Sayers OCR

H i e flow

es A. T.. Sayers, BSc (E11g), MSc, PhD, C Eng, MIMechE Departl11el1t ofM echclnical Engineering University of Cape To"vn

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HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

Published by

McGRA W-HILL Book Company (UK) Limited Shoppenhangers Road, Maidenhead, Berkshire, England, SL6 2QL. Telephone Maidenhead (062H) 23432 Cables MCGRA WHILL MAIDENHEAD Telex 848484 Fax 0628 35895

British Library Cataloguing in Publication Data Sayers, A. T. (Anthony Terence), /946Hydraulic and compressible now turbomachines. 1. Tu rbomach inery I. Title 621.8 ISBN 0-07-707219-7 Library of Congress Cataloging-in-Publication Data Sayers, A. T., 1946Hydraulic and compressible now turbomachines / A. T. Sayers. p. em. Includes bibliographical references (p ISBN 0-07-707219-7 1. Turbomachines. I. Title H9-1368H TJ267.S26 1990 CIP 621.406-dc20 Copyright © 1990 McGraw-Hili Book Company (UK) Limited. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. 12345

93210

Typeset by Thomson Press (India) Limited, New Delhi and printed and bound in Great Britain by Page Bros (Norwich) Ltd

To HANNAH WILLIAM GILES

CONTENTS

Preface Symbols 1

2

xi xiii

Introduction 1.1 Definition 1.2 Units 1.3 Dimensional analysis 1.4 Prototype and model efficiency 1.5 Dimensionless specific speed 1.6 Basic laws and equations Exercises Solutions

1 1 4 4 11 12 16 21

lIydruulic punlps 2.1 Introduction 2.2 Centrifugal pUlnps 2.3 Slip factor 2.4 Centrifugal purnp characteristics 2.5 Flo\v in the discharge casing 2.6 Cavitation in pUlnps 2.7 Axial now IHlnlp 2.8 PUlnp and systcln 111atching Exercises Solutions

31 31

23

33 37

40 48 53 56 63 68

70 vii

viii

CONTENTS

3

Hydraulic turbines 3.1 Introduction 3.2 Pelton wheel 3.3 Radial flow turbine 3.4 Axial flow turbine 3.5 Cavitation in turbines Exercises Solutions

88 88 89 96 103 106 107 109

7

Centrifugal compressors and fans 4.1 Introduction 4.2 Inlet velocity limitations 4.3 Pre-whirl and inlet guide vanes 4.4 Mach number in the diffuser 4.5 Centrifugal compressor characteristic Exercises Solutions

131

References

296

Index

297

Axial now compressors and fans 5.1 Introduction 5.2 Compressor stage 5.3 Reaction ratio 5.4 Stage loading 5.5 Lift and drag coefficients 5.6 Blade cascades 5.7 Blade efficiency and stage efficiency 5.8 Three-dimensional flow 5.9 Multi-stage performance 5.10 Axial flow com pressor characteristics Exercises Solutions

180 180 182

4

5

6

Axial now steam and gas turbines 6.1 Introduction 6.2 Turbine stage 6.3 Stator (nozzle) and rotor losses 6.4 Reaction ratio 6.5 Effect of reaction ratio on stage efficiency 6.6 Blade types 6.7 Aerodynamic blade design 6.8 Multi-stage gas turbines Exercises Solutions

CONTENTS

131 138 141 141 142 146

148

187 188 189 191 199

200 . 202 204 206 208 227 227

229

232 236

239 240 244 245 247

249

Radial now gas turbines 7.1 Introduction 7.2 Velocity diagrams and thermodynamics of flow 7.3 Spouting velocity 7.4 Turbine efficiency 7.5 Dimensionless specific speed Exercises Solutions

ix

275

275 277 278

279 281

282 283

PREFACE

This book has arisen from a collection of lecture notes compiled for an undergraduate course in turbo machines. The subject of turbomachines covers both hydraulic and compressible flow machines, and, while many specialized books concerning specific types of machines are available, the author was not able to find a book covering both hydraulic and compressible flow machines at a suitable level. The book is aimed at the undergraduate and diploma student, and introduces ternlS and concepts used in turbolnachinery. It is not an exhaustive text on the subject, nor is it intended as a design text, although many design parallleters are given. It should rather be used as an introduction to the more spccialized texts. Thc book aSSUlllCS that the student has followed a course of basic fluid mechanics and thermodynamics, but mathelnatical manipulations are minilnal. Each chapter attcrnpts to be self-contained with regard to a particular type of machine, and it is suggested that the lecturer supplements the text with worked exalnples during the lecture and tutorial sessions, when the maxinlunl understanding of the subject is derived. To aid in this, some worked exercises are supplicd at the end of each chapter. In a book of this naturc, great patience is required in the typing and correcting of thc lnanuscript. In this regard I would like to convey my great appreciation to Mrs Iris von Bentheim and Mrs Lyn Scott, who jointly performed this task. For photographic reproduction, I would like to thank Mt Vernon Appleton, while for rnuch of the draughting of the diagrams, I thank nly wife, Susan, \vithout whose constant encouragelnent this textbook would not have been writtcn. Finally, nlY thanks must go to the many students who have over the years helped to correct the notes and tutorial problems during their undergraduate studies. A.T.S. xi

SYMBOLS

A a

b C Co COA CDC CDS CL

Cp Cs Cv c

D

dH E e F 9

H

h

I

area velocity of sound depth or height of blade absolute velocity drag coefficient annulus drag coefficient tip clearance drag coefficient secondary loss drag coefficient lift coefficien t specific heat at constant pressure spollting velocity nozzle velocity coefficient blade chord diameter drag force hydraulic diameter power per unit \veight of flow radius of inscribed circle force gravitational acceleration head head loss specific enthalpy blade height impeller constant incidence angle

m2

m/s m

m/s

J/kg K m/s

m N m W/{N/s) m

N

rnls 2 m m

J/kg rn J/kg deg xiii

xiv

SYMBOLS XV

SYMBOLS

L

I M n1

N NPSH Ns

N sp P

p

p

Q q R

Re r

SP s T

U

u V v

W y

Z

lift force length span Mach number mass mass flow ra te nun1ber of stages speed net positive suction head dilTIensionless specific speed dilTIensionless po\ver specific speed power power coefficient pressure heat transfer to systel11 VOIUlTIe flow rate specific heat transfer leakage now rate radius eye tip radius gas constant Reynolds nUlnber radius eye hub radius specific speed blade pitch specific entropy temperature tin1e torque tangential velocity internal energy per unit n1ass velocity volume velocity \\'ork done per second relative velocity pressure loss coefficient nUlnber of blades elevation or potential head

P

N n1 n1

r

kg kg/s rpl11 111

rev or rad rev or rad

\V

~-...

y lJ

:r

e

(

'1 (J

).

11

p (J

Pa (Je

W 3

js Jjkg In 3 js 111

(Js

¢ ljJ

In 111

Jjkg K

w

In 111

Subscripts

radjs 111

Jjkg K K s N nl ITIjS Jjkg n1js n1 3 Injs W

Injs

a

atm av b c D d f fp g H h

In nl

Greek (J.

P'

angle of absolute velocity vector angle of relative velocity vector

blade angle circulation ratio of specific heats deviation deflection enthalpy loss coefficient stagger angle efficiency camber angle work done factor absolute viscosity density cavitation paralTIeter solidity ratio critical ca vi tat ion paralneter slip factor flow coefficient power input factor stage loading coefficient head coefficient angular velocity

dcg dcg

N nom

axial atlTIosphcric average blade casing compressor design concli tion draft tube friction pipe friction gate hydraulic hub impeller inlet leakage at nlean dialTIeter 111echanical nozzle norninal

deg m 2 /s deg deg deg deg Pa s kg/m 3

radjs

·:

~~:

xvi

SYMBOLS

0

p R

r r reI s

suc

t-s t-t u v vap x

outlet overall total (stagnation) conditions projected area rotor at tip radial runner at hub based on relative velocity isentropic shaft static suction tip turbine total-to-static total-to-total unit quantities volumetric vapour tangential

CHAPTER

ONE INTRODUCTION

1.1 \DEFINITION A turbomachine can be described as any device that extracts energy from or imparts energy to a continuously moving stream of fluid, the energy transfer being carried out by the dynamic action of one or more rotating blade rows. The dynamic action of the rotating blade rows sets up forces between the blade row and fluid, while the components of these forces in the direction of blade motion give rise to the energy transfer between the blades and fluid. By specifying that the fluid is moving continuously, a distinction is drawn between the turbomachine and the positive displacement machine. In the latter, the fluid enters a closed chamber, which is isolated from the inlet and outlet sections of the machine for a discrete period of time, although work may be done on or by the fluid during that time. The fluid itself can be a gas or a liquid, and the only limitations that we shall apply are that gases (or steam) are considered perfect and that liquids are Newtonian. The general definition of the turbomachine as used above covers a wide range of machines, such as ship propellers, windmills, waterwheels, hydraulic turbines and gas turbines, and is therefore rather loose for the purposes of this text. We will limit ourselves to a consideration of only those types of turbomachines in which the rotating member is enclosed in a casing, or shrouded in such a way that the streamlines cannot diverge to flow around the edges of the impeller, as would happen in the case of an unshrouded windmill or aerogenera tor. The types of machines falling into our defined category and which will be considered in detail in succeeding chapters are listed in Table 1.1 and fall into

2

HYDRAULIC AND COMPRESSIULE FLO\V TURUOMACIlINES

Table 1.1 Types of turbolnachincs Turbomachines in which \Vork is done by lluid

\Vork is done

Axial llow hydraulic turbinc

Centrifugal pump

Oil

lluid

Radial llow hydraulic turbinc

Axial llow pump

Mixed llow hydraulic turbine

Ccntrifugal comprcssor

Axial llow gas turbine

Axial llow comprcssor

Pelton wheel hydraulic turbine

Radial llow fan

one of t\VO classes depending on whether \vork is done by the fluid on the rotating member or whether work is done by the rotating 111elnber on the fluid. Types of turbolllachines can also be defined as to the l11anner of fluid I1l0Vement through the rotating Inelnber. If the flow is essentially axial with no radial movement of the strealnlines, then the machine is classed as an axial flow I1lachine; whereas if the flow is essentially radial, it is classed as a radial flow or centrifugal machine. Other special types of turbolllachines exist, e.g. the Minto \vheel or Baki turbine, but they will not be considered in this text. Considering the two classes of Inachines listed in Table 1.1, some broad generalizations l11ay be rnade. The first is that the left-hand colulnn consists of machines in which the fluid pressure or head (in the case of a hydraulic machine) or the enthalpy (in the case of a cOlnpressible flo\v nlachine) decreases from inlet to outlet, whereas in the right-hand COIUlllI1 are listed those machines which increase the head or enthalpy of the fluid flowing through them. This decrease or increase in head, \vhen 111ultiplied by the weight flow per unit tilne of fluid through the Inachine, represents the energy absorbed by or extracted frolll the rotating blades, \vhich are fixed onto a shaft. The energy transfer is effected ill both cases by changing the angular momentUI11 of the fluid. It might therefore be reasonable to assume that ditTerent types of turbolnachine would exhibit differing shapes of blades and rotating 1l1enlbers, and this indeed is the casc, as is S110\Vn in Fig. i.1. In addition, because turbomachines have developed historically at different times, nan1es have been given to certain parts of the l11achincs as well as to different types of lnachines, and these are no\v dcfined. Turbine. A nlachine that produces power by expanding a continuously flowing fluid to a lower pressure or head; the po\ver output is usually expressed in kW. PUI1Zp. A n1achine that increases the pressurc or head of a flowing liquid, and which is usually expressed in k Pa or 111.

Centrifugal compressor

Kaplan turbine

Radial flow fan

Francis turbine

Centrifugal pump

Axial flow pump

Steam turbine

Axial flow compressor Figure 1.1 Types and shapes of turhomachines

3

4

HYDRAULIC AND COMPRESSlilLE FLOW TURBOMACHINES

Fan. A term used for machines imparting only a small pressure rise to a continuously flowing gas, usually with a density ratio across the machine of less than 1.05 such that the gas may be considered to be incompressible; pressure increase is usually expressed in mm of water. Compressor. A machine imparting a large pressure rise to a continuously flowing gas with a density ratio in excess of 1.05. Impeller. The rotating member in a centrifugal pump or centrifugal compressor. Runner. The rotating member of a radial flow hydraulic turbine or pump. Rotor. The rotating member of an axial flow gas or steam turhine; sometimes called a disc. Diffuser. A passage that increases in cross-sectional area in the direction of fluid flow and converts kinetic energy into static pressure head; it is usually situated at the outlet of a compressor. Draught tube. A difTuser situated at the outlet of a hydraulic turbine. Volute. A spiral passage for the collection of the diffused nuid ora compressor or pump; in the hydraulic turbine the volute serves to increase the velocity of the fluid before entry to the runner.

1.2 UNITS The units used will be those of the Sf system, the basic units being the kilogram, metre and second.

INTRODUCTION

Q

p

I

[

1-:- --"----; I

t

~

I

I gIll ,

1_-_-_-_---,,- - -

)

-1

N

ffilIffillILIITIIl I

:

t_~ - - -7-----Control surface

5

Flow rate. Q (mJ/s) Speed, N (rev/s) Power, P (W) Energy difference across turhine, fI H (N mlkg) Fluid density, p(kg/m J ) Fluid viscosity, ,u(Pa s) Diameter, f) (m)

,- 2-_:

!

Q Figure 1.2 Hydraulic turhine control volume

volume represents a turbine or diameter D, which develops a shaft power P at a speed of rotation N, then \ve could say that the power output is a function of all the other varia bles, or p= f(p,N,jl,D,Q,({j//))

(1.1)

In Eq. (1.1), f means a function of' and g, the acceleration due to gravity, has 4

be~n co.mbined with /-/ to form the energy per unit mass instead of energy per

1.3 DIMENSIONAL ANALYSIS The large number of variables involved in describing the performance characteristics of a turbomachine virtually demands the use of dimensionless analysis to reduce the variables to a number of manageable dimensional groups. Dimensional analysis also has two other important uses: firstly, the prediction of a prototype performance from tests conducted on a scale model; and secondly, the determination of the most suitable type of machine on the basis of maximum efficiency for a specified range of head, speed and flow rate. Only a brief description of the method used for forming dimensionless groups and their application to model testing for turbomachines will be given here, the generalities of the subject usually being covered in a course of fluid mechanics.

unIt weIght. We now assume that Eq. (1.1) may be written as the product of all the variables raised to a power and a constant, such that (1.2) If each variable is expressed in terms of its fundamental dimensions, mass M, length L and time T, then, for dimensional homogeneity, each side of Eq. (1.2) must have the same powers of the fundamental dimensions so the indices of M, Land T can be equated to form a series of simul;aneous equations. Thus 2

(ML /T 3 ) = const(M/L 3 t(1/T)h(M/LTY(L)d(L 3 /TY(L 2 /T 2 )f and equating the indices we get M

1.3.1 Hydraulic Machines Figure 1.2 shows a control volume through which an incompressible fluid of density p flows at a volume flow rate of Q, which is determined by a valve opening. The head difTerence across the control volume is H, and if the control

(1.3)

L T

l=a+c

2 = - 3a - c + d + 3e + 2f -3= -b-c-e-2f

There are six variables and only three equations. It is therefore possible to solve for three of the indices in terms of the remaining three. Solving for a, b

6

HYDRAULIC AND COMPRESSlBLI: F!.O\V TURIHJMACIIINI:S

and d in tenns of c, e and

I

INTRODUCTION

7

\ve get (/==I-c /) ==

3- c- e-

2.r

d == 5 - 2c - 3e - 2./

Substituting for

Q,

P/f)N:'D~)

band d in Eq. (1.2),

P == const[pl-CN3-c-e-2f/l"Ds-2c-3e-2fQt'(OJ-I)f]

and collecting like indices into separate brackets, P == canst [(pN 3 DS ) {jLI plY D2y( QIN D 3 y(O I I I N 21)2)f]

(1.4)

The second term in the brackets will be recognized as the in verse of the Reynolds number and, since the value of c is unknown, this term can be inverted and Eq. (1.4) may be \vritten as PlplV 3D s = const[(pND 2jpY(QIND 3)t!(g}IIN 2D 2)f] (1.5) Each group of variables in Eq. (1.5) is truly dimensionless and all are used in hydraulic turbomachinery practice. Because of their frequent use, the groups are known by the following nan1es: PjpN 3 DS == j5 3

QIND ==¢ gHIN 2 D2 == l/J

the power coefficient the OO\V cocfficient the head coefficient

2

The tern1 pN D IlL is equivalent to the Reynolds nun1ber Re == p VDIIL, since the peripheral velocity V is proportional to N D. Hence Eq. (1.1) rnay be rewritten as

P ==

f(Re, ¢, l/J)

(a)

Figure 1.3 Performance characteristics of hydraulic machines drawn in terms of dimensionless groups: (a) hydraulic turbine; (b) hydraulic pump

characteristics of any other combination of P, N, Q and H for a given machine or for any other geon1etrically similar machine of different diameter. Since these groups are dilnensionless, they may be divided or multiplied by themselves to forn1 other dimensionless groups depending on the type of test being carried out, and it therefore follows that while in this particular case solutions for Q, band d were found in terms of c, e and f, other solutions could have been determined which give different dimensionless groups. Each set of groups taken together is correct, although they will of course be related by differently shaped curves. For the turbine, the hydraulic efficiency is defined as Power delivered to runner Power available to runner

tl == ------------------- ----- ----

(1.6)

which states that the power coefficient of a hydraulic 1l1achine is a function of Reynolds nUlnber, flow coefficient and head coefficient. I t is not possible to say what the functional relationship is at this stage, since it n1ust be obtained by experiment on a particular prototype machine or model. In the case of a hydraulic lnachine, it is found that the Reynolds nurnber is usually very high and therefore the viscous action of the fluid has very little effect on the power output of the machine and the power coefficient ren1ains only a function of t/J and ¢. To see how j5 could vary with ¢ and l/J, let us rcturn to Fig. 1.2. To determine the relationship bet\vecn P, l/J and 4), the head across the machine can be fixed, as is usually the case in a hydroelectric installation. For a fixed value of inlet valve opening, the load on the Illachinc is varied while the torque, speed and Oow ratc are Ineasured. Froln these lneasurelnents, the power may be calculated, and P and ¢ plotted against l/J. Typical dimensionless characteristic curvcs for a hydraulic turbine and pUIUp are shown in Figs 1.3a and 1.3b, respectively. These curves are also the

(b)

(1.7)

== PlpgQH

Then substituting for P and rearranging gives tl == P(lv D 3IQ)(N 2D 2IgH) ==

PI¢l/J

(1.8)

For a pump (1.9)

1.3.2 Model Testing Many hydraulic machines are so large that only a single unit might be required, as for example a hydraulic turbine in a hydroelectric installation producing many megawatts (MW) of power. Therefore, before the full-size

8

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACIIINES

INTRODUCTION

machine is built, it is necessary to test it in model forn1 to obtain as much information as possible about its characteristics. So that \ve may accurately transpose the results obtained from the model to the full-size machine, three criteria ~ust be met. The first is that the nlodel and prototype must be geometrically similar; that is, the ratio of all lengths bet ween the model and prototype must be the sarne. The second requirement is that of kinematic similarity, where the velocities of the nuid particles at corresponding points in the model and prototype must be related through a fixed ratio. The third requirement is that of dynamic similarity, where the forces acting at corresponding points must be in a fixed ratio between rnodel and prototype. For a geometrically similar model, dynamic similarity implies kinematic similari ty. In order to ensure the above criteria, the values of the dimensionless groups in Eq. (1.5) must remain the same for both the model and the prototype. Therefore if the curves shown in Fig. 1.3 had been obtained for a completely similar model, these same curves would apply to the full-size prototype machine. It can then be seen that these curves apply to any size machine of the same family at any head, now rate or speed.

Pill

T

9

T

POI

02s

02

01

(b)

(a)

L·

Figure 1.4 Compression and expansion in compressible now machines: (a) turbine; (b) compressor

p = p/RT and it therefore hecomes supernuous since we already have T and P as variables, so deleting density, and cornbining R with T, the functional

relationship can be written as {'02

=

f{po l ' RTo l ' R T o2 , 111, N, D, II)

and writing P02 as a product of the terms raised to powers,

1.3.3 Compressible Flow Machines

P02

Not all turbomachines use a liquid (hydraulic nuid) as their nuid medium. Gas turbines and axial now compressors are used extcnsivcly in the jet engines of aircraft where the products of combustion and air respectively are the working fluids, while many diesel engines use centrifugal corn pressors for supercharging. To accommodate the compressibility of these types of nuids (gases), some new variables must be added to those already nlcntioned in the case of hydraulic machines, and changes must be made in some of the definitions used. With compressible now machines, the parameters of importance are the pressure and tempcrature increase of the gas in a compressor and the pressure and temperature decrease of the gas in the turbine plotted as a function of the mass flow rate of the gas. In Fig. 1.4, the T -s charts for a compression and expansion process arc shown. In isentropic flow the outlet conditions of the gas are at 02s whereas the actual outlet conditions arc at 02. The subscript 0 refers to total conditions and 1 and 2 refer to the inlet and outlet points of the gas respectively. The s refers to constant entropy. Now the pressure at the outlet, P02' can be written as a function of the following variables: (1.10) Here the pressure ratio P02/POI replaces the head H in the hydraulic machine, while the mass flow rate m (kg/s) replaces Q. However, by examining Eq. (1.10) we can see that, using the equation of state, the density may be written as

= const [(PO 1Y'{RTo1 t(RTo2 r(nl)d(Nt(D)f(jl)gJ

(1.11)

Putting in the basic dimensions (M/L T 2 )

= consl [(MIL T 2 y'(L 2 /T 2 )h(L 2 /T 2 Y-(M/T)d( l/Tt(L)f{M/L T)g]

Equating indices

1 =a+d+u

M

- 1 = - a + 21> + 2c + j' - 0 - 2 = - 2a - 21> - 2c - d - e - g

L T

and solving for a, hand .1' in terms of d,

C, l!

and g we obtain

a=l-d-g h = d/2 - c - e/2 f = e - 2d - g Substitute for

G,

band

f

+ 0/2

in Eq. (1.11), then

P02 = const [P6t d- 9{R To 1 )d /2 -c - el2 + g/2(R T o2 Ynl dN eDe - 2d - g jlgJ

= const X

X POI

{(RTo2/RToIY[m(RTol)1/2/POID2Jd[ND/(RTol)1/2Je

[,u(RToI) 1/2/PO I DJg}

(1.12)

Now if the last term in the brackets in Eq. (1.12) is multiplied top and bottom by (RT~I)1/2 and noting that POl/RTol equals POI' then jlRTol/POl {RTo1 )1/2D = jL/(RTol)1/2pOlD

10

INTRODUCTION

HYDRAULIC AND COMPRESSIBLE FLO\V TURBOMACIIINLS

But the units of (R To 1 )1 /2 are LIT, \vhich is a veloci ty, and therefore the last term in brackets is expressible as a Reynolds nUlnber. T'hus the functional relationship may be written as P02/PO 1

== f((R T 02 / R To 1)' (lll(R To 1) 1/2 /po 1D2 ), (N 1)1(/~ To 1)1/2), He) (1.13)

The exact form of the function (1.13) must be obtained by experimental measurements taken from model or prototype tests. For a particular nlachine using a particular fluid, or for a I110del using the SaI11e fluid as the prototype, R is a constant and may be eliminated. The Reynolds nUlllbcr is in Inost cases so high and the flow so turbulent that changes in this panuneter over the usual operating range may be neglected. However, where large changes of density

surgelin0

O.9~

~

take place, a significant reduction in Re can occur, and this must then be taken into account. For a particular constant-dianleter Inachine, the diameter D may be ignored and therefore, in view of the above considerations, function (1.13) becomes (1.14)

where it should be noted that some of the terms are now no longer dimensionless. It is usual to plot P02/PO 1 and T 02 /To 1 against the mass flow rate parameter I1l rb il2 /POI for different values of the speed parameter N/T6 1/ for a particular machine. But for a family of machines, the full dimensionless groups of Eq. (1.13) 1l1Ust be used if it is required to change the size of the machine or the gas contained. The term ND/(RT01 )1/2 can be interpreted as the Mach-number effect. This is because the impeller velocity V ex N D and the acoustic velocity a OI ex (RTo d l/2 , while the Mach number M == V/a Ol ' Typical performance curvcs for an axial flow compressor and turbine are shown in Figs 1.5 and 1.6. r

1.4 PROTOTYPE AND MODEL EFFICIENCY Increasing N / T~i2

I

L------'

Before leaving this introduction to the use of dimensionless groups, let us look at the relationship between the efficiency of the model and that of the prototype, assunling that thc similarity laws are satisfied. We wish to build a Il10del of a prototype hydraulic turbine of efficiency YIp' Now froln siInilarity laws, denoting the model and prototype by subscripts m and p respcctively,

(b)

(a)

Figure 1.5 Axial flow compressor characteristics: (a) pressure ratio; (b) efficiency

\

J

; \ ~

. IncreaslI1g N /T~i2

\~ (a)

H p/(N pD p )2 == fl m /(N m DnJ2

or

Hp/H m== (N p /N m )2(D p /D m )2

Qp/NpD~ == QIll/NmDI~l

or

Qp/Qm

Pp/N~D~

or

Pm/P p == (N m /N p )3(D m /D p )5

==

Pm/NI~lD~l

==

(N p /N m )(D p /D m )3

Now

Choking

~I

11

Choking mass flow

fT"' Power transferred from fluid ' T ur b Inc e IClcncy == - - - - - - - - - - Fluid power available

/

,

== P/pgQH

:I :I :I :/ I

!

(1.15)

Therefore 11m/ l 1p == (P mI P p)(Qp/Qm)(H p/ H m) == 1

r (b)

Figure 1.6 Axial flow gas turbine characteristics: (a) pressure ratio; (b) efficiency

and the efficiencies of the model and prototype are the same providing the similarity laws are satisfied. In practice, the two are not the same due to scaling eITects, such as relative surface roughness, slight Reynolds-number changes and Mach-number effects at higher blade speeds.

12

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACIIINES

INTRODUCTION

1.5 DIMENSIONLESS SPECIFIC SPEED We have seen in Sec. 1.3 that the curves showing the functional relationship between dimensionless groups for a particular machine also apply to machines of the same family (similar design), providing the similarity laws are obeyed when changing to a smaller-diameter machine, at perhaps a difTerent speed and head. It is therefore possible to obtain curves of many difTerent types of machines, and to use these curves to select a machine design for a particular operating requirement. Typical curves that might be obtained for difTerent types of hydraulic pumps are shown in Fig. 1.7, where it is seen that each machine type lies in a well-defined region of head and flow coefficients, it being possible in some cases to choose two or more impeller types for a specific flow coefficient. There are of course an infinite number of designs that could be produced, but for each design only one point exists on its characteristic curve where the efficiency is at a maximum. Thus for each design of pump unique values of ¢ and t/J exist at the maximum efficiency point. In the case of turbines, the unique values would be P and ¢ at maximum efficiency. The specifications for a pump design are usually expressed in terms of a required head H, at a flow rate of Q and speed N, the speed being specified since motors are usually only available in fixed speed intervals. No mention has been made concerning the diameter or type of machine, both of which must be determined. For the best design point, constant values of (Po and t/JD will exist corresponding to the maximum efficiency point, or and

If the diameter is elinlinated from these t\vo eq uations, then

D = {Ull/l/If))II2/N

and

(/)0

or

= QN 2 {t// O /[]H)3 /2 ( 1.16)

I

0

I

1.

1

I

i

1-

I

0.91-

I

> 0.63

O1

3

/s

0.8

~

;;.. u

c

0

'u H: w

0'7~ I

t/J 0 = ~J Ii/ N 2 D 2 0.6

Radial 0.5

0.4 '--'--_--'--_L-I- - l . . 1 - J - - ' - - - - - - - - L - - _ - - - L - _ - - L - - - - L - - . J _ . . L - . . . L - _ L J 0.2 0.3 0.4 0.5 2 4

Axial

Dimensionless specifIc speed, N,! (rad)

Centrifugal Fi~ure

1.7 Characteristic curves for various pump designs

13

Mixed flow

Figure 1.8 Variation of hydraulic pump impeller design

Axial

14

HYDRAULIC AND COMPRESSIBLE FLOW' TURBO~tA(,lllj\;FS

N s is known as the dimensionless specific speed, the units being revolution or radians depending on the units of IV, and I11Ust nol be confused \vith specific speed. Since D was eliminated at the InaxiiTIul11 efficiency point, the dil11ensionless specific speed acts as a design paralneter, which indicates the type of machine that should be used for the given N, Hand Q. Equation (1.16) shows that a pun1p with a high N s will have a low head and high flow rate, and implies an axial flow pUlnp \vith a large s\vallo\ving capaci ty. 1\ hnv IV s ilnplies a high head and low flow rate, and a centrifugal type of pUlllp. Figure I.g shows the variation of N s with pUlnp in1peller type, and indicates the optil11UIn efficiencies to be ex pected. In practice, N s is often expressed as NQ l 12/H 3 /-+, the y being dropped since it is a constant, and the resulting value of N s will therefore be ditTerent. It may also be found that consistent sets of units are not always llsed for N, Q and H, so that when a value of IV s is expressed, it should be ensured that the definition being used is kno\vn. In this text the SI systelll \vill be llsed and N s will therefore be dimensionless. However, as a point of reference, conversion factors arc listed in Table 1.2 so that the reader n1ay calculate the din1ensionless specific speed froln specific speeds using Q, IV and H in other units. The fluid contained is water and, where quoted, gpIn are US gallons per 111inute, ft is foot, cfs arc cubic feet per second, and hp is horsepower. Tenns that are often used in hydraulic OO\V lnachines are those of unit head, unit speed, unit po\ver and unit quantity. They arose fr0l11 the need to be able to COInpare hydraulic nlachines tested under a set of standard conditions. In turbine work, the speed, power output and flow rate are detennined for a turbine operating under an assul11ed unit head of say I In or 1ft, its efficiency renlaining constant. For instance, consider a turbine tested under a head HI and speed N 1 rpn1. Then fron1 Eg. (1.6), for any other speed and head, fIlllvi

==

INTRODUCTION

Putting H 2

==

1 (unit head) then N 2 == NIl J-/ ~ /2 == N u

(1.17)

and this is the unit speed of the turbine. Unit quantities for Q and P may be similarly obtained to give

(1.18)

and

(1.19) For a turbine, the din1ensionless specific speed is found by a procedure similar to that for pUlnps except that D is eliminated from P and t/J to yield what is often referred to as the power specific speed, N sp , where N sp == N P l /2I p l/2(gH)5/4

(1.20)

Figure 1.9 shows typical hydraulic turbine runner shapes for difTerent specific speeds along with their optin1um or design efficiencies. A wide range of rotor designs from low to high values of specific speed for both hydraulic and cOInpressible flow machines are shown in Figs 1.10 and 1.11, where it will be noted that low-specific-speed machines have large diameters and high-specific-speed machines have small diameters. In general,

~ ~ 0.6

1.1

!7::J 1.6

0.98

Pelton wheel

H 21lv ~

0.94

I

or

. \ . FranCIS turbmes

~

~

g

\

Axial flow turbines

0.90

v

T>

~

(J.J

Table 1.2 Conversion factors for specific speed Specific speed

Dimensionless specific speed. N s (rad)

SP\ = SP 1 = SP 3 =

rpm(cfs)\j2/ftJ!~

Ns

rpm(mJ/s)\·l/mJ·~

N~=SP2/5J

rpm(gpm)\!2/ftJ·~

SP~ = rpm(hp)\/2/fl~:~

SP s =

rpm(m~tric hp)\/2/1ll~,4

=

N~ =

Ns Ns

= =

SI\/129 SP 3/2730 SP ~/42 Sl\/I X7

15

0.82

o

2 3 Dimensionless specific speed. N. (rad)

4

Figure 1.9 Variation of hydraulic turbine runner design with dimensionless specific speed

16

HYDRAULIC AND COMPRESSInLE FLOW TURnOMACHINES

0.05

0.10

0.20

Ns (rad) 1.0

I

I ..I - - - - - - - - - - + i - I

Pelton wheel -I multi-jet Centrifugal pumps

I-

5.0

2.0

Francis turbines slow normal fast

Pelton wheel single jet I

I-

0.50

I..

10.0

20.0

Kaplan -I turbines

PI~ope~l~r

turbines Mixed flow Propeller ·1' .1 1 -I (radial flow) pumps pumps Radial compressors II 'I and fans Axial flow compressors, blowers

'I

Axial flow

_\

(I)

and ventilators

steam and gas turbines

Figure 1.10 Correlation or rotor designs with dimensionless specific speed (collrtesy Wyss Ltd)

(~r

Escher

(4)

(5)

(6)

the smaller the diameter the lower will be the cost orthe rnachine, and therefore the design usually aims for the highest possible specific speed.

1.6 BASIC LAWS AND EQUATIONS The basic laws of thermodynamics and fluid mechanics are used in turbomachines although they are usually arranged into a more convenient form. All or some may be used under any set of circumstances and each will be brieny dealt with in turn.

(9)

(8)

1.6.1 Continuity For steady flow through the control volume, the mass now rate m remains constant. Referring to Fig. 1.12, (1.21) where the velocity vectors C 1 and C 2 are perpendicular to the cross-sectional areas of flow A 1 and A 2' In compressible now nlachines the mass now (kg/s) is used almost exclusively while in hydraulic machines the volume now rate Q (m 3 Is) is preferred.

1.6.2 Steady Flow Energy Equation (First Law of Thermodynamics) For steady flow through a system control volume, where the heat transfer rate to the system from the surroundings is Q and the work done by the system on

(10) (I) Pelton wheel

(2) Pelton wheel (3)- Francis turbine (4) Steam turbine (5) Centrifugal pump (6) Radial compressor

(11 )

(12)

N,. 0.05 (7' Steam turbine 0.13 (8) Steam turbine 0.38 (9) Centrifugal pump 0.4 (10) Gas turbine 0.54 (t I) Radial compressor 0.54 (t 2) Axial compressor

Nfl,

0.54 1.07 1.07 1.18 1.34 1.6

Figure 1. t t Some rotor designs and their dimensionless specific speeds 17

INTRODUCTION 19

YL

x

(14)

(13)

Control volume

/ Figure 1.12 Control volume for linear momentum

the surroundings is W, then (16)

(17)

(18)

Q - W = In[(P2jP2 - PI/PI) + (C~ - Ci)j2 + g(Z2 - Z 1) + (U 2 - u1)J

(1.22)

where pjp = pressure energy per unit mass (Jjkg), C 2 /2 = kinetic energy per unit mass (Jjkg), u == internal energy of the fluid per unit mass (J/kg), gZ = potential energy per unit mass (J/kg), m = mass flow rate (kgjs), W = work done on surroundings ( + ve) (W) and Q = heat transfer to system ( + ve) (W). In words, Eq. (1.22) states that in steady flow through any region:

(21)

(20)

(19)

Heat added to Shaft work done Increase in Increase in fluid per unit - by the fluid per = pressure energy + kinetic energy mass unit mass per unit mass per unit mass Increase in

Increase in

+ potential energy + internal energy per unit mass

Ns (13) (14) (15) (16) (17) (18)

Steam turbine Francis turbine Francis turbine Mixed flow pump Kaplan turbine Axial cOInpressor

Figure 1.11 18

(collld)

The steady flow energy equation applies to liquids, gases and vapours as well as to real fluids having no viscosity. It may be simplified in many cases because many of the tcrms are zero or cancel with others, and this will be shown in the rclevant scctions.

(23)

(22)

1.6 1.72 2.14 2.14 2.41 2.41

N8

( I tJ) (20) (21) (22) (23)

Axial compressor Propeller pump Axial blower Propeller pump Kaplan turbine

per unit mass

3.21 3.21 4.82 5.36 5.36

1.6.3 Newton's Second Law of Motion This law states that the SUIll of all the forces acting on a control volume in a particular direction is cq ual to the rate of change of In0I11entum of the fluid across the control YOIUI11C. With reference to Fig. 1.12, Newton's second law may be written as (1.23)

20

INTRODUCTION

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

21

causes the power developed by a turbine to be less than the ideal isentropic power developed and why the work input to a pump is greater than the isentropic or ideal work input (Fig. 1.4). I n theory the entropy change might also be zero for an adiabatic process b~t it is impossible in practice. For a reversible process the second law is expressed as dqjT = f1s

---.--

(1.26)

where dq = heat transfer per unit mass (Jjkg), T = absolute temperature at which heat transfer occurs (K) and f1s = entropy change (Jjkg K). In the a bsence of motion, gravity and any other effects, Eq. (1.22) has no potential or kinetic energy terms, and so or

dq - dw

= du

where the units are Jjkg. Substituting for dq and rearranging, Figure 1.13 Control volume for angular momentum

f1s = dqjT = (dll

Equation (1.23) applies for linear momentum. However, turbomachines have impellers that rotate, and the power output is expressed as the product of torque and angular velocity, and therefore angular momentum is the more useful parameter. Figure 1.13 shows the movement of a fluid particle from a point A to a point B while at the same time moving from a radius r 1 to '2' If the tangential velocities of the fluid are C x1 and C x2 respectively, then the sum of all the torques acting on the system is equal to the rate of change of angular momentum,

IT = m('2

C x2

-'1 C

If the machine revolves with angular velocity

ITw

=

m(U 2 C x2

-

x .)

(JJ

then the power is

U l C x.)

+ dw)jT

Putting dw = p dv, where dv is an infinitesimal specific volume change, then T ds Defining specific enthalpy as It = II

=

dll

+ Pdv

(1.27)

+ pv and substituting for du in Eq. (1.27),

T ds = dh - v dp

( 1.28)

and this equation IS used extensively in the study of compressible flow machines. Tn the following chapters, lise will be nlade of the concepts discussed in this introduction. This chapter should have acted as a reminder of the many separate concepts learned in thermodynamics and fluid mechanics, and has shown how these two separate subjects combine to form the subject of turbomach inery.

For a turbine W=m(U 1 C x1

-

U 2 C x2 »O

(1.24)

and is known as Euler's turbine equation. For a pump W = m(U 2Cx2

-

U 1 C x .) > 0

(1.25)

which is Euler's pump equation.

1.6.4 Entropy (Second Law of Thermodynamics) This law states that, for a fluid undergoing a reversible adiabatic process, the entropy change is zero, while for the same fluid undergoing an adiabatic or other process, the entropy increases from inlet to outlet. It is this fact that

EXERCISES 1.1 A radial now hydraulic turbine is required to be designed to produce 30 MW under a head of 14 m at a speed of95 rpm. A geometrically similar model with an output of 40 kW and a head of 5 m is to be tested under dynamically similar conditions. At what speed must the model be run, what is the required impeller diameter ratio between the model and prototype and what is the volume now rate through the model if its efficiency can be assumed to be 90 per cent? 1.2 The performance curves of a centrifugal pump are shown in Fig. 1.14. The impeller diameter is 127 mm and the pump delivers 2.831/s at a speed of 2000 rpm. If a 102 mm diameter impeller is fitted and the pump runs at a speed of 2200 rpm, what is the volume now rate? Determine also the new pump head. 1.3 An axial now compressor is designed to run at 4500 rpm when ambient atmospheric conditions are 101.3 kPa and 15"C. On the day when the performance characteristic is obtained, the atmospheric temperature is 2Y'C. What is the correct speed at which the compressor must run?

22

HYDRAULIC AND COMPRESSI13LE FLO\V TURBOMACIIINES

INTRODUCTION

23

turbines with a specific speed of 180 rpm are investigated. The normal running speed is to be 50 rpm in both schemes. Determine the dimensionless specific speeds and compare the two proposals insofar as the number of machines are concerned, and estimate the power to be developed by each machine. The units in either installation are to be of equal power and the efficiency of each type may be assumed to be 0.9.

20

1.9 A customer approaches a salesman with a particular pump requirement and is quoted for an axial now pump of rotor diameter 152.4 mm. Running at a speed of980 rpm, the machine is said to deliver 0.283 m 3 /s of water against a head of9.1 m at an efficiency of 85 per cent. Are the claims of the salesman realistic?

16

12

1.10 A Francis turbine runs at I HO rpm under a head of 146 m with an efficiency of 93.5 per cent. Estimate the power output of the installation.

SOLUTIONS

4

OL.-.-_ _L.-.-_--I._ _

o

--.L._ _--.L._ _----L-_ _--.l...-_ _----L-_ _-.J

1.0 2.0 3.0 Volume now rate, Q x 10 3 (01 3 /s)

4.0

Exercise 1.1 Equating head, flow and power coefficients for the model and prototype and noting that the density of the fluid remains the same, then, if subscript 1 refers to the prototype and subscript 2 to the model, where Pl = P2

Figure 1.14

Then If an entry pressure of 60 k Pa is obtained at the point where the normal ambient condition mass now would be 65 kg/s, calculate the mass now obtained in the test. 1.4 Specifications for an axial now coolant pump for one loop of a pressurized water nuclear reactor are:

Head 85 m Flow rate 20000 mJ/h Speed 1490 rpm Diameter 1200 mm Water density 714 kg/nl" Power 4 MW (electrical) The manufacturer plans to build a model. Test conditions limit the available electric power to 500 kW and flow to 0.5 nl J/s of cold water. If the model and prototype elliciencies arc 'assumed equal, find the head, speed and scale ratio of the model. Calculate the dimensionless specific speed of the prototype and confirm that it is iuentical with the model.

Also

Then

~: =(:: )1/2(Z:)=

1.5 A pump with an available driven speed of 800 rpm is requireu to overcome a IJD m head while pumping 0.2 m J Is. What type of pump is required and what power is required? 1.6 A reservoir has a head of 40 manu a channel leading from the reservoir permits a now rate of 34 m 3 /s. If the rotational speed of the rotor is 150 rpm, what is the most suitable type of turbine to use? 1.7 A large centrifugal pump contains liquid whose kinematic viscosity may vary between 3 and 6 times that of water. The dimensionless specific speed of the pump is 0.1 HJ rev and it is to discharge 2 m 3 /s of liquid against a total head of 15 Ill. Determine the range of speeds and test heads for a one-quarter scale model investigation of the full size pump if the model uses water. 1.8 In a projected low-head hydroelectric scheme, 10000 ft J/s of water arc a vailable under a head of 12ft. Alternative schemes to usc Francis turbines having a specific speed of 105 rpm or Kaplan

Therefore equating the diulnetcr ratios

(

0.266 -Nl).\i~ -_( -5 )1/2(N - 1) N2 14 N2

or N 2)2/5 = 2.25 ( N~

C4Y/2(ZJ 5

24

INTRODUCTION

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

whence

N 2 == 2.25 5 / 2 x 95

and diameter D are dropped to yield Eq. (1.14). Considering first the speed parameter,

Model speed == 721.4 rpm D 2 == 0.266 (~)3/5

D1

25

lV l

N2

JT

JT

Ol

02

+ 25 )1 /2 + 15

273 N 2 == 4500 ( 273

721.4

Model scale ratio == 0.079

Correct speed == 4577 rpm

. Power output M o d eI efliIClency == - - - - - - Water power input

Considering no\v the mass now parameter,

40 x 10 3 0.9==--pgQH Q==0.9

X

40 x 10 3 10 3 x 9.81 x 5

n1 2

Mass now obtained == 37.85 kg/s

Model volume flow == 0.906 m 3Is Exercise 1.2 Assuming dynamic similarity exists between the first and second sized pumps, we equate the flow coefficients. Thus

== 65 x ( -60- )(288)1/2 -101.3 298

Exercise 1.4 Using Eq. (1.5), equate the head po\ver and now coefficien ts for the model and prototype. Then

or 2.83 2000 X 127 3

N

1 ( 20 000 ) IV 2 == 0.5 x 3600

2200 x 102 3

2 ) 3

D;

=11.11(~:r

Solving we get Q2 == 1.61 ljs

(D

Also

~=(Nl)3(~.)5(~)

From Fig. 1.14 at Q1 == 2.83ljs (2.83 x 10- 3 m 3 Is) and 2000 rpm the head H 1 is 14 m and equating head coefficients for both cases gives

P2

N2

D2

P2

Substitute for (N II N 2); then

4 0.5

and substituting 9.81 x 14 (2000 x 127)2

9.81 xH 2 (2200 x 102)2

D

(

Solving we get H 2 == 10.9 m of water

Exercise 1.3 The dimensionless groups of Eq. (1.13) may be used here but since the same machine is being considered in both cases the gas constant R

714) =(11.11)3 (DD: )9(DD: )5( 1000

2)4

D;

8 =(11.11)J x O.714

Scale ratio D2 1D 1 = 0.3 Then N l/lV 2 = 11.11 Speed ratio N 2/ N 1== 3.3

X

(0.3)3

26

HYDRAULIC AND COMPRESSI13LE FLOW TUR130MACIiINES

INTRODUCTION

27

supplied to the shaft we divide by the efficiency,

Also

Shaft power required = 3.59/0.80 Shaft power = 4.49 kW Exercise 1.6 We have

Turbine power = pgQli Head ratio H 2 /H I = 1.0

=1000x9.81 x34x40

= 13.3 MW From Eq. (1.16) the dinlcnsionlcss spccific specd is given by NQI/2 N=-S (gH)3/4

Power specific speed is given by Eq. (1.20) N

For the prototype

Np 1 / 2 sp

=----

P1/2(gH)5/4

150 x (13.3 x 10 6 )1/2 60 X (1000)1/2 x (9.81 X 40)5/4

- = 2rr x 1490 x (20000)1/2(~_)3/4(~)3/4 N sl 60 3600 9.81 85

= 0.165 rev (1.037 rad)

= 2.37 rad 1490

N s2 = 2rr x

60

(0.5)1/2 x 3.3 x (9.81 x 85)3/4

= 2.35 rad

Therefore taking rounding errors into account ~he __~ii_nlensionlcss specific speeds of both model and prototype are thc sanlC.

bc the most suitable From Fig. 1.10 it is seen that the Francis turbine would _.-._--------choice for this application. Exercise 1.7 Since the viscosity of the liquids used in the model and prototype vary significantly, equality of Reynolds nUI11ber in Eq. (1.5) n1ust apply for dynamic similarity. Let subscripts 1 and 2 apply to the prototype and model respectively. Equating Reynolds nUI11ber

Exercise 1.5 Froln Eq. (1.16) NQ1/2

N

N2D~

V1

V2

!!_~ = ~ (~)2

= ----------

(yH)J/4

S

N 1Di

N1

800 (0.2) 1/2 =-x 60 (9.81 X 1.83)J/4

P = pyQli = 1000 x 9.8 1 x 0.2 x I. 83 =

3.59 kW

This is the power delivered to the water and to get the power that must be

D2

For the liquid with viscosity three tinlcS that of watcr N 2 42 - =-= 5.333 N1 3

= 0.683 rev (4.29 rad) For the given flow rate Fig. 1.8 shows that a propcller or axial flow pump is required and that an efficiency of about 80 per ccnt can be expectcd. Therefore the power required is

V1

Equating flow cocfficicnts

Q2=N 2 Ql N I

(D )J 2

D1

5.333 =43=0.0833

28

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

Equating head coefficients

INTRODUCTION

Dimensionless specific speed of Francis turbine =

105/42

=

2.5 rad

Dimensionless specific speed of Kaplan turbine

= 180/42 = 4.3 rad From Eq. (1.16) N 1Qt N s1 = (gH d3/4 /2

N1 =

These values may be checked against those values in Fig. 1.10. Converting to SI units 10 000 ft3 Is = 283.17 m 3Is

0.183 (9.81 x 15)3/4 21 / 2

12ft=3.66m Turbine efficiency =

= 5.47 revls N 2 = 5.47 x 5.33

P 0.9 = pgQH

Model speed = 29.16 revls

P = 0.9 x 1000 x 283.17 x 3.66 x 9.81

Q2 = 2 x 0.0833 Model flow rate = 0.166 m 3 H 2 = 15 x 1.776

Model head = 26.67 m

= 9150kW This is the total power delivered by all the turbines. Now where N is in revls

Similarly for the case when the prototype viscosity is six times that of water IN 2 = 14.58 revls

For the Francis turbine 50 x 2 x 1t X p 1/ 2 25- - - - - - - - - - . - 60 x (1000)1/2(9.81 X 3.66)5/4

,H 2 = 6.67m For one-quarter scale model 14.58 < model speed < 29.16 rev Is 6.67 m < model head < 26.67 m

Exercise 1.8 The dimensionless specific speed is obtained from the conversion

factors for specific speed given in Sec. 1.5. In this case for the non-SI units used N sp = SPI42

Power deli vered '1 bl Power aval a e

whence P=1761kW . . Total power required Number of FranCIS turbInes = - - - - - - - Power per machine 9150 1761 = 5.19 (say 6 machines)

29

30

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

For the Kaplan turbine 4.3

CHAPTER ==

50 x 2 x n x 60 x (1000)1/2(9.81

pl/2

X

TWO

3.66)5/4

HYDRAULIC PUMPS

whence P == 5209.7 kW

NUInber of Kaplan turbines

==

9150 5209.7

==

1.76 (say 2)

Exercise 1.9 From Eq. (1.16) for dilllensioniess specific speed

NQ1/2 N s == --3-'4 rev (y 11) I

2 x IT x 980 X 0.283 1/ 2 " -----~-rau 60 x (9.81 x 9.1 )3/4 == 1.88

2.1 INTRODUCTION

rad

Referring to Fig. 1.10 it is seen that axial flow pUln~s only begin .at a dimensionless specific spced of approximately 2.0 rad. 1t IS therefore unlIkely that the salesman's clainls are realistic. A suitable pUlnp would be of the tllixed flow type which gives the stated efficiency at the required flow rate and calculated ditllcnsiol1less spccific speed. Exercise 1.10 Using Fig. 1.9 the Francis turbine has an efficiency of 93.5 per cent at a dimensionless specific speed of 2.0 rad. Fron1 Eq. (1.20) the dimensionless power specific speed is N

sp

Npl/2 pi /2 (y I-I)5/4

==----

whence ,

p1 / 2

==

2.0 x (1000) 1/2 x (9.81

146)5/4 x 60

X --~------------

180 x 2 x n

== 29 563

and P==874MW

The term 'hydraulics' is defined as the science of the conveyance of liquids through pipes. Most of the theory applicable to hydraulic pumps has been derived using water as the fluid medium but this by no means precludes the use of other liquids. Two types of pUlnps commonly used arc centrifugal and axial flow types, so named because of the general nature of the fluid flow through the impeller. Both work on the principle that the energy of the liquid is increased by imparting tangcntial acceleration to it as it flows through the pump. This energy is supplied by the ilnpeller, which in turn is driven by an electric motor or SOlne other drive. In order to ilnpart tangential acceleration to the liquid, rows of curved vanes or blades move transversely through it and the liquid is pushed sideways as it tnoves over the vanes as well as retaining its original forward component of velocity. Figure 1.1 showed typical centrifugal and axial flow pun1p itnpellcrs, while between these two extremes lie mixed flow pumps, which are a cOlnbination of centrifugal and axial flow pumps, part of the liq uid flow in the illl peller being axial and part radial. The centrifugal and axial flow pumps will be dealt with in turn in the following sections. However, before considering the operation of each type in detail, we will look at a general pumping system, which is common to both types. This is shown in Fig. 2.1 where a pump (either axial or centrifugal) pumps liq uid from a low to a high reservoir. At any point in the system, the pressure, elevation and velocity can be expressed in tenns of a total head measured from a datum line. For the lower reservoir the total head at the free surface is H A and is equal to the elevation of 31

32

HYDRAULIC AND COMPRESSIBLE FLOW TlJRBOMACflINI:S

r-

It,o

I

+

HYDRAULIC PUMPS

then

D

4-

I I

33

Pump total inlet head F

--r-----~~

P um p tot alOll tic thead

=

p)Prl

+ Vt /2q + Zj

= Po / f> Y +

V l~ /2 r!

+Z

0

Total head developed by pump = [(Po - Pi)/ puJ + [( V~ - V~)/2gJ

,I

=

H

+ (Zo -

Zi)

(2.1)

Total energy line

~

..I

This is the head that would be llsed in Eq. (I. t 2) for determining the type of pump that should be selected, and the term ~manometric head' is often used. The static head H s is the vertical distance bctween the two levels in the reservoirs and from Fig. 2.1 it can be secn that for the pipeline H = Hs

+ I losses

I I

It is worth noting here that, for the san1e size inlet and outlet diameters, Vo and Vi are the same, and in practice (2 0 - 2 j ) is so small in comparison to (Po - Pi)/pg that it is ignored. It is therefore not surprising to find that the static pressure head across th~ pun1p is often used to describe the total head developed by the pump.

2.2 CENTRIFUGAL PUMPS

Figure 2.1 Diagram of a pumping system

the free surface above the datum line since the velocity and static gauge pressure at A are zero. The liquid enters the intake pipe causing the head loss hin , with the result that the total head line drops to point B. As the fluid flows from the intake to the inlet flange of the pump at elevation Zj, the total head drops further to the point C due to pipe friction and other losses /tfi. The fluid enters the pump and energy is imparted to it, which raises the total head to point D at the pump outlet. Flowing from the pump outlet to the upper reservoir, friction and other losses account for a total head loss hfo down to point E, where an exit loss hout occurs when the liquid enters the upper reservoir, bringing the total head at the upper reservoir to point F at the free surface. If the pump total inlet and outlet heads are measured at the inlet and. outlet flanges respectively, which is usually the case for a standard pump test,

Figure 2.2 shows the three important parts of a centrifugal pump: (1) the impeller, (2) the volute casing and (3) the dilTuser ring. The diffuser is optional and mayor may not be present in a particular design depending upon the size and cost of the pump. The impeller is a rotating solid disc with curved blades standing out vertically from the face of the disc. The tips of the blades are sometimes covered by another flat disc to give shrouded blades; otherwise the blade tips are left open and the casing of the pump itself forms the solid outer wall of the blade passages. The advantage of the shrouded blade is that flow is prevented from leaking across blade tips from one passage to another. As the impeller rotates, the fluid that is drawn into the blade passages at the impeller inlet or eye is accelerated as it is forced radially outwards. In this way, the static pressure at the outer radius is much higher than at the eye inlet radius. The fluid has a very high velocity at the outer radius of the impeller, and, to recover this kinetic energy by changing it into pressure energy, diffuser blades mounted on a diffuser ring may be used. The stationary blade passages so formed have an increasing cross-sectional area as the fluid moves through them, the kinetic energy of the fluid being reduced while the pressure energy is further increased. Vaneless diffuser passages may also be utilized.

34

HYDRAULIC AND COMPRESSIBLE FLOW TUR130~tACIiINES

HYDRAULIC PUMPS

Stationar) diffuser vanes

35

y Vr

= C r (radial velocity of fluid)

Diffuser -

-,

~

Impeller

't---z

y~

With diffuser

Volutc

_

(axial velocity of fluid)

.......

L-

VIJ

= C (tangential I

velocity of fluid)

_

-+-----+~

\Vithout difTuser Figure 2.2 Ccntrifugal PUIllP cOlllponcnts

Finally, the fluid 1110ves fron1 the difTuser bladcs into thc volute casing, which collects it and conveys it to the pUInp outlct. SOll1CtilllCS only the volute casing exists without the difTL1ser~ however, son1e pressure recovery will take place in the volute casing alone. In dealing with the theory of hydraulic pU1l1pS, a nUlllbcr of assuInptions will be made. At any point within the blade passages the l1uid velocity will in general have three components, one each in the axial, radial and angular directions as indicated in Fig. 2.3. The velocity may then be written as a function of the three components V == f(r, 0, :)

However, we will assume that the following hold: 1. There are an infinite number of blades so closely spaced that avlao = O. That is, there is no flow in the blade passage in the tangential direction and Vo ==

o.

.

2. The impeller blades are infinitely thin, thus allowing the pressure ddTerence across them, which produces torque, to be replaced by tangential forces that act on the fluid. 3. The velocity variation across the \vidth or depth of thc ill1pcller is zero and hence vloz = O. 4. The analysis will be confined to conditions at the ilnpcller inlet and outlet, and to the angular n10111enturn change bctwecn thesc t\\'o stations. No account is taken of the condition of the fluid bctween thcse two stations.

a

Figure 2.3 Cylindrical coordinates for a centrifugal pump

5. It is assumed that at inlct the l1uid is moving radially after entering the eye of the pump. Assumptions 1 and 2 1l1Can that the velocity is a function of the radius only, V = f(r), and now with thesc assu111ptions the velocity vectors at inlet and outlet of the impeller can be drawn and the theoretical energy transfer determined. Figure 2.4 shows a ccntrifugal pump impeller with the velocity triangles drawn at inlet and outlet. The blades are curved between the inlet radius r 1 and outlet radius r 2 , a particle of fluid moving along the broken curve shown. /31 is the angle subtended by the blade at inlet, measured froln the tangent to the inlet radius, while /32 is the blade angle measured from the tangent at outlet. The fluid enters the blade passages with an absolute velocity C 1 and at an angle (Xl to the ilnpeller inlet tangential velocity vector U}, where U 1 = wr 1 , w being the angular velocity of the impeller. The resultant relative velocity of flow into the blade passage is W} at an angle {1'} to the tangent at inlet. Similarly at outlet the relative velocity vector is W 2 at angle /3~ from the tangent to the blade. By subtracting the impeller outlet tangential velocity vector U 2' the absolute velocity vector C 2 is obtained, this being set at angle (1.2 from the tangent to the blade. I t is seen that the blade angles at inlet and outlet do not equal the relative now angles at inlet and outlet. This is for a general case, and unless otherwise stated (see slip factor, Sec. 2.3), it will be assumed

36 HYDRAULIC AND COMPRESSrBLE FLOW TURBOMACIlINES

HYDRAULIC PUMPS

But by using the cosine rule, W 2

=

U2

+ C2 -

(fIC\

cos~\ =(Uf -

U 2C2

COSCf. 2

37

2UCcosa, then

vvi + Cf)/2

and

= (U~ -

vv~

+ C~)/2

and suhstituting into Eq. (2.]) gives C2

W'l

Cr'l

E = [(U~ - Uf)

/ I

I

~

/

I

1ft

,

C

I~

+ (Wi

- W~)J/2g

(2.4)

Q = 2rrr i C,l h t = 2rrr 2 C,2 h 2

I I I

I

Cf)

The terms in Eq. (2.4) may now be exatnined in turn. (C~ - Cf)/2(J represents the increase of kinetic energy of the fluid across the impeller, (U~ - Uf)/2(J represents the energy used in setting the fluid into circular motion about the impeller axis and (Wi - W~)/2g is the gain of static head due to a reduction of the relative velocity within the impeller. The flow rate is

a2

W

+ (C~ -

where C, is the radial component of the absolute velocity and is perpendicular to the tangent at inlet and outlet while h is the width of the blade (in the z direction). It is usually the case that C 1 = C,I and hence Ct. 1 = 90°. In this case C xl = 0, where e x1 is the component of the inlet absolute velocity vector resolved into the tangential direction. W xand ex are often respectively called the relative and absolute whirl components of velocity. When 131 = f3 /l ' this is referred to as the 'no-shock condition' at entry. In this case the fluid moves tangentially onto the blade. When 132 = f3~ there is no fluid slip at the exit.

V'

PI

VI

I

2.3 SLIP FACTOR

Figure 2.4 Velocity triangles for centrifugal pump impeller

that the inlet and outlet blade angles are equal to their corresponding relative flow angles. From Euler's pump equation (Eq. (1.25)), the work done per second on the fluid per unit weight of fluid flowing is E=W/mg=(U2Cx2-UICxd/y

(J/sperN/sorm)

(2.2)

where C x is the component of absolute velocity in the tangential direction. E is often referred to as the Euler head and represents the ideal or theoretical head developed by the impeller only. Now and Thus (2.3)

It was stated in the previous section that the angle at which the fluid leaves the impeller, fi~, may not be the same as the actual blade angle f32' This is due to fluid slip, and it occurs in both centrifugal pumps and centrifugal compressors, and manifests itself as a reduction in C x2 in the Euler pump equation. One explanation for slip is that of the relative eddy hypothesis. Figure 2.5 shows the pressure distribution built up in the impeller passages due to the motion of the blades. On the leading side of a blade there is a high-pressure region while on the trailing side of the blade there is a lowpressure region, the pressure changing across the blade passage. This pressure distribution is similar to that about an aerofoil in a free stream and is likewise associated with the existence of a circulation around the blade, so that on the low-pressure side the fluid velocity is increased while on the high-pressure side it is decreased, and a non-uniform velocity distribution results at any radius. Indeed, the flow may separate from the suction surface of the blade. The mean direction of the flow leaving the impeller is therefore f3~ and not 132 as is assumed in the zero-slip situation. Thus Cx2 is reduced to C~2 and the

38

HYDRAULIC AND COMPRESSIBLE FLO\V TURBOMACIiINES

HYDRAULIC PUMPS

39

2e /sin /1 2

-- - - - Ideal --Actual

Rela ti ve edd y ,/ ,/

Figure 2.6 The relative eddy hdween impeller blades

blades is 2nr 2 /Z if we have 2 blades of negligible thickness. This may be approximated to 2e/sin {>2 and upon rearrangement

~(tJ

+

e = (nr 2/2) sin fJ2 ~Cx = (U 2/2r2)(nr 2 sin

Figure 2.5 Slip and velocity distribution in centrifugal pump impdler blades

=

difference

~CX

= C~2/Cx2 =

as

Cx 2

(2.5)

Stodola 1 proposed the existcnce of a relativc eddy within the blade passages as shown in Fig. 2.6. He proposed that if a frictionless fluid passes through the blade passages it will, by definition, have no rotation; therefore at the outlet of the passage rotation should be zero. No\v the ilnpel1er has an angular velocity w so that relative to the ill1peller thc fluid 111Ust have an angular velocity - w in the blade passages to cOlnply with the zero-rotation condition. If the radius ofa circle thatlnay be inscribed bet\\'een two successivc blades at outlet and at a tangent to the surfaces of both blades is e, then the slip is given by ~CX =

(U 2 n sin fJ 2)/2

Now referring back to Fig. 2.5 for the no-slip condition

is defined as thc slip. Slip factor is defined as Slip factor

f32)

we

Now the ilnpeller circul11ference is 2n1' 2 and therefore the distance between

= U2

-

Cr 2 co t {1 2

and substituting into Eq. (2.5) gives Slip factor

~Cx)/Cx2

=

(C x2 -

=

I - (U 2n sin {>2)/[Z( U 2

=

I -(nsin{J2)/{Z[I-(C r2 /U 2)cot{J2]}

-

C r2 cot fJ2)]

(2.6)

For purely radial blades, which are often found in a centrifugal compressor, {12 will be 90 and the Stodola slip factor becomes C

'

()s

= 1 - n/Z

(2.7)

The Stodola slip factor equation gives best results when applied in the

40

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACIIINJ':S

HYDRAULIC PUMPS

41

range 20° < [32 < 30°. Other slip factors are named after Ruseman 2 as = [A - B( CrZ / U 2) cot [32J/[ 1 - (C r2 / U 2) cot /J 2J

(2.8)

where A and B are functions of /i 2 , Z and 1'2/1'1' and are best used in the range 30° < f32 < 80 u . The Stanitz J slip factor given by (Js=

I-O.6]rr/{Z[1 -(C r2 /U 2 )cotI1 2 J}

(2.9)

is best used in the range RO° < 1/2 < 90°.

/

When applying a slip factor, the Euler pUlnp equation (Eq. (1.25)) becomes

Leakagc now hctwccn blade and casing. q

(2.10)

Typically, slip factors lie in the region of 0.9, while slip occurs even if the fluid is ideal.

2.4 CENTRIFUGAL PUMP CHARACTERISTICS In Sec. 1.3 dimensionless groups were used to express the power, head and flow relationships for a hydraulic machine. A well-designed pump should operate at or near the design point and hence near its maximum efficiency, but the engineer is often required to know how the pump will perform at off-design conditions. For instance, the head against which the pump is operating may be decreased, resulting in an increase in mass flow rate. However, before examining this aspect of ofT-design performance, we will look at the losses occurring in a pump and the differing efficiencies to which these losses give rise, whether or not the pump is working at the design point. We will then examine the efTects of working at the off-design condition.

Figure 2.7 Lcakagc and recirculation in a centrifugal pump

associated with the flow rate Qi through the impeller, and so the impeller power loss is ex pressed as (2.11 )

However, while the flow through the impeller is Qi' this is not the flow through the outlet or inlet flange of the machine. The pressure difTerence between impeller tip and eye can cause a recirculation ofa small volume offluid q, thus reducing the flow rate at outlet to Q as shown in Fig. 2.7, and then (2.12)

If Hi is the total head across the impeller, then a leakage power loss can be defined as PI

2.4.1 Pump Losses The shaft power, Ps or energy that is supplied to the pump by the prime mover is not the same as the energy received by the liquid. Some energy is dissipated as the liquid passes through the machine and the mechanism of this loss can be split up into the following divisions. 1. Mechanical friction power loss, Pm due to friction between the fixed and rotating parts in the bearing and stuffing boxes. 2. Disc friction power loss, Pi due to friction between the rotating faces of the impeller (or disc) and the liquid. 3. Leakage and recirculation power loss, PI due to a loss of liquid from the pump or recirculation of the liquid in the impeller. 4. Casing power loss, Pc Impeller power loss is caused by an energy or head loss hi in the impeller due to disc friction, flow separation and shock at impeller entry. This loss is

= pgHiq

(N m/s)

(2.13)

Equation (2.12) shows that when the discharge valve of the pump is closed, then the leakage flow rate attains its highest value. In flowing from the impeller outlet to the pump outlet flange, a further head loss he takes place in the difTuser and collector, and since the flow rate here is Q, then a casing power loss may be defined as Pc = pgQh c

(N m/s)

(2.14)

Summing these losses gives P~ = Pm

+ py(hiQi + hcQ + Hiq + QH)

(2.15)

where the total head delivered by the pump is defined as in Fig. 2.1 and Eq. (2.1). Anum ber of efficiencies are associated with these losses: . Overall efficiency

=

Fluid po\ver developed by pump Sf r '. t 1a t power Inpu

42

HYDRAULIC PUMPS

HYDRAULIC AND COMPRESSIBLE FLO\V TURUOMACIIINI:S

43

or '/0

Casing efficiency

(2.16)

== IJyQII/P s

-q

Fluid power at casing outlet

== ----~-

- - - - - - - -~-

Fluid po\ver at ~~~~_in_g o~t_l~_t ~~. developed by ilnpeller - Leakage loss

I Useful Iluid power

Vl

or

.2

Vl

.2

(2.17)

PfJQII / pyQi Ii == H / fl i

bO COd

~

...:w:

Fluid po\ver at iinpeller exit

'cC'd

~

Fluid power supplied to in1pellcr

:E

--------~

COd

..c:

--~~-~--

11 Hi

iU

u

Impeller efficiency ==

B

N

Vl

==

I'

~

== F~iJ-i)-o~~er-

'Ie

Q

J

0

A

Fluid power at casing inlet

I-

I

u 11)

Fluid power at in11?~~I~:~~~~__.. _ Fluid power developed by inlpeller + Impeller loss

I

E

or (2.18)

Flow rate through pUInp Volumetric efficiency == , Flow rate through IInpeller

L

M K

Jz(,

Casing loss 5%

G

Impeller loss 5%

11 1

F

Mechanical loss 10 %

C

0

Flow rate (m:l/s)

Figure 2.8 Losses in a centrifugal pump

or 'Iv == Q/(Q

+ q)

(2.19)

Fluid power supplied to the iInpeller Mechanical efficiency == - - - - - , - - - - - - - - Po\ver Input to the shaft or

next loss to be accounted for is the leakage loss pgHjq represented by rectangle DJKI, and finally the casing loss pghcQ represented by rectangle MLGK is removed. This leaves us with rectangle JBLM, which represents the fluid power output or power developed by the pump pgQH.

(2.20)

Therefore

2.4.2 The Characteristic Curve (2.21)

Euler's pump equation (Eq, (1.25)) gives the theoretical head developed by the pump, but if it is assumed that there is no whirl component of velocity at entry then C xl == 0 and the act uul theoretical head developed is

A hydraulic efficiency Inay be defined as Actual head developed by pUInp l1H == Theoretical head developed by iInpeller

== H/(H i + hJ

E == U 2 C x2/g == (Hi (2.22)

where the theoretical head (Hi + hi) is that obtained froIn Euler's equation (Eq. (1.25)) and '1H == 'li'lc' Figure 2.8 shows how each of the po\ver losses arc subtracted from the initial input power. The rectangle OAllC represents the total power input to the shaft while OADEFC is equivalent to the ITIechanical power loss. The impeller loss pgQihj is next relnoved and is represented by rectangle EFGI. The

+ hi)

(2.23)

and if slip is accounted for, Eg. (2.23) becomes EN == CJsE == CJs(H i + hj)

Now C x2

== U 2 - W x2 == U 2

-

C r2 cot fJ 2

== U 2 - (Q/ A) co t {J 2

(2.24)

44

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

HYDRAULIC PUMPS

45

according to (2.27)

where Qn is the design nO\V rate. The friction losses are accounted for in the form hr = K 4 Q2

Equations (2.27) and (2.28) are plotted in Fig. 2.9 and the sum of them is subtracted froIll the curve of Eq. (2.26) to give the final characteristic. This curve is called the head-now characteristic of the pump.

E

Hydraulic losses '"

'"

H

'" '" '"

(2.28)

-',., EN = Err..

2.4.3 Effect of Flow Rate Variation

- ----

-, II r

h8hO C'k

Figure 2.9 Centrifugal pump characteristic

where A is the flow area at the periphery of the impeller and C, is perpendicular to it. Thus from Eq. (2.23) the energy per unit weight of flow becomes E = U 2[U 2 -(Q/A)cotfJ2J/g

and since U 2' fJ2 and A are constants, then

E=K 1 -K 2 Q

(2.25)

A pump is usually designed to run at a fixed speed with a design head and flow rate and these conditions would normally occur at the maximum efficiency point. I-Io\vever~ it is not always the case in practice that the operating point lies at the design point. 'rhis rnay be due to a pipeline being partially blocked, a valve jammed partially closed or poor matching of the pump to the piping system. Also in general a variable-speed rnotor is not available to correct for any deviation from the design condition, so that in what follows it is assumed that the speed of the pump remains constant. Figure 2.10 shows the velocity diagrams that pertain for three possible flow rates: normal design now rate, increased nO\V rate and decreased flow rate. When the flow rate changes, C r2 changes, and since U 2 is constant and the blade outlet angle fJ 2 is constant (assuming rJ~ = fJ 2)' the magnitude of W 2 and C 2 must change along with the angle (12' Since the etTective energy transfer E depends on C x2 , then E will change accordingly. Thus a reduction in Qgives an increase in C x2 ' while an increase in Q gives a reduction in C x2 ' It follows that, should the head against which the pump operates be momentarily increased, E and therefore C x2 increase and Q decreases to give the new operating point at the increased head. Similarly a reduction in the operating head gives an increase in Q,

and this equation may be plotted as the straight line shown in Fig. 2.9, If slip is taken into account, it is seen from Eq. (2.9) that as C,2 increases (and hence Q) then as decreases, thus reducing the value of E in Eq. (2.25) to

l

(2.26)

The loss due to slip can occur in both a real and an ideal fluid, but in a real fluid account must also be taken of the shock losses at entry to the blades, and the friction losses in the casing and impeller vanes, or indeed at any point where the fluid is in contact with a solid surface of the pump. At the design point the shock losses are zero since the fluid would move tangentially onto the blade, but on either side of the design point the head loss due to shock increases

~

Reduced flow

~

Design flow

Figure 2.10 EITect of now rate variation on outlet velocity

~

Increased flow

46

HYDRAULIC PUMPS

HYDRAULIC AND COMPRLSS(BLF (''L()\V TURBOMACII(l':ES

47

H

.

-'J\

.,:,\

--

-'\,

- , . ItJ (t)

- - ---- --.- 11 = {/ (jl'l = 90°)

----------------Vl Reduced flow

Design now

Increased flow ia

Figure 2.11 Effect of flow rate variation on inlet velocity

At the inlet the efTect of now rate change is to cause eddies on the suction surface of the blade for a reduced nO\\1 rate and on the pressurc surface of the blade for an increased flow rate. The dcsign condition is the ~no-shock' condition, which corresponds to the now QD in Fig. 2.9. The corresponding velocity diagrams can be seen in Fig. 2.11. In all cases it is assuI11cd that Cxt is zero.

Q

Figure 2.13 Theoretical characteristics for varying outlet blade angle

Writing E as a head, (2.29)

H = a- bQ

2.4.4 Effect of Blade Outlet Angle

and for pumps [3 2 typically lies between 15° and 90°.

The characteristic curve will also be afTected by the blade angle at outlet, the three types of blade settings being back ward-facing, forward-facing and radial blades. Figure 2.12 shows clearly the velocity triangles for each case with C x ! =0.

Case (ii). Radial blades, [J 2 = 90°

P,H

Case (i). Backward-facing blades, [1 2 < 90°

(2.30)

H=a

Head

Forward-facing

Therefore or E = (U ~ / {j) - (Q U 2co t [3 2/{j A ) Radial - Power

Backward Forwarding-facing vanes

Radial vanes

Radial

Forward

Back ward-facing vanes

Figure 2.12 Centrifugal pump outlet velocity triangles for varying blade outlet angle

Q Figure 2.14 Actual characteristics for varying blade outlet angle

48

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACIfINES

Case (iii). Forward-facing vanes, H

Ii 2 > 90 =

HYDRAULIC PUMPS

49

Cl

a + hQ

(2.31 )

0

where fJ2 would be typically 140 for a multi-bladed centrifugal fan. These equations are plotted in Fig. 2.13 as characteristics and they revert to their more recognized curved shapes (for the reasons previously discussed) as shown in Fig. 2.14. For both radial and forward-facing blades the power is rising continuously as the flow rate is increased. In the case of back ward-facing vanes the maximum efficiency occurs in the region of maximum power, and if, for some reason, Q increases beyond Qn, this results in a power decrease and therefore the motor used to drive the pump may be safely rated at the maximum power. This is said to be a self-limiting characteristic. In the case of the radial and forward-facing vanes, if the pump motor is rated for maximum power, then it will be under-utilized most of the time, and extra cost will have been incurred for the extra rating, whereas if a smaller motor is employed rated at the design point, then if Q increases above Qo the motor will be overloaded and may fail. It therefore becomes more difficult to decide on a choice of motor for these latter cases.

Impeller

_

Volute of increasing cross section

Figure 2.15 Simple scroll collector

volute

or

where P = radial force (N), /-1 = head (m), D 2 = peripheral diameter (m), B 2 = impeller width (m) and K = constant determined from the following equation for a particular value of Q: (2.33)

2.5 FLOW IN THE DISCHARGE CASING The discharge casing is that part of the casing following the impeller outlet. It has two functions: (i) to receive and guide the liquid discharged from the impeller to the outlet ports of the pump, and (ii) to increase the static head at the outlet of the pump by reducing the kinetic energy of the liquid leaving the impeller. These two functions may be called collector and diffuser functions. The former function may be used alone while the latter can occur either before or after the collector function. In addition diffusion can take place in a vaned or vaneless diffuser.

A cross section of the volute casing is shown in Fig. 2.16. The circular section is adopted to reduce the losses due to friction and impact when the fluid hits the casing walls on exiting from the impeller. Of the available kinetic energy at impeller outlet, 25-30 per cent may be recovered in a simple volute.

2.5.2 Vanelcss Diffuser Diffusion takes place in a parallel-sided passage and is governed by the principle of conservation of angular momentum of the fluid. The outlet

2.5.1 Volute or Scroll Collector A simple volute or scroll collector is illustrated in Fig. 2.15 and consists of a circular passage of increasing cross-sectional area. The advantage of the simple volute is its low cost. The cross-sectional area increases as the increment of discharge increases around the periphery of the impeller and it is found that a constant average velocity around the volute results in equal pressures around the pump casing, and hence no radial thrust on the shaft. Any deviation in capacity (flow rate) from the design condition will result in a radial thrust, which if allowed to persist could result in shaft bending. Values of radial thrust are given by the empirical relationshi p 4 (2.32)

Figure 2.16 Section through volute casing

50

HYDRAULIC AND COMPRESSIBLE I''LD\V TURBOMACIlINLS

HYDRAULIC PUMPS

51

where b is the width of tile diffuscr passage perpendicular to the peripheral area of the inlpeller and is usually the same as the impeller width. Letting the subscripted variables represent conditions at the impeller outlet and the unsubscripted variablcs represent conditions at any radius r in the vaneless difTuser, then froln continuity

Diffuser passage

;I'

b

/

/ /

rbpC r == r 2 b 2 P2 Cr2

f Free vortex now in difTuser passage

or (2.34) If frictionless flow is assuillcd, then by the conservation of angular momentum C x ==

w

C x2 r2 /r

But C x » C r (usually) and therefore the absolute velocity C is approximately equal to Cx or

Figure 2.17 Vaneless diffuser passage

(2.35)

tangential velocity is reduced as the radius increases, while the radial component of absolute velocity is controlled by the radial cross-sectional area of flow b. A vaneless difTuser passage is shown in Fig. 2.17. With reference to Fig. 2.18 the size of the diffuser n1ay be deternlined as follows. The mass flow rate n1 at any radius r is given by HJ ==

pAC, == 2nrhpC,

From Eq. (2.35), for C to be slnall, which is what we are trying to achieve, then r n1ust be large and thereforc, for a large reduction in the outlct kinetic energy, a diffuser with a large radius is required. For an incompressible fluid, the inclination of the absolute velocity vector to the radial line remains constant at all esince at the outlet from the impeller (Fig. 2.18) tan (I.~ == C x2/Cr2 == constant --= tan Y.' since rC r is constant froln the constant mass flow rate requirement, and Cxr is constant from the conservation of angular momentulTI requirement. Thus the flow in the diffuser remains at a constant inclination (I.' to radial lines, the flow path tracing out a logarithlnic spiral, and if for an incremental radius dr the fluid moves through angle dO, then from Fig. 2.18 r dO == (tan a') dr

Integrating, (2.36) Putting a' == 78° and (r/r 2 ) == 2, the change in angle of the diffuser is almost 180°, giving rise to a long flow path, which may result in high frictional losses, which in turn gives a low efficiency. So it is seen that the length of the diffuser must be balanced by the pressure recovery that is required and an optimum point is usually found based on either economic or hydraulic friction loss considerations.

2.5.3 Vaned Diffuser Figure 2.18 Logarithmic spiral now in vaneless space

The vaned diffuser shown in Fig. 2.19 is able to diffuse the outlet kinetic energy at a much higher rate, in a shorter length and with a higher efficiency than the

52

HYDRAULIC PUMPS

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

Casing

Collector

\

53

Throat of diffuser passage

----. I Diffuser

~--i-I-----i

i

Delivery

I

I

1/

I ..----.-+---,'

/

__

I I

I I Impeller

-~~

Impeller

Figure 2.20 Head rise across a centrifugal pump

The collector and diffuser operate at their maximum efficiency at the design point only. Any deviation from the design discharge will alter the outlet velocity triangle and the subsequent flow in the casing. Figure 2.20 shows the contribution of each section of the pump to the total head developed by the pump.

2.6 CAVITATION IN PUMPS Figure 2.19 A vaned diffuser

vaneless diffuser. This is very advantageous where the size of the pump is important. A ring of difTuser vanes surrounds the inlpeller at the outlet, and after leaving the impeller the fluid moves in a logarithmic spiral across a short vaneless space before entering the difTuser vanes proper. Once the fluid has entered the diffuser passage, the controlling variable on the rate of diffusion is the divergence angle of the difTuser passage, which is of the order of 8-10° and should ensure no boundary-layer separation along the passage walls. The number of vanes on the diffuser ring is subject to the following considerations:

1. The greater the vane number, the better is the diffusion but the greater is the friction loss. 2. The cross section of the diffuser channel should be square to give a maximum hydraulic radius (cross-sectional area/channel perimeter). 3. The number of difTuser vanes should have no common factor with the number of impeller vanes. This is to obviate resonant or sympathetic vibration.

Cavitation is a phenomenon that occurs when the local absolute static pressure of a liquid falls below the vapour pressure of the liquid and thereby causes vapour bubbles to form in the main body of liquid, that is the liquid boils. When t-he liquid flows through a centrifugal or axial flow pump, the static pressure (suction pressure) at the eye of the impeller is reduced and the velocity increases. There is therefore a danger that cavitation bubbles may form at the inlet to the itnpeller. When the fluid moves into a higher-pressure region, these bubbles collapse with tremendous force, giving rise to pressures as high as 3500 atm. Local pitting of the impeller can result when the bubbles collapse on a metallic surface, and serious damage can occur from this prolonged cavitation erosion, as shown in Fig. 2.21. Noise is also generated in the form of sharp cracking sounds when cavitation takes place. Referring to Fig. 2.1, cavitation is most likely to occur on the suction side of the pump between the lower reservoir surface and the pump inlet since it is in this region that the lo\vest pressure will occur. A cavitation parameter (J is defined as Pump total inlet head above vapour pressure Head developed by pump

(J=------------------

54

HYDRAULIC AND COMPRESSIULE FLOW TURUOMACHINES

HYDRAULIC PUMPS

55

3% head reduction H

Figure 2.21 Cavitation erosion in centri-

Q

fugal pump impl.:lkr

and at the inlet flange (J

=

(Pi/PU + V; /2y - Pvap/PY)I II

(2.37)

where all pressures are absolute. l~hc nunlerator of Eq. (2.37) is a suction head and is called the net positive suction head (IV PSH) of the pUlllp. It is a measure of the energy a vailable on the suction side of the punlp. Every pump has a critical cavitation nlJlnber (J\:, which can only be determined by testing to find the Illininlum value of N PSiJ before cavitation occurs. Various 111cthods exist for determining the point of cavitation inception, and (Jc, and therefore the minimunl N PSil required by the pump, will depend upon the criteria chosen to define (J c as well as the conditions under which the test is carried nut. One n1ethod is to deterIlline the normal head-flow characteristic of the pUlnp and then to repeat the test with the inlet to the pump progressively throttled so as to increasc the resistance to flow at the inlet. It will be found that for difTercnt throttle valve settings the performance curve will fall away from the nonnal operating curve at various points and one definition of the occurrence of IniniIl1Uln N PSI-I is the point at which the head H drops below the normal operating characteristic by some arbitrarily selected percentage, llsually about 3 per cent. At this condition, static inlet pressure Pi and inlet velocity Vi are ITIeasured, and (Je is then calculated fronl Eq. (2.37). The mininlum required N PSH or (Je may then be plotted for the difTerent degrees of inlet throttling to givc a curve of (Je versus flow coefficient (Fig. 2.22). In Fig. 2.1, the energy loss bct\veen the free surface (A) and the inlet side of the pUlnp (i) is given by thc steady flow energy equation as Energy at A - Energy at i = Energy lost bctween A and i

NPSH Measured

~~~:;:;:;~NPSH I I I

•

I

I

I

I I I

I

I

Figure 2.22 Critical N PSH plotted on the pump characteristic

placed at the lower reservoir surface, then Z A is also zero and Eq. (2.38) becomes PAl pg

= p) pg + Vf 12y + ilsu\:tion

where Z i

+ h in + hfi =

H suction

Substituting for p) pg in Eq. (2.37) gives (J

== (PAl pg - Pvapl pg - H suction)1 H

(2.39)

Providing (J is above (J c' cavitation will not occur, but, in order to achieve this, it may be necessary to decrease Hsuction by decreasing Zi and in some cases the pump may have to be placed below the reservoir or pump free surface, i.e. negative Zi' especially if hfi is particularly high due to a long inlet pipe. Thus when the pump is connected to any other inlet pipe system, (J as determined from Eq. (2.39) may be calculated and providing (J(available) > O"e(required) then cavitation will be avoided.

Wri ting the energy in tern1S of heads (PAl py

+ vi/2y + ZA) = (pJ pg + vf /20 + Zi) + (h in + hfi )

(2.38)

where (h in + /lfi) represents the losses. No\v VA cquals zero and, if the datum is

2.6.1 Suction Specific Speed It is reasonable to expect that the efficiency will be dependent not only upon the flow coefficient but also upon another function due to cavitation. The

56

HYDRAULIC PUMPS

HYDRAULIC AND COMPRESSIRLE FLOW TURROMACHINES

57

other function is the dimensionless suction specific speed and is defined as N suc = NQl/2/[g(NPSH)]3/4

Thus

r,

=

(2.40)

!(¢, NsuJ

(2.41)

2

It is found from experiments that the inception of cavitation occurs at constant values of N slIC and empirical results sho\v that N suc ~ 3 for N in rad/s, Q in m 3 /s and g(NPSH) in m 2 /s 2 • The cavitation parameter may also be determined by dividing the dimensionless specific speed by the dimensionless suction specific speed:

View on

X-X

Ns/N suc = [NQl/2/(gH)3/4]/{NQl/2/[{J(NPSH)]3/4} = (N PSH)3/4 / H 3/4

= (J~/4

c~

(2.42)

Also from the similarity laws NPSH 1 /NPSH 2 =(N 1 /N 2)2(D 1 /D 2)2

=

(Jl/(J2

Outlet guide vane

2.7 AXIAL FLOW PUMP An axial flow pump consists of a propeller type of impeller running in a casing with fine clearances between the blade tips and the casing walls. In the absence of secondary flows, fluid particles do not change radius as they move through the pump; however, a considerable amount of swirl in the tangential direction will result unless means are provided to eliminate the swirl on the outlet side. This is usually done by fitting outlet guide vanes. The flow area is the same at Stationary inlet guide vanes

Stationary outlet guide vanes

\

1\ 1\

1\

Figure 2.23 An axial now pump

I

~-~

1~4_ _ U~

--- - ------

---J

= VI = V

L

Figure 2.24 Axial now pump velocity triangles

inlet and outlet and the maximum head for this type of pump is of the order of 20 m. It may be seen in Fig. 1.7 that the dimensionless specific speed of axial flow pumps lies at the right-hand side of the pump spectrum, its characteristics being one of low head but high capacity. The usual number of blades lies between two and eight, with a hub diameter/impeller diameter ratio of 0.3-0.6. In many cases the hlade pitch is fixed but nl0st large hydroelectric units have variable-pitch blades to allow for load variations. Figure 2.23 shows an axial flow pump impeller. The section through the blade at X-X is sho\vn enlarged with the inlet and outlet velocity triangles superimposed in Fig. 2.24. Ii will be noticed that the blade has an aerofoil section and that the inlet relative velocity vector WI does not impinge tangentially but rather the blade is inclined at an angle of incidence i to the relative velocity vector WI' This is similar to the angle of attack of an aerofoil in a free stream. It is assumed that there is no shock at entry and that the fluid leaves the blade tangentially at exit. Changes in the condition of the fluid take place at a constant mean radius; therefore U 1 = U 2 = U = (or

58

HYDRAULIC PUMPS 59

HYDRAULIC AND COMPRESSIBLE FLOW TURBOr-.tACHINES

the fluid at exit frOITI the blade is relatively small, resulting in a low kinetic energy loss. An axial flo\v pump therefore tends to ha ve a higher hydraulic cfficiency than the centrifugal pump.

Assuming also a constant flow area from inlet to outlet

and noting that the flow area is the annulus fanned bet \veen the hub and the blade tips, then we may write 111

= pCan(R~

- R~)

From Eq. (2.2), (2.43) and for maximum energy transfer Cxl = 0, i.e. ~l = 90° and C 1 = C a , the absolute flow velocity at inlet being axial for nlaxinlulll energy transfer. Now or C x2 = U - C a cotfJ 2

Hence substituting for C x2 in Eg. (2.43) with C.d transfer or head is

= 0,

the Inaxinlunl energy

U( U - Cu cot fJ 2)/0

f, =

(2.44)

For constant energy transfer, Eq. (2.44) applies over the whole span of the blade from hub to tip; that is it applies at any radius r between R l and R h · For E 2 to be constant over the whole blade length it is obvious that, as U increases with radius, so an equal increase in UCacot fJ2 nlust take place and since Ca is constant then cot fJ 2 must increase, and the blade must therefore be twisted as the radius changes. Strictly speaking the work done per unit weight of flow through an annulus of thickness dr should be considered and this then integrated across the whole flow area from the hub to the tip

2.7.1 Blade Element Theory An axial flow pump impeller may have a large number of blades spaced closely together or a few blades spaced far apart, while for mechanical strength considerations, the blade chord will vary from hub to tip. The peripheral distance betwecn similar points on two adjacent blades is the pitch, and the ratio blade chord/blade pitch at a given radius is known as the solidity ratio a: a = cis

(2.46)

It is therefore possible to have high- or low-solidity blades, an impeller with a low number of blades ilnplying a low solidity. Where the blades have a low solidity, flow interference frOlTI one blade to the next is low and the blade may be considered to be acting alone in a free stream and is analysed as such. However, for high-solidity blades implying very closely spaced blades, the flow between the blades will be greatly influenced by the adjacent blades and we must resort to cascade data for an analysis of the forces acting on them. Since axial flow pump impellers invariably have less than six blades, it is usual to consider only isolated blade clement theory for them and this is now briefly

E = WlIng = U( U - Cu cot fJ2)/Y

or But the incren1ental nlass flow rate dl1Z is dl1z = p(2nr) (CJ dr and U = wr. Therefore RI

W = 2npC(JuJ

J

r 2 (evr - C el cot f~ 2) dr

(2.45)

'\. '\.

Rh

Equation (2.45) can only be integrated if the relationship between fJ2 and r is known. For design purposes it is usual to select conditions for use in Eq. (2.44) at the mean radius (R h + R()/2 along the blade. The whirl cOll1ponent ilnparted to

~

..... '-.1 __

A

I

-1

Figure 2.25 Circulation around an isolated blade

60

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

HYDRA ULIC PUM PS

described, the treatment of cascade analysis being reserved for the section on axial flow compressors, to which machines it is more appropriately applied. Consider the circulation r around the control surface of the isolated blade shown in Fig. 2.25, where the lengths AB and CD are the blade pitches at inlet and outlet and AD and BC bisect the flow passages between adjacent blades. The circulation is given by the line integral around ABCD and may be evaluated by summing the individual circulations comprising the circuit, such that

r

AIlCO

=

f

Vds=

r

Vds 2

+ i~ Vdl+

t

D

Vds l

+

I:

while

Jc

V dS I = -

eXIS I

Hence (2.48) For a number of blades shown in Fig. 2.26 the circulation around each blade may be determined, and the total circulation about Z blades is the sum of the individual circulations, remembering that along a line such as BG the circulation for one blade is positive (anticlockwise) while for the adjacent blade it is negative (clockwise). Therefore for Z blades the total circulation is

r ADEH == zr b b

(2.49)

is the circulation around a single blade.

ZS2 = 2nr 2

Therefore From Eq. (2.43),

E=

wzr b/2ny

CL = I'D

r

and

JD

and

where

d )

(2.50)

The Kutta loukowski law (L = p U or,J for lift per unit span on an aerofoil may now be used, where U 0 is the free stream velocity. Dividing this by 0.5p U 6c gives the lift coefficient

= - fA V dl

e V dl

r = Z(S2Cx2 -.'\\ C. But

(2.47)

where the circulation is positive anticlockwise. But

IB

Substitution into Eq. (2.48) gives the circulation for the whole impeller as

Hence Vdl

61

r

b

/0.5U oc

where c is the chord of the impeller blade. Since the appropriate free stream velocity for flo\v over the blade is the relative velocity W instead of U 0' and since this is different at blade inlet and outlet, the appropriate relative velocity is usually defined as H/~

= Cc; + [(W.d + Wx2 )/2J 2 = C; + [cot fi 1 + cot 112)/2J 2

Therefore

r h == a.5CI. W,XlC,

and substituting this into Eq. (2.50) E = 0.5coZC L Wooc/2ng

(2.51)

Values of lift coefficients for differing blade profiles may be determined from readily available tables and charts 5 and an estimate for E obtained. 2.7.2 Axial Flo,,, Pump Characteristics

o

C

/--- --7 -

/

/

/

I

.L

/

/

/

/

/ /

F

I

/

/

/

/

/

/

/

/

L

A

I

I

/

E

B

--7--- ---7

-

/1)1 1/) 11)/ /1 /

/

-

/

/

G

/

Axial flow pump design has evolved empirically and it is only in relatively modern times that aerofoil theory has been applied. Nevertheless, efficiencies of over 90 per cent were achieved using empirical data and it would seem that aerodynanlic design has not itnproved the efficiencies by much. Typical head--now, power and efficiency curves arc shown in Fig. 2.27. A steep negative slope is evident on the head and power curves at low flow rates. This can be explained by considering Eq. (2.44). For a given blade design at fixed speed with axial now at inlet

/ ./

/

/

H

Figure 2.26 Circulation around a number of isolated blades

E = U(U - CLIent /1 2 )/U

Now Q is proportional to C a and therefore dE/dC a ex dE/dQ ~ - U cot 11 2

HYDRAULIC PUMPS

For axial flow at inlet, ti 2 is relatively small, and thus for a given pump at a given speed the head- flow relationship has a steep negative slope. The power curve is similarly very steep, the power requirement at shut-ofT being perhaps 2-2.5 times that required at the design point. This makes for a very expensive electric motor to cover the eventuality of low flow rates and so the fixed-blade axial flow pump is limited to operation at the fixed design point. Variable-flow machines may be designed employing variable blade stagger or setting angles. Here the blade angle is adjusted so that the pump runs at its maximum efficiency at all loads and also reduces the shut-off power requirement. Figure 2.28 shows the efTect of changing blade stagger angle. In Fig. 2.27 the power and head curves are seen to enter a region of instability at about 50 per cent of the design flow rate. This is due to Ca becoming increasingly slnall and thereby increasing the angle of incidence of flow onto the blade until separation and stalling of the blade occurs. The further head rise at even lo\ver flow rates and the consequent power increase is due to recirculation of the fluid around the blade fronl the pressure side to the suction side and then up onto the pressure side of the next blade. An increased blade stagger angle will once again reduce this recirculation and thereby the power req uiremen t.

N", = 4 rad

Head

,, ,

,,

,,

,,

,,

,,

100 = design point

,

/', ",

Power

,, "

' ...... __

.... /

-

.....

I

Effici,ency

o

20

40

60

80

100

120

63

140

Percentage of design flow rate

2.8 PUMP AND SYSTEM MATCHING

Figure 2.27 Axial flow punlp characteristics

Efficiency lJ,H

Loci of maximum efficiencies

\

\

\ \

----~

Increasing stagger angle

\ \ \ \

\ \ \ \

\

It has been shown that a hydraulic pump has a design point at which the overall efficiency of operation is a maximum. However, it may happen that the pipe system in which the pump is being used is unsuited to the pump and a different pump with a more suitable characteristic is required. This section will examine how a pump and a pipe system may be nlatched to each other, the effect of changing the pump speed and dialneter, and finally the efTect of connecting pum ps in series and parallel. Consider the pipe system in Fig. 2.1. On the suction side the losses expressed in terms of standard loss coefficients are the sum of the minor losses hin

/Head

,,-

\

", '--,.,

h·In

= ~kVtj2f1 i...J I ~

and the friction loss

_/

hfi =

4llj V; j2yd

j

where f is the Darcy friction factor, I is the length of the inlet pipe and dj its diaIl1eter. Thus the total head loss is j

/ /'

Increasing stagger angle

Q Figure 2.28 Changing blade stagger angle on an axial flow pump 62

On the delivery side the

SUlll

of the bend, friction and exit losses that must be

64

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

HYDRAULIC PUMPS

overcome is

Ito = 4f/o V;/2gd o + Ik V; /2g Finally, the liquid must be moved from the lower reservoir to the upper reservoir through the static head H s ; hence the total opposing head of the pipe system that must be overcome in order to move the fluid from the lower to upper reservoir is H

= H s + Ito + hi

(2.52)

No.w from th.e continuity equation (Eq. (1.21)) the flow rate through the system IS proportIonal to the velocity. Thus the resistance to flow in the form of friction losses, head losses, etc., is proportional to the square of the flow rate and is usually wri tten as System resistance

= K

Q2

(2.53)

It is a measure of the head lost for any particular flow rate through the system. ~fany.parameter in the system is changed, such as adjusting a valve opening, or InsertIng a new bend, etc., then K will change. The total system head loss of Eq. (2.52) therefore becomes H = H s + KQ2

(2.54)

and if this equation is plotted on the head-flow characteristic, the point at which Eq. (2.54) intersects the pump characteristic is the operating point, and this mayor may not lie at the duty point, which usually corresponds to the de~ign point and maximum efficiency. The closeness of the operating and duty pOInts depends on how good an estimate of the expected system losses has been made. In Fig. 2.29 the system curve is superimposed on the H-Q characteristic.

65

It should be noted that if there is no static head rise of the liquid (e.g. pumping in a horizontal pipeline betwecn two rcservoirs at the same elevation) then H s is zero and the system curve passes through the origin. This has implications when speed and diameter changes take place. Because of the flatness of rotodynanlic pump characteristics, a poor estimate of the system losses can seriously affect the flow rate and head~ whereas in posi tive displacement pumps, the ]i -Q curve is almost vertical and, even if the head changes substantially, the flow rate stays almost constant.

2.8.1 Effect of Speed Variation Consider a pump of fixed diameter pumping liquid with zero static lift. If the characteristic at one speed lv 1 is known, then it is possible to predict the corresponding characteristic at speed N 2 and also the corresponding operating points. Figure 2.30 shows the characteristic at speed N l' For points A, Band C the corresponding head and flows at a new speed N 2 are found thus. We have and Similarly and ]-f 1/ N~ = H 2/ N~

(2.56)

Applying Eqs (2.55) and (2.56) to points A, Band C and letting the corresponding points be A', B' and C', and

1J,H

or

H 2 oc Q~

Design point

(2.57)

H System resistance

c

/ /

/'

Operating point

Q

Figur

=

fJoo

=

Ca/W00 sin - 1 (40/56.26)

= 45.3°

Q/A

then

i = 3.8°

Now

2

G:~) C:2

Y Q2

= (4 x 0.005 x 1000 x 16)Q2 2 x 9.81

X

rr 2 x 0.2 5

= 5164Q2 m (Q in m 3 /s)

82 HYDRAULIC AND COMPRESSIBLE FLO\V TURBOMACIIlNES 70

HYDRAULIC PUMPS

70

83

. pgQH Power to dnve pump = - -

Etllciency

Yf

60

10

60

3

X

9.81 x 145 x 40.2

60 x 60 x 0.625 0

50

__ 40

40

g

t::0 (J

!

"'0 ~

25.4kW

Exercise 2.9 (a) Figure 2.39 shows the head-flow and efficiency characteristics plotted for the speed of 750 rpm. Since water is being transferred between reservoirs of the same water level, then from Eq. (2.53), System resistance = KQ2

~,

0

:r:

=

50

(j

c

30

it)

30 'u ~ iJ.l

20

10

K = 35/25 2 = 0.056 Therefore the system head loss at the difTerent flow rates may be calculated:

20

//

Pump characteristic

Solving for K at the point given:

10

Q(m 3 jmin)

o

7

14

21

2X

35

System loss (m)

o

2.74

1I

24.7

43.9

6S.6

42

49

S6

The system resistance curve is now drawn (note that it passes through zero) and the head and now read ofT at point A. The corresponding efficiency is read off at point n. At the operating point

OIL..---......L.----..L l...--_ _---l...-_ _---J o o 50 100 150 200 250 Volume flow rate. Q (m:l/h)

Figure 2.38 Pump and system characteristics

Q == 26 In 3/ n1 in Ii == 38.3 m 1/ == 81 per cent

System resistance H = 32 + 5164Q2 m

System resistance, H (m)

(b) Sum of the head losses and static head is given by Eq. (2.54):

o

46

92

138

184

32.0

32.8

35.4

39.6

45.5

230 53.1

The operating point is at the intersection of the pump characteristic and system resistance curves. At the operating point A in Fig. 2.38

H==H s +KQ2

The head losses may be written as 4flv 2 v2 Head losses == - - + k exil -

2gd

2g

v2

+ kenlrY-2--

9

) v2 4 x 93 x 0.004 + 1 +0.5 0.45 2g

Q = 145 m 3 /h

=

H = 40.2m

== (3.31 + 1 + 0.5) -2

(

v2

9

The efficiency corresponding to the flow rate of 145 111 3/h is '7 = 62.5 per cent

v2

== 4.81 2g

84

HYDRAULIC AND COMPRESSIBLE FLOW TURROMACIfINES

50

HYDRAULIC PUMPS

Including the static Jift

,100

System Joss = 15

System characteristic with zero static head

40

80

I

I

... --- .../..... I I

, I

",

\

____________

\

t I

I I

30

,, I

~

t

g

I I

,,

"0

t

~

I

:t

~

---------j

I

t I I

Q(m 3 jmin) lI(m)

o 15

7 15.1)

14

21

28

15.52

16.18

17.11

35 IH.)

42 19.75

49 21.47

70

The new system resistance curve is drawn noting that it begins at H = 15 m. The operating point is at point C and the corresponding efficiency at point D. At the operating point

60

c d)

U

50 ], >-. u

H

c::

d)

= 20.4m

Tj

40

20

+ 9.69Q2 m

The head Joss is now dctcrrnincd for thc various flo\v rates.

90

I I I

85

~

'1 = 68.4 per cent

I

E\ ~ \

30

I

,

Power absorbed = pgQH '1 3 10 X 9.81 x 45 x 20.4

I

\

\

,

I

--,I

975 rpm \ \

10

-20

I I I

,

0.684 x 60

I

\ 975 rpm

.....--I I I

10

=219.4kW

o

(c) Since we have static lift, it is necessary to construct part of the characteristic at the new speed of 900 rpm. The corresponding points for the new impeller and the new speed are found from Eg. (1.6):

I I I I

o

'£.-_----L_ _----l....-_------L'..L.-_--L_ _-L-_..l.....-..l.....----.J

o

10

20 30 40 50 Volume flow rate (m 3 /min)

60

Figure 2.39 Pump characteristics at 750 and 900 rpm

and whence 975) (0.51)3 Q2 = Ql ( 750 0.7

Now v= Q/A and substituting for v 2

4.81 X 4 Head losses = [ 2 x 9.81 (n x 0.45 2)2

= 9.69Q2 m

l

= 0.503Ql 2

Q

975 x 0.51)2 H =}-I 2 1 ( 750 x 0.7

86

HYDRAULIC AND COMPRESSIBLE FLOW TURUOMACHINES

0

QI

14

7

21

28

35

HYDRAULIC PUMPS

42

49

56

Q2

0

3.5

7.1

10.6

14.2

17.7

21.3

24.8

28.3

HI

40

H2

36

40.6 36.5

40.4 36.4

39.3 35.4

38 34.2

33.6 30.24

25.6 23.0

14.5 13.1

0

87

30

0

The new characteristic is dra\vn and also the efficiency curve by nloving the corresponding values of efficiency horizontally across. The operating point is at E and the corresponding efficiency at G. At the operating point

25

20

Pumps in series

g :t:

15 -d ~ 0

::r:

, \

11== 16.51n

10

, .....

'--

A .....

"-

" '\B

'/ == 62.4 per cen t Power absorbcd ==

\

System curve

23.75 x 16.5 x 10J x 9.81

-- -

0.624 x 60

---0

== 102.7 kW Exercise 2.10 The single pun1p characteristic is plotted in Fig. 2.40 along with the characteristics for the pun1ps connccted in parallel and series. Since the san1e pUlnp is used in both cascs, for the series connection the flow rate through the two pumps remains the saIne while the head is doubled and for the parallel connection the head across the purnps rcrnains the same while the flow rate is doubled.

'\ '\

~

Single pump

0

0.2

0.4

" 1.0 0.6 0.8 Volume flow rate, Q (m 3 /s)

H(m)

H(m)

1.4

From Eq. (2.54) for the system curve H==H s +KQ2

But H s == O. Therefore H == 0.48 2 K

0 25.2

0.136 18.9

0.233 19.58

0.311 18.14

0.388 15.22

0.466 10.9

0.608

and K == 42.32

0

The system characteristic is parabolic and may now be drawn in for various heads and flow rates and the point B gives the operating point for the single pump within the system.

Parallel connection Q(m 3 /s)

1.2

Figure 2.40 Axial now pump characteristics when connected in series and parallel

Series connection Q(m 3 js)

Pumps in parallel

'\

5

0

0.272

O.4()6

0.622

0.776

12.6

9.45

9.79

9.07

7.61

0.932 5.45

1.216 0

At point A both connections give the sanlC head and flow and the systen1 characteristic must pass through this point and zero since there is no static lift. At operating point A in Fig. 2.40

System now rate (m 3 js) 0

0.1

0.2

0.3

0.4

0.5

0.6

System head loss (01)

0.42

1.69

3.81

6.77

10.58

15.23

0

The single pump operates at point B:

Q == 0.41 m 3 Is

3

Q == 0.48 n1 /s

H == 9.75 III

H == 7m

HYDRAULIC TURBINES

89

Table 3.1 Operating range of hydraulic turbines CHAPTER

THREE HYDRAULIC TURBINES

3.1 INTRODUCTION Turbines are used for converting hydraulic energy into electrical energy. The capital costs of a hydroelectric power scheme (i.e. reservoir, pipelines, turbines, etc.) arc higher than thermal stations but they have many advantages, some of which are:

1. 2. 3. 4. 5. 6.

High efficiency Operational flexibility Ease of maintenance Low wear and tear Potentially inexhaustible supply of energy No atmospheric pollution

The main types of turbines used these days are impulse and reaction turbines. The predominant type of impulse machine is the Pelton wheel, which is suitable for a range of heads of about 150-2000 m. One of the largest single units is at the New Colgate Power Station, California, with a rating of 170 MW. Reaction turbines are of two types: 1. Radial or mixed flow 2. Axial flow Of the radial flow type the Francis turbine predominates, a single unit at Churchill Falls having a power output of 480 MW with a head of 312 m. Two 88

Pelton Wheel Dimensionless srccific 0.05-0.4 speed (rad) Head (01) 100-1700 Maximum rower (k\V) 55 Best efficiency (%) 9J Regulation Spear nozzle and mechanism dcncctor rlate

Francis turbine

Kaplan turbine

0.4-2.2

1.8-4.6

80-500

up to 400 30

40 94

94

Guide vanes

Blade stagger

types of axial now turbines exist, these being the propeller and Kaplan turbines. The former has fixcd blades whereas the latter has adjustable blades. Table 3.1 summarizcs the head, power and efficiency values that are typical but by no means maxima for each type of turbine. A reversiblc pump-turbine can operate as either a pump or a turbine and is used in pump-storage hydroelectric schemes. At times of low electricity demand (e.g. during the night) cheap electricity is used to pump water from the low- to the high-level rescrvoir. This watcr may then be used during the day for power generation during peak periods, when the unit runs as a turbine in the reverse direction. One of the largest pump-storage schemes in the world is at Cabin Creek in Colorado, where each turbine generates 166 MW with a head of 360 m. In the sections that follow, each type of hydraulic turbine will be studied separately in terms of the velocity triangles, efficiencies, reaction and method of operation.

3.2 PELTON WHEEL The Pelton wheel turbine is a pure impulse turbine in which a jet of fluid issuing from a nozzle impinges on a succession of curved buckets fixed to the periphery of a rotating wheel, as in Fig. 3.1, where four jets are shown. The buckets deflect the jet through an angle of between 160 and 165 0 in the same plane as the jet, and it is the turning of the jet that causes the momentum change of the nuid and its reaction on the buckets. A bucket is therefore pushed away by the jet and the next bucket moves round to be similarly acted upon. The spent water falls vertically into the lower reservoir or tailrace and the whole energy transfer from nozzle outlet to tailrace takes place at constant pressure. Figure 3.2 shows a large Pelton wheel with its buckets. A diagram of a Pelton wheel hydroelectric installation is shown in Fig. 3.3. The water supply is from a constant-head reservoir at elevation H 1

90

HYDRAULIC AND COMPRESSIBLE rU)\V TURBOMACIII~LS

Figure 3.2 Pelton wheel (courtesy Escher Wyss Ltd)

Figure 3.1 Elements of a Pelton wheel turbine

of

(('(wries)' (~j' /:"sc},el' ~V)'ss Lid)

above the centre-liIle of the jet. A shallow-slope pressure tunnel extends from the reservoir to a point alITIOst vertically above the location of the turbine. A pipe of almost vertical slope called the penstock joins the end of the pressure tunnel to the nozzle, while a surge tank is installed at the upper end of the penstock to damp out flow control pressure and velocity transients. It is emphasized that, compared with the penstock, the pressure tunnel could be extremely long, its slope is extrcI11cly shallo\v and it should undergo no large pressure fluctuations caused by inlet valve flow control. The penstock must be protected against the large pressure fluctuations that could occur between the nozzle and surge tank, and is usually a single steel-lined concrete pipe or a steel-lined excavated tunnel. At the turbine end of the penstock is the nozzle, which converts the total head at inlet to the nozzle into a water jet with velocity C 1 at atll10spheric pressure. The velocity triangles for the flow of fluid onto and off a single bucket are shown in Fig. 3.4. If the bucket is brought to rest, then subtracting the bucket speed U 1 from the jet velocity C 1 gives the relative fluid velocity It'l onto the bucket. The angle turned through by the jet in the horizontal plane during its passage over the bucket surface is Y. and the relative exit velocity is W z . If the

Surge tan_;-- ~ / Pressure tunnel

I

!

I

I

,I

I

II

i

h,

I

H' :I

Penstock

Datum

H

1J

Figure 3.3 Pelton wheel hydroelectric installation 91

92

HYDRAULIC AND COMPRESSIBLE FLOW TURBOMACHINES

t' "

I

V

t·

WI

-r

)

)

j.

-1:

I

93

" \

\ /

Theoretical

/7

J

Ct

HYDRAULIC TURBINES

Actual

zJ

C2 Figure 3.4 Velocity triangles for a Pelton wheel

V

0.5

bucket speed vector U 2 is added to W2 in the appropriate direction, the absolute velocity at cxit, C 2' results. It should be rcalizcd that the component C x2 of C 2 can bc in the positive or negative x direction depending on the magnitude of U. From Euler's turbine equation (Eq. (1.24)) W/nl

= U IC xl

+ WI) + [W2 cos(180° -

'1 -

Assuming no loss of relative velocity due to friction across the bucket surface = W2 ), then -

WI cosa)

Therefore E= U(C I

-

U)(1-cosa)/g

(3.1)

the units of E being watts per newton per second weight of flow. Equation (3.1) can be optimized by differentiating with respect to U. Thus dE/dU = (1 - cos a)(C I

-

Eq. (3.1) becorl1cs

a) - UJ}

(WI

W/n1 = U(W1

Efficiencies and jet speed ratio of a Pelton wheel

-

U)(t - kcosa)/y

where k is the relative velocity ratio W 2 /W I • If the hydraulic efficiency is defined as

and since in this case C x2 is in the negative x direction, W/m = U{(U

3.~

E = U(C I

U 2 C x2

-

Figure

2U)/g = 0

II -

=

Energy transferred -_.-Energy available injet

E/(Ci/ 2g)

(3.4)

then if C( = 180°, the maximum hydraulic efficiency is 100 per cent. In practice, the deflection angle is in the order of t 60-165° to avoid interference with the oncoming jet and 1JH is accordingly reduced. Figure 3.5 shows the theoretical efficiency as a function of speed ratio. The overall efficiency is lower than the theoretical as well as having a reduced speed ratio at maximum efficiency. This is due to pipeline and nozzle losses, which will be discussed in a later section.

for a maximum, and then

3.2.1 Pelton Wheel Load Changes or (3.2)

Substituting back into Eq. (3.1) gives E max

=

Ci( 1 - cos a)/4y

(3.3)

In practice, surface friction of the bucket is present and W 2 #- WI' Then

Hydraulic turbines are usually coupled directly to an electrical generator and, since the generator must run at a constant speed, the speed U of the turbine must remain constant when the load changes. It is also desirable to run at maximum efficiency and therefore the ratio U/C I must stay the same. That is, the jet velocity must not change. The only way left to adjust to a change in turbine load is to change the input water power. The input water power is given by the product pgQH' but H' is constant

94

HYDRAULIC AND COMPRESSIBLE I·'LO\-\, TURBOMACIIINLS

HYDRAULIC TURllINES

Fl0~g_~_~ __ *_~~_ High load

95

when expressed as kinetic energy per unit weight of flow. Now .. " . Energy at end of pipeline PipelIne tranSrl11SSl0n effiCiency = - - -- -- ------. Energy available at reserVOIr or

Low load

I/lrans

= (H 1 -

hdlH 1

= HIH 1

(3.5)

and

Deflector plate in normal

Position~", " '

<

,,

.. . Nozzle enlclency

Energy at nozzle outlet

= ------- ---- - -

----

Energy at nozzle inlet \\

Bu~kel trajectory

,

-----~~--'==~~-~--.-==-~~~-.: --

Fully J 0.5, Fig. 5.7b shows the diagram skewed to the right since f32 > aI' and the static enthalpy rise in the rotor is greater than in the stator. The static pressure rise is also greater in the rotor than the stator. If R < 0.5 the diagram is skewed to the left as in Fig. 5.7c, and static enthalpy and pressure rises are greater in the stator than in the rotor. A reaction ratio of 50 per cent is usually chosen so that the adverse pressure gradient over the stage is shared equally by the stator and rotor. This decreases the likelihood of boundary-layer separation in both the stator and rotor blades and is the condition for maximum temperature rise and efficiency.

5.4 STAGE LOADING If the power input is divided by the term nlU 2 , a dimensionless coefficient ljJ, called the stage loading factor, results:

C L = LIO.5pW~A

where the blade area is the product of the chord c and the span I, and putting

D

---...,u WI

/ /

/ /

L

/

/

/

/ /

/

/

Figure 5.8 Lift and drag forces on a compressor rotor blade

/

w2

190

AXIAL FLOW COMPRESSORS AND FANS

HYDRAULIC AND COMPRESSIBLE FLOW TURUOMACllINES

191

5.6 BLADE CASCADES

fJ:x

~\r----~~~FI I

",

The previous sections have concentrated on relating the required energy transfer or stage work to the blade inlet and outlet angles for both the rotor and stator. The next requirement is to decide on the blade shape that will give the required stage work at the maximum efficiency along with the minimum of pressure loss. In Sec. 2.7.1 use was made of blade element theory to relate the blade lift coefficient to the energy transfer across the impeller of an axial flow pump (Eq. (2.61)), the blades of axial flow pumps and hydraulic turbines being of low solidity. Axial flow C0l11preSSor (and gas turbine) blading is of high solidity, with the result that the gas flow around a blade is affected by the flow around adjacent blades. In order to obtain information on the effect of different blade designs on air flow angles, pressure losses and expected energy tran.sfer across blade rows, one must resort to cascade wind tunnels and cascade theory. A cascade is a row of geometrically similar blades arranged at equal distflnces from each other and aligned to the flow direction as shown in Fig. 5.10. The row of blades is installed on a turntable at the end of a wind tunnel channel such that the angle of incidence of the blades with respect to the approaching air ITIay be varied. Vertical traverses between successive blades may then be made with pitot tubes, and yaw meters to determine pressure losses and air flow angles. Figure 5.10 is known as a linear cascade and can be imagined as a row of compressor blades unwound from the rotor to form the

I

I : I "

I

L

,I I

',R~

I I

I

"

I 'I

I I I

',I

t-...

I' I

"

I I

_-l __ ~1 D

Figure 5.9 Resolving blaJe forces into the direction of rotation

Woo == Ca/cos f3 00 then

F x == pC;cIC L sec fJoo[1 + (CD/CL)tan {J ocJ/2

The power delivered to the air is given by UF x == nl(h 03

-

11 01 )

== pC)s(h 03

-

(W)

ho 1)

where the flow through one blade passage of width s has been considered. Therefore

l/J == (11 03

-

h01 )/U 2

== F:jpClllsU == (C1JU)(cls) sec fJtr'(C L + CD tan {J,xJ/2 == ¢(c/s) sec {3 (C L + CD tan {J cxJ/2 00

(5.15)

For rnaximun1 efficiency, the Il1Can flow angle fJ ct.' is usually about 45° and substituting for this into Eq. (5.15), the optin1un1 bladc loading factor t/f becomes opt l/JOPI == ( P02 because no work is done in the cascade and the flow is irreversible. Eq uation (5.21) will be written as

+ (SPa/COS Ct.~) The first two ternlS on the RHS are equal and therefore disappear to leave

(5.22)

D=sj)ocOS~l-

where ~P = (P2 - PI) and Po = (Po 1

-

P02)'

The summation of all forces acting on the air in the control volume in the x and y directions must equal the rate of change of momentum of the air in these directions. Considering first the y direction, since Cll is constant, there is no velocity change from 1 to 2 in the y direction and consequently no momentum change. Hence for a unit length of blade,

Dividing the drag by 0.5 pC~c gives the drag coefficient C n = 2(s/c)(po/pC~,) cos aX)

But Coo = Ca/cos aa, and

C n = 2(s/(')(iJo/pC~)(cosJ ~,x./COS2 ad

Therefore -

s~fJ/cos

(5.26)

and L(I

+ Dsina oo = - PC ll S(C x2 - Cxd

~P

L = (p c,; s/ ens ~,., )( tan (/. t - tan C( 2) - {L tan (/.7' - (s/cos C(oo)[p(Ci - C~)/2 - Po]} tan a oo

(5.23)

a,t,

In the x direction the velocity changes from C xt to C x2 ' and noting that these are in the negative x direction, Lcosa,t,

= C 1 COS CXl~ thus substituting in Eq. (5.25) gives

A similar procedure Inay be followed for C L by substituting for D and in Eq. (5.24) to give

L sin a'/, - D cos a,t, - s~r = 0

D = L tan a ,x.)

ClI

(5.25)

+ tan 2 Ct.'xJ = (p('(;s/cos Ct.oo)(tan a 1 - tan ( 2 ) + (spC:'l;/2 cos (/.-. u c: v

60

tt:

40

'u

283

HYDRAULIC AND COMPRESSIBLE FLO\V TURBOMAClllNES

90° radial flow gas turbine

U.J

0.01

10.0

1.0 0.1 Dimensionless specific speed, Ns (rad)

= 2.11(C 2 /C s)O.5(A 2 /A r)o.5

30000 rpm 1.8 kg/m 3

7.3 The design data of a proposed inward radial now exhaust gas turbine are as follows:

Figure 7.5 Variation of efficiency with dimensionless specific speed

Thus substituting for Q2/NDi in Eg. (7.19) gives N s = 0.336(C 2/CJo. 5 (A 2 /A r )o.5

0.447 0.707

Determine: (a) the dimensionless specific speed of the turbine (b) the volume flow rate at impeller outlet and ' (c) the power developed hy the turbine.

20 0

90 mm 62 mm 25 mm

rev rad

(7.21 ) (7.22)

In practice 0.04 < (C 2jC s )2 < OJ 0.1 < (A 2 IA r ) < 0.5

Stagnation pressure at inlet to nozzles p Stagnation temperature at inlet to no~'zl~~o' T 00 S . tat~c pressure at exit from nozzles, PI StatIC temperature at exit from nozzles ' T 1 S tatic pressure at exit from rotor p Static temperature at exit from T Stagnation temperature at exit from Ratio r2.vlr I • 02 Rotational speed, N

r~t02r

'rot~r ~

700 kPa 1075 K 510 kPa 995 K

350 kPa 918 K 920 K 0.5 26000 rpm

The flow into the rotor is: purcl y rd. d"la I an d at exit. the flow is axial at all radii C a l l ' . cu ate. (a) the total-to-statlc efficiency of the turbine, (b) the outer diameter of the rotor (c) the enthalpy loss coefficient for' the nozzle and rotor rows (d) the blade outlet angle at the mean diameter {J • d ., (e) the total-to-total efficiency of the turbine. lav an 7.4 Using the data of exercise 7.3· n· .. now 0 f exhaust gas availahle to the turbine is 2.66 kg/s. Ie mass Calculate:

Then 0.3