C:\BOOKS\PUMBOOK\PUMBOOK.DVI

Hydrodynamics

of Pumps

Christopher E. Brennen

California Institute of Technology

Pasadena, California

Concepts ETI, Inc.

and

Oxford University Press

Concepts ETI, Inc.

P. O. Box 643, Norwich, Vermont 05055, USA

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1994 Concepts ETI, Inc.

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Preface

This book is intended as a combination of a reference for pump experts, and

a monograph for advanced students interested in some of the basic problems

associated with pumps. It is dedicated to my friend and colleague Allan Acosta,

with whom it has been my pleasure and privilege to work for many years.

But this book has other roots as well. It began as a series of notes prepared

for a short course presented by Concepts ETI, Inc., and presided over by another

valued colleague, David Japikse, the president of Concepts ETI, Inc. Another

friend, Yoshi Tsujimoto, read early versions of the manuscript, and made many

valuable suggestions. My thanks to all my other friends in turbomachinery re-

search and the pump industry with whom it was my pleasure to be associated,

including Dara Childs, Paul Cooper, Nick Cumpsty, Jules Dussourd, Tony East-

land, Arpad Fay, Jim Fenwick, S. Gopalakrishnan, Ed Greitzer, Loren Gross,

Gene Jackson, Terry Jones, Kenjiro Kamijo, Kiyoshi Minemura, Bill Morgan,

Hideo Ohashi, Sheldon Rubin, Peter Runstadler, Ed Ruth, Bruno Schiavello,

Helmut Siekmann, Henry Stinson, Walt Swift and a host of others. Moreover,

it was a privilege to have worked on turbomachinery problems with a group of

talented students at the California Institute of Technology including Sheung-

Lip Ng, David Braisted, Javier Del Valle, Greg Hoffman, Curtis Meissner, Ed-

mund Lo, Belgacem Jery, Dimitri Chamieh, Douglas Adkins, Norbert Arndt,

Ronald Franz, Mike Karyeaclis, Rusty Miskovish, Abhijit Bhattacharyya, Adiel

Guinzburg and Joseph Sivo.

Finally, none of this would have been possible without Doreen’s encourage-

ment, love, and companionship and that debt is beyond words.

Christopher Earls Brennen

California Institute of Technology, July 1994.

PREFACE

Contents

Preface

Nomenclature

1INTRODUCTION

1.1 SUBJECT............................... 15

1.2 CAVITATION ............................ 15

1.3 UNSTEADYFLOWS ........................ 16

1.4 TRENDSINHYDRAULICTURBOMACHINERY........ 17

1.5 BOOKSTRUCTURE ........................ 18

REFERENCES............................ 19

2 BASIC PRINCIPLES

2.1 GEOMETRICNOTATION ..................... 21

2.2 CASCADES.............................. 25

2.3 FLOWNOTATION ......................... 27

2.4 SPECIFIC SPEED .

......................... 28

2.5 PUMPGEOMETRIES ....................... 29

2.6 ENERGYBALANCE ........................ 31

2.7 IDEALIZED NONCAVITATING PUMP PERFORMANCE . . . 34

2.8 SEVERALSPECIFICIMPELLERSANDPUMPS........ 35

REFERENCES............................ 38

3 TWO-DIMENSIONAL PERFORMANCE ANALYSIS

3.1 INTRODUCTION .......................... 39

3.2 LINEAR CASCADE ANALYSES

.................. 39

3.3 DEVIATIONANGLE ........................ 43

3.4 VISCOUSEFFECTSINLINEARCASCADES.......... 45

3.5 RADIAL CASCADE ANALYSES . . . . . .

........... 47

3.6 VISCOUSEFFECTSINRADIALFLOWS ............ 51

REFERENCES............................ 53

CONTENTS

4 OTHER FLOW FEATURES

4.1 INTRODUCTION .......................... 57

4.2 THREE-DIMENSIONALFLOWEFFECTS ........... 57

4.3 RADIAL EQUILIBRIUM SOLUTION: AN EXAMPLE . . . . . 60

4.4 DISCHARGEFLOWMANAGEMENT .............. 65

4.5 PREROTATION ........................... 67

4.6 OTHERSECONDARYFLOWS .................. 71

REFERENCES............................ 74

5 CAVITATION PARAMETERS AND INCEPTION

5.1 INTRODUCTION .......................... 77

5.2 CAVITATIONPARAMETERS................... 77

5.3 CAVITATIONINCEPTION..................... 80

5.4 SCALINGOFCAVITATIONINCEPTION ............ 84

5.5 PUMPPERFORMANCE ...................... 84

5.6 TYPESOFIMPELLERCAVITATION .............. 87

5.7 CAVITATIONINCEPTIONDATA ................ 92

REFERENCES............................ 99

6 BUBBLE DYNAMICS, DAMAGE AND NOISE

103

6.1 INTRODUCTION ..........................103

6.2 CAVITATION BUBBLE DYNAMICS

...............103

6.3 CAVITATIONDAMAGE ......................108

6.4 MECHANISM OF CAVITATION DAMAGE . . .

........111

6.5 CAVITATIONNOISE ........................114

REFERENCES............................120

7 CAVITATION AND PUMP PERFORMANCE

123

7.1 INTRODUCTION ..........................123

7.2 TYPICALPUMPPERFORMANCEDATA............123

7.3 INDUCERDESIGNS ........................130

7.4 INDUCERPERFORMANCE....................131

7.5 EFFECTSOFINDUCERGEOMETRY..............135

7.6 ANALYSES OF CAVITATION IN PUMPS . . . .

........138

7.7 THERMAL EFFECT ON PUMP PERFORMANCE . . . . . . . 141

7.8 FREESTREAMLINEMETHODS.................148

7.9 SUPERCAVITATINGCASCADES.................152

7.10PARTIALLYCAVITATINGCASCADES .............154

7.11 CAVITATION PERFORMANCE CORRELATIONS . . . . . . . 161

REFERENCES............................162

8 PUMP VIBRATION

169

8.1 INTRODUCTION ..........................169

8.2 FREQUENCIESOFOSCILLATION................172

8.3 UNSTEADYFLOWS ........................175

8.4 ROTATINGSTALL .........................178

CONTENTS

8.5 ROTATINGCAVITATION .....................181

8.6 SURGE ................................182

8.7 AUTO-OSCILLATION .......................185

8.8 ROTOR-STATORINTERACTION:FLOWPATTERNS ....189

8.9 ROTOR-STATORINTERACTION:FORCES ..........192

8.10DEVELOPEDCAVITYOSCILLATION..............196

8.11 ACOUSTIC RESONANCES . .

..................198

8.12BLADEFLUTTER .........................199

8.13POGOINSTABILITIES.......................202

REFERENCES............................203

9 UNSTEADY FLOW IN HYDRAULIC SYSTEMS

209

9.1 INTRODUCTION ..........................209

9.2 TIMEDOMAINMETHODS ....................210

9.3 WAVEPROPAGATIONINDUCTS................211

9.4 METHODOFCHARACTERISTICS ...............214

9.5 FREQUENCYDOMAINMETHODS ...............216

9.6 ORDEROFTHESYSTEM.....................217

9.7 TRANSFERMATRICES ......................218

9.8 DISTRIBUTEDSYSTEMS .....................219

9.9 COMBINATIONSOFTRANSFERMATRICES .........220

9.10PROPERTIESOFTRANSFERMATRICES ...........221

9.11SOMESIMPLETRANSFERMATRICES.............224

9.12FLUCTUATIONENERGYFLUX .................226

9.13NON-CAVITATINGPUMPS ....................230

9.14CAVITATINGINDUCERS .....................233

9.15SYSTEMWITHRIGIDBODYVIBRATION...........241

REFERENCES............................242

10 RADIAL AND ROTORDYNAMIC FORCES

245

10.1INTRODUCTION ..........................245

10.2NOTATION..............................246

10.3 HYDRODYNAMIC BEARINGS AND SEALS . . . . . .

....250

10.4BEARINGSATLOWREYNOLDSNUMBERS .........251

10.5 ANNULUS AT HIGH REYNOLDS NUMBERS . . . . . .

....256

10.6SQUEEZEFILMDAMPERS....................257

10.7 TURBULENT ANNULAR SEALS . . . . . .

...........258

10.8LABYRINTHSEALS ........................264

10.9 BLADE TIP ROTORDYNAMIC EFFECTS

...........265

10.10STEADYRADIALFORCES ....................267

10.11EFFECTOFCAVITATION ....................275

10.12CENTRIFUGALPUMPS......................275

10.13MOMENTSANDLINESOFACTION ..............280

10.14AXIALFLOWINDUCERS.....................282

REFERENCES............................283

CONTENTS

INDEX

288

Nomenclature

Roman Letters

Pipe radius

Cross-sectional area

ijk

Coefficients of pump dynamic characteristics

[

]

Rotordynamic force matrix

Cross-sectional area ratio

Breadth of passage or flow

[

]

Rotordynamic moment matrix

Chord of the blade or foil

Speed of sound

Rotordynamic coefficient: cross-coupled damping

Interblade spacing

Specific heat of liquid

Compliance

Rotordynamic coefficient: direct damping

Drag coefficient

Lift coefficient

Coefficient of pressure

pmin

Minimum coefficient of pressure

Ratio of blade thickness to blade spacing

Impeller diameter or typical flow dimension

Diffusion factor

Determinant of transfer matrix [

]

Specific internal energy

Energy flux

Young’s modulus

Friction coefficient

Force

Acceleration due to gravity

Component of

in the

direction

Specific enthalpy

Blade tip spacing

NOMENCLATURE

Pitch of a helix

Total specific enthalpy

∗

Piezometric head

Total head rise

(

s, θ, t

)

Clearance geometry

Acoustic impulse

I, J

Integers such that

ω/

Ω=

I/J

Pump impedance

Square root of

−

Rotordynamic coefficient: cross-coupled stiffness

Thermal conductivity of the liquid

Rotordynamic coefficient: direct stiffness

Gas constant

Pipe length or distance to measuring point

Lift

Inertance

Axial length

Latent heat

Mass flow rate

Rotordynamic coefficient: cross-coupled added mass

Mass of gas in bubble

Constant related to the drag coefficient

Constant related to the lift coefficient

Moment

Mach number,

u/c

Rotordynamic coefficient: direct added mass

Coordinate measured normal to a surface

Specific speed

(

)

Cavitation nuclei number density distribution function

NPSP

Net positive suction pressure

NPSE

Net positive suction energy

NPSH

Net positive suction head

Pressure

Radiated acoustic pressure

Total pressure

Partial pressure of gas

Sound pressure level

Vapor pressure

Power

Vector of fluctuating quantities

Volume flow rate (or heat)

Rate of heat addition

Radial coordinate in turbomachine

Radial dimension in turbomachine

Bubble radius

NOMENCLATURE

Resistance

Cavitation nucleus radius

Reynolds number

Coordinate measured in the direction of flow

Solidity

Surface tension of the saturated vapor/liquid interface

Suction specific speed

Inception suction specific speed

Fractional head loss suction specific speed

Breakdown suction specific speed

Slip factor

Time

Temperature or torque

Transfer matrix elements

[

]

Transfer matrix based on ̃

[

∗

]

Transfer matrix based on ̃

[

]

Pump transfer matrix

[

]

System transfer matrix

Velocity in the

directions

Velocity vector

Fluid velocity

∞

Velocity of upstream uniform flow

Fluid velocity in non-rotating frame

Volume or fluid velocity

Fluidvelocityinrotatingframe

Rate of work done on the fluid

Elevation

Common factor of

and

Number of rotor blades

Number of stator blades

Greek Letters

Angle of incidence

Thermal diffusivity of liquid

Angle of relative velocity vector

Blade angle relative to cross-plane

Wave propagation speed

Geometric constant

Deviation angle at flow discharge

Clearance

Eccentricity

Angle of turn

NOMENCLATURE

Efficiency

Angular coordinate

Camber angle

∗

Momentum thickness of a blade wake

Thermal term in the Rayleigh-Plesset equation

Inclination of discharge flow to the axis of rotation

Bulk modulus of the liquid

Dynamic viscosity

Kinematic viscosity

Density of fluid

Cavitation number

Cavitation inception number

Fractional head loss cavitation number

Breakdown cavitation number

Choked cavitation number

Thoma cavitation factor

Thermal parameter for bubble growth

Geometric constants

Blade thickness

Flow coefficient

Head coefficient

Head coefficient at zero flow

Radian frequency of whirl motion or other excitation

Bubble natural frequency

Ω

Radian frequency of shaft rotation

Subscripts

On any variable,

Initial value, upstream value or reservoir value

Value at inlet

Value at discharge

Component in the axial direction

Pertaining to the blade

∞

Value far from the bubble or in the upstream flow

Value in the bubble

Critical value

Design value

Equilibrium value

Value for the gas

Value at the inlet hub

Value at the discharge hub

Components of vector

NOMENCLATURE

Pertaining to a section,

, of the hydraulic system

Saturated liquid value

Meridional component

Mean or maximum value

Nominal conditions or pertaining to nuclei

Components normal and tangential to whirl orbit

Pertaining to the pump

Component in the radial direction

Component in the

direction

Value at the inlet tip

Value at the discharge tip

Saturated vapor value

Component in the

direction

Component in the

direction

Component in the circumferential (or

) direction

Superscripts and other qualifiers

On any variable,

Mean value of

complex conjugate of

Complex amplitude of

Time derivative of

Second time derivative of

∗

Rotordynamics: denotes dimensional

{

}

Real part of

{

}

Imaginary part of

NOMENCLATURE

Chapter 1

INTRODUCTION

1.1 SUBJECT

The subject of this monograph is the fluid dynamics of liquid turbomachines,

particularly pumps. Rather than attempt a general treatise on turbomachines,

we shall focus attention on those special problems and design issues associated

with the flow of liquid through a rotating machine. There are two characteristics

of a liquid that lead to these special problems, and cause a significantly different

set of concerns than would occur in, say, a gas turbine. These are the potential

for cavitation and the high density of liquids that enhances the possibility of

damaging unsteady flows and forces.

1.2 CAVITATION

The word cavitation refers to the formation of vapor bubbles in regions of low

pressure within the flow field of a liquid. In some respects, cavitation is similar

to boiling, except that the latter is generally considered to occur as a result of

an increase of temperature rather than a decrease of pressure. This difference in

the direction of the state change in the phase diagram is more significant than

might, at first sight, be imagined. It is virtually impossible to cause any rapid

uniform change in temperature throughout a finite volume of liquid. Rather,

temperature change most often occurs by heat transfer through a solid bound-

ary. Hence, the details of the boiling process generally embrace the detailed

interaction of vapor bubbles with a solid surface, and the thermal boundary

layer on that surface. On the other hand, a rapid, uniform change in pressure

in a liquid is commonplace and, therefore, the details of the cavitation process

may differ considerably from those that occur in boiling. Much more detail on

the process of cavitation is included in later sections.

It is sufficient at this juncture to observe that cavitation is generally a malev-

olent process, and that the deleterious consequences can be divided into three

categories. First, cavitation can cause damage to the material surfaces close

CHAPTER 1. INTRODUCTION

to the area where the bubbles collapse when they are convected into regions of

higher pressure. Cavitation damage can be very expensive, and very difficult

to eliminate. For most designers of hydraulic machinery, it is the preeminent

problem associated with cavitation. Frequently, one begins with the objective

of eliminating cavitation completely. However, there are many circumstances

in which this proves to be impossible, and the effort must be redirected into

minimizing the adverse consequences of the phenomenon.

The second adverse effect of cavitation is that the performance of the pump,

or other hydraulic device, may be significantly degraded. In the case of pumps,

there is generally a level of inlet pressure at which the performance will decline

dramatically, a phenomenon termed cavitation breakdown. This adverse effect

has naturally given rise to changes in the design of a pump so as to minimize

the degradation of the performance; or, to put it another way, to optimize the

performance in the presence of cavitation. One such design modification is the

addition of a cavitating inducer upstream of the inlet to a centrifugal or mixed

flow pump impeller. Another example is manifest in the blade profiles used

for supercavitating propellers. These supercavitating hydrofoil sections have

a sharp leading edge, and are shaped like curved wedges with a thick, blunt

trailing edge.

The third adverse effect of cavitation is less well known, and is a consequence

of the fact that cavitation affects not only the steady state fluid flow, but also

the unsteady or dynamic response of the flow. This change in the dynamic

performance leads to instabilities in the flow that do not occur in the absence

of cavitation. Examples of these instabilities are “rotating cavitation,” which

is somewhat similar to the phenomenon of rotating stall in a compressor, and

“auto-oscillation,” which is somewhat similar to compressor surge. These insta-

bilities can give rise to oscillating flow rates and pressures that can threaten the

structural integrity of the pump or its inlet or discharge ducts. While a complete

classification of the various types of unsteady flow arising from cavitation has

yet to be constructed, we can, nevertheless, identify a number of specific types

of instability, and these are reviewed in later chapters of this monograph.

1.3 UNSTEADY FLOWS

While it is true that cavitation introduces a special set of fluid-structure inter-

action issues, it is also true that there are many such unsteady flow problems

which can arise even in the absence of cavitation. One reason these issues may

be more critical in a liquid turbomachine is that the large density of a liquid

implies much larger fluid dynamic forces. Typically, fluid dynamic forces scale

Ω

where

is the fluid density, and Ω and

are the typical frequency

of rotation and the typical length, such as the span or chord of the impeller

blades or the diameter of the impeller. These forces are applied to blades whose

typical thickness is denoted by

. It follows that the typical structural stresses

in the blades are given by

Ω

/τ

, and, to minimize structural problems, this

quantity will have an upper bound which will depend on the material. Clearly

1.4. TRENDS IN HYDRAULIC TURBOMACHINERY

this limit will be more stringent when the density of the fluid is larger. In many

pumps and liquid turbines it requires thicker blades (larger

)thanwouldbe

advisable from a purely hydrodynamic point of view.

This monograph presents a number of different unsteady flow problems that

are of concern in the design of hydraulic pumps and turbines. For example,

when a rotor blade passes through the wake of a stator blade (or vice versa),

it will encounter an unsteady load which is endemic to all turbomachines. Re-

cent investigations of these loads will be reviewed. This rotor-stator interaction

problem is an example of a local unsteady flow phenomenon. There also exist

global unsteady flow problems, such as the auto-oscillation problem mentioned

earlier. Other global unsteady flow problems are caused by the fluid-induced

radial loads on an impeller due to flow asymmetries, or the fluid-induced ro-

tordynamic loads that may increase or decrease the critical whirling speeds of

the shaft system. These last issues have only recently been addressed from a

fundamental research perspective, and a summary of the conclusions is included

in this monograph.

1.4 TRENDS IN HYDRAULIC

TURBOMACHINERY

Though the constraints on a turbomachine design are as varied as the almost

innumerable applications, there are a number of ubiquitous trends which allow

us to draw some fairly general conclusions. To do so we make use of the affinity

laws that are a consequence of dimensional analysis, and relate performance

characteristics to the density of the fluid,

, the typical rotational speed, Ω, and

the typical diameter,

, of the pump. Thus the volume flow rate through the

pump,

, the total head rise across the pump,

, the torque,

,andthepower

absorbed by the pump,

, will scale according to

Qα

Ω

(1.1)

Hα

Ω

(1.2)

TαρD

Ω

(1.3)

PαρD

Ω

(1.4)

These simple relations allow basic scaling predictions and initial design esti-

mates. Furthermore, they permit consideration of optimal characteristics, such

as the power density which, according to the above, should scale like

ρD

Ω

One typical consideration arising out of the affinity laws relates to optimizing

the design of a pump for a particular power level,

, and a particular fluid,

This fixes the value of

Ω

. If one wished to make the pump as small as

possible (small

) to reduce weight (as is critical in the rocket engine context)

or to reduce cost, this would dictate not only a higher rotational speed, Ω,

but also a higher impeller tip speed, Ω

2. However, as we shall see in the

next chapter, the propensity for cavitation increases as a parameter called the

CHAPTER 1. INTRODUCTION

cavitation number decreases, and the cavitation number is inversely proportional

to the square of the tip speed or Ω

4. Consequently, the increase in tip

speed suggested above could lead to a cavitation problem. Often, therefore, one

designs the smallest pump that will still operate without cavitation, and this

implies a particular size and speed for the device.

Furthermore, as previously mentioned, the typical fluid-induced stresses in

the structure will be given by

Ω

/τ

, and, if

Ω

is fixed and if one

maintains the same geometry,

D/τ

, then the stresses will increase like

−

as the size,

, is decreased. Consequently, fluid/structure interaction problems

will increase. To counteract this the blades are often made thicker (

D/τ

decreased), but this usually leads to a decrease in the hydraulic performance of

the turbomachine. Consequently an optimal design often requires a balanced

compromise between hydraulic and structural requirements. Rarely does one

encounter a design in which this compromise is optimal.

Of course, the design of a pump, compressor or turbine involves many factors

other than the technical issues discussed above. Many compromises and engi-

neering judgments must be made based on constraints such as cost, reliability

and the expected life of a machine. This book will not attempt to deal with

such complex issues, but will simply focus on the advances in the technical data

base associated with cavitation and unsteady flows. For a broader perspective

on the design issues, the reader is referred to engineering texts such as those

listed at the end of this chapter.

1.5 BOOK STRUCTURE

The intention of this monograph is to present an account of both the cavitation

issues and the unsteady flow issues, in the hope that this will help in the design of

more effective liquid turbomachines. In chapter 2 we review some of the basic

principles of the fluid mechanical design of turbomachines for incompressible

fluids, and follow that, in chapter 3, with a discussion of the two-dimensional

performance analyses based on the flows through cascades of foils. A brief

review of three-dimensional effects and secondary flows follows in chapter 4.

Then, in chapter 5, we introduce the parameters which govern the phenomenon

of cavitation, and describe the different forms which cavitation can take. This is

followed by a discussion of the factors which influence the onset or inception of

cavitation. Chapter 6 introduces concepts from the analyses of bubble dynamics,

and relates those ideas to two of the byproducts of the phenomenon, cavitation

damage and noise. The isssues associated with the performance of a pump

under cavitating conditions are addressed in chapter 7.

The last three chapters deal with unsteady flows and vibration in pumps.

Chapter 8 presents a survey of some of the vibration problems in pumps. Chap-

ter 9 provides details of the two basic approaches to the analysis of instabilites

and unsteady flow problems in hydraulic systems, namely the methods of so-

lution in the time domain and in the frequency domain. Where possible, it

includes a survey of the existing information on the dynamic response of pumps

REFERENCES

under cavitating and non-cavitating conditions. The final chapter 10 deals with

the particular fluid/structure interactions associated with rotordynamic shaft

vibrations, and elucidates the fluid-induced rotordynamic forces that can result

from the flows through seals and through and around impellers.

REFERENCES

Anderson, H.H.

Centrifugal pumps.

The Trade and Technical Press Ltd., Eng-

land

Balje, O.E. (1981).

Turbomachines. A guide to design, selection and theory

John Wiley and Sons, New York.

Csanady, G.T. (1964).

Theory of turbomachines.

McGraw-Hill, New York.

Eck, B. (1973).

Fans.

Pergamon Press, London.

Jakobsen, J.K. (1971).

Liquid rocket engine turbopumps.

NASA SP 8052.

Kerrebrock, J.L. (1977).

Aircraft engines and gas turbines.

MIT Press.

Stepanoff, A.J. (1957).

Centrifugal and axial flow pumps.

John Wiley and

Sons, Inc.

CHAPTER 1. INTRODUCTION

Chapter 2

BASIC PRINCIPLES

2.1 GEOMETRIC NOTATION

The geometry of a generalized turbomachine rotor is sketched in figure 2.1, and

consists of a set of rotor blades (number =

) attached to a hub and operating

within a static casing. The radii of the inlet blade tip, inlet blade hub, discharge

blade tip, and discharge blade hub are denoted by

,and

respectively. The discharge blade passage is inclined to the axis of rotation at

an angle,

, which would be close to 90

◦

in the case of a centrifugal pump, and

much smaller in the case of an axial flow machine. In practice, many pumps and

turbines are of the “mixed flow” type , in which the typical or mean discharge

flow is at some intermediate angle, 0

<θ<

◦

The flow through a general rotor is normally visualized by developing a

meridional surface (figure 2.2), that can either correspond to an axisymmetric

Figure 2.1: Cross-sectional view through the axis of a pump impeller.

CHAPTER 2. BASIC PRINCIPLES

stream surface, or be some estimate thereof. On this meridional surface (see

figure 2.2) the fluid velocity in a non-rotating coordinate system is denoted by

(

) (with subscripts 1 and 2 denoting particular values at inlet and discharge)

and the corresponding velocity relative to the rotating blades is denoted by

(

). The velocities,

and

,havecomponents

and

in the circumfer-

ential direction, and

and

in the meridional direction. Axial and radial

components are denoted by the subscripts

and

. The velocity of the blades is

Ω

. As shown in figure 2.2, the flow angle

(

) is defined as the angle between

the relative velocity vector in the meridional plane and a plane perpendicular

to the axis of rotation. The blade angle

(

) is defined as the inclination of the

tangent to the blade in the meridional plane and the plane perpendicular to the

axis of rotation. If the flow is precisely parallel to the blades,

. Specific

values of the blade angle at the leading and trailing edges (1 and 2) and at the

hub and tip (

and

) are denoted by the corresponding suffices, so that, for

example,

is the blade angle at the discharge tip.

At the leading edge it is important to know the angle

(

)withwhichthe

flow meets the blades, and, as defined in figure 2.3,

(

)

−

(

)

(2.1)

This angle,

, is called the incidence angle, and, for simplicity, we shall denote

the values of the incidence angle at the tip,

(

), and at the hub,

(

and

, respectively. Since the inlet flow can often be assumed to be

purely axial (

(

and parallel with the axis of rotation), it follows that

(

)=tan

−

(

Ω

), and this can be used in conjunction with equation 2.1

in evaluating the incidence angle for a given flow rate.

The incidence angle should not be confused with the “angle of attack”, which

is the angle between the incoming relative flow direction and the chord line (the

line joining the leading edge to the trailing edge). Note, however, that, in an

axial flow pump with straight helicoidal blades, the angle of attack is equal to

the incidence angle.

At the trailing edge, the difference between the flow angle and the blade

angle is again important. To a first approximation one often assumes that the

flow is parallel to the blades, so that

(

). A departure from this

idealistic assumption is denoted by the deviation angle,

(

), where, as shown

in figure 2.3:

(

)

−

(

)

(2.2)

This is normally a function of the ratio of the width of the passage between the

blades to the length of the same passage, a geometric parameter known as the

solidity which is defined more precisely below. Other angles, that are often used,

are the angle through which the flow is turned, known as the

deflection angle

−

, and the corresponding angle through which the blades have turned,

known as the

camber angle

and denoted by

−

Deviation angles in radial machines are traditionally represented by the

slip

velocity

θs

, which is the difference between the actual and ideal circumferential

2.1. GEOMETRIC NOTATION

Figure 2.2: Developed meridional surface and velocity triangle.

velocities of the discharge flow, as shown in figure 2.4. It follows that

θs

=Ω

−

cot

(2.3)

This, in turn, is used to define a parameter known as the

slip factor

,where

−

θs

Ω

−

(cot

−

cot

)

(2.4)

Other, slightly different “slip factors” have also been used in the literature; for

example, Stodola (1927), who originated the concept, defined the slip factor as

−

θs

Ω

−

cot

). However, the definition 2.4 is now widely used. It

CHAPTER 2. BASIC PRINCIPLES

Figure 2.3: Repeat of figure 2.2 showing the definitions of the incidence angle

at the leading edge and the deviation angle at the trailing edge.

Figure 2.4: Velocity vectors at discharge indicating the slip velocity,

θs

follows that the deviation angle,

, and the slip factor,

, are related by

−

cot

−

cot

−

)

(2.5)

where the flow coefficient,

, is defined later in equation 2.17.

2.2. CASCADES

Figure 2.5: Schematics of (a) a linear cascade and (b) a radial cascade.

2.2 CASCADES

We now turn to some specific geometric features that occur frequently in dis-

cussions of pumps and other turbomachines. In a purely axial flow machine,

the development of a cylindrical surface within the machine produces a

linear

cascade

of the type shown in figure 2.5(a). The centerplane of the blades can

be created using a “generator”, say

∗

(

), which is a line in the

−

plane.

If this line is rotated through a helical path, it describes a helicoidal surface of

the form

∗

(

(2.6)

where

is the “pitch” of the helix. Of course, in many machines, the pitch is

also a function of

so that the flow is turned by the blades. If, however, the

pitch is constant, the development of a cylindrical surface will yield a cascade

with straight blades and constant blade angle,

. Moreover, the blade thickness

is often neglected, and the blades in figure 2.5(a) then become infinitely thin

lines. Such a cascade of infinitely thin, flat blades is referred to as a

flat plate

cascade

It is convenient to use the term “simple” cascade to refer to those geometries

for which the blade angle,

, is constant whether in an axial, radial, or mixed

flow machine. Clearly, the flat plate cascade is the axial flow version of a simple

cascade.

Now compare the geometries of the cascades at different radii within an axial

flow machine. Later, we analyse the cavitating flow occurring at different radii

CHAPTER 2. BASIC PRINCIPLES

(see figure 7.35). Often the pitch at a given axial position is the same at all

radii. Then it follows that the radial variation in the blade angle,

(

), must

be given by

(

)=tan

−

tan

(2.7)

where

is the blade angle at the tip,

In a centrifugal machine in which the flow is purely radial, a cross-section of

the flow would be as shown in figure 2.5(b), an array known as a

radial cascade

In a

simple

radial cascade, the angle,

, is uniform along the length of the

blades. The resulting blade geometry is known as a logarithmic spiral, since it

follows that the coordinates of the blades are given by the equation

−

(2.8)

where

=cot

and

are constants. Logarithmic spiral blades are therefore

equivalent to straight blades in a linear cascade. Note that a fluid particle in a

flow of uniform circulation and constant source strength at the origin will follow

a logarithmic spiral since all velocities will be of the form

C/r

where

is a

uniform constant.

In any of type of pump, the ratio of the length of a blade passage to its

width is important in determining the degree to which the flow is guided by the

blades. The solidity,

, is the geometric parameter that is used as a measure

of this geometric characteristic, and

can be defined for any

simple

cascade as

follows. If we identify the difference between the

coordinates for the same point

on adjacent blades (call this ∆

) and the difference between the

coordinates

for the leading and trailing edges of a blade (call this ∆

), then the solidity

for a simple cascade is defined by

∆

cos

(2.9)

Applying this to the linear cascade of figure 2.5(a), we find the familiar

c/h

(2.10)

In an axial flow pump this corresponds to

πR

,where

is the chord

of the blade measured in the developed meridional plane of the blade tips. On

the other hand, for the radial cascade of figure 2.5(b), equation 2.9 yields the

following expression for the solidity:

sin

(2.11)

which is, therefore, geometrically equivalent to

c/h

in the linear cascade.

In practice, there exist many “mixed flow” pumps whose geometries lie be-

tween that of an axial flow machine (

= 0, figure 2.1) and that of a radial

machine (

π/

2). The most general analysis of such a pump would require

a cascade geometry in which figures 2.5(a) and 2.5(b) were

projections

of the

2.3. FLOW NOTATION

geometry of a meridional surface (figure 2.2) onto a cylindrical surface and onto

a plane perpendicular to the axis, respectively. (Note that the

marked in

figure 2.5(b) is not appropriate when that diagram is used as a projection). We

shall not attempt such generality here; rather, we observe that the meridional

surface in many machines is close to conical. Denoting the inclination of the

cone to the axis by

, we can use equation 2.9 to obtain an expression for the

solidity of a simple cascade in this conical geometry,

sin

(2.12)

Clearly, this includes the expressions 2.10 and 2.11 as special cases.

2.3 FLOW NOTATION

The flow variables that are important are, of course, the static pressure,

,the

total pressure,

, and the volume flow rate,

. Often the total pressure is

defined by the total head,

/ρg

. Moreover, in most situations of interest in

the context of turbomachinery, the potential energy associated with the earth’s

gravitational field is negligible relative to the kinetic energy of the flow, so that,

by definition

ρv

(2.13)

(2.14)

Ω

−

Ω

(2.15)

using the velocity triangle of figure 2.2. In an incompressible flow, the total

pressure represents the total mechanical energy per unit volume of fluid, and,

therefore, the change in total pressure across the pump,

−

, is a fundamental

measure of the mechanical energy imparted to the fluid by the pump.

It follows that, in a pump with an incompressible fluid, the overall character-

istics that are important are the volume flow rate,

, and the total pressure rise,

ρgH

,where

−

)

/ρg

is the total head rise. These dimensional char-

acteristics are conveniently nondimensionalized by defining a head coefficient,

−

)

/ρR

Ω

gH/R

Ω

(2.16)

and one of two alternative flow coefficients,

and

Q/A

Ωor

Q/A

Ω

(2.17)

where

and

are the inlet and discharge areas, respectively. The discharge

flow coefficient is the nondimensional parameter most often used to describe

the flow rate. However, in discussions of cavitation, which occurs at the inlet to

a pump impeller, the inlet flow coefficient is a more sensible parameter. Note

CHAPTER 2. BASIC PRINCIPLES

that, for a purely axial inflow, the incidence angle is determined by the flow

coefficient,

(

)

−

tan

−

(

r/R

)

(2.18)

Furthermore, for a given deviation angle, specifying

fixes the geometry of

the velocity triangle at discharge from the pump.

Frequently, the conditions at inlet and/or discharge are nonuniform and one

must subdivide the flow into annular streamtubes, as indicated in figure 2.2.

Each streamtube must then be analysed separately, using the blade geometry

pertinentatthatradius. Themassflowrate,

, through an individual stream-

tube is given by

πρrv

(2.19)

where

is a coordinate measured normal to the meridional surface, and, in the

present text, will be useful in describing the discharge geometry.

Conservation of mass requires that

have the same value at inlet and

discharge. This yields a relation between the inlet and discharge meridional

velocities, that involves the cross-sectional areas of the streamtube at these two

locations. The total volume flow rate through the turbomachine,

,isthen

related to the velocity distribution at any location by the integral

πrv

(

)

(2.20)

The total head rise across the machine,

, is given by the integral of the

total rate of work done on the flow divided by the total mass flow rate:

(

)

−

(

))

ρg

πrv

(

)

(2.21)

These integral expressions for the flow rate and head rise will be used in later

chapters.

2.4 SPECIFIC SPEED

At the beginning of any pump design process, neither the size nor the shape

of the machine is known. The task the pump is required to perform is to use

a shaft rotating at a frequency, Ω (in

rad/s

), to pump a certain flow rate,

(in

) through a head rise,

(in

). As in all fluid mechanical formula-

tions, one should first seek a nondimensional parameter (or parameters) which

distinguishes the nature of this task. In this case, there is one and only one

nondimensional parametric group that is appropriate and this is known as the

“specific speed”, denoted by

. The form of the specific speed is readily deter-

mined by dimensional analysis:

Ω

(

)

(2.22)

2.5. PUMP GEOMETRIES

Though originally constructed to allow evaluation of the shaft speed needed

to produce a particular head and flow, the name “specific speed” is slightly

misleading, because

is just as much a function of flow rate and head rise

as it is of shaft speed. Perhaps a more general name, like “the basic perfor-

mance parameter”, would be more appropriate. Note that the specific speed is

a size-independent parameter, since the size of the machine is not known at the

beginning of the design process.

The above definition of the specific speed has employed a consistent set

of units, so that

is truly dimensionless. With these consistent units, the

values of

for most common turbomachines lie in the range between 0

1and

0 (see below). Unfortunately, it has been traditional in industry to use an

inconsistent set of units in calculating

. In the USA, the

is dropped from

the denominator, and values for the speed, flow rate, and head in

rpm

gpm

and

are used in calculating

. This yields values that are a factor of 2734

larger than the values of

obtained using consistent units. The situation is even

more confused since the Europeans use another set of inconsistent units (

rpm

,headin

, and no

) while the British employ a definition similar to the

U.S., but with Imperial gallons rather than U.S. gallons. One can only hope

that the pump (and turbine) industries would cease the use of these inconsistent

measures that would be regarded with derision by any engineer outside of the

industry. In this monograph, we shall use the dimensionally consistent and,

therefore, universal definition of

Note that, since

and

were separately nondimensionalized in the def-

initions 2.16 and 2.17,

can be related to the corresponding flow and head

coefficients by

cos

−

(2.23)

In the case of a purely centrifugal discharge (

π/

2), the quantity within the

square brackets reduces to 2

πB

Since turbomachines are designed for specific tasks, the subscripted

will

be used to denote the design value of the specific speed for a given machine.

2.5 PUMP GEOMETRIES

Since the task specifications for a pump (or turbine or compressor or other

machine) can be reduced to the single parameter,

, it is not surprising that

the overall or global geometries of pumps, that have evolved over many decades,

can be seen to fit quite neatly into a single parameter family of shapes. This

family is depicted in figure 2.6. These geometries reflect the fact that an axial

flow machine, whether a pump, turbine, or compressor, is more efficient at high

specific speeds (high flow rate, low head) while a radial machine, that uses the

centrifugal effect, is more efficient at low specific speeds (low flow rate, high

head). The same basic family of geometries is presented quantitatively in figure

2.7, where the anticipated head and flow coefficients are also plotted. While the

CHAPTER 2. BASIC PRINCIPLES

Figure 2.6: Ranges of specific speeds for typical turbomachines and typical pump

geometries for different design speeds (from Sabersky, Acosta and Hauptmann

1989).

existence of this parametric family of designs has emerged almost exclusively

as a result of trial and error, some useful perspectives can be obtained from

an approximate analysis of the effects of the pump geometry on the hydraulic

performance(seesection4.3).

Normally, turbomachines are designed to have their maximum efficiency at

the design specific speed,

. Thus, in any graph of efficiency against spe-

cific speed, each pump geometry will trace out a curve with a maximum at its

optimum specific speed, as illustrated by the individual curves in figure 2.8.

Furthermore, Balje (1981) has made note of another interesting feature of this

family of curves in the graph of efficiency against specific speed. First, he cor-

rects the curves for the different viscous effects which can occur in machines

of different size and speed, by comparing the data on efficiency at the same

effective Reynolds number using the diagram reproduced as figure 2.9. Then,

as can be seen in figure 2.8, the family of curves for the efficiency of different

types of machines has an upper envelope with a maximum at a specific speed

of unity. Maximum possible efficiencies decline for values of

greater or less

2.6. ENERGY BALANCE

Figure 2.7: General design guidelines for pumps indicating the optimum ratio

of inlet to discharge tip radius,

, and discharge width ratio,

for various design specific speeds,

. Also shown are approximate pump per-

formance parameters, the design flow coefficient,

, and the design head coef-

ficient,

(adapted from Sabersky, Acosta and Hauptmann 1989).

than unity. Thus the “ideal” pump would seem to be that with a design specific

speed of unity, and the maximum obtainable efficiency seems to be greatest at

this specific speed. Fortunately, from a design point of view, one of the spec-

ifications has some flexibility, namely the shaft speed, Ω. Though the desired

flow rate and head rise are usually fixed, it may be possible to choose the drive

motor to turn at a speed, Ω, which brings the design specific speed close to the

optimum value of unity.

2.6 ENERGY BALANCE

The next step in the assessment of the performance of a turbomachine is to

consider the application of the first and second laws of thermodynamics to such

devices. In doing so we shall characterize the inlet and discharge flows by their

pressure, velocity, enthalpy, etc., assuming that these are uniform flows. It is

understood that when the inlet and discharge flows are non-uniform, the analysis

actually applies to a single streamtube and the complete energy balance requires

integration over all of the streamtubes.