachine, or may result in fatal errors.

47. II.4.1.7. Caviation in Turbomachinary
Signs of cavitation in pumps (or turbines):
Noise,
Vibration,
Lowering of the head-discharge and efficiency curves.
Cavitation is encountered at the inlet of pumps, at the exit of turbines where the pressure is low.
The net positive suction head (NPSH) is used to designate the potential for cavitation.

48. II.4.1.7. Caviation in Turbomachinary
The NPSH of the system calculated (available) should be higher than the NPSH required in order to prevent cavitation.
NPSHa > NPSHR
NPSH is defined as,
where s denotes the suction side of the pump and is the vapor pressure of the liquid.

49. II.4.1.7. Caviation in Turbomachinary
For the following figure Bernoulli equation can be written between the points sr and s. P
reservoir s sr zs

50. II.4.1.7. Caviation in TurbomachinaryP
reservoir s sr zs
Eeservoir surface is opened to the atmosphere.
By selecting the reservoir surface as the reference
= head loss due to frictional losses at the inlet piping.

51. II.4.1.7. Caviation in Turbomachinary
Substracting term from both sides,
Notice that the left hand side is NPSH, as a result the NPSH becomes,
For a given system NPSH condition should be satisfied by exceeding a required number. As can be interpreted from the above equation the NPSH decreasesn by the increase in pump elevation and total friction.

52. II.4.1.7. Caviation in Turbomachinary
Paramters affecting caviation:
Velocity and Flow rate 
2. Pump Geometry
Change of the Impeller Diameter
Properties of the Fluid
Suction Piping
6. Pump Material

53. Example: A conventional centrifugal pump has a specific speed of Ns = to pump Q = 0.85 m3/s to H = 165m at its design point. Pa=100 kPa Hs
For water Pv = bar
For N (rpm) consistent with data
Hs = ? if Sreq = 3.2
Solution :
Eliminating NQ1/2, we have
Thoma Cavitation Coefficient

54. Thus,
But,
assuming no loss in suction pipe Hs
For no cavitation, pump has to be placed below tail water level. (Hs is negative)