COMPRESSOR
`AERODYNAMICS
`
`N.A.Cumpsty
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`Compressor aerodynamics
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`Compressor
`aerodynamics
`
`N.A. Cumpsty
`Department of Engineering
`University of Cambridge
`
`KRIEGER PUBLISHING COMPANY
`Malabar, Florida
`2004
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`We are gra
`
`Education Linu,~u’, the original publisher of Compressor
`for permission. ~to reprgdffce the typography and design of their edition.
`
`Original Edition 1989
`Reprinted 1996, 1997 and 1998
`Reprint Edition 2004 w/new Preface, Introduction and Updated Bibliography
`
`Printed and Published by
`KRIEGER PUBLISHING COMPANY
`KRIEGER DRIVE
`MALABAR, FLORIDA 32950
`
`Copyright © 1989 by Longman Group UK Limited (Pearson Education Limited)
`Transfered to Author
`Reprinted by Arrangement.
`
`All rights reserved. No part of this book may be reproduced in any form or by any
`means, electronic or mechanical, including information storage and retrieval syste_ms
`without permission in writing from the publisher.
`No liability is assumed with respect to the use of the information contained herein.
`Printed in the United States of America.
`
`FROM A DECLARATION OF PRINCIPLES JOINTLY ADOPTED BY A
`COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COM-
`MITTEE OF PUBLISHERS:
`This publication is designed to provide accurate and authoritative information
`in regard to the subject matter covered. It is sold with the understanding that the
`publisher is not engaged in rendering legal, accounting, or other
`professional service. If legal advice or other expert assistance is required, the
`services of a competeni professional person should be sought.
`
`Library of Congress Cataloging-in-Publication Data
`
`Cumpsty, N. A.
`Compressor aerodynamics / N.A. Cumpsty.
`p. era.
`Includes bibliographical references and index.
`Reprint. Originally published: Hariow, Essex, England : Longman Scientific &
`Technical, 1989.
`ISBN 1-57524-247-8 (alk. paper)
`1. Compressor--Aerodynamics. I. Title.
`
`TJ267.5.C5C86 2004
`621.5v 1--dc22
`
`10 9 8 7 6 5 4 3 2
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`2003069481
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`Contents
`
`Preface to the 2004 Krieger Reprint
`Preface
`Acknowledgements
`Notation
`Introduction. to the 2004 Krieger Reprint
`
`1
`
`Useful basic ideas
`
`¯ Introduction
`1.1
`112 Blades and flow
`1.3 Work input into compressors
`1.4 Dynamic scaling
`1.5
`Losses
`1.6 Efficiency
`
`2 - General design considerations
`
`Introduction
`2.1
`The axial compressor
`2.2
`The radial compressor
`2.3
`2.4 The matching of multistage compressors
`
`3
`
`Throughflow on the hub-casing surface and some
`aspects of flow in three dimensions
`
`Introduction
`3.1
`3.2 Approximations applicable to axial compressors: simple
`radial equilibrium
`Early developments
`3.3
`3.4
`Practical methods for the meridional flow
`3.5 Applications of streamline curvature methods in axial
`compressors
`3.6 Mixing in multistage axial flow compressors
`3.7 Axial compressor off-design trends
`3.8
`Flow chart -- use of a streamline curvature method in
`analysis mode for an axial compressor
`
`1
`1
`4
`11
`21
`34
`
`46
`
`46
`47
`62
`78
`
`93
`
`93
`
`97
`102
`106
`
`114
`121
`126
`
`129
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`4 Blade-to-blade flow for axial compressors with
`subsonic inlet flow
`
`Introduction
`4.1
`The effect of blade shape
`4.2
`Loading limits for blade rows
`4.3
`The selection of incidence
`4.4
`The prediction of deviation
`4.5
`The determination and prediction of losses
`4.6
`The effect of Reynolds number on blade performance
`4.7
`4.8 The effect of inlet Mach number on blade performance
`4.9 Concluding remarks
`
`132
`
`132
`140
`149
`159
`168
`171
`176
`180
`i91
`
`5 Blade-to-blade flow for axial compressors with
`supersonic inlet flow 194
`
`Introduction
`5.1
`5.2 Choked flow with attached shocks -- ’unique
`incidence’
`5.3 Operation with detached shocks
`5.4 ~hock structure and the nature of flow in supersonic
`rotors
`5.5 Losses in supersonic blading
`5.6 The design process for supersonic blades
`
`6
`
`The centrifugal impeller
`
`194
`
`198
`205
`-.
`209
`214
`217
`
`220
`
`220
`Introduction
`6.1
`223
`The flow pattern in impellers
`6.2
`6.3 Calculation methods and predictions of flow in impellers 236
`245
`Slip and the estimation of slip factor
`6.4
`249
`6.5
`Loss in impellers
`254
`6.6 Design choices for the impeller
`
`7
`
`The diffuser of the centrifugal compressor
`
`7.1
`Introduction
`7.2
`The nonuniform flow from the impeller
`7.3 The’vaneless diffuser
`7.4
`The vaned diffuser
`7.5
`The volute or scroll
`
`Viscous effects in compressors
`
`8.1
`8.2
`
`Introduction
`Three-dimensional viscous flows in compressors
`
`266
`
`266
`269
`276
`285
`301
`
`310
`
`310
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`8.3 Axial blade boundary layers
`Flow in the endwall regions of axial compressors
`8.4
`8.5 Viscous effects in centrifugal compressors
`
`9
`
`Stall and surge
`
`Introduction
`9.1
`Instability and the inception of stall
`9.2
`Post stall behaviour
`9.3
`9.4 The flow in the rotating stall cell
`Stability enhancement: casing treatment
`9.5
`
`10 Vibration and noise
`
`Introduction
`10.1
`10.2 Vibration
`Mechanical vibration modes
`Forced vibration
`Flutter
`Supersonic unstal!ed flutter
`10.3 Noise
`Scales and rating of noise
`Elementary acoustics
`Compressor and fan noise
`Non-aeronautical aspects of compressor noise
`Acoustic treatment
`
`11 Design, Measurement and computation
`
`Introduction
`11.1
`11.2 Understanding and design
`11.3 Experimental techniques
`11.4 Mathematical techniques
`
`320
`331
`356
`
`359
`
`359
`369
`391
`398
`401
`
`410
`
`410
`410
`412
`415
`417
`422
`428
`429
`431
`440
`455
`457
`
`459
`
`459
`459
`461
`466
`
`Appendix Blade profile families for axial compressors 479
`
`Bibfiography
`
`Additional Bibliography for the 2004 Krieger Reprint
`
`Index
`
`484
`
`505
`
`513
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`Notation
`
`There are very many notations in use for the consideration of turbomachines
`and it is just about impossible to evolve a system which has no duplication
`of symbols without recourse to excessive use of subscripts. It is hoped that
`the system adopted will represent a reasonable compromise which is fairly
`transparent and agrees with that generally used for the topic being discussed.
`The overlap that does exist here (for example rn denotes mass flow rate and
`meridional distance) should not confuse the reader too much. There are other
`inconsistencies (as Emerson wrote, a foolish consistency is the hobgoblin of
`little minds) but this should not be too irritating. The list given is not an
`exhaustive one and various additional symbols are introduced throughout the
`book.
`
`General points
`
`Throughout the book all angles are measured from the meridional flow direc-
`tion, which reverts to the axial direction for the blade-to-blade flow in axial
`machines and the radial direction towards the outlet of centrifugal compressors.
`(The terms radial or centrifugal compressor are used as alternatives without
`any implied difference; both are in common use).
`The velocity magnitude and direction in the relative or rotating frame of
`reference are denoted by W and/3 whilst in the absolute or stationary frame
`of reference they are denoted by V and oz.
`As is common in British and American practice for compressors, a conven-
`tion of positive and negative signs for flow or blade angles is not used; angles
`are taken a~ positive and the appropriate sense adopted.
`The word stagnation is normally used, as for stagnation enthalpy
`ho = h + V2/2, and not the word total. The usage total-to-static, as in total-
`to-static efficiency, is so widespread and the corresponding term based on
`stagnation so much harder to say that this is retained.
`The outer diameter of axial machines is sometimes called the tip. This may
`be ambiguous, for the tip of the stators is at the hub. The word casing is
`therefore preferred for the outer diameter. The phrase hub-tip ratio is so
`cornmon that this is occasionally used in place of hub-casing ratio for the
`ratio of the hub diameter divided by the casing diameter.
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`Variables commonly used
`
`Geometric variables
`
`b
`
`D
`
`d,
`g
`h
`rn
`r,R
`
`s
`
`s
`
`passage width in spanwise direction, used for centrifugal compressors
`blade chord
`diameter
`staggered gap, pitch resolved normal to the flow direction
`blade height, used mainly for axial compressors
`distance in meridional direction dm =x/(dx2+dr2), dx/Vx =dr/Vn
`distance in the radial direction
`blade pitch
`distance along streamline ds =x/(dx2 +dr2 + r2d02),
`dx / Vx = dr/Vn = rdO/V0
`
`blade thickness
`tip clearance
`distance in axial direction
`distance in the pitchwise direction
`distance normal to x and y
`solidity c/s
`
`Angles Relating to Blading (see Fig. 4:1)
`
`angle between a blade filament and the radial direction in axial
`view (blade lean)
`stagger (angle of chord line measured from the axiaH" direction
`camber
`angle in the circumferential direction
`blade inlet angle (measured from the axial~- direction)
`blade outlet angle (measured from the axialt direction)
`blade lean in radial compressors
`
`Flow variables
`Stationary frame of reference
`eq
`flow inlet angle (measured from the axial1" direction)
`flow outlet angle (measured from the axial’~ direction)
`or2
`inlet flow velocity
`V~
`outlet flow velocity
`V2
`
`Rotating frame of reference
`
`flOW inlet angle (measured from the axial1" direction)
`flow outlet angle (measured from the axial~ direction)
`
`~ For radial and mixed flow machines, angles are measured from the meridional direction rather
`than the axial direction. For axial machines when the meridional streamlines are inclined at a
`substantial angle to the axial direction, the angles are also sometimes referred to the meridional..
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`inlet flow velocity
`outlet flow velocity
`
`Subscripted velocities
`tangential component of velocity into blade row
`Vol
`radial component of velocity into blade row
`VRl
`axial component of velocity into blade row
`Vx~
`... likewise for other velocities, V2, W etc
`meridional component velocity, Vm=~/(Vx2 + V~)
`
`Vm
`
`Special
`i
`
`A
`
`angles
`incidence (angle between inlet flow direction and blade inlet
`direction, i=cq-xt or i=f3~-X~ for stator or rotor
`respectively)
`angle of attack (angle between inlet flow direction and the chord
`line, A =oq-~ or A =/~-~)
`deviation (angle between outlet flow angle and blade outlet
`angle, 6=Ot2--X2 or 6=~2--X_2)
`inclination of meridional streamline to axial direction
`inclination of meridional streamline to axial direction (used for
`radial machines)
`
`General
`
`variables
`
`A
`a
`a*
`
`AVDR
`b
`B
`
`C
`
`streamtube cross-sectional area
`velocity of sound
`velocity of sound at condition when flow sonic (similarly p*,p*
`etc.)
`axial velocity-density ratio P2Vx2/Pl Vx|
`streamtube depth measured normal to two-dimensional surface
`blockage, 1-(mass flow + mass flow across same section in
`ideal flow)
`,velocity of sound (Chapter 10)
`dissipation coefficient or integral
`skin friction coefficient, rw/(-~pU2)
`
`specific heat capacity at constant pressure
`staticpressure rise coefficient, (p-p 0/(Pot -P ~)
`Lieblein’s diffusion factor
`flow function, m(cpTo)~/2/Apo
`specific enthalpy
`specific stagnation enthalpy, h + V2/2
`specific rothalpy, h + W2/2 - U2/2
`acoustic wavenumber
`mass flow rate
`Mach number
`
`DF
`F
`h
`
`ho
`I
`k
`m
`M
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`N
`N
`P
`Po
`
`R
`R
`$
`T
`
`U
`
`number of blades
`angular velocity rev/min
`static pressure
`stagnation pressure, sometimes termed total pressure
`volume flow rate
`gas constant
`degree of reaction
`specific entropy
`static temperature
`stagnation temperature, sometimes termed total temperature
`blade speed
`boundary layer thickness
`boundary layer displacement thickness
`force deficit thickness
`efficiency
`boundary layer momentum thickness
`ratio of specific heat capacities cp/cv
`swirl parameter Vo/VR, used for radial° compressors
`acoustic wavelength
`dynamic viscosity
`kinematic viscosity,
`density
`slip factor, (absolute whirl velocity + ideal absolute whirl
`velocity)
`shear stress
`flow coefficient; Vx/U for axial, different definitions for radial
`compressors
`velocity potential
`stream function
`loading, Aho/U2
`loss coefficient, Apo/(Po~ --p
`angular velocity
`vorticity
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`2 General design, considerations
`
`2.1
`
`Introduction
`
`In the design of any compressor the initial decisions on the layout and duty
`determine to a large extent problems to be encountered and the level of effi-
`ciency to be_achieved. It needs to be recognized that the single most important
`design decision is the choice of stage loading, usually meaning the pressure
`rise in relation to the number of stages and the rotational speed. If, for exam-
`plel unduly high loading is required of one or more components it is probable
`that no subtlety of design will render the overall performance satisfactory. Great
`skill and extensive commercial databases may be involved in making the initial
`decisions, steering the choice between ambitious goals and safely realizable
`ones. Occasionally the preliminary design is not given the serious attention
`it deserves and the results may be catastrophic.
`The decision to have an axial or a radial compressor (radial compressors
`are very often termed centrifugal) is one of the basic preliminary decisions
`of this section and this excludes a potentially wide class of mixed flow machines.
`The mixed flow compressor is rarely used, probably because of the limited
`experience and data existing for it, although it would seem to have a very natural
`niche. Amongst the problems of the mixed flow compressor compared to the
`axial or radial is weight, with the mixed flow machine coming out longer than
`the radial but of similar massive construction. The diffuser downstream of
`the mixed flow impeller has also been found to be a problem, with performance
`well down on what was expected of the radial machine. With sufficient effort
`and appropriate design there is reason to expect that the diffuser performance
`could be greatly improved.
`The decision to choose either an axial or a radial compressor rests on many
`factors, not least the experience of the company building the machine. For
`aircraft propulsion the high flow rate per unit area of the axial is a big advan-
`tage but when the blade height becomes very small the advantage swings to
`the radial: helicopter engines usually employ radial compressors and they have
`even been proposed for the later stages of large jet engines with very high
`pressure ratios. Highly loaded radial compressors seem to have generally lower
`efficiencies than axial machines, but this is not altogether clear. In cases where
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`The axial compressor 47
`
`the impeller can be precision cast, such as those in turbochargers for automotive
`use, the .simplicity of the radial compressor means that it has a huge cost
`advantage over the axial. In the remainder of this chapter it will be assumed
`¯ that the decision of axial versus radial has been made.
`It is a common factor with all compressors that when several stages are used
`together in series there is a serious problem of matching the stages so that
`the outlet flow from one stage is acceptable to the next. This becomes more
`acute as the overall pressure ratio across the machine increases because of
`the large density changes that result. Because the pressure ratio, and therefore
`the density ratio, is roughly proportional to the square of the rotational speed
`it is a common difficulty to match multistage machines at both the full design
`speed and at reduced speed, leading to many problems not least that of starting
`the compressor or engine. This aspect of compressors is considered in this
`chapter and is illustrated with reference to the particular problems of axial
`compressors.
`What follows in this chapter are some fairly simple ideas relating to the
`overall performance of compressors with the treatment being essentially one-
`dimensional. This begins with the axial and then moves on to the radial com-
`pressor. The final section of the chapter is the elementary consideration of
`stage matching for axial compressors.
`
`2.2 The axial compressor
`
`In the preliminary design calculations are usually performed at a mean radius,
`called the pitchline in some work. Refinements may be introduced to assess
`the blade loadings at hub and casing, particularly if the ratio of the hub and
`casing radii is low. (NB: This is sometimes referred to as the hub-tip ratio.
`Here the word tip will not be used because of its possible ambiguity; for a
`stator cantilevered inwards, does tip refer to the hub or the casing end of the
`blade? The word casing will be preferred instead.) Criteria have to be chosen
`for satisfactory blade loading, pressure rise at the walls and maximum Mach
`number.
`The blade loading is now usually assessed by diffusion factor or alternatively
`equivalent diffusion ratio, both derived by Lieblein and described in Chapter
`4. Here diffusion factor will be used. Essentially this relates empirically the
`peak velocity on the suction surface of the blade to the velocity at the trailing
`edge, with one component due to the one-dimensional deceleration of the flow
`and the second due to the turning of the flow. The term related to the turning
`introduces the blade solidity. For a simple two-dimensional geometry diffu-
`sion factor reduces to
`~,
`
`DF = 1
`
`V~ + A Vo
`Vl
`2aV~
`
`(2.1)
`
`where V~ and Vz are the average velocities into and out of a blade row in a
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`48 General design considerations
`
`frame of reference fixed to the blade, A Vo is the change in Whirl velocity in
`the row and a is the solidity, equal to blade chord/blade pitch. Values of DF
`in excess of 0.6 are thought to indicate blade stall and a value of 0.45 might
`be taken as a typical design choice. Over the last few years attention has been
`focussed more on the endwall region as the limit for loading and the weight
`given to the diffusion factor has decreased.
`The criterion to be adopted for endwall loading or pressure rise is less clear,
`mainly because the fluid mechanics is still not understood. Methods analogous
`to that produced by de Hailer (1953) are still current and this will be discussed
`more in later chapters, de Hailer deduced that the velocity out of a blade should
`not be less than about 0.75 times the inlet velocity if the performance is to
`be satisfactory. This is equivalent to requiring that the static pressure rise at
`the wall should not exceed about 0.44 times the dynamic pressure into a blade
`row. The de Hailer criterion has not been found to be entirely satisfactory.
`More recently Koch (1981) has published a method which relates stage pressure
`rise capability to the mean height (i.e. mid-span) solidity averaged over the
`stage; it is based on a large number of measurements in multistage compressors
`and will be discussed, more fully in Chapter 9. The most, common method of.
`assessing what is acceptable loading at the wall is probably by reference back
`to previous designs by the same manufacturer, it is now very rare for an
`organization to be designing an axial compressor for the first time! The general
`view seems to be that a stage pressure rise not exceeding about 0.4pU2 is
`reliable.
`In looking at the trends in multistage compressor design it is very helpful
`to take advantage of the results given by Wisler (1988) in a comprehensive
`setof lecture notes, relating mainly the work of his company, General Elec-
`tric. These will be referred to many times in this chapter.
`The limit on maximum Mach number is flexible and depends to a large extent
`on the balance between high efficiency and high pressure ratio per stage being
`sought. The loss in efficiency with Mach number is nowhere near as serious
`as was once thought. Inlet relative Mach numbers of 1.4 are now common
`at the tips of first-stage rotors _in multistage compressors for aircraft and the
`flow may even be slightly supersonic into the third stage. As the Mach number
`is increased the operating range reduces, i.e. the difference between the mass
`
`Table 2.1 Compressor developments by General Electric .,
`
`Year
`
`Designation
`
`late 50s
`1969
`1974
`1982
`
`CJ805/J79
`CF6-50
`CFM56
`E3 engine
`
`Design
`pressure
`ratio
`
`12.5
`13.0
`12
`23
`
`Number
`of stages
`
`Corrected
`tip speed (m/s)
`
`17
`14
`9
`10
`
`291
`360
`396
`456
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`The axial compressor 49
`
`Adiabatic
`efficiency
`
`÷++
`
`+ +
`f~ (
`97.5°/o 100
`
`102.
`
`x× ~"
`~" f
`/,.,+,~ /,,-
`95
`,x’~8~2
`.585 87.5 90 92.5
`4- ~3
`,+ ~ 80
`70
`
`,
`
`I
`
`Corrected
`design speed
`’ I
`
`-f~f~l
`50 60
`I
`=
`
`I
`=
`
`i
`
`I
`
`I
`
`I
`Design
`point
`
`97.5
`
`Corrected
`
`92.5
`
`0.9
`
`0.8
`
`0.7
`
`0.6 -
`
`Pc2 25
`
`Pc1
`
`20
`
`15
`
`10
`
`5
`
`0
`
`0
`
`I
`10
`
`82.5
`
`I
`20
`
`I
`_ 30
`
`I
`40
`Inlet corrected mass flow kg/s
`
`I
`50
`
`60
`
`Fig. 2.1 The pressure ratio and efficiency characteristics of the General Electric E3
`compressor. (Published with permission, Courtesy of General Electric Co.)
`
`flow for choke and surge is reduced. An important reason for keeping the speeds
`0f industrial compressors down is to maintain the widest possible operating
`range. The numbers given in Table 2.1 are taken from Wisler (1988) and show
`the trend for much higher tip speeds from one manufacturer of jet engines,
`General Electric, but similar trends would be found for other companies as
`well as for land-based machines.
`The pressure rise-mass flow and efficiency-mass flow characteristics for
`the compressor of the E3 engine are shown as Fig. 2.1, the solid lines being
`from tests of a compressor rig and the crosses from engine tests. Just prior
`to surge at 102.4 per cent speed the very high pressure ratio, of 29:1 was
`achieved. It is interesting that the engine performance is better than the rig,
`partly because the Reynolds number was higher but mainly because the tip
`clearances were smaller for the engine. The compressor was designed with
`six variable stagger stator rows but only four were used for the performance
`map shown. The peak adiabatic efficiency corresponds to a polytropic effi-
`ciency of 90.4 per cent, a high value, and evidently .the high pressure ratio
`per stage does not have to be bought at the expense of low efficiency. A
`photograph comparing the rotors for the E3 compressor with that of the much
`earlier CJ805/J79 is shown in Fig. 2.2 from which the very much higher solidity
`and lower aspect ratio of the more recent compressor is very obvious.
`Decisions have to be taken regarding the blade chord and the number of
`blades. Increasing the chord reduces the aspect ratio (height/chord) and in-
`creases solidity (chord/pitch) for the same annulus and number of blades. Both
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`50 General design considerations
`
`(a)
`
`(b)
`
`Fig. 2.2 Comparison of (a) CJ805/J79 rotor (late 1950s) Po2/Poi = 12.5; 17 stages
`and (b) E3 rotor (early 1980s), Po2/Pol = 23; 10 stages. Note lower aspect ratio and
`higher solidity of newer machine. (Published with permission, Courtesy of General
`Electric Co.)
`
`these trends are evident in Fig. 2.3, taken from Wisler (1988). The rise in
`solidity and fall in aspect ratio can both’ be attributed in the main to a rise
`in chord length. With these trends for aspect ratio and solidity there is the
`striking rise in pressure rise per stage and the increase in the overall press_ure
`ratio possible and utilizable for a single compressor. It should be emphasized
`that the single most important decision in the design process is the choice of
`a realistic stage loading. An over-ambitious choice may lead to untold problems
`later with little possibility of actually achieving the combination of efficiency,
`pressure ratio, mass flow and range originally intended.
`Back in the 1950s it was believed that the trend would be towards high aspect
`ratio blades to give a short compressor, mainly, it seems, because the blade
`behaviour well away from the endwalls was comparatively well understood
`and this was the direction of development which consideration of the blades
`seems to indicate. The trend was reversed mainly because large chord blades
`are more effective in the endwall regions and it is these regions which are
`crucial in determining both the efficiency and the stall point. High aspect ratio
`blades were long and thin and had atrocious vibration problems. The change
`towards low aspect ratios was not the result of an understanding of the processes
`involved but consideration of the trends for performance of different designs.
`Wennerstrom (1986) has described the catastrophic effect of adopting high
`aspect ratio blading.
`There are several performance goals to be compared, in particular pressure
`rise, efficiency and operating range (operating range might be defined as the
`ratio of the difference between maximum and minimum mass flow to the design
`value). The evidence suggests that for a good compressor near the design point
`efficiency tends to be slightly lower if the solidity is on the high side (and
`the aspect ratio low) but the pressure rise and operating range are greater.
`The major trend over the last 30 years has shown a rise in efficiency but a
`more marked rise in overall pressure rise as Fig. 2.4, from Freeman and
`Dawson (1983), shows for Rolls-Royce compressors.
`There are special problems that arise from combining stages to form
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`The axial compressor 51
`
`Stage
`average
`solidity
`
`Average
`aspect
`ratio
`
`Average
`loading
`AP/(Po- P)i.
`
`Spool
`pressure
`ratio
`
`1.6
`
`1.4
`
`1.2
`
`1.0
`
`4
`3
`2
`1
`
`0.4
`
`25
`
`20
`15
`
`10
`~ ~ ~
`5
`1950 1960 1970 198~
`Year
`
`Fig. 2.3 The trend in compressor geometry (solidity and aspect ratio) and in perform-
`ance (stage loading and spool pressure ratio) with time. (From Wisler, 1988)
`
`I
`
`0.9
`Polytropic
`efficiency
`
`0.8
`
`30 Overall o~
`pressure rati
`PR 20 f
`10
`
`0 I I I
`1950 1960 1970 1980
`
`Fig. 2.4 The variation in overall pressure ratio and in polytropic efficiency for gas
`turbine compressors. (From Freeman and Dawson, 1983)
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`52 General design considerations
`
`multistage compressors, usually referred to as matching, and this is considered
`later in this chapter. The ability to handle the matching of compressors and
`the operation of several rows of variable stagger stator blades has made pos-
`sible the very large increase over the years in the pressure ratio for a single
`compressor spool which is illustrated by Fig. 2.3.
`Increasing Mach number by increasing rotational speed can lead to
`mechanical problems. The limiting condition for a compressor with large
`pressure ratio is normally reached at the rear hub; this is largely a materials
`problems connected with the high temperatures. High solidity blading exacer-
`bates the problem-because of its greater mass of blade metal. A maximum
`hub rotational speed of about 380 m/s may be taken as a guide, but this is
`not a firm boundary because the choice of more expensive materials or the
`use of a heavier disc would, at a price, allow some increase.
`Increased rotational speed makes it possible to increase the flow per unit
`area; Freeman and Dawson (1983) show that it is now possible to have an
`efficient compressor giving a high stage pressure, ratio while passing a flow
`approaching 90 per cent of that which would choke the empty annulus at inlet.
`With the emphasis on blade design for axial compressors it is easily over--
`looked that the overall meridional flowpath (that is the flowpath in a longitudinal
`cross-section showing axial and radial components) has a crucial effect on the
`design and the performance of a compressor. The aerodynamic problems are,
`for example, greatly relieved if the hub radius can increase from front to back,
`whilst they are made worse if the annulus area is too large towards exit. Deci-
`sions taken at the preliminary stage in laying down the annulus shape and
`choosing the inlet and outlet radii can effectively determine whether a com-
`pressor will be satisfactory or not and may be far more influential than
`subsequent decisions regarding the blade shape.
`Fundamental to all of the aerodynamic design are the basic decisions of an
`aerodynamic nature. At the blade mid-height (sometimes known as the pitch-
`line radius) a choice must be made for the local flow coefficient ~b = Vx/U and
`the stage loading ~=Aho/U2 (or alternatively z~p0/p U2). Sometimes the
`degree of reaction R =z~hrotor/Ahstage (or the equivalent in terms of static
`pressure rise) is treated as important too. Such decisions are separate from
`choice of solidity, blade section etc., although solidity does have a marked
`effect on the choice of loading. A fascinating report of the preliminary design
`of a multistage compressor has been published by Wisler et al. (1977). Here
`the interactions between different design decisions are demonstrated with the
`advantage of the realistic estimates and extensive database available to a large
`company.
`
`Parametric study for a repeating axial stage
`The decisions on aerodynamic design take into account amongst many other
`things the compressibility of the flow, and this will be considered in later
`chapters. However, many of the trends in most stages of a multistage
`compressor are not related to compressibility and the flow can be understood
`adequately by treating it as incompressible. In this section some parametric
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`The axial compressor 53
`
`studies will be used to demonstrate the effect of various choices of design
`variables. The blading is two-dimensional, so that all endwall effects are
`ignored. A simple axial stage will be considered with identical velocities in
`and out; such a stage is often called a repeating stage.
`In the parametric study the diffusion factor will be taken as the measure
`of blade loading and comparisons will be made varying flow coefficient ~,
`blade loading ¢ and degree of reaction R as independent variables. Some
`estimate of efficiency is made utilizing data measured in two-dimensional
`cascades by Lieblein (discussed in Chapter4). This shows that as a reasonably
`good approximation the blade profile loss is given by
`
`co- Pot- Poz _ 0.007. 2a (2.2)
`P01 -- P 1 cosot2
`
`where a=c/s is solidity and ot2 is the flow angle out of the blade row. For
`incompressible or low Mach number flow it is possible to write the dynamic
`pressure in the form
`
`Pot - P~ = 1/2 pV~. -
`
`In calculating the stage loss the individual contributions of rotor and stator
`are added so that the efficiency r/can be. written in terms of the loss in stag-
`nation pressure in the stator and rotor row as
`
`useful work work input-losses
`-
`work input
`work input
`
`=l -
`
`(Ap0rotor q"Apostator)
`
`pAho
`
`(2.3)
`
`where Ah0 is the stage enthaply rise and Ap0 are the losses.
`Because the efficiency r/is often nearly equal to unity it is more helpful to
`consider
`
`1--r/ = AP0stat°r + Ap0r°t°r
`pAh 0
`
`The velocity triangles for the examples considered are shown in Fig. 2.5. The
`
`u
`
`Fig. 2.5 The velocity triangles for an axial rotor row
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`54 General design considerations
`
`restrictions adopted mean that
`
`Vx~ = Vx2 = vx3,
`
`where station 3 is at stator outlet, and a number of expressions suited to the
`parametric study are readily found. For example the relative velocity into the
`rotor is given by
`
`w~, = v~ + w~,
`
`so that
`
`(Wl/U)2 = ~ 2 + (Wol/U)2
`
`(2.4)
`
`The degree of reaction can be written
`
`R = w~’- w~ _ W~, - w~
`2z~ho 2 U( We~ - W¢2)
`
`and since Vxl = Vx2 it follows that
`
`w~,-w~,=w~,-w~
`
`and the degree of reaction is given by
`
`R = (We~ + We2)/2U
`
`(2.6)
`
`Rearranging equation 2.6 gives
`
`2RU = We-~ + W~
`
`whilst from the definition for the stage loading ~
`
`~U = Wo~ - w~.
`
`So that on rearranging
`
`W0~ =(2R +if)U/2 and Wo2=(2R - ~)U/2.
`
`(2.7)
`
`The relative velocity into the rotor can ~en be written, using this equation
`and 2.4, as
`
`(W~/U)~ = 6~ + (if/2 + R)~.
`
`(2.8)
`