Hydrodynamics of Pumps

HYDRODYNAMICS

OF PUMPS

Christopher Earls

Brennen

OPEN

1994

Also available as a bound book from Concepts NREC, White River

Junction, VT

Published in 1994 by Concepts NREC and Oxford University

Press

ISBN 0-933283-07-5 (Concepts NREC)

ISBN 0-19-856442-2 (Oxford University Press)

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Contents - Hydrodynamics of Pumps

HYDRODYNAMICS OF PUMPS

Christopher Earls Brennen

Preface

Nomenclature

CHAPTER 1.

INTRODUCTION

1.1

Subject

1.2

Cavitation

1.3

Unsteady Flows

1.4

Trends in Hydraulic Turbomachinery

1.5

Book Structure

References

CHAPTER 2.

BASIC PRINCIPLES

2.1

Geometric Notation

2.2

Cascades

2.3

Flow Notation

2.4

Specific Speed

2.5

Pump Geometries

2.6

Energy Balance

2.7

Idealized Noncavitating Pump Performance

2.8

Several Specific Impellers and Pumps

References

CHAPTER 3.

TWO-DIMENSIONAL PERFORMANCE

ANALYSIS

3.1

Introduction

3.2

Linear Cascade Analyses

3.3

Deviation Angle

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3.4

Viscous Effects in Linear Cascades

3.5

Radial Cascade Analyses

3.6

Viscous Effects in Radial Flows

References

CHAPTER 4.

OTHER FLOW FEATURES

4.1

Introduction

4.2

Three-dimensional Flow Effects

4.3

Radial Equilibrium Solution: an Example

4.4

Discharge Flow Management

4.5

Prerotation

4.6

Other Secondary Flows

References

CHAPTER 5.

CAVITATION PARAMETERS AND INCEPTION

5.1

Introduction

5.2

Cavitation Parameters

5.3

Cavitation Inception

5.4

Scaling of Cavitation Inception

5.5

Pump Performance

5.6

Types of Impeller Cavitation

5.7

Cavitation Inception Data

References

CHAPTER 6.

BUBBLE DYNAMICS, DAMAGE AND NOISE

6.1

Introduction

6.2

Cavitation Bubble Dynamics

6.3

Cavitation Damage

6.4

Mechanism of Cavitation Damage

6.5

Cavitation Noise

References

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Contents - Hydrodynamics of Pumps

CHAPTER 7.

CAVITATION AND PUMP PERFORMANCE

7.1

Introduction

7.2

Typical Pump Performance Data

7.3

Inducer Designs

7.4

Inducer Performance

7.5

Effects of Inducer Geometry

7.6

Analyses of Cavitation in Pumps

7.7

Thermal Effect on Pump Performance

7.8

Free Streamline Methods

7.9

Supercavitating Cascades

7.10

Partially Cavitating Cascades

7.11

Cavitation Performance Correlations

References

CHAPTER 8.

PUMP VIBRATION

8.1

Introduction

8.2

Frequencies of Oscillation

8.3

Unsteady Flows

8.4

Rotating Stall

8.5

Rotating Cavitation

8.6

Surge

8.7

Auto-oscillation

8.8

Rotor-Stator Interaction: Flow Patterns

8.9

Rotor-Stator Interaction: Forces

8.10

Developed Cavity Oscillation

8.11

Acoustic Resonances

8.12

Blade Flutter

8.13

POGO Instabilities

References

CHAPTER 9.

UNSTEADY FLOW IN HYDRAULIC SYSTEMS

9.1

Introduction

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9.2

Time Domain Methods

9.3

Wave Propagation in Ducts

9.4

Method of Characteristics

9.5

Frequency Domain Methods

9.6

Order of the System

9.7

Transfer matrices

9.8

Distributed Systems

9.9

Combinations of Transfer Matrices

9.10

Properties of Transfer Matrices

9.11

Some Simple Transfer Matrices

9.12

Fluctuation Energy Flux

9.13

Non-cavitating Pumps

9.14

Cavitating Inducers

9.15

System with Rigid Body Vibration

References

CHAPTER 10.

RADIAL AND ROTORDYNAMIC FORCES

10.1

Introduction

10.2

Notation

10.3

Hydrodynamic Bearings and Seals

10.4

Bearings at Low Reynolds Numbers

10.5

Annulus at High Reynolds Numbers

10.6

Squeeze Film Dampers

10.7

Turbulent Annular Seals

10.8

Labyrinth Seals

10.9

Blade Tip Rotordynamic Effects

10.10

Steady Radial Forces

10.11

Effect of Cavitation

10.12

Centrifugal Pumps

10.13

Moments and Lines of Action

10.14

Axial Flow Inducers

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References

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Preface - Hydrodynamics of Pumps - Christopher E. Brennen

HYDRODYNAMICS OF PUMPS

Christopher Earls Brennen

Preface to the original hardback edition

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 research and the pump industry with whom it was my

pleasure to be associated, including Dara Childs, Paul Cooper, Nick Cumpsty, Jules

Dussourd, Tony Eastland, 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, Edmund 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 encouragement, love, and

companionship and that debt is beyond words.

Christopher Earls Brennen, California Institute of Technology.

July 1994

Preface to the Japanese translation

by Yoshinobu Tsujimoto

published by Osaka University Press

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Preface - Hydrodynamics of Pumps - Christopher E. Brennen

I am greatly honored that Prof. Yoshi Tsujimoto has chosen to prepare this Japanese

translation of ``Hydrodynamics of Pumps'' for he is a gentleman and a scholar who has my

greatest admiration. Ever since we first met some 20 years ago, Yoshi and I have enjoyed

a very valuable exchange of ideas and developed a deep mutual respect. Indeed, his

feedback was very important to me during the preparation of the original English edition.

Consequently, I am in the enviable position of knowing, with confidence, that this

Japanese edition will be an

improvement

on the original.

It also seems appropriate in this preface to acknowledge the major contributions which

Japanese scientists and engineers have made to our current understanding of the

hydrodynamics of pumps. In the modern era, we are all guided by the multitude of

seminal ideas of Hideo Ohashi and I would like to express my deep graditude and respect

to Professor Ohashi for the help he has given to so many younger engineers throughout

the world. Many other Japanese pump researchers have had an important influence on my

thinking and I would like to acknowledge, in particular, the help and inspiration given by

Kenjiro Kamijo, Kiyoshi Minemura, Okitsugu Furuya, Hiroharu Kato, Jun-ichi

Kurokawa, among others.

Finally I would like to express my gratitude to Concepts ETI, Inc., to Oxford University

Press and to Osaka University Press for their help in bringing the idea of this translation to

reality.

Christopher Earls Brennen, Pasadena, Calif.

August 1997

Preface to the Internet edition

Though my conversion of "Hydrodynamics of Pumps" from the hardback book to HTML

is rough in places, I am so convinced of the promise of the internet that I am pleased to

offer this edition freely to those who wish to use it. This new medium clearly presents

some advantages and some disadvantages. The opportunity to incorporate as many color

photographs as I wish (and perhaps even some movies) is a great advantage and one that I

intend to use in future modifications. Another advantage is the ability to continually

correct the manuscript though I will not undertake the daunting task of trying to keep it up

to date. A disadvantage is the severe limitation in HTML on the use of mathematical

symbols. I have only solved this problem rather crudely and apologize for this roughness

in the manuscript.

In addition to those whom I thanked earlier, I would like to express my thanks to my

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academic home, the California Institute of Technology, for the help in providing me the

facilities to effect this conversion and to Concepts NREC for their permission to place this

entire book on the internet.

Christopher Earls Brennen, Pasadena, Calif.

Oct.2003

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Nomenclature - Hydrodynamics of Pumps - Christopher E. Brennen

HYDRODYNAMICS OF PUMPS

Christopher Earls Brennen

Nomenclature

ROMAN LETTERS

Pipe radius

Cross-sectional area

ijk

Coefficients of pump dynamic characteristics

[A]

Rotordynamic force matrix

Cross-sectional area ratio

Breadth of passage or flow

[B]

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

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Young's modulus

Friction coefficient

Force

Acceleration due to gravity

Component of

in the

direction

Specific enthalpy

Blade tip spacing

Pitch of a helix

Total specific enthalpy

Piezometric head

Total head rise

H(s,

,t)

Clearance geometry

Acoustic impulse

I,J

Integers such that

Ω

=I/J

Pump impedance

Square root of

-1

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

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Coordinate measured normal to a surface

Specific speed

N(R

)

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

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

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[T]

Transfer matrix based on

[T*]

Transfer matrix based on

[TP]

Pump transfer matrix

[TS]

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

Fluid velocity in rotating frame

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

Efficiency

Angular coordinate

Camber angle

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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

{1,2,3}

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

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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

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,

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Mean value of

or complex conjugate of

Complex amplitude of oscillating

Time derivative of

Second time derivative of

Rotordynamics: denotes dimensional

{Q}

Real part of

{Q}

Imaginary part of

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Chapter 1 - Hydrodynamics of Pumps - Christopher E. Brennen

HYDRODYNAMICS OF PUMPS

Christopher Earls Brennen

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

boundary. 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 malevolent process,

and that the deleterious consequences can be divided into three categories. First, cavitation

can cause damage to the material surfaces close 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

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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 instabilities 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 interaction 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 like

ρΩ

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

ρΩ

, and, to minimize structural problems, this

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

) than would be advisable from a purely

hydrodynamic point of view.

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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. Recent 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

rotordynamic 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,

, and

the power absorbed by the pump,

, will scale according to

Ω

......

(1.1)

Ω

......

(1.2)

Ω

......

(1.3)

Ω

......

(1.4)

These simple relations allow basic scaling predictions and initial design estimates.

Furthermore, they permit consideration of optimal characteristics, such as the power

density which, according to the above, should scale like

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

D/2

. However, as we

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

the cavitation number decreases, and the cavitation number is inversely proportional to the

square of the tip speed or

Ω

. 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

ρΩ

, and, if

Ω

is fixed and if one maintains the same geometry,

, then the stresses will increase like

-4/3

as the size,

, is decreased. Consequently,

fluid/structure interaction problems will increase. To counteract this the blades are often

made thicker (

is 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 engineering 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 issues 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

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Chapter 1 - Hydrodynamics of Pumps - Christopher E. Brennen

presents a survey of some of the vibration problems in pumps.

Chapter 9 provides details

of the two basic approaches to the analysis of instabilites and unsteady flow problems in

hydraulic systems, namely the methods of solution 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 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., England

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.

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Chapter 2 - Hydrodynamics of Pumps - Christopher E. Brennen

HYDRODYNAMICS OF PUMPS

Christopher Earls Brennen

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

, R

, and

, respectively. The discharge blade

passage is inclined to the axis of rotation at an angle,

, which would be close to

90°

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,

<90°

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

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

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

v(r)

(with subscripts 1 and 2 denoting

particular values at inlet and discharge) and the corresponding velocity relative to the

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Chapter 2 - Hydrodynamics of Pumps - Christopher E. Brennen

rotating blades is denoted by

w(r)

. The velocities,

and

, have components

and

the circumferential 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

(r)

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

(r)

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 (

and

) and at the hub and tip (

and

) are denoted by the corresponding

suffices, so that, for example,

bT2

is the blade angle at the discharge tip.

Figure 2.2 Developed meridional surface and velocity triangle.

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Chapter 2 - Hydrodynamics of Pumps - Christopher E. Brennen

At the leading edge it is important to know the angle

(r)

with which the 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,

)

, by

and

respectively. Since the inlet flow can often be assumed to be purely axial (

(r)=v

and

parallel with the axis of rotation), it follows that

(r)=tan

-1

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.

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.

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

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