Turbomachinery Lecture Notes - KTH

Turbomachinery Lecture Notes KTH Course MJ2429

Part I: Turbomachinery for Incompressible Fluids

Damian M. Vogt KTH Heat and Power Technology

 Damian Vogt, 2012 All rights reserved

www.exploreenergy.eu

www.kth.se

compedu.net

Turbomachinery Lecture Notes

last years. It looks like a book but it is much more. It is conceived as an animated interactive document that contains recorded lecture units, calculation exercises and access to remote laboratory exercises to name a few. In other words, this learning material adds new dimensions to your learning experience and by this supercharges your learning. It is my intention to have you understanding the basic concepts of turbomachinery in a truly genuine way and to have you employing them in your future career with confidence.

Preface

T

urbomachines are exciting machines. They propel aircrafts, drive machines, move fluids, supercharge, compress, expand and are essentially found in most applications that involve the conversion of energy. Their area of application is vast ranging from miniature sized cooling fans in computers over modern large bypass ratio turbofans to gigantic steam turbines providing more than one million shaft horse powers for use in power generation. Turbomachines all work on the same principle, which is by changing swirl momentum. This is also where they get their name from: “turbo” in latin means “swirl”. A turbomachines is in other words nothing else than a machine that changes swirl momentum, be it on a small or on a large scale. Has this teased your curiosity? If so, then you have come to the right place to learn more about these fascinating machines.

I have organized this learning material such that I first take you through turbomachines for incompressible applications (pumps and hydro turbines) and thereafter to turbomachines for compressible applications (compressors and gas/steam turbines). Looking forward to ignite your fascination for these types of machines,

My name is Damian Vogt and I am the developer of this learning material as well as the main lecturer in the KTH MJ2429 Turbomachinery Course. What you hold in your hands is the first version of a new type of learning material that I have developed based on experiences gained in teaching over the

Damian Vogt Associate Professor (Docent)

I


About This Learning Material

various short learning units giving a feeling of virtual classroom attendance. The learners will follow the lecturer’s writing on a whiteboard while hearing the corresponding spoken explanations.

he present learning material that you hold in your hands is a new type of material that non only contains text but also recorded lecture units and links to online self-assessment tests. The scope of this material is to provide new dimensions in learning, going beyond traditional lecture notes and add a dimension in which virtually are brought into the classroom – anywhere and at any time. The written sections in this document are kept brief on purpose such as to allow for additional activities. It is therefore not primarily to be understood as a replacement for a textbook and instead presents a structured and enhanced presentation of lecture notes.

In addition, a number of self-study problems are posed throughout the learning material such as to give the learners the possibility to apply their newly gained knowledge. These problems are of similar types of the ones made available in the classroom problem solving that are provided if the present learning material is part of a taught course.

T

Animations and tools are included to give the learner a different means of understanding and to provide the possibility for own experience. Where available, remote laboratory exercises give the learners the possibility to acquire test data on real hardware and by this apply their knowledge on real cases. Furthermore, links or references to articles are included such as to link the taught subject to available material and by this provide additional reading for interested learners.

What is New with This Learning Material? The novelty of this learning material is that it goes beyond traditional “static” books and in addition includes animated sections. These animated sections are recorded lecture units that give the learner the possibility to learn by following

Last, class-attending learners will have access to a discussion forum in which several of the aspect can be discussed with other learners as well as the instructor and course assistants.

II


Remote laboratory exercises are included where available and are highlighted as follows:

Most items are linked to animated and/or interactive content via the item symbol. Recorded lecture units are included throughout the learning material and are highlighted as follows:





Lecture Unit

This lecture unit covers the theory or practical application of a specific subject.



Animations or Tools

This animation or tool gives you of a different viewpoint.

Link or Reference

This link or reference gives you further reading in open literature on a specific subject

Indications to the Checkpoints are included throughout the learning material and are highlighted as follows:

Self-study problems are included throughout the learning material and are highlighted as follows:



This remote laboratory exercise gives you the possibility to acquire real test data interactively on a test facility on a specific subject.

Links or references to articles are included throughout the learning material and are highlighted as follows:

Animations or tools are included throughout the learning material and are highlighted as follows:



Remote Laboratory Exercise



Self-Study Problem

This self-study problem allows you to get handson experience.

III

Checkpoint

This indicates a checkpoint of the learning of the subject, which must be understood for ensuring the successful mastering of this subject.


the pump rather than using speed-control? Yet another level would be the component level. It is here where you “dive” into the basics of the subject that you study, as for example turbomachinery in our case. How do I need to design a component such that it operates at maximum efficiency? How does it affect the way I need to manufacture the pump?

A Note on Sustainability

S

ustainability is an often-used word these days and you probably (and hopefully) have come across it in your previous education. Sustainability is something that we want you to work towards once you enter the phase as a “productive” engineer after your education.

As it gets apparent from these examples, sustainability is much related to environmental impact in the sense of ecofriendliness. This is surely an important aspect but it is not all. Sustainability equally applies to the resources used to provide a service in terms of material, energy and economic resources used. In other words, you might very well be able to design a super-efficient pump, but will it be sustainable from an overall perspective. Just assume that you foresee the use of an exotic material for your super-efficient pump, which is hugely expensive to produce and which leaves a massive environmental footprint. Will your super-efficient pump still be the most sustainable solution from an overall perspective?

But what do we actually mean by “sustainability”? There are plenty of definitions out there and what I give you here is my personal view of me seeing you as an engineer working towards sustainability. I define sustainability as a state in which you perform something that impacts on the environment in a way that it allows you to continue for a given period. The duration of this period might vary from short durations to longer extending over several generations. This “something” that I talk about above might for example be the operation of a turbomachine. When we bring in sustainability aspects, we can include them on several levels. One possible level is the system perspective, in which you question the reason for a given service to be performed. Do we need to pump up water to a given height? Can it be solved differently? Another level, still on the system perspective, would be to question the way a specific service is performed. Do we use a rotodynamic pump? Shall we throttle

What I want you to adopt is a general attitude of responsibility. You as a future engineer are responsible to propose solutions that are sustainable from an overall point of view. By mastering the basics in a subject such as turbomachinery, you will have the tools to recognize dependencies and propose such solutions. Be proud to have chosen this career path and get ready to make a difference!

IV


Contents Preface About This Learning Material What is New with This Learning Material? A Note on Sustainability

I

Conservation of Momentum

6

Leonhard Euler

II

Pumps

IV

8 9 11

Pumping Systems

12

Classification of Pumps

16

Pump Elements

17

X

Pump Types

19

X

Pump Velocity Triangles

21

VII

List of Figures

IX

Lecture Units

5

Euler’s Turbine Equation

II

Nomenclature List of Complementary Material

Conservation of Energy

Tools

XI

Design Parameters

27

Animations

XI

Constructional Aspects of Pumps

28

Self-Study Problems

XI

Pump Characteristics

30

Remote Laboratory Exercises

XI

Pump Operating Point

32

Links and References

XII

Pump Power

34

Checkpoints

XII

Pump Efficiency

35

Coordinate System and Views

1

Affinity Laws

36

Turbomachines for Incompressible Fluids

4

Serial and Parallel Operation of Pumps

40

4

Harmful Effects

41

4

Preliminary Design of Pumps

44

Review of Basic Laws Conservation of Mass

V


Hydro Turbines

47

Turbine Systems

48

Application of Euler Turbine Equation to Turbines

49

Turbine Elements

50

Turbine Types

54

Summary of Equations

55

Index

58

References

60

VI


Nomenclature Symbol

Denotation

Unit

A

Surface area

2

m

b

Impeller outlet width

m

c

Absolute velocity

m/s

d

Diameter

m

E

Internal energy

J

F

Force

N

g

Gravitational constant

m/s

h

Enthalpy

J/kg

H

Head

m

m

Mass flow rate

Kg/s

M

Moment

Nm

p

Pressure

Pa

Q

Heat energy

J

2

Q

Flow rate

m3/s

r

Radius

m

u

Tangential velocity

m/s

w

Relative velocity

m/s

W

Mechanical energy

J

z

Height coordinate

m

ρ

Density

kg/m3

ω

Rotational speed

rad/s

η

Efficiency

-

Subscripts

VII

0

Total

0

Inlet stator (turbine)

1

Inlet impeller

2

Outlet impeller


3

Outlet stator (pump)

c

Total in absolute frame of reference

f

friction

n

Normal

p

Pressure

r

Radial component

s

Static

v

Velocity

tot

Total

w

Total in relative frame of reference

x

Axial component

θ

Tangential component

VIII


Figure 20. Pump operating point Figure 21. Pump head and efficiency Figure 22. Effect of speed regulation Figure 23. Effect of impeller trim (diameter change) Figure 24. Cavitation phenomenon Figure 25. Pump and system NPSHs Figure 26. Example of a turbine system Figure 27. Hydro turbine stage denotations (axial) Figure 28. Hydro turbine stage denotations (centripetal) Figure 29. Example of schematic turbine rotor blade row Figure 30. Examples of profiled turbine rotor blade row Figure 31. Pump types and their specific speeds

List of Figures Figure 1. Schematic turbomachine showing the turbomachinery coordinate system 1 Figure 2. Axial view of a schematic turbomachine 2 Figure 3. Side view (cut) of a schematic turbomachine 2 Figure 4. Unwrapped view of the stream surface in a turbomachine (no change in swirl) 3 Figure 5. Unwrapped view of the stream surface in a turbomachine (change in swirl) 3 Figure 6. Example of a pumping system 12 Figure 7. Typical system characteristic 13 Figure 8. Example Pump system 14 Figure 9. Pump classification 16 Figure 10. Pump stage denotations (axial) 17 Figure 11. Pump stage denotations (centrifugal) 17 Figure 12. Examples of schematic pump rotor blade row 18 Figure 13. Examples of profiled pump rotor blade row 18 Figure 14. Pump types and their specific speeds 20 Figure 15. General concept of absolute and relative velocity 21 Figure 16. Impeller velocity triangles 22 Figure 17. Concept of relative eddy (adapted from [1]) 25 Figure 18. Effect of slip on velocity triangle at impeller exit 26 Figure 19. Dependence of the pump characteristics from blade metal angle 31

IX

33 35 38 39 42 43 48 50 51 51 52 54


Lecture Unit 17: Different Shapes of Pump Flow Channels 24 Lecture Unit 18: Change of Total Head in Stator and Rotor Blade Rows 25 Lecture Unit 19: Dependence of Head Coefficient from Flow Coefficient 27 Lecture Unit 20: Main Components of Axial Pumps 29 Lecture Unit 21: Main Components of Radial Pumps 29 Lecture Unit 22: Separation Phenomena in Pump 29 Lecture Unit 23: Tip Leakage Loss 29 Lecture Unit 24: Minimizing Leakage Losses 29 Lecture Unit 25: Pictures of Pumps 30 Lecture Unit 26: Pump Operating Characteristics 31 Lecture Unit 27: Pump Operating Point 33 Lecture Unit 28: Pump Power 34 Lecture Unit 29: Pump Efficiency 35 Lecture Unit 30: Pump Operation at Off-Design 36 Lecture Unit 31: Pump Operation at Variable Speed 36 Lecture Unit 32: Pump Affinity Laws 37 Lecture Unit 33: Pump Operation at Varying Speed 39 Lecture Unit 34: Pump Operation after Diameter Change 39 Lecture Unit 35: Reason for Cavitation 41 Lecture Unit 36: Analysis of Cavitation Phenomenon 41 Lecture Unit 37: Implosion of Vapor Bubbles during Cavitation 42 Lecture Unit 38: Measures against Cavitation 43 Lecture Unit 39: NPSHr and NPSHa 43

List of Complementary Material Lecture Units Lecture Unit 1: Conservation of Mass Lecture Unit 2: Conservation of Energy Lecture Unit 3: Conservation of Momentum Lecture Unit 4: Introduction to Euler’s Turbine Equation Lecture Unit 5: Analysis of Euler’s Turbine Equation (1/2) Lecture Unit 6: Analysis of Euler’s Turbine Equation (2/2) Lecture Unit 7: Pumping System Lecture Unit 8: Analysis of Pumping Systems Lecture Unit 9: Total Head of a Pumping System Lecture Unit 10: Different Forms of Energy in a Pumping System Lecture Unit 11: Concept of Frame of Reference Lecture Unit 12: Flow Deviations in Turbomachine Components Lecture Unit 13: Pump Velocity Triangles Lecture Unit 14: Influence of Flow Channel Shape on Velocities Lecture Unit 15: Influence of Blade Shape on Velocities Lecture Unit 16: Flow at Various Spanwise Positions

5 6 7 9 9 9 12 13 15 15 22 23 23 24 24 24

X


Lecture Unit 40: Preliminary Design of a Pump 45 Lecture Unit 41: Euler’s Equation for Hydro Turbines 49 Lecture Unit 42: Deviation of Flow in Turbines and Pumps 50 Lecture Unit 43: Blade Shapes and Flow Passages in Turbines and Pumps 53 Lecture Unit 44: Flow Direction at Rotor Inlet 53 Lecture Unit 45: Types of Hydro Turbines 54

Self-Study Problems Self-Study Problem 1: Fluid Forces on a Blade Row 7 Self-Study Problem 2: Euler’s Turbine Equation 10 Self-Study Problem 3: Different Forms of Energy 11 Self-Study Problem 4: Pumping System 14 Self-Study Problem 5: Deviation of Flow in Blade Rows 19 Self-Study Problem 6: Velocity Triangles 23 Self-Study Problem 7: Design Parameters 28 Self-Study Problem 8: Pump Off-Design Operation 32 Self-Study Problem 9: Pump Operating Point 34 Self-Study Problem 10: Affinity Laws 39 Self-Study Problem 11: Preliminary Design of a Pump 45 Self-Study Problem 12: Turbine System 48 Self-Study Problem 13: Flow Deviation in a Turbine 49 Self-Study Problem 14: Deviation of Flow in a Turbine Blade Row 52

Tools Tool 1: Preliminary Design of a Pump

45

Animations Animation 1: Absolute and Relative Streamlines in a Pump Rotor 18 Animation 2: Absolute and Relative Streamlines in a Turbine Rotor 52

Remote Laboratory Exercises Remote Laboratory Exercise 1: Off-Design Operation of Pumps 38 Remote Laboratory Exercise 2: Serial and Parallel Operation of Pumps 41

XI


Checkpoint 3: Absolute and Relative Flow Paths Checkpoint 4: Pump Velocity Triangles Checkpoint 5: Slip Checkpoint 6: Design parameters Checkpoint 7: Pump Operating Characteristics Checkpoint 8: Pump Operating Point Checkpoint 9: Pump Power Checkpoint 10: Pump Efficiency Checkpoint 11: Affinity Law Checkpoint 12: Preliminary Design of a Pump Checkpoint 13: Types of Hydro Turbines

Links and References Reference 1: Pump Manufacturer (ITT Flygt) Reference 2: Pump Manufacturer (SULZER) Reference 3: Pump Design in Industry Reference 4: Hydro Turbine Manufacturer (VOITH)

30 30 45 54

Checkpoints Checkpoint 1: Euler equation Checkpoint 2: Classification of Pumps

10 16

XII

19 23 26 27 31 34 35 36 38 46 54


Coordinate System and Views

Circumferential direction Radial direction

Flow exits

U

nderstanding the coordinate system and views is of paramount importance when dealing with turbomachines. Turbomachines are rotating machines, hence we use a cylindrical coordinate system that is aligned with the machine axis.

Axial direction

To start off with, consider a turbomachine as a tube in which you have an inner and an outer wall. These walls are usually referred to as hub and casing respectively. The hub and casing walls bound the flow channel and give it an annular shape. The flow enters the turbomachine on one side and exits on the other side. As the flow passes through the turbomachine, the cross section of the annular flow channel will most probably vary. Also, as you will learn in this course, the swirl of the flow is changed such as to add or extract energy from the fluid.

Flow enters Hub

Casing

Figure 1. Schematic turbomachine showing the turbomachinery coordinate system

The axial view is defined by viewing the machine in positive axial direction. Hence, one would see the axial cross-section in such a view, see the Figure 2.

Consider a schematic turbomachine as the one depicted below

1


Radial direction

Circumferential direction

Meridional direction

Casing

Radial direction Flow enters

Flow exits

Hub

Hub

Casing

Axial direction Figure 2. Axial view of a schematic turbomachine Figure 3. Side view (cut) of a schematic turbomachine

The side view is defined by viewing a cut through the machine in an axial-radial plane. The cross-section that one would see is referred to as the meridional cross-section. The direction of the mean radius is referred to as the meridional direction. For constant mean radii, this is then the same as the axial direction. An example of a turbomachine side view is included in Figure 3.

In order to address the flow in detail inside a turbomachine, an unwrapped view of a stream surface is used. For a 1D analysis, this would be the stream surface on the reference radius. This view is the defined by the meridional (or axial) direction and the unwrapped circumferential coordinate ( r ⋅ θ ) as included in Figure 4 and Figure 5.

2


Streamlines

Streamlines Meridional direction Unwrapped circumferential direction ( )

Unwrapped circumferential direction ( )

Meridional direction

Streamlines

Figure 4. Unwrapped view of the stream surface in a turbomachine (no change in swirl)

Streamlines

Figure 5. Unwrapped view of the stream surface in a turbomachine (change in swirl)

3


For steady process, the mass in a control volume is constant

Turbomachines for Incompressible Fluids

over time (

∂m = 0 ) thus ∂t

∑ m = 0

T

Eq. 2

i

urbomachines for incompressible fluids are machines that use a working fluid, which features constant density. Water is an example of such a fluid but there are other examples such as oil, liquid fuels or other types of liquids. Depending on the type of machine, these turbomachines are classified into pups or turbines. In brief, a pump is a turbomachine that adds energy to a system, whereas a turbine extracts energy from a system. Before we analyze these two types of machines more closely, we first review some basic laws.

Mass flow rate through boundary m = ρ ⋅ c n ⋅ A

Eq. 3

Conservation of mass for control volume featuring one inand one outflow and assuming incompressibility ρ1 = ρ 2 c n,1 A1 = c n,2 A2

Eq. 4

The indexes “1” and “2” refer to inlet and outlet of the control volume respectively as depicted below.

Review of Basic Laws Conservation of Mass The sum of mass flow rates over all system boundaries equals to change in mass in control volume

∑ i

m =

∂m ∂t

Note: •

Eq. 1

4

The velocity is in the above case inversely proportional to the cross section


• •

For a steady flow process the conservation of energy per unit time is regarded, i.e. conservation of power

A smaller cross section means that the velocity will be greater A larger cross section means that the velocity will be smaller



dE = m ⋅ (dh0 + gdz ) = Q − W

Where dh0 denotes the change in total enthalpy and the term

Lecture Unit 1: Conservation of Mass

gdz change

in specific potential energy. As we now deal with liquid flows the latter term cannot be neglected. Furthermore a change in static enthalpy in liquids is rather pressure than temperature dependent as is the case for gases leading to

This lecture unit covers the conservation of mass.

dh =

Conservation of Energy The first law of thermodynamics applied to closed process, i.e. system taken through a complete cycle

∫ (dQ − dW ) = 0

dp

Eq. 8

ρ

The change in internal energy can therefore be rewritten as

Eq. 5

de =

Change in internal energy during change in state from one point to another in the cycle dE = dQ − dW

Eq. 7

dp

ρ

+

dv 2 + gdz 2

Eq. 9

, which contains the same contributions as the total head introduced further above apart from the fact that the friction head is not addressed specifically. This can be expressed by

Eq. 6

de = H tot ⋅ g

5

Eq. 10




, where H tot =

p 3 − p1 v 3 2 − v1 2 + + z 3 − z1 2g ρg

Eq. 11

The energy balance in a hydraulic turbomachine therefore writes as W = −m ⋅ H tot ⋅ g

•

This lecture unit covers the conservation of energy.

Conservation of Momentum Note that in a steady flow process the momentum is entirely due to a change in flow velocity as the mass flow rate is constant, as well as a pressure contribution. This is expressed in a general form by

Eq. 12

Note: •

Lecture Unit 2: Conservation of Energy

F = m (c1 − c2 ) + p1 A1 − p2 A2

For H tot > 0 , which for example is the case if the pump gives an increase in pressure, velocity or head, the work is negative  work absorbing machine (pump) In contrary, if the pressure, velocity or head over the machine decreases then H tot < 0 indicating the the work is positive  work producing machine (turbine)

Eq. 13

For a rotodynamic turbomachine, it is the conservation of moment of momentum (i.e. the momentum in circumferential direction) that is of interest. Given the rotational symmetry, the pressure forces cancel out yielding M z = r ⋅ Fθ = m (r1cθ 1 − r2 cθ 2 )

Eq. 14

Note: •

6

From the perspective of the fluid the forces are acting as pressure forces ( F = p ⋅ A ).


•

A change in velocity indicates a change in pressure  remember Bernoulli’s equation for incompressible

axial direction

1 2

•

circumferential direction

fluids: p0 = p + ρ ⋅ c 2 = const. From the perspective of the turbomachine the pressure forces on the fluid are yielding a resultant reaction force (actio=reactio).

 

Lecture Unit 3: Conservation of Momentum

This lecture unit covers the conservation of momentum.

flow

The following is given: • • •

Self-Study Problem 1: Fluid Forces on a Blade Row

Mass flow rate 50kg/s Circumferential velocity at blade row inlet 10m/s Circumferential velocity at blade row outlet -40m/s

Assuming that there are 15 blades in this blade row determine the following:

Consider a non-rotating blade row as the one that is depicted below (unwrapped view). The blade row is used to deviate the flow or in other words change the swirl of the flow. The axial flow velocity is not changed.

• • •

7

Total circumferential force on all blades Circumferential force on one blade Total torque on blade row


Substituting r ⋅ ω by the tangential speed u and eliminating m yields

Answer the following questions: • • •

Has the total energy in the flow changed? Has the static pressure in the fluid changed from inlet to outlet of the blade row? If so, why? Has the flow velocity changed from inlet to outlet of the blade row? If so, why?

H=

Note: •

At this point the conservation of energy and the conservation of moment of momentum shall be combined. The mechanical work per unit time ( power) equals the product of moment and rotational speed

•

Eq. 15

•

Thus the conservation of energy can be related to the conservation of momentum as follows −m ⋅ H tot ⋅ g = m ⋅ (r1cθ 1 − r2 cθ 2 ) ⋅ ω

Eq. 17

The above equation is referred to as Euler’s turbomachine equation or Euler’s turbine equation.

Euler’s Turbine Equation

W = M z ⋅ ω

1 (u 2 cθ 2 − u1cθ 1 ) g

Eq. 16

8

A change in total head is equivalent to a change in tangential flow speed and/or tangential engine speed For engines with little change in mean radius u1 ≈ u 2 (e.g. axial pumps, axial turbines) the change in total head is entirely due to change in tangential flow speed  H tot ≈ u ⋅ ∆cθ g  blades are bowed For engines with large change in mean radius (e.g. radial engines) the change in total head is to a large degree due to the change in radius  H tot ≈ ∆u ⋅ cθ g  centrifugal effect, possibility for larger change in enthalpy




Petersburg, where he remained until his death. Euler's prolific output caused a tremendous problem of backlog: the St. Petersburg Academy continued publishing his work posthumously for more than 30 years. Euler married twice and had 13 children, though all but five of them died young.

Lecture Unit 4: Introduction to Euler’s Turbine Equation

This lecture unit gives an introduction to the Euler’s Turbine Equation.

Leonhard Euler Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century and one of the most prolific of all time; his publication list of 886 papers and books fill about 90 volumes. Remarkably, much of this output dates from the the last two decades of his life, when he was totally blind. Though born and educated in Basel, Switzerland, Euler spent most of his career in St. Petersburg and Berlin. He joined the St. Petersburg Academy of Sciences in 1727. In 1741 he went to Berlin at the invitation of Frederick the Great, but he and Frederick never got on well and in 1766 he returned to St.

9



Lecture Unit 5: Analysis of Euler’s Turbine Equation (1/2)



Lecture Unit 6: Analysis of Euler’s Turbine Equation (2/2)

This lecture unit gives the first part of an analysis of Euler’s turbine equation.

This lecture unit gives the second part of an analysis of Euler’s turbine equation.


 

Checkpoint 1: Euler equation

Explain how energy can be added to or extracted from flow by deviating the flow. What else than deviation is needed?

Self-Study Problem 2: Euler’s Turbine Equation

Assume that we have a pump with axial inflow (hence, the circumferential velocity at impeller inlet is zero). What is the circumferential flow speed needed for a total head of 200m and a tangential rotor speed of 20m/s? Assuming that the axial flow velocity is constant in the rotor and that it amounts to 10m/s, by how many degrees does the flow need to be deviated in the rotor? In what direction (with respect to the direction of rotation) does the flow need to be deviated?

10


The friction head reflects the losses in a system and is commonly expressed in meters. To choose an appropriate pump for a given installation all the above heads need to be accounted for as follows

Pumps

P

umps are used to increase the total energy in a fluid. Whereas compressors are working with gaseous fluids pumps are working with liquid fluids. The increase in energy in a pump is commonly referred to as total dynamic head H tot measured in meters. The total dynamic head, or short “head”, can be used to increase pressure (pressure head), overcome a height difference (static head), accelerate the flow (velocity head) or overcome a friction head in a system (i.e. friction losses), which can be expressed by the following expressions Pressure head

p − p1 Hp = 2 gρ

Eq. 18

Static head

H s = h2 − h1

Eq. 19

Velocity head Friction head

Hv =

v 2 2 − v1 2 2g

H f = hf

H tot = H p + H s + H v + H f



Eq. 22

Self-Study Problem 3: Different Forms of Energy

Think about how energy can be experienced in various ways in a pumping system. How does pressure energy translate into kinetic energy for example? Can you have the equal amount of energy in a system in which the fluid is not moving as in a moving system?

Eq. 20 Eq. 21

11


Each system has its characteristics. The system characteristics tells us how the head changes with flow rate. Both the parts of the system on the suction side and the discharge side contribute to the overall system characteristics.

Pumping Systems A pumping system denotes a system, in which a pump is used to add energy to the system. The system consists of pipes on the suction and the discharge side of the pump as well as eventually valves, reservoirs and other devices. An example of a pumping system is depicted in Figure 6.

flow

pump



flow

reservoir

Figure 6. Example of a pumping system

12

Lecture Unit 7: Pumping System

This lecture unit gives an introduction to a pumping system and explains the various types of heads.


Figure 1 depicts an example of a typical system characteristics, i.e. head versus flow rate.

other hand would have roughly equally large contributions from all aforementioned heads.



H [m] system

Hs Q [m3/s] Figure 7. Typical system characteristic

A pump supplying water to a reservoir at high altitude will consequently have a dominant contribution from the static head. In the same consideration a pump in a fire fighter application will have its main contribution from the velocity head. Pumps in district heating and cooling systems on the

13

Lecture Unit 8: Analysis of Pumping Systems

This lecture unit provides an analysis of pumping systems.




The following is given:

Self-Study Problem 4: Pumping System

This self-study problem allows you to get handson experience on analyzing a part of a pumping system.

• • • • • •

Consider the part of a tubing system upstream of a pump as the one depicted below.

static pressure p1 100.0 kPa volume flow rate Q 5.0 l/s pipe inner diameter d1 49.0 mm height unit H 6.0 m ratio of pipe inner diameters d2/d1 0.83 equal pipe inner diameters at positions 1 and 3

Furthermore • • •

fluid density 1000.0 kg/m3 gravitational constant g 9.81 m/s2 neglect friction and separation

Determine the following: • • • Figure 8. Example Pump system

14

flow velocity at point 1 static pressure at point 2 static pressure at point 3




Lecture Unit 9: Total Head of a Pumping System



Lecture Unit 10: Different Forms of Energy in a Pumping System

This lecture unit provides an analysis of the total head of a pumping system. It goes hand-in-hand with the self-study problem above and prepares for the conservation of energy.

This lecture unit gives you an overview of different forms of energy in a pumping system.

15


Classification of Pumps



Depending on the method at which energy is transferred to the fluid pumps can be classified into rotodynamic, positive displacement (or short “displacement” only) and special effects pumps. An overview of this classification is included in figure 2. In the current document rotodynamic pumps only are treated. Pumps

Rotodynamic

Special effects

Displacement

• Ejector Axial flow

• Electromagnetic

Reciprocating • Piston

Mixed flow

• Diaphragm Rotary

Centrifugal

• Vane • Screw • Gear

Figure 9. Pump classification

16

Checkpoint 2: Classification of Pumps

Give a classification of main types of pumps. What is the key difference between rotodynamic and displacement pumps?


Pump Elements 3

A rotodynamic pump may consist of one or several stages. A stage includes a rotor and a stator as depicted schematically in figure 3. The figure shows a cross section of a machine in the axial-radial plane. Note that these machines are axisymmetric.

stator

2 1 rotor

1

rotor

2

stator

3

Figure 11. Pump stage denotations (centrifugal)

The rotor is often referred to as impeller as it gives the flow impetus (i.e. momentum). The stator is also called diffuser as it diffuses (i.e. decelerates) the flow. Commonly three control sections are identified in a stage as follows 1 2 3

Figure 10. Pump stage denotations (axial)

17

rotor inlet rotor outlet, stator inlet (also called “interface”) stator outlet


Rotor and stator are so-called blade rows. A blade row is a row of blades and is used to guide the flow in a specific way. As it has been shown above by means of the Euler equation, it is the deviation of the flow in the absolute frame of reference that matters in a turbomachine. Hence, the blade rows are designed such that they yield a certain deviation of the flow at a given operating point. Examples of schematic blade rows for pump and turbine rotors are included below. Absolute streamlines

Absolute streamline

Relative streamline

Relative streamlines

Figure 13. Examples of profiled pump rotor blade row

 Figure 12. Examples of schematic pump rotor blade row

18

Animation 1: Absolute and Streamlines in a Pump Rotor

Relative

This animation shows the absolute on relative streamlines in a pump rotor.




Pump Types

Self-Study Problem 5: Deviation of Flow in Blade Rows

Different types of pumps can be classified by their specific speed, which is defined as follows

Draw qualitatively the deviation of the flow in a rotor blade row of a pump and a turbine (absolute and relative streamlines). With respect to the rotation of the rotor, in which direction is the flow deviated? Can you make any statement on the change of swirl momentum for these two cases? How would the following blade rows look like? • • •

ωs = ω

What do you think could be the benefit of a high change in swirl momentum?

and pumps).

Eq. 23

(gH )0.75

In imperial units the specific spead is defined the same way however with different units. The specific speed is thereby referred to as Ns. A comparison of the units used is included below in table 1.

High change in swirl momentum Low change in swirl momentum No change in swirl momentum



Q 0 .5

Parameter SI rad/s ω Q m3/s H m

Checkpoint 3: Absolute and Relative Flow Paths

Imperial rpm gpm ft

Table 1. Comparison of units in specific speeds for SI and Imperial definition

Sketch and explain the absolute and relative flow paths in turbomachinery blade rows (turbines

The figure below depicts an organization of pumps depending on their specific speed. Note that high flow rates lead to high

19


specific speeds whereas high heads tend to decrease the specific speed. Centrifugal pumps are therefore located at low values whereas axial pumps feature high specific speeds.

Figure 14. Pump types and their specific speeds

20


Pump Velocity Triangles Velocity triangles are used to describe the kinematics of the flow in a turbomachine. As it has been shown previously a change in head is due to a change in tangential velocity components times the tangential speed at the respective location. For that purpose it is essential to know the velocities in the absolute and relative frame of reference.

c

•

cx=wx

β u

Note •

w

α

cΘ

Absolute frame of reference: non-rotating, fixed with respect to ground Relative frame of reference: rotating with rotor, i.e. the frame of reference if you as an observer sit on the rotor

wΘ

Figure 15. General concept of absolute and relative velocity

Absolute and relative flow velocities are related as follows:

It is the velocity of the frame of reference (here the tangential speed of the rotor u at the respective position) that relates absolute and relative velocities. Absolute velocities are commonly denoted by “c” whereas relative velocities are denoted by “w”. The general concept is illustrated in figure 5.

wx = c x wx = c x wθ = cθ − u

21

Eq. 24 Eq. 25 Eq. 26




Lecture Unit 11: Concept of Frame of Reference

Radial direction

3

This lecture unit introduces you to the concept of various frames of reference.

u2 2 1

Note the following: • • • •

Absolute and relative axial components are identical Circumferential components that point in the same direction as the tangential speed are positive Circumferential component that point against the tangential speed must be treated as negative Flow angles behave in the same way; in the above figure α would consequently be positive whereas β would be negative

c2

w2

ω

u1 Centrifugal pump

w1

c1 Axial direction

The velocity triangles of the impeller in a centrifugal compressor stage are depicted in Figure 16.

Figure 16. Impeller velocity triangles

22






Lecture Unit 12: Flow Deviations in Turbomachine Components

Learn about how the flow is deviated in turbomachine components and why.

Self-Study Problem 6: Velocity Triangles

This self-study exercise is concerned with determining the various velocity components in a velocity triangle.

The following velocity triangle is given:





Lecture Unit 13: Pump Velocity Triangles

Learn about velocity triangles in pumps.

Checkpoint 4: Pump Velocity Triangles

• • •

Draw velocity triangles at impeller inlet and outlet for an axial and a radial pump.

Absolute flow direction axial Axial flow velocity 25m/s Tangential rotor speed 40m/s

Determine the following: • •

23

Relative flow angle Relative flow speed


The velocities in a blade row are heavily affected by the shape of the flow channel as well as the shape of the blades in a blade row. It is the task of turbomachinery design engineers not only to match a design point but also to choose these shapes such that the turbomachine is operating at maximum efficiency.

 channel.



As the blade rows are three-dimensional objects, the flow velocities do not only change along the blade in a blade row but also from hub to casing (i.e. along blade span). In the present course, we exclusively focus on 1D considerations, which means that we work with one representative velocity per control stations and disregard changes in spanwise direction.

Lecture Unit 14: Influence of Flow Channel Shape on Velocities

This lecture unit illustrates how the velocities in a blade row are affected by the shape of the flow

Lecture Unit 15: Influence of Blade Shape on Velocities

This lecture unit illustrates how the velocities in a blade row are affected by the shape of blades.

24



Lecture Unit 16: Flow at Various Spanwise Positions



Lecture Unit 17: Different Shapes of Pump Flow Channels

This lecture unit introduces you to the variability of the flow along blade span.

This lecture unit introduces to different flow channel shapes.


Note that the directions at impeller inlet and outlet feature different orientation for centrifugal machines; whereas at the inlet the components are axial-circumferential, at impeller outlet the directions are radial-circumferential. Furthermore it is important to stress that the inflow to the impeller will be axial unless there are inlet guide vanes. This means that the circumferential component at impeller inlet equals zero leading to the following simplified Euler equation u c H = 2 θ2 g



onto the relative main flow direction. This relative eddy establishes in a blade passage due to the rotation of the impeller as sketched in figure 7. For a given rotation a counter-rotating eddy will establish in a blade passage (figure 16a). This eddy affects the outflow from the blade passage such that the relative outflow is deflected against the direction of rotation (figure 16b).

Eq. 26

Lecture Unit 18: Change of Total Head in Stator and Rotor Blade Rows

Learn about how the total head changes in stator and rotor blade rows. Figure 17. Concept of relative eddy (adapted from [1])

At impeller exit the relative flow leaves the blade approximately at the blade metal angle. In reality a phenomenon called slip leads to the relative outflow being deflected against the direction of rotation. The underlying physical phenomenon is the superposition of a relative eddy

A practical consequence of the slip is that the relative circumferential component at impeller exit is reduced against the direction of rotation. This in turn implies that the absolute

25


circumferential component is reduced as well, which leads to a reduction in pump head. The effect of slip on the velocity triangle is graphically expressed in figure 8.

Ideal outflow is thus represented by σ=1 whereas real outflow features slip factors of σ 0  forward sweep

theoretical

β2= 0  radial blades real



β2< 0  backwards sweep Φ

Figure 19. Dependence of the pump characteristics from blade metal angle

31

Checkpoint 7: Pump Operating Characteristics

Explain the operating characteristics of a pump and relate it to the changes in velocity triangles.




Self-Study Problem 8: Pump Off-Design Operation

Pump Operating Point

This self-study exercise is with determining the off-design operation of a pump.

The pump operating characteristics give us a picture of how the pump head changes with changing flow rate but of it own it does not yet tell us at which operating point a pump will run. An operating point is first established when a pump is connected to a system, i.e. a consumer. As outlined further above the system characteristics can be described by pressure head, velocity head, static head and friction head. Following the same consideration as for the pump characteristics a system characteristics can be established such that H sys = f (Q sys ) where

The following is given: • • • •

Head at design point 250m Axial flow velocity at design point 10m/s Reference radius 0.25m Rotational speed 2400rpm

H sys = H p, sys + H v, sys + H s, sys + H f , sys

Answer the following: •

•

•

While the pump speed is kept constant, the operating point is changed such that the axial flow velocity is increased by 30%. What is the head at the off-design point? Assuming that the pump had a reference radius of 0.4m while the other parameters are unchanged, what would the head be at off-design operation? Do the results differ for these two cases? If so, explain why.

Eq. 34

In this expression H s, sys is independent of the flow rate.

The operating point of a pump connected to a certain system then yields from H sys = H pump

This is expressed graphically in Figure 20.

32

Eq. 35


H [m]

Operating point

When determining the operating of a pump connected to a system, the problem to be solved is to find the intersection between two curves. The ways of solving this problem depends on the form, in which these curves are available. The following ways can be listed:

System head

•

Pump head Hs

• Q [m3/s] Figure 20. Pump operating point



•

Lecture Unit 27: Pump Operating Point

This lecture unit teaches aspects of the operating point of a pump.

33

Curves are available graphically: the intersection can be determined graphically, i.e. by drawing one curve on top of the other curve in the same diagram. Curves are available as data series: the intersection can be determined using a piecewise linear approach. This means that the intersection between two straight lines connecting the two closest lines is determined. Curves are available as mathematical expressions: in this case the intersection can be determined analytically by subtracting the two equations from each other. As the pump and system characteristics are most probably are parabolic, a quadratic equation needs to be solved.






Self-Study Problem 9: Pump Operating Point

This self-study exercise is concerned with determining the operating point of a pump when connected to a system.

The power needed to achieve a head in a fluid at a given operating point is expressed by

[m]

The pump characteristics is given by: H = 41.28 − 1.42 ⋅ Q − 0.16 ⋅ Q 2

P = Q⋅H ⋅g ⋅ρ

[m]

Determine the following:

•

Eq. 36

Note that this expression contains the same elements as when dealing with compressors where P = m ⋅ ∆h0 , as m = Q ⋅ ρ and ∆h0 = H ⋅ g .

η = 21.8 + 14.5 ⋅ Q − 1.4 ⋅ Q 2 [%]

•

Determine the operating of a pump connected to a system.

Pump Power

The system curve is defined as follows: H sys = 3.4 + 0.7 ⋅ Q + 0.29 ⋅ Q 2

Checkpoint 8: Pump Operating Point



Operating point of the pump when connected to the system Efficiency of the pump at this operating point

34

Lecture Unit 28: Pump Power

This lecture unit discusses power requirements of pumps.




Similar to the pump head characteristic the efficiency peaks around a certain flow rate, i.e. the so called “best point”. To either side the efficiency decreases as illustrated in figure 11.

Checkpoint 9: Pump Power

Determine the power that a pumps need when running at a specific operating point.

H [m] Pump Efficiency A number of efficiencies can be defined for pumps as follows: •

• •

Pump efficiency

Hydraulic efficiency: compare actual head increase to theoretical head increase obtained from Euler equation  accounts for friction and hydraulic losses in pump Volumetric efficiency: compare actual volume flow to theoretical volume flow  accounts for internal leakage and backflow Mechanical efficiency: compare actual power supplied by motor to power received by impeller  accounts for mechanical friction power losses

Pump head Hs Q [m3/s] Figure 21. Pump head and efficiency



These efficiencies are combined to a total efficiency as follows η tot = η hyd ⋅η vol ⋅η mech =

Q⋅H ⋅g ⋅ρ M motor ⋅ ω

Eq. 37

35

Lecture Unit 29: Pump Efficiency

This lecture unit discusses the change of efficiency with change in pump operating point.




Better solutions are achieved by regulating the pump, which can be done by either regulating the speed or as a one-time measure by reducing the impeller diameter (also called “trimming”). The laws that describe how the pump characteristics change upon either type of these regulations are called “affinity laws”.

Checkpoint 10: Pump Efficiency

Determine the efficiency of a pumps need when running at a specific operating point based on an efficiency-volume flow diagram.



Affinity Laws In very few cases it is possible to find a pump that will yield a certain operating point. Often either the system or the pump must be regulated. A straightforward way to regulate the system is to include an adjustable valve, which leads to an additional and variable system head. This solution is however not that energy-efficient as the head over the valve must be considered as lost.



Lecture Unit 31: Pump Operation at Variable Speed

Learn about off-design operation of pumps at variable speed.

The question that we start off with is how head, flow rate and power change upon a) change in rotational speed and b) change in impeller diameter. To answer this it is necessary to express these parameters in terms of the regarded variables as follows

Lecture Unit 30: Pump Operation at OffDesign

H=

This lecture unit introduces to pump operation at off-design. , as

36

ψ ⋅u22 g

=

ψ ⋅ d 2 2 ⋅ω 2 4g

Eq. 38


u2 =

d2 ω 2

•

Eq. 39

• Similar Q = φ ⋅ A2 ⋅ u 2 =

φ ⋅ π ⋅ d 2 2 b2ω 2

Eq. 40

Practically this means that similar points on pump curves at various rotational speeds lie on parabolic lines emerging from the origin ( H ∝ Q 2 ) in case of speed regulation, see figure 12. In case of change in impeller outlet diameter the similar points of different curves will lie on straight lines emerging from the origin ( H ∝ Q ) as depicted in figure 13. Similar in this context means that the operation of the pump is comparable, i.e. that the pump runs at the same efficiency.

, as A2 = π ⋅ d 2 ⋅ b2

Eq. 41

Fractional changes of head and flow rate yield from H A d A2ω A2 = H B d B 2ω B 2

Eq. 42



And Q A d A 2ω A = Q B d B 2ω B

Change in rotational speed (diameter constant): H ∝ω2, Q ∝ω  P ∝ω3 Change in diameter (rotational speed constant): H ∝ d2, Q ∝ d2  P ∝ d4

Eq. 43

From these relations the following conclusions can be drawn

37

Lecture Unit 32: Pump Affinity Laws

This lecture unit introduces to pump affinity laws.




Remote Laboratory Exercise Design Operation of Pumps

1:

Off-

This remote laboratory exercise gives you the possibility to acquire real test data interactively on a test facility that operates pump in off-design manner.



Checkpoint 11: Affinity Law

Explain the operating speed needs to be changed for achieving a certain operating point. From an energy-usage point-of-view, does variable pump speed have an advantage over throttling?

Figure 22. Effect of speed regulation

38




Lecture Unit 34: Pump Operation after Diameter Change

This lecture unit presents the application of the affinity law and teaches how a radial pump operates after changed impeller exit diameter.



Self-Study Problem 10: Affinity Laws

This self-study exercise is concerned with determining the off-design operation of a pump by applying the affinity laws.

The operating characteristics of a pump running at 3000rpm is given by Figure 23. Effect of impeller trim (diameter change)



H = 31.3 + 0.62 ⋅ Q − 0.11 ⋅ Q 2 [m]

η = −64.6 + 26.5 ⋅ Q − 1.28 ⋅ Q 2 [%]

Lecture Unit 33: Pump Operation at Varying Speed

he pump is connected to a system that is described by the following equation:

This lecture unit presents the application of the affinity law and teaches how a pump operates at changed speed.

H sys = 6.63 + 0.02 ⋅ Q + 0.1 ⋅ Q 2 [m]

39


Note: In all equations Q is indicated in m3/s.

Serial and Parallel Operation of Pumps

Determine the following:

Pumps can be operated in arrangements, i.e. several pumps can be integrated into a system and be operated simultaneously. The motivation for operating pumps in arrangements is to be able to achieve other operating points that otherwise would not have been possible to achieve with just one single pump (of the same type).

• • • •

•

Operating point of the system when connected to the pump (Q, H) Power requirement at operating point Value of pump peak efficiency and flow rate at which it occurs (current pump curve) Pump speed if we would like to run the pump at peak efficiency while still maintaining the same flow rate as the original operating point Pump operating point (Q, H) when running pump at peak efficiency as described above as well as power required

Assume that two identical pumps (pump A and pump B) are operated in a certain arrangement. The following identities apply:

Parallel operation:

The gravitational constant is g 9.81 m/s2. Working medium is water with a density of 1000kg/m3.

Qtot = Q A +Q B

Eq. 44

H tot = H A = H B

Eq. 45

Serial operation:

40

Qtot = Q A =Q B

Eq. 46

H tot = H A + H B

Eq. 47






Remote Laboratory Exercise 2: Serial and Parallel Operation of Pumps

This remote laboratory exercise gives you the possibility to acquire real test data interactively on a test facility that operates pump in series of in parallel.

One of the most harmful effects of machines working with liquid fluids (pumps, hydro turbines and propellers) is cavitation. Cavitation denotes a phenomenon at which the saturation pressure of the fluid is reached. Recall that the saturation pressure is dependent on temperature as listed in table 2.

0 0.6

30 4.2

50 12.3

This lecture unit explains why cavitation occurs.

A sharp pressure decrease (i.e. under-pressure) as it for example might be the case at the inlet of a pump or in certain regions on the impeller thus can lead the flow to evaporate locally having small vapor bubbles formed locally. The harmful effect itself occurs first upon subsequent pressure increase that forces the vapor bubbles to collapse. As the vapor density is some order of magnitudes lower than the one of the liquid state the collapse induces an implosion yielding micro jets at extremely high pressures. These pressures are so high that implosions that occur in vicinity of surfaces can destroy the material locally. Figure 14 shows a sketch of the cavitation phenomenon.

Harmful Effects

T [C] p [kPa]

Lecture Unit 35: Reason for Cavitation

100 101.3



Table 2. Dependency of saturation pressure from temperature

41

Lecture Unit 36: Analysis of Cavitation Phenomenon

This lecture unit includes an analysis of the cavitation phenomenon.


1



2

Lecture Unit 37: Implosion of Vapor Bubbles during Cavitation

This lecture unit discusses in detail the reason for the harmful nature of cavitation, i.e. the implosion of vapor bubbles.

3

A measure that is used when designing a pump application for cavitation-free operation is the so-called “net positive suction head” or NPSH. The NPSH is a value in meter and indicates, what minimum head is allowed at the pump inlet to avoid cavitation. Its usage is as follows:

4

• •

•

The pump manufacturers specify a required NPSH, or short NPSHr. Similar to the pump operating characteristics it is a curve dependent on the flow rate From the system layout and pump placement an available NPSH, or short NPSHa, can be determined. This is the head present at pump inlet for a given operating point. The criteria applied for avoiding cavitation is finally NPSHa>NPSHr.

Figure 24. Cavitation phenomenon

To be able to avoid (or remedy) cavitation it is necessary to understand how the NPSHs are affected by different

42


operating parameters and how it can be changed. An overview is included below: •

•

•

•

For direct assessment of the risk of cavitation, the NPSH curves are often included in operating diagrams as done in Figure 25.

High suction height of pump Leads to: Low NPSHa Avoid by: Reducing suction height of pump (e.g. low placement of pump) High inflow losses Leads to: Low NPSHa Avoid by: Increasing inflow pipe High liquid temperature Leads to: High NPSHr Avoid by: Reduce liquid temperature High pump speed Leads to: High NPSHr Avoid by: Reduce pump speed

Upper flow rate to avoid cavitation

H [m] Pump curve

NPSHr NPSHa Q [m3/s] Figure 25. Pump and system NPSHs



Lecture Unit 38: Measures against Cavitation



This lecture unit teaches what measures that can be taken to avoid cavitation.

43

Lecture Unit 39: NPSHr and NPSHa

This lecture unit introduces the parameters of Net Positive Suction Head (NPSH).


Preliminary Design of Pumps The preliminary design of pumps includes the steps of determining a meridional flow channel and suitable blade row geometries such that a given operating point as well as a given operating characteristics is achieved. A note regarding this shall be made upfront: preliminary design in specific and design in general is such that there is not just ONE correct solution. Instead, various solutions might fulfill the requested task while being slightly different though fully viable. You will often reach a point at which you need to take decisions or make choices. When making choices, say for example choosing the mean radius at pump inlet, do not be afraid of making a choice. This does not mean that you should make blind choices. Think first what boundary conditions are given (i.e. what your design space is) and then make a choice, which you find suitable. Preliminary design is often about re-iterating and starting all over again. It is therefore appropriate to use simple and transparent techniques. Here, we use 1D analysis for this task.

•

•

•

Below, a suitable way for performing the preliminary design of a pump is presented. •

Step 1: Preparation: establish full awareness of the task that the pump needs to solve. What is the design

•

44

operating point? Is the pump likely to be operated at off-design condition? If so, you also need to design for off-design characteristics. The design target will be volume flow rate and head. Step 2: Meridional flow channel: draft the meridional flow channel. Here you already need to make the choice of the pump type in terms of axial, mixed flow or radial. Again, you might achieve the targeted operating points with any of these types, but the pump will look substantially different. High head pumps are usually of the radial type whereas high volume flow pumps are usually of the axial type. See also the diagram on specific speed above. Step 3: Determine the velocity triangle at rotor inlet. If the pump does not feature an inlet guide vane, the flow will enter the rotor axially (in the absolute frame of reference). Step 4: Determine the velocity triangle at rotor outlet. Note that the relative blade angle at rotor outlet will determine the operating characteristics of the pump. If the characteristics is not to satisfaction, you might need to go back to step 2 above and modify the meridional flow channel. Step 5: Decide whether you wish to include a stator. The stator will not change the total energy content of




the fluid downstream of the rotor but it can change in which form the energy is available (pressure head, velocity head). For example, if you design an axial pump and the flow shall leave the pump without swirl (i.e. axial flow direction), then you need to include a stator. Determine the velocity triangle at stator outlet.



The preliminary design of the pump is now finished as you have the velocity triangles at rotor inlet, rotor outlet and stator outlet. These give you the pump operating point as well as give you indications of the off-design behavior of the pump.

 

Reference 3: Pump Design in Industry

Get an impression of how pump design is performed in industry.

Self-Study Problem 11: Preliminary Design of a Pump

This self-study exercise is concerned with the preliminary design of a pump. It presents a representative task in that you are given a required design point and are asked to design a pump that fulfills these requirements while fulfilling a set of boundary conditions.

Lecture Unit 40: Preliminary Design of a Pump

This lecture unit teaches how the preliminary design of a pump is performed.

The following shall be fulfilled: • • • • •

Tool 1: Preliminary Design of a Pump

This tool gives you the possibility to perform the preliminary analysis of a simple pump (runs in MS Excel)

45

Design speed 2900 rpm. Total head at design point 200 m Volume flow rate at design point 30 l/s Maximum impeller tangential speed u2,max 100m/s Maximum meridional velocity at impeller outlet 15m/s


•

Axial inflow to the impeller

The gravitational constant is g 9.81 m/s2. Working medium is water with a density of 1000kg/m3.

Determine the following: • • •

•

Overall geometry including mean radii at impeller inlet and outlet Complete velocity triangles at impeller inlet and outlet Pump characteristics (simplified off-design characteristics). What is the head at zero volume flow rate? Draft drawing of the pump



Checkpoint 12: Preliminary Design of a Pump

Explain how the preliminary design of a pump is performed and be able to perform it yourself.

46


Hydro Turbines

Static head

H

ydro turbines are so-to-say the counterpart of pumps; whereas pumps are used to increase the total energy in a fluid, hydro turbines are used to extract energy from a fluid and by this decrease the total energy. We could almost go so far and take a pump and run it the opposite way, i.e. apply high-pressure fluid on the pump’s discharge side and extract mechanical energy from the pump shaft. The pump would run as a turbine, maybe not very efficient and at a low power density, but basically it would work. The sections below give you the necessary background to understand the basics of hydro turbines.

Velocity head Friction head

•

Eq. 48

,where Hp =

p 2 − p1 gρ

v 2 2 − v1 2 2g

H f = hf

Eq. 51 Eq. 52

A turbine decreases the total head in a system. This implies that there is high-energy fluid available at the inlet of a turbine and that the fluid leaves the turbine with reduced energy content. Depending on the application, the primary contribution of the high-energy source might be different:

•

Pressure head

Hv =

Eq. 50

The friction head reflects the losses in a system and is commonly expressed in meters.

The different forms of energy that were listed initially in the pump section apply equally to turbines. The total energy in the fluid is measured by a total head that composes of various forms of energy as follows: H tot = H p + H s + H v + H f

H s = h2 − h1

Eq. 49

47

Hydro turbine driven by high-velocity fluid which results from a great difference in elevation: the primary high energy source is static head that is transformed into velocity head by flow acceleration. Hydro turbine that is driven by the flow in a river: the primary high energy source is velocity head.


Turbine Systems



A turbine system denotes a system, in which a turbine is used to extract energy from a fluid. The system consists of pipes (or ducts) on the pressure and the discharge side of the turbine as well as eventually valves, reservoirs or other devices. An example of a turbine system is included in Figure 26.

Self-Study Problem 12: Turbine System

This self-study exercise is concerned a turbine system.

Consider a turbine system as the one depicted below. The turbine is connected to an upper reservoir on the pressure side and discharges to a lower reservoir.

reservoir

flow flow hydro turbine

The following is given: • • • •

reservoir Figure 26. Example of a turbine system

48

Static pressure p1 80kPa Volume flow rate at pos 2 10m3/s Height H 400m Pipe diameter at turbine inlet d2 0.6m


• •

Pipe diameter d3=d2 Static pressure p3 105kPa

H=

The gravitational constant is g 9.81 m/s2. Working medium is water with a density of 1000kg/m3. Friction can be neglected.

Eq. 53

As a turbine is extracting energy from a fluid, the change in total head gets negative. Compared to pumps, turbines therefore feature an opposite change in swirl; the flow is deviated against the direction of rotation in the rotor.

Determine the following: • • • •

1 (u 2 cθ 2 − u1cθ 1 ) g

Flow speed at turbine inlet Total head at turbine inlet Difference in total head over turbine Maximum possible power produced

Application of Euler Turbine Equation to Turbines By applying the Euler’s turbine equation on a turbine, the change in total head is related to change in swirl inside the turbine. The application is identical to the pump keeping in mind that the Euler’s turbine equation is always applied over the rotating part of the machine. The Euler’s turbine equation is given by



Lecture Unit 41: Euler’s Equation for Hydro Turbines



Self-Study Problem 13: Flow Deviation in a Turbine

This lecture unit introduces to the application of the Euler’s equation on hydro turbines.

This self-study exercise is concerned with the deviation of flow in a hydro turbine.

Assume that you were to design a hydro turbine extracting a certain amount of total head from a flow. First consider the case where you do not have a stator upstream of the rotor. Consequently, the flow will enter the rotor axially. At what

49


direction will the flow exit the rotor? Will the direction be affected by the amount of energy that you extract?

machines in the axial-radial plane. Note that these machines are axisymmetric.

Now, assume that you include a stator upstream of the rotor. What deviation of flow would you propose to achieve in this stator such as to maximize the energy extraction in this turbine?

In case of axial machines (Kaplan turbines), the rotational axis is oriented either horizontally or vertically. In case of centripetal machines (Francis turbines), the rotational axis is in most cases oriented vertically. There are however also cases in which these machines are aligned horizontally, especially if two turbines are arranged in parallel on the same shaft (backto-back turbines, also referred to as camel-back turbines).



Lecture Unit 42: Deviation of Flow in Turbines and Pumps

0

In this lecture unit, the deviation of flow in turbines is related to the ones in pumps.

stator

1

rotor

2

Turbine Elements Other than rotodynamic pumps, a hydro turbine usually consists of one single stage only. The reason for this is that a single stage turbine can be designed to take care of a very big change in total head, in extreme cases several thousands of meters of head. Similar to pumps, a turbine stage consists of a stationary and a rotating part referred to as stator and rotor as included in fig. The figures below show cross sections of

Figure 27. Hydro turbine stage denotations (axial)

50


0

Rotor and stator are so-called blade rows. A blade row is a row of blades and is used to guide the flow in a specific way. As it has been shown above by means of the Euler equation, it is the deviation of the flow in the absolute frame of reference that matters in a turbomachine. Hence, the blade rows are designed such that they yield a certain deviation of the flow at a given operating point. Examples of schematic blade rows for pump and turbine rotors are included below.

1

stator

rotor

2 Absolute streamlines

Relative streamlines

Figure 28. Hydro turbine stage denotations (centripetal)

The rotor is often referred to as runner. If a stator is included it is sometimes called “nozzle” or “guide vane” as it accelerates and deviates the flow. Commonly three control sections are identified in a stage as follows 0 1 2

stator inlet stator outlet, rotor inlet rotor outlet

Figure 29. Example of schematic turbine rotor blade row

51




Relative streamline

Self-Study Problem 14: Deviation of Flow in a Turbine Blade Row

Draw qualitatively the deviation of the flow in a rotor blade row of a turbine (absolute and relative streamlines). With respect to the rotation of the rotor, in which direction is the flow deviated? Can you make any statement on the change of swirl momentum? How would the following blade rows look like?

Absolute streamline

• • •

High change in swirl momentum Low change in swirl momentum No change in swirl momentum

What do you think could be the benefit of a high change in swirl momentum?

Figure 30. Examples of profiled turbine rotor blade row

What would be the maximum achievable change of momentum? How would such a turbine look like?



Animation 2: Absolute and Streamlines in a Turbine Rotor

Relative

This animation shows the absolute on relative streamlines in a turbine rotor.

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

Lecture Unit 43: Blade Shapes and Flow Passages in Turbines and Pumps



Lecture Unit 44: Flow Direction at Rotor Inlet

This lecture unit provides a comparative analysis of the schematic shape of blades in pump s and turbines and makes the link to the shape of flow passage.

This lecture unit is concerned with the direction of the flow at rotor inlet and explains the importance for turbine operation.

53


Turbine Types



Different types of turbines can be classified by their total head. The figure below depicts an organization of turbines depending on their head capacity and volume flow rate capacity. Note that depending on the head capacity, different types of turbines are used.

 

Lecture Unit 45: Types of Hydro Turbines

This lecture unit introduces to various types of hydro turbines.

Checkpoint 13: Types of Hydro Turbines

List the different types of hydro turbines and explain in what situations you would use them.

Reference 4: Hydro Manufacturer (VOITH)

Turbine

This reference links you to the VOITH web site where you can find brochures of hydro turbines. VOITH is a leading hydro turbine manufacturer.

Figure 31. Pump types and their specific speeds

54


Summary of Equations General Conservation of energy Pressure head

Bernoulli equation

H tot = H p + H v + H s + H f = const

Hp =

p 2 − p1 ρg

Mass balance

v 2 2 − v1 2 2g

Velocity head

Hv =

Static head

H s = h2 − h1

Friction head

H f = hf

p

ρ

+

v2 + gh = const 2

m = A ⋅ c n ⋅ ρ

, where A cross section, cn velocity normal to this cross section, ρ density

Momentum balance (axial)

Fx = m ⋅ (c x1 − c x 2 ) + p1 A1 − p 2 A2

, where A cross section (normal to axial), cx axial velocity, p static pressure, m mass flow rate

, where p static pressure, v velocity, h height coordinate, ρ density, g gravitational constant

Momentum balance (circumferential) Note: indices 1 and 2 refer to two arbitrary points in an isoenergetic part of the system.

Fθ = m ⋅ (cθ 1 − cθ 2 )

Euler turbomachine equation H tot ⋅ g = u 2 cθ 2 − u1cθ 1

55


Affinity Laws Load line (speed regulation)

H = k ⋅Q2

Affinity law (speed regulation)

H 1  N1   = H 2  N 2 

2

Q1 N = 1 Q2 N 2 P1  N 1   = P2  N 2 

Load line (trimming of radial pump)

H = k ⋅Q

Affinity law (trimming of radial pump)

H 1  d1 = H 2  d 2

Q1  d 1   = Q 2  d 2  3

P1  d1  =  P2  d 2 

, where H head, Q Volume flow rate, N pump speed, P pump power

  

2

2

4

, where H head, Q Volume flow rate, d pump impeller outlet diameter, P pump power

56


Trigonometry

c

α

tan α =

cθ cx

cos α =

cx c

sin α =

cθ c

cx

cΘ

Algebra Quadratic equation

H = a ⋅Q2 + b ⋅Q + c

General solution (H=0)

Q1, 2 =

− b ± b2 − 4⋅a ⋅c 2⋅a

57


Index absolute frame of reference, 21 affinity laws, 36 axial view, 1 blade metal angle, 31 casing, 28 cavitation, 41 classification of pumps, 16 clearance, 28 conservation of energy, 5 conservation of mass, 4 conservation of momentum, 6 cross section, 17 cylindrical coordinate system, 1 design parameters, 27 deviation of flow, 19 efficiency, 35 Euler’s turbine equation, 8 flow coefficient, 27 frame of reference, 22 Francis, 50 Friction head, 11 head coefficient, 27 hydro turbines, 47 impeller, 28 implosion, 41 inlet flange, 28

inlet guide vane, 28 Kaplan, 50 leakage flow, 29 Leonhard Euler, 9 meridional direction, 2 micro jet, 41 NPSHa, 42 NPSHr., 42 off-design performance, 30 outlet flange, 28 parallel operation, 40 Pressure head, 11 pump characteristics, 30 pump operating point, 33 pump power, 35 pumping system, 12 relative eddy, 25 relative frame of reference, 21 reservoir, 12 rotor, 17 serial operation, 40 side view, 2 slip, 26 spanwise direction, 24 specific speed, 19 Static head, 11

58


stator, 17, 28 sustainability, IV system characteristics, 13 total head, 11 turbine system, 48 turbine types, 54

turbomachinery coordinate system, 1 unwrapped view, 3 Velocity head, 11 velocity triangles, 21 volute, 28

59


References [1]

Dixon, S.L., 1998, "Fluid Mechanics and Thermodynamics of Turbomachinery", Fourth edition, Butterworth-Heinemann, Woburn, MA, USA, 1998, ISBN 0-7506-7059-2

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