Department of Mechanical and Aerospace EngineeringCARLETON UNIVERSITY

MECH 5401

TURBOMACHINERY

SUPPLEMENTARY COURSE NOTES

S.A. SjolanderJanuary 2010

CARLETON UNIVERSITYDepartment of Mechanical & Aerospace Engineering

MECH 5401 - TurbomachineryCOURSE CONTENTS

Week

1 Introduction. Review of similarity and non-dimensional parameters. Ideal versus non-ideal gases. Velocity triangles.

2 Energy considerations and Steady Flow Energy Equation. Angular momentum equation. Eulerpump and turbine equation. Definitions of efficiency.

3 Preliminary design: meanline analysis at design point. Stage loading considerations. Bladeloading and choice of solidity. Degree of reaction.

4 Correlations for performance estimation at the design point for: axial compressors, axial turbinesand centrifugal compressors. Approximate off-design performance: compressor maps and turbinecharacteristics.

5 Two-dimensional flow in turbomachinery. Spanwise flow effects. Simple radial equilibrium. Free-vortex and forced-vortex analysis.

6 Actuator disc concept. Application to blade-row interactions. Through-flow analysis: governingequations and computational implementation; role in design.

7 Blade-to-blade flow. Blade profile design considerations: boundary layer behaviour and diffusionlimits; significance of laminar- to turbulent-flow transition.

8 Three-dimensional flows in turbomachinery. Governing equations. Role of Computational FluidDynamics (CFD) in turbomachinery design and analysis. Limitations of CFD.

9 Compressible flow effects: choking in turbomachinery blade rows; shock waves in transoniccompressors and turbine; shock-induced boundary layer separation; limit load in axial turbines. Effects of compressibility on losses and other flow aspects.

10 Unsteady flows in turbomachinery. Fundamental role of unsteadiness. Significance of wake-blade interaction. Approximate analysis of unsteady behaviour of compression systems: dynamicsystem instability (surge); factors affecting compressor surge.

11 Current issues in turbomachinery aerodynamics. Very high loading for weight and blade-countreduction. Effects of gaps, steps, relative wall motion and purge flow on blade passage flows.

12 Passive and active flow control to extend range of performance. Aero-thermal interactions. Multi-disciplinary optimization.

S.A. SjolanderJanuary 2010

Department of Mechanical and Aerospace EngineeringCARLETON UNIVERSITY

MECH 4305 - Fluid Machinery

TABLE OF CONTENTS

Page

1.0 INTRODUCTION

1.1 Course Objectives1.2 Positive-Displacement Machines vs Turbomachines1.3 Types of Turbomachines

2.0 NON-DIMENSIONAL PARAMETERS AND SIMILARITY

2.1 Dimensional Analysis - Review2.2 Application to Turbomachinery

2.2.1 Non-Dimensional Parameters for Incompressible-Flow Machines2.2.2 Effect of Reynolds Number2.2.3 Performance Curves for Incompressible-Flow Turbomachines2.2.4 Non-Dimensional Parameters for Compressible Flow Machines2.2.5 Performance Curves for Compressible-Flow Turbomachines

2.3 Load Line and Operating Point2.4 Classification of Turbomachines - Specific Speed2.5 Selection of Machine for a Given Application - Specific Size2.6 Cavitation

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS ANDTHERMODYNAMICS

3.1 Steady-Flow Energy Equation3.2 Angular Momentum Equation3.3 Euler Pump and Turbine Equation3.4 Components of Energy Transfer3.5 Velocity Diagrams and Stage Performance Parameters

3.5.1 Simple Velocity Diagrams for Axial Stages3.5.2 Degree of Reaction3.5.3 de Haller Number3.5.4 Work Coefficient3.5.5 Flow Coefficient3.5.6 Choice of Stage Performance Parameters for Design

3.6 Efficiency of Turbomachines

3.6.1 Incompressible-Flow Machines3.6.2 Compressible-Flow Machines

4.0 AXIAL-FLOW COMPRESSORS, FANS AND PUMPS

4.1 Introduction4.2 Control Volume Analysis for Axial-Compressor Blade Section

4.2.1 Force Components4.2.2 Circulation

4.3 Idealized Stage Geometry and Aerodynamic Performance

4.3.1 Meanline Analysis4.3.2 Blade Geometries Based on Euler Approximation4.3.3 Off-Design Performance of the Stage4.3.4 Spanwise Blade Geometry

4.4 Choice of Solidity - Blade Loading Limits4.5 Empirical Performance Predictions

4.5.1 Introduction4.5.2 Blade Design and Analysis Using Howell’s Correlations4.5.3 Blade Design and Analysis Using NASA SP-36 Correlations

4.6 Loss Estimation for Axial-Flow Compressors

4.6.1 Blade Passage Flow and Loss Components4.6.2 Loss Estimation Using Howell’s Correlations4.6.3 Loss Estimation Using NASA SP-36 Correlations4.6.4 Effects of Incidence and Compressibility4.6.5 Relationship Between Losses and Efficiency

4.7 Compressor Stall and Surge

4.7.1 Blade Stall and Rotating Stall4.7.2 Surge

4.8 Aerodynamic Behaviour of Multi-Stage Axial Compressors4.9 Analysis and Design of Low-Solidity Stages - Blade-Element Methods

5.0 AXIAL-FLOW TURBINES

5.1 Introduction5.2 Idealized Stage Geometry and Aerodynamic Performance5.3 Empirical Performance Predictions

5.3.1 Flow Outlet Angle5.3.2 Choice of Solidity - Blade Loading

5.3.2.1 Zweifel Coefficient5.3.2.2 Ainley & Mathieson Correlation

5.3.3 Losses

6.0 CENTRIFUGAL COMPRESSORS, FANS AND PUMPS

6.1 Introduction6.2 Idealized Stage Characteristics6.3 Empirical Performance Predictions

6.3.1 Rotor Speed and Tip Diameter6.3.2 Rotor Inlet Geometry6.3.3 Rotor Outlet Width6.3.4 Rotor Outlet Metal Angle - Slip6.3.5 Choice of Number of Vanes - Vane Loading6.3.6 Losses

7.0 STATIC AND DYNAMIC STABILITY OF COMPRESSION SYSTEMS

7.1 Introduction7.2 Static Stability7.3 Dynamic Stability - Surge

Appendix A: Curve and Surface Fits for Howell’s Correlations for Axial Compressor BladesAppendix B: C4 Compressor Blade ProfilesAppendix C: Curve and Surface Fits for NASA SP-36 Correlations for Axial Compressor BladesAppendix D: NACA 65-Series Compressor Blade ProfilesAppendix E: Curve and Surface Fits for Kacker & Okapuu Loss System for Axial TurbinesAppendix F: Centrifugal Stresses in Axial Turbomachinery Blades

Department of Mechanical and Aerospace EngineeringCARLETON UNIVERSITY

MECH 4305 - Fluid Machinery

Recommended Texts

S.L. Dixon, Fluid Mechanics, Thermodynamics of Turbomachinery, 5th ed., Elsevier Butterworth-Heineman, 2005.

A short, inexpensive book which covers all the major topics, but sometimes a little too briefly. Somewhat short on design information and data. Clearly written.

H.I.H. Saravanamuttoo, G.F.C.Rogers, H. Cohen, and P.V. Straznicky, Gas Turbine Theory, 6th ed.,Pearson Education, London, 2008.

About gas turbine engines generally, but there are useful chapters on the three types ofturbomachines which are used most often in these engines: axial and centrifugal compressors andaxial turbines. These chapters contain methods and correlations which can be used in preliminaryaerodynamic design.

D. Japikse and N.C. Baines, Introduction to Turbomachinery, Concepts-NREC Inc./Oxford UniversityPress, 1994.

A recent book published for use with a short course offered by Concepts-NREC, a company inVermont which develops courses on various turbomachinery topics for industry. Reasonablygood. One of the few books on turbomachinery fluid mechanics which also addresses mechanicaldesign aspects (centrifugal stress, creep, durability, vibrations etc.).

B. Lakshminarayana, Fluid Dynamics and Heat Transfer of Turbomachinery, Wiley, New York, 1996.

A hefty, recent book written by the head (recently deceased) of turbomachinery research at PennState University. The emphasis is on more advanced topics, particularly computationaltechniques. Brief and somewhat weak on fundamentals and the concepts used in preliminarydesign. For these reasons, not well suited as a companion to this course. However, someonecontinuing in turbomachinery aerodynamic design will probably want to have a copy of the bookin his/her personal library.

Additional Readings

The Library has a number of older textbooks on turbomachinery in which you may find materialof interest: see for example the books by Vavra, Csanady and Balje. The following books are ones I havefound particularly useful over the years. Some of them cover topics discussed in the present course whileothers extend the material to topics which are beyond its scope. D.G. Shepherd, Principles of Turbomachinery, Macmillan, Toronto, 1956.

A deservedly popular text book in its day. Now out of print, as well as somewhat out-of-date. Nevertheless, it contains a lot of useful material and very lucid discussions on most topics itcovers.

The following two, relatively short books were written by the man who subsequently helped tofound the Whittle Turbomachinery Laboratory at Cambridge University. He spent a number ofyears as its Director. Good discussion of the design techniques which were current at the time(and which still play a part in the early stages of design). Lots of practical engineeringinformation. They remain in-print thanks to an American publisher who specializes in reprintingclassic technical books which remain of value.

J.H. Horlock, Axial Flow Compressors, Fluid Mechanics and Thermodynamics, Butterworth, London,1958, (reprinted by Krieger).

J.H. Horlock, Axial Flow Turbines, Fluid Mechanics and Thermodynamics, Butterworth, 1966, (reprintedby Krieger).

The next book is by a more recent Director of the Whittle Laboratory. In the Preface he explicitlydisclaims any intention to present design information. However, it presents a detailed, relativelyup-to-date discussion of the physics of the flow in axial compressors, which is still very useful.

N.A. Cumpsty, Compressor Aerodynamics, Longman, Harlow, 1989.

The following book on radial machines (both compressors and turbines) is also publishedpublished by Longman, like Cumpsty and Cohen, Rodgers & Saravanamuttoo. It is the leastsatisfactory of the three, and is apparently going out of print. Nevertheless, worth being aware ofsince most other available books on radial turbomachinery are quite old and rather out-of-date.

A. Whitfield and N.C. Baines, Design of Radial Turbomachines, Longman, Harlow, 1990.

To the extent that they present design information, the books by Horlock and Cumpsty reflectlargely British practice. The North American approach to axial compressor design was developedby NASA (then called NACA) through the 1940's and 50's. The results are summarized in thefamous SP-36, and many axial compressors continue to be designed according to it.

NASA SP-36, “Aerodynamic Design of Axial Compressors,” 1956.

AGARD, the scientific arm of NATO, organizes conferences, lecture series and specialist courseson many aerospace engineering topics, including turbomachinery aerodynamics. The followingare two particularly useful publications which have come out of this activity.

A.S. Ucer, P. Stow and Ch. Hirsch eds., Thermodynamics and Fluid Mechanics of Turbomachinery,Martinus Nijhoff, Dordrecht, Vol. I and II, 1985.

AGARD-LS-167, Blading Design for Axial Turbomachines, 1989.

NOMENCLATURE FOR TURBOMACHINES

ENERGY TRANSFER TO THE FLUID

ENERGY TRANSFER FROM THE FLUID

FansBlowers

TurbinesTurbo-expandersWind mills/Wind turbines

Gases

LiquidsIncompressible flow

Compressible flow

Both

Compressors

Propellers

Pumps

Gases

Both

lift forcedrag force

LD

⎛

⎝⎜

⎞

⎠⎟ = ⎛

⎝⎜⎞⎠⎟

model prototype

P RT a RT= =ρ γ

2.0 NON-DIMENSIONAL PARAMETERS AND SIMILARITY

2.1 DIMENSIONAL ANALYSIS - REVIEW

Non-dimensional parameters allow performance data to be presented more compactly. They can alsobe used to identify the connections between related flows, such as the flow around a scale “model” and thataround the corresponding full-scale device (sometimes called the “prototype”).

Two flows are completely similar (“dynamically similar”) if all non-dimensional ratios are equal forthe two flows. This includes geometric ratios, which are needed for “geometric similarity”. For example, ifthe flows around two geometrically-similar airfoils are dynamically similar, then

Similarly for other force ratios, velocity ratios, etc.

For a given case there is only a limited number of independent non-dimensional ratios: these are the“criteria of similarity”. If the criteria of similarity are equal for two flows, all other non-dimensional ratioswill also be equal, since they are dependent on the criteria of similarity.

Finding Criteria of Similarity:

(1) List all the independent physical variables that control the flow of interest (based on experience,judgment, physical insight etc.). For example, consider again the airfoil flow. Assume that the flowis compressible and the working fluid is a perfect gas.

For a particular airfoil shape, the flow is completely determined by:c - chordα - angle of attackU - freestream velocityρ - fluid densityμ - fluid viscosityR - gas constanta - speed of soundγ - specific heat ratio

Note that the pressure and temperature are not quoted. For a perfect gas,

Thus, by specifying a, γ and R, we have implicitly specified T. Similarly, with ρ and R specified, andT implicitly specified, then P is implicitly specified through the perfect gas law. Therefore, for ourparticular choice of independent variables, P and T are just dependent variables. All other quantities,such as the lift, L, and drag, D, likewise depend uniquely on the values of the independent variables.

U

L

D

M

c

α

ρμ

ρμ

× × × =

× × ×

1

3

U cU c

ML

LTM

LT

L

Ua

Mach number M= ,

( )DU c

DU c

or D

U cC

M LT

LM

TL L L

D× × ××

= ≡

× × ×

1 1 11 1

21

2 22

2

3 2

2

ρ ρ ρ

(2) Form non-dimensional groups from the independent variables.

Buckingham’s Π Theorem gives the number of independent non-dimensional ratios which exist:

If n = no. of independent physical variablesr = no. of basic dimensions (eg. Mass, Length, Time, Temp. (θ), etc.)

Then (n - r) criteria of similarity exist

eg. for the airfoil n = 8r = 4 (M, L, T, θ)ˆ (n - r) = 4

ie. there are 4 criteria of similarity

Form the criteria of similarity by inspection, or using dimensional analysis.

eg. for the airfoil, we can non-dimensionalize the density as follows:

which is clearly the Reynolds number, Re

α is already non-dimensional and can be used directly as a criterion ofsimilarity

γ is also already non-dimensional

Thus, for the airfoil 4 suitable criteria of similarity are: Re, M, α, and γ. If these are matched betweentwo geometrically similar airfoils, the two flows will be dynamically similar.

(3) All other non-dimensional ratios are then functions of the criteria of similarity.

Take each dependent variable in turn and non-dimensionalize it using the independent variables.eg. for the drag of airfoil (per unit span), D

then ( )C f MD = Re, , ,α γ

CC

LD

L

D

=

Similarly for all other dependent non-dimensional ratios (CL, Cm, etc.).

Any non-dimensional ratios we develop could also be combined, by multiplication, division etc., toform other valid non-dimensional ratios. This does not provide any new information, simply a rearrangementof known information. However, the resulting ratios may be useful alternative ways of looking at theinformation. For example, for the airfoil, having derived CD and CL then

is another valid (and in fact useful) non-dimensional parameter.

2.2 APPLICATION TO TURBOMACHINERY

2.2.1 Non-Dimensional Parameters for Incompressible-Flow Machines

For now, consider just pumps, fans, and blowers. Hydraulic turbines will be discussed briefly inSection 2.4.

For a given geometry, the independent variables that determine performance are usually taken as.

D - machine size (usually rotor outside diameter)ρ - fluid densityμ - fluid viscosityN (or ω) - machine speed; revs or rads per unit timeQ - volume flow rate through the machine

Note that the choice of independent variables is somewhat arbitrary. One way to visualize what arepossible independent variables and what are dependent variables is to imagine a test being conducted on themachine in the laboratory. The variables which, when set, fully determine the operating point of the machineis then one possible set of independent variables. In the laboratory test, one might set the rotational speed (bycontrolling the drive motor) and the flow rate (by throttling at the inlet or outlet ducts). With N and Q set, thehead or pressure rise produced or power absorbed are then dependent functions of the characteristics of themachine. Alternatively, if the throttling valve is adjusted to produce a particular pressure rise, then we losecontrol over the flow rate and it becomes a dependent variable. The independent variables listed above are themost common choices for incompressible flow machines that raise the pressure of the fluid. All othervariables are then dependent. For example

W&Q

Q

D

N

ΔH - total head rise across machine (or sometimes, total pressure rise)- shaft power absorbed by the machine&W

T - torque absorbed by the machineη - efficiency of the machine

Applying Buckingham Π Theorem:

n = 5 r = 3 (M, L, T) n - r = 2 (ie. are 2 criteria of similarity)

Form the criteria of similarity:

(1) Flow rate:

This is known as the flow coefficient, capacity coefficient or flow number

(2) Fluid properties (specifically, viscous effects):

ie. the Reynolds numberρμ

ρμ

N DD N D=

2

All other non-dimensional ratios or coefficients then depend on these two criteria of similarity.

For power coefficient (non-dimensional work per unit time)

then&

,WN D

f QN D

N Dρ

ρμ3 5 3

2

=⎛

⎝⎜

⎞

⎠⎟

Obviously, μ rather than ρ could have been used to cancel the M appearing in . It can easily be shown that&Wthe resultant power coefficient would be the one derived here multiplied by the Reynolds number.

( )( )

ΔH f D N Q

W f D N Q etc

=

=1

2

, , , ,& , , , , .

ρ μ

ρ μ

QN D

QN D

LT

TL

× × =

× ×

1 1

11

3 3

3

3

&&

WN D

WN D

M LTT

T LM L

× × × =

⎛

⎝⎜

⎞

⎠⎟

× × ×

1 1 1

11

3 5 3 5

2

2 3 3

5

ρ ρ

Next consider the total head rise, ΔH, across the machine. By definition, the total head H is given by

H Pg

Vg

z

static head dynamichead elevation head

= + +

= + +ρ

2

2

and H can be interpreted physically as the mechanical energy content per unit weight. However, the energycontent is more commonly expressed on a per unit mass basis:

gH mechanical energy per unit mass=

We therefore create a non-dimensional head coefficient as follows:

Sometimes the head rise ΔH is simply written H. As with the power coefficient, the head coefficient is adependent function of the two criteria of similarity:

The g is also sometimes dropped to give H/N2D2, but the head coefficient is then dimensional and will takedifferent values in different systems of units.

A corresponding total pressure coefficient can be obtained from

since ρgΔH has units of pressure.

Using the conventional definitions, efficiency is already non-dimensional. For pumps, fan andblowers, the efficiency is usually defined as:

η pumpuseful power transferred to fluid

input powerfluid powershaft power

= =

g HN D

g HN D

LT

L TL

ΔΔ

× × =

× ×

1 1

11

2 2 2 2

2

2

2

g HN D

orgH

N Df Q

N DN DΔ

2 2 2 2 3

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟,

ρμ

g HN D

g HN D

PN D

Δ Δ Δ2 2 2 2

02 2= =

ρρ ρ

andfluid power mass flow rate mechanical energy change per unit mass

m g HQg H

= ×

= ×

=

& Δ

Δρ

Thus

Similarly, for turbines:

ηρ

ρ

pumpQg H

WQ

N Dg H

N D

WN D

Flow Coefficient Head CoefficientPower Coefficient

=

=

⎛

⎝⎜

⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

=×

Δ

Δ

&

&

3 2 2

3 5

ηρturb

shaft powerfluid power

WQg H

Power CoefficientFlow Coefficient Head Coefficient

= =

=×

&

Δ

2.2.2 Effect of Reynolds Number

We have shown that in general for incompressible flow:

The flow in most turbomachines is highly turbulent. Therefore, most frictional effects are due toturbulent mixing. Viscosity has a minor direct effect and losses tend to vary slowly with Re: recall from theMoody chart that in pipe flow the friction factor varies much more slowly with Re for turbulent flow than forlaminar flow. Thus, if the Reynolds numbers are high and the differences in Re are not too large between themachines being compared, Re is often neglected as a criterion of similarity. We can then use, as anapproximation

Where Re variations can not be neglected, a number of empirical relations have been proposed forcorrecting for the effect of Re on efficiency. These corrections typically take the form

where ReM is the smaller of the two values of the Reynolds number and n varies with the type of machine andReynolds number level. For example, the ASME Power Test Code (PTC-10, 1965) suggests the followingvalues:

n = 0.1 for centrifugal compressorsn = 0.2 for axial compressors

if ReM 105, where Re = ND2/ (ie. the tip Reynolds number). Note that (1) indicates that efficiency improveswith increasing Re.

g HN D

WN D

etc fns QN D

N D

fns QN D

∆2 2 3 5 3

2

3

,

, , . ,

, Re

ρη ρ

µ=

=

g HN D

WN D

etc fns QN D

only∆2 2 3 5 3,

, , .ρ

η =

11

−−

=

ηη

P

M

M

P

nReRe

(1)

Reynolds Number Based on Blade Chord

Taken from: AGARD-LS-167

2.2.3 Performance Curves for Incompressible-Flow Turbomachines

Relationships such as

(neglecting Re)g HN D

f QN D

∆2 2 3=

imply that if we test a family of geometrically-similar, incompressible-flow machines (different sizes, differentspeeds etc.), the resulting data will fall on a single line if expressed in non-dimensional form. For example, thenon-dimensional coefficients for a pump of fan might appear as follows (we will discuss later why the curveswill have the particular trends shown):

The thick curves are used to suggest variations which could be due to the neglected Re effects, and perhapssome secondary effects which were not included in the original list of independent parameters (e.g. mildcompressibility effects for a fan or blower). The dashed line indicates the likely "design point": the preferredoperating point, since the efficiency is best there.

Because of the universality of the performance curves, the tests could be conducted for a singlemachine and the results used to predict the performance of geometrically similar machines of different sizes,different operating speeds, and even with different working fluids.

Note again that there is flexibility in the choice of dependent and independent parameters. See P.S. #1Q 1 for the form of non-dimensional parameters which are often used for hydraulic turbines.

Coe

ffici

ents

Likely "Design Point"

3DNQ

22DNHg∆

η

53DNW

ρ

2.2.4 Non-Dimensional Parameters for Compressible-Flow Turbomachines

We now develop the criteria of similarity for compressible-flow turbomachines. Assuming theworking fluid is a perfect gas, a suitable list of independent variables which control performance is as follows:

N D maorT

Por R, , & , , , , ,

01

01

01

01

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟ρμ γ

where

= mass flow rate (rather than Q as measure of flow rate)&m

(stagnation speed of sound)a RT01 01= γˆ could use T01 rather than a01

(perfect gas)P RT01 01 01= ρ

ˆ can use ρ01 or P01, as convenient

(N.B. temperatures and pressures must be absolute values)

Then from the Buckingham Π Theorem:

n = 8 r = 4 (M, L, T, θ) n - r = 4 (4 criteria of similarity)

By inspection, the 4 independent coefficients are:

(1) speed parameter (effectively the tip Mach number)NDa01

(2) flow parameter (effectively the axial Mach number)&m

D aρ012

01

(3) or we could use againμDm& Re

=1 ρ

μ01

2N D

(4) specific heat ratio (which is already non-dimensional)γ =CC

p

v

All other performance coefficients are then functions of these four coefficients (as always, geometricalsimilarity is assumed).

Dependent performance coefficients:

The main change from incompressible-flow machines is in the form of the pressure changecoefficient. Instead of the head or total pressure coefficient, we conventionally use the pressure ratio:

PP

P machine outlet total pressure02

0102 =

Then

PP

WN D

etc fns NDa

ma D

02

01 013 5

01 01 012,

&, , . ,

&, Re,

ρη

ργ=

⎛

⎝⎜

⎞

⎠⎟ (1)

The form of the independent coefficients used here is very general. The main assumption that hasbeen made is that the working fluid is a perfect gas. We can make use of some of the perfect gas expressionsto rewrite the independent parameters in a somewhat more convenient form:

(1) Speed coefficient:

NDa

NDRT

NT

DR01 01 01

= =γ γ

(2) Flow coefficient:

& & &ma D

mP

RTRT D

m TP

RDρ γ γ01 01

201

0101

2

01

012

1= =

Then (1) can be written

PP

WN D

etc fns NT

DR

m TP

RD

02

01 013 5

01

01

012

1,&

, , . ,&

, Re,ρ

ηγ γ

γ=⎛

⎝⎜⎜

⎞

⎠⎟⎟ (2)

This is the form of the parameters that is appropriate for the most general case, where we are relating theperformance of geometrically-similar, compressible-flow turbomachines of different sizes and operating withdifferent working fluids (both of which are perfect gases).

In practice, the parameters are often simplified somewhat according to specific circumstances.

In many cases, the same working fluid (eg. air) will be used for both the model and prototype. Thus,R and γ are often known constants and it is somewhat tedious continually to have to include them in thecalculation of the coefficients. If we then omit the known, constant fluid properties we can write:

PP

WN D

etc fns N DT

m TP D

02

01 013 5

01

01

012,

&, , . ,

&, Re

ρη =

⎛

⎝⎜⎜

⎞

⎠⎟⎟ (3)

This form of the coefficients is suitable for relating geometrically-similar machines with different sizes butwith the same working fluid. Note that by assuming the same working fluid, we have reduced the number ofcriteria of similarity by one. The main disadvantage to this form of the coefficients is that the speed and flowcoefficients are now dimensional and we must specify what system of units we are working in.

If the performance curves are intended to represent the performance of a particular machine operatingat different inlet conditions, then D is a known constant and is often omitted:

PP

WN D

etc fns NT

m TP

02

01 013 5

01

01

01

,&

, , . ,&

, Reρ

η =⎛

⎝⎜⎜

⎞

⎠⎟⎟ (4)

This is the form of the independent coefficients typically used to present the performance characteristics of thecompressors and turbines for gas turbine engines.

As with incompressible-flow machines, it is sometimes possible to neglect Re as a criterion ofsimilarity (by the same arguments used in Section 2.2.2). Note that the speed and flow coefficients are againdimensional.

2.2.5 Performance Curves for Compressible-Flow Turbomachines

If we can neglect the Reynolds number effects, Eqns. (3) and (4) indicate that our performance curveswill take the form:

PP

f NDT

m TP D

etc02

011

01

01

012=

,

.

Thus, whereas our performance tests for the incompressible-flow machines led to a single curve for eachdependent performance coefficient, for compressible-flow machines we will obtain a family of curves.

The resulting performance diagrams for compressible-flow compressors and turbines would then lookas follows (again, we will discuss the reasons for the detailed shape of the characteristics later in the course):

(a) Compressor ("Compressor Map")

Implicitly, this map applies for one value of some reference Reynolds number. If the effects of Re can not beneglected, then we would have to generate a series of such graphs, each one containing the performance datafor a different value of the reference Re.

01

02

PP

01TDN

01TDN

201

01

DPTm

INCREASING

CHOKING

SURGE LINE(UPPER LIMIT OF

STABLE OPERATION)

LINE OF CONSTANT

(b) Turbine Characteristic:

In a gas turbine engine, the pressure ratio developed by the compressor is applied across the turbine atthe hot end of the engine. The mass flow rate swallowed by the turbine and its power output are thendependent functions of the turbine characteristics. That is, as far as the turbine is concerned the pressure ratiois imposed and is effectively an independent parameter. When presenting performance data, we generally plotindependent parameters on the “x axis” and dependent parameters on the “y axis”, as was done on thecompressor map. By this argument, the turbine characteristic should be presented as:

and this is in fact the way turbine characteristics are generally presented in the gas turbine business.

201

01

DPTm

01

02

PP

STATORS CHOKED

LINES OF CONSTANT01TDN

01TDN

CONSTANT

201

01

DPTm

01

02

PP

NASA 8-Stage Research Axial Compressor

2.3 LOAD LINE AND OPERATING POINT

The performance diagrams discussed in the earlier sections present a wide range of conditions atwhich the machine can operate. For example, the compressor in the last section can operate stably at any pointto the right of the surge line. The precise point at which a turbomachine actually operates depends on the loadto which it is connected.

(a) The simplest case is a compressor or pump connected to a passive load (e.g. pipe line with valves,elbows etc.). At the steady-state operating point we must have:

(1) (or, for compressible flow, )Q Qmachine load= & &m mmachine load=

(2) (or )Δ ΔH Hmachine load= Δ ΔP Pmachine load0 0, ,=

Thus, the operating point is where the machine and load , or , characteristicsΔH vs Q ΔP vs m0 &intersect.

e.g. Suppose a pump is supplying flow to a pipe line. The head drop along the pipe varies with V2

(or Q2), as determined from the friction factor (e.g. Moody chart) and the loss coefficients of any othercomponents in the pipe system. The resulting ΔH vs Q variation is known as the load line for thesystem. The head rise produce by the pump is a function of the flow rate and the rotational speed. Then if the pump is run at N1, the operating point will be A, etc.

(b) For a gas turbine engine, the operating points of the compressor and turbine are determined bycompressor/turbine matching conditions (a propulsion nozzle will also influence operating points - seeSaravanamuttoo et al., Ch. 8 & 9).

ΔH

Q

N1

N2

N3

LOAD LINE

PUMP CHARACTERISTICSAT CONSTANT SPEED

A

B

C

COMPRESSOR

COMBUSTOR

TURBINE

fuelm&

Cm&Tm&

outW&CW&

For the simple shaft-power engine shown, the matching conditions would be:

& & &

& & &

m m mN N

W W W

T C fuel

C T

T C out

= +

=

= +

(c) In hydro-power installations, total head across the turbine is imposed by the difference in elevationbetween reservoir and tailwater pond (minus any losses in the penstock). Since

&W gQ HT T= η ρ Δ

to produce varying power (according to electrical demand), it is necessary to vary the equilibrium Q, at fixedΔH. Furthermore, since the electricity must be generated at fixed frequency, we do not have the option ofvarying N to achieve different operating points. The solution to this is to vary the geometry of the machine. This can be done with variable inlet guide vanes or with variable rotor blade pitch.

ΔH

Q

β1 β2 β3

CONSTANT SPEED LINES -SAME SPEED,

DIFFERENT BLADE SETTINGS

LOAD LINENEGLECTING FRICTION

LOAD LINEINCLUDING FRICTION

(with N in revs/s in the coeffcients)

g ∆H⋅

N2 D2Head coefficient:

Q

N D3⋅Flow coefficient:

RPMN 1750:=Pump speed:

cmD 30:=Pump diameter:

0 0.002 0.004 0.006 0.008 0.010

1

2

3

4Pump Characteristics

Flow Coefficient

Hea

d C

oeff

icie

nt

The pump has the characteristics shown in the plot, and the following information applies to the pump:

m2/sν 10 6−:=Viscosity (water):

mL 125:=Pipe length:

(smooth)mmdpipe 50:=Pipe diameter:

WATER

PUMP

K = 1 (EXIT LOSS)

6 m.

VALVE K = 1, K=10

K = 0.9

K = 0.5 (ENTRY LOSS)

K = 0.9

K = 0.9

K = 0.9

K = 0.9

A pump is connected to the piping system shown. What flow rate of water will be pumped for the two valve settings?

EXAMPLE (Section 2.3):

2.4 CLASSIFICATION OF TURBOMACHINES - SPECIFIC SPEED

Neglecting Reynolds number effects, for a given family of geometrically-similar incompressible-flowturbomachines the efficiency is a function of one criterion of similarity only. Normally we use the flowcoefficient as the independent parameter. That is

η =⎛

⎝⎜

⎞

⎠⎟f Q

N Donly3

Thus, the maximum 0 will occur for this family (say family A) at some particular value of Q/ND3. Foranother family of machines, the maximum 0 might occur at a different value of Q/ND3. We could thereforeclassify turbomachines according to the value of Q/ND3 at which they produce the best efficiency. Then if weknew the value of Q/ND3 that we required in a given application, we would choose the machine that gives thebest value of efficiency at that value of Q/ND3. Unfortunately, this idea presupposes that we know thediameter of the machine. In general, this will not be the case. We therefore look for an alternative parameterto Q/ND3 that does not involve the size of the machine to use as a basis for classifying families ofturbomachines.

We can always form valid new non-dimensional parameters by combining existing ones. Combinethe flow and head coefficients to eliminate D:

( )

QN D

g HN D

NQ

g H

3

12

2 2

34

12

34

⎛

⎝⎜

⎞

⎠⎟

⎛⎝⎜

⎞⎠⎟

=∆ ∆

Following convention, we then define

( )Ω

∆= ωQ

g H

12

34

where T is in radians/s so that S is truly non-dimensional. Conceptually, we could then plot the efficienciesof various families of turbomachines against S (rather than Q/ND3) and note the value of S at which eachfamily achieves its best 0. This value of S is known as the specific speed for that family of machines. Thenext figure (taken from Csanady) shows the values of specific speed that are observed for various types ofturbomachines:

3DNQ

ηFAMILY A

FAMILY B

A number of more detailed summaries of specific speed have been presented over the years. Unfortunately, the non-dimensional form of the specific speed has not been used consistently. The followingtable can be used to convert between the various definitions used:

AREA OF APPLICATION SPECIFIC SPEED EQUIVALENT S

FANS, BLOWERS ANDCOMPRESSORS (BRITISH UNITS) N

RPM cfs

ftS1 3

4

= Ω =N S1

129

PUMPS (AMERICAN

MANUFACTURERS)N

RPM USgpm

ftS2 3

4

= Ω =N S2

2730

HYDRAULIC TURBINES(BRITISH UNITS) N RPM HP

ftS 3 5

4

= Ω =N S3

42(IF WORKING FLUID IS WATER)

HYDRAULIC TURBINES(METRIC UNITS) N

RPM metric HP

mS4 5

4

= Ω =N S4

187(IF WORKING FLUID IS WATER)

FANS, BLOWERS ANDCOMPRESSORS (METRIC UNITS) N

RPM m s

mS5

3

34

= Ω =N S5

53

Several plots showing the specific speeds for various classes of machines are given on the next pages. In addition to giving the values of specific speed, the plots can also be used for initial estimates of theefficiencies that can be expected. These efficiencies apply for machines that are well-designed, correctly sizedfor their applications, and operating at their design points.

Hydraulic turbines are usually characterized according to their output power rather than the flow rate. Since shaft power output is related to the flow rate by

&W Qg Ht t= η ρ ∆

we can rewrite the specific speed as

( ) ( )Ω

∆ ∆= =

ω ω

η ρ

Q

g H

W

g H34

54

&

In practice, 0, D and g are usually dropped, and T is replaced by N (usually in RPM). Thus, the "powerspecific speed" normally used with hydraulic turbines is

N N W

HS =

&

∆54

The following figure (from Shepherd, 1956) shows the variation of the power specific speed for hydraulicturbines of different geometries.

The plots shown above were based on data that is as much as 50 years old. One might expect thatover time the efficiency of all types of machines would improve as a result of the application improved designtools such as computational fluid dynamics. This is illustrated in the following figure which shows thevariation of efficiency with specific speed for compressors. The baseline data, taken from Shepherd (1956),dates from 1948 or earlier. Japikse & Baines (1994) compared more recent compressor data with the plot fromShepherd and concluded that efficiencies had improved noticeably since Shepherd’s time. They also projectedthat there would be further improvements by 2000, as shown in the figure.

Specific Speed, NS

Effic

ienc

y,η

101 102 1030.4

0.5

0.6

0.7

0.8

0.9

1

Shepherd (1956): 1948 DataJapikse & Baines (1994): 1990 DataJapikse & Baines (1994): 2000 Projected

Positive-DisplacementMachines

CentrifugalMachines

Axial-FlowMachines

10 20 40 60 80 100 200 400 600 1000

2.5 SELECTION OF MACHINE FOR A GIVEN APPLICATION - SPECIFIC SIZE

The selection starts from the required “duty”: the conditions at which it is intended to operate:

For pumps, compressors N, Q and H (or P0) are typically specified.For turbines N, and H (or P0) are typically specified. W

In practice, a precise value of N may not be known, but it is often constrained to specific values by the factthat, for example, electrical motors come with certain maximum speeds according to the number of poles. There may also be mechanical constraints (e.g. maximum tip speed, because of centrifugal stressconsiderations). Often the selection process will involve varying the speed to get a specific speed whichresults in good efficiency.

From the duty, one can work out the specific speed and then use the figures in Sec. 2.4 to select anappropriate type of machine. However, the efficiencies shown on the figures will be achieved only if themachine is well-designed and correctly sized. Size is important because:

(a) if machine is too small: high flow velocities, and since frictional losses vary as 0.5V2 (and withgases, shocks can occur), the efficiency will be poor;(b) if machine is too big: low velocities, low Reynolds numbers, boundary layers will be thick andmay separate, again reducing the efficiency; also, machine will be expensive.

In Sect 2.4, we noted that for a given family of machines the peak occurs for a particular Q/ND3. In effect,having chosen a suitable machine, knowing Q and N, we want to pick D to get the appropriate Q/ND3. However, efficiency data for turbomachines has not in fact been correlated in this form. Instead of usingQ/ND3, we define a new parameter, the "specific size" :

( )∆

∆=

D g H

Q

14

The specific size for a given machine is then the value of at which it achieves its best efficiency. The valueof depends on the machine type (i.e. ) and to some degree on its detailed design. However, in the early1950s Cordier examined the data for a wide range of well-designed, actual machines, and found that correlated quite well with alone: the correlation is summarized in the Cordier diagram (see over). Summarizing:

To get best efficiency for a specified duty:

(1) Select the machine type such that its is

( )Ω

∆=

ω Q

g Hduty

34

(2) From , read from the Cordier diagram and size the machine such that

( )D g H

Qduty

∆∆

14

=

2.5 SELECTION OF MACHINE FOR A GIVEN APPLICATION - SPECIFIC SIZE

The selection starts from the required “duty”: the conditions at which it is intended to operate:

For pumps, compressors N, Q and )H (or )P0) are typically specified.For turbines N, and )H (or )P0) are typically specified. &W

In practice, a precise value of N may not be known, but it is often constrained to specific values by the factthat, for example, electrical motors come with certain maximum speeds according to the number of poles. There may also be mechanical constraints (e.g. maximum tip speed, because of centrifugal stressconsiderations). Often the selection process will involve varying the speed to get a specific speed whichresults in good efficiency.

From the duty, one can work out the specific speed and then use the figures in Section 2.4 to select anappropriate type of machine. However, the efficiencies shown on the figures will be achieved only if themachine is well-designed and correctly sized. Size is important because:

(a) if machine is too small: there will be high flow velocities, and since frictional losses vary as0.5DV2 (and with gases, shocks can occur), the efficiency will be poor;(b) if machine is too big: there will be low flow velocities, low Reynolds numbers, boundary layerswill be thick and may separate, again reducing the efficiency; also, the machine will be expensive.

In Section 2.4, we noted that for a given family of machines the peak 0 occurs for a particular Q/ND3. Ineffect, having chosen a suitable machine, knowing Q and N, we want to pick D to get the appropriate Q/ND3. However, efficiency data for turbomachines has not in fact been correlated in this form. Instead of usingQ/ND3, we define a new parameter, the "specific size" ):

( )∆

∆=

D g H

Q

14

The specific size for a given machine is then the value of ) at which it achieves its best efficiency. The valueof ) depends on the machine type (i.e. S) and to some degree on its detailed design. However, in the early1950s Cordier examined the data for a wide range of well-designed, actual machines, and found that )correlated quite well with S alone: the correlation is summarized in the Cordier diagram (see over). Summarizing:

To get best efficiency for a specified duty:

(1) Select the machine type such that its S is

( )Ω

∆=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ω Q

g Hduty

34

(2) From S, read ) from the Cordier diagram and size the machine such that

( )D g H

Qduty

∆∆

14

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

Example (Section 2.5):

A small hydraulic turbine is to deliver a power of 1000 kW. The total head available is 6 m. and the turbine is directly connected to an electrical generator which is to deliver power at 60 Hz.(a) What is the required flow rate?(b) Determine a suitable type, size and speed for the turbine.

2.6 CAVITATION

If the local absolute static pressure falls below the vapour pressure of a liquid, it will boil, formingvapour cavities or bubbles. This is known as cavitation. When the bubbles collapse, brief, very high forcesare created which can cause rapid erosion of metal surfaces. Cavitation will also cause significantperformance deterioration. Thus, cavitation should be avoided.

Cavitation is a danger on the low-pressure ("suction") side of the machine: the inlet for pumps, theoutlet for turbines.

Define the Net Positive Suction Head (NPSH):

H H hsv abs v= −

where Habs is the absolute total head at the suction side of the machine, defined as

HP

gV

gabsabs

suction side

= +⎡

⎣⎢

⎤

⎦⎥ρ

2

2

where Pabs is the absolute value of the static pressure and V is the fluid velocity, both on the lower pressure orsuction side of the machine. hv is the head corresponding to the vapour pressure of the liquid,

hP

gvvap=ρ

Note: Habs is not the usual total head H since it does not include the elevation term. In fact Habs = P0/ρg.At the minimum pressure point on the suction side of the machine, the local static head will be less than thetotal head, Habs, but directly related to it. Thus, the onset of cavitation will occur for some critical, positivevalue of Hsv.

1

2

01P

( )Tf

1PSVgHρ

2P

212

1 Vρ

vP

222

1 VρP

01P

1P2P

vP

222

1 VρcriticalSVgHρ

o

We non-dimensionalize Hsv to obtain the "suction specific speed", S

( )S

Q

gHsv

=ω

34

For a given machine there will then be some critical value of S ( = Si, “i” for cavitation “inception”),corresponding to the critical value of Hsv, at which cavitation will start. If

S < Si

then there is no cavitation. The higher the value of Si, the more resistant the machine is to cavitation.

The value of Si can be found experimentally by holding Q and N constant (i.e. Q/ND3 constant) whilereducing the pressure on the suction side of the machine and observing the ΔH or η behaviour. For example,for a pump a valve in the intake pipe can be used to reduce gradually the inlet total head while an outlet valvecan be used to maintain the constant the flow rate. Plot the results versus the resulting values of S:

At cavitation inception, the blade passages fill with vapour and ΔH and η drop drastically.

The value of Si depends in the detailed design of the machine (e.g. surface curvatures in the low-pressure section of the blade passage). However, for machines which have been properly designed to avoidcavitation it has been found that the values of Si are fairly similar:

For pumps: Si . 2.5 - 3.5 N.B.: near the design pointFor turbines: Si . 3.5 - 5.0

Recall that a higher value of Si means a machine more resistant to cavitation.

The Thoma Cavitation Parameter, σ, is also sometimes used:

σ =H

Hsv crit

Δ

SSi

ΔHη

3DNQ

DATA FOR CONSTANT

INCEPTION

where is the critical value of : that is, the value at cavitation inception. However, the value ofHsv crit Hsvσ will vary with the details of the design of the machine. This can be illustrated by considering two pumpimpellers that have identical inlet geometries:

If the pumps are run at the same rotational speeds and flow rates, the flow in the inlet region will be identical. Thus, they should cavitate at the same values of Hsv. Then since

( )S

Q

gHsv

=ω

34

it follows that the two machines have the same critical value of S: Si1 = Si2. However, the two rotors do nothave the same value of ΔH. In fact, the larger rotor will produce a significantly larger ΔH because of itshigher tip speed (ΔH varies as (ND)2, as implied by the form of the head coefficient; see also later sections). Thus, at cavitation

σ σ11

12

2

2

= > =H

H

H

Hsv crit sv crit, ,

Δ Δ

since ΔH1 < ΔH2. Consequently, the Thoma parameter should be used only within a geometrically-similarfamily of machines. For example, a critical value of σ determined from model tests can be used to predict theconditions for the onset of cavitation in another member of the same family.

Since cavitation is a significant danger to the machine, checking for cavitation should be a normalpart of selecting a hydraulic machine for a particular duty.

D1

D2

12

EXAMPLE (Section 2.6): In Section 2.5 we selected a hydraulic turbine for the following service: W = 1000kW, H = 6 m. An axial-flow (propeller or Kaplan) turbine was chosen, with a diameter of 2.7 m, a flow rate of 18.9 m3/sec and running at 180 RPM. What is the maximum height above the tailwater level that thisturbine can be installed if cavitation is to be avoided? The draft tube is a length of diffusing duct at the exit of the turbine. Assume that the draft tube has an outlet area of 6 m2 and the outlet is 3 m below the turbine. The water is at 20 oC for which Pv = 2.3 kPa. Patm = 101.3 kPa. Assume that the tailpond is large compared with the draft tube outlet so that the flow is effectively being dumped into a very large reservoir atthe draft tube outlet.

TAIL POND

6m

h3m

DRAFT TUBE OUTLET

1 2

m& m&

Q&

shaftW&

E dm dQ W E dmshaft1 2∫ ∫ ∫+ + =& & & &

& & & &mE Q W mEshaft1 2+ + = (1)

E u P C gz

thermal mechanical

h C gz

= + + +⎛

⎝⎜

⎞

⎠⎟

+

= + +

ρ

2

2

2

2

(2)

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICSAND THERMODYNAMICS

3.1 STEADY-FLOW ENERGY EQUATION

Consider a control volume containing a turbomachine:

For steady flow, conservation of energy can be written

Rate of energy flow into CV + Rate of energy addition inside = Rate of energy flow out of CV

If the energy content is the same for all fluid entering or leaving the CV (or using mean values) SFEE can bewritten

where = mass flow rate of fluid&mE = energy per unit mass for fluid

= rate of heat transfer to the machine&Q

= shaft power into the machine&Wshaft

The energy content of the fluid includes thermal and mechanical components:

where u = internal thermal energy per unit mass (= CvT)P/ρ = flow work (“pressure energy”) per unit massC = absolute velocity of fluidC2/2 = kinetic energy per unit massgz = potential energy per unit massh = P/ρ + u = enthalpy per unit mass

( )

& &

&

W m hC

hC

m h h

shaft = +⎛

⎝⎜

⎞

⎠⎟ − +

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −

222

112

02 01

2 2

(3a)

&

&

WmorQ

uP C

gz uP C

gzshaft =⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ + +⎛

⎝⎜

⎞

⎠⎟ − + + +

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥ρ

ρ ρ22 2

2

2 11 1

2

12 2(4)

u ug

H total head loss due to friction inside the machineL2 1−

= = " "

( )&W Q g H H H

Q g H Q g Hshaft L

L

= − +

= +

ρ

ρ ρ2 1

Δ(5)

For a turbomachine at steady state, the flow is essentially adiabatic, . For gases, we usually&Q = 0neglect potential energy changes. Then SFEE can be written

whereh h C stagnation enthalpy

C T for perfect gasesP

0

2

0

2= + =

=

For general non-uniform flows, we would write

For incompressible flow , temperature (i.e. internal energy, u) changes only due to frictional heating,since ρ is constant and we have already assumed the process is adiabatic. In order to separate the frictionaleffects from other effects, we retain the internal energy separate from the flow work:

It is also common to write

The total head is a measure of the total mechanical energy content of the fluid

H total head Pg

Cg

z= = + +ρ

2

2

Then for an incompressible-flow compression machine (eg. a pump or blower) (4) can be written

ΔH = H2 - H1 is the total head rise that appears in the fluid between the inlet and outlet of the machine. It isthe ΔH which was used in the head coefficient, (gΔH/N2D2), and ρQgΔH is what was referred to earlier as the“fluid power”.

& & &W h dm h dmshaft = −∫ ∫02

01

(3b)

We defined the efficiency of a pump or blower as

η pumpfluid powershaft power

=

thenη ρ

ρ ρpumpL

L

Qg HQg H QgH

HH

=+

=+

ΔΔ

Δ

1

1

(6)

As shown later, we have ways to estimate the various contributions to HL (eg. frictional losses at the walls varyas V2). We can then use (6) to estimate the resulting efficiency of the machine.

For incompressible-flow expansion machines (i.e. turbines),

&W Q g H Q g Hshaft L= −ρ ρΔ

since the friction inside the machine now reduces the shaft power output compared with the fluid powerreleased by the fluid, as given by ρQgΔH. We then define turbine efficiency

ηturbineshaft power out

fluid power=

Efficiency is discussed further in Section 3.6.

T r C dm r C dmout in

0 = × − ×

T rC dm rC dmwout

win

= − (7)

( ) ( )T m rC m rCw out w in= −

3.2 ANGULAR-MOMENTUM EQUATION

The energy transfer between the fluid and the machine occurs by tangential forces exerted on the fluidas it interacts with the rotor blades. Although forces are also exerted between the fluid and the stators(stationary blades), no energy transfer occurs since there is no displacement associated with the forces - thus,stators can only redistribute energy among its components.

The angular form of Newton’s second law (the angular-momentum equation) governs the interaction(see earlier courses for derivation):

Torque applied to fluid in CV = outflow of angular momentum - inflow of angular momentum

The torque about the axis of rotation of the machine is then

where r = radial distance from the axisCw = tangential component of absolute velocity

Or using mean values

3.3 EULER PUMP AND TURBINE EQUATION

We will use the following nomenclature in this and the subsequent sections:

C = absolute velocityW = relative velocity (as seen in the rotating frame of reference)U = blade circumferential speed ( = ωr)

Subscripts:

a = axial component (of velocity) (subscript x also used)r = radial componentw = "whirl" (circumferential or tangential) component (subscripts t and θ also used)

Angles:

α = absolute velocityαN = stator blade metal anglesβ = relative velocityβN = rotor blade metal angles

The datum for all angles is the main flow direction: axial in axial-flow machines, radial in radial-flowmachines.

Sign conventions: The question of signs only arises with reference to velocity components andangles in the tangential direction. Unfortunately, there is not much consistency in the use of signs in theturbomachinery literature. When needed, we will use the following conventions:(i) Tangential components of velocity are positive if they are in the same direction as the blade speed, U.(ii) The signs of angles are consistent with the sign convention for the tangential velocity components.

ROTOR

STATORS

U

C

W

U

β (−)

α (+) β (−)

α (+)

β (−)

T rC dm rC dmw w= −∫ ∫& &

2 1(7)

Consider again the general turbomachinery rotor

The torque applied to the fluid as it passes through the rotor is given by (7):

The torque is supplied at the shaft, transmitted through the disk and blades, and applied by the blades to thefluid in the form of a tangential force. The corresponding shaft power is

&W Tshaft = ω

and multiplying through by ω in (7)

& & &

& &

W r C dm r C dm

UC dm UC dm

shaft w w

w w

= −

= −

∫∫∫∫

ω ω12

12

(8)

where U = rω is the blade speed.

But the SFEE also relates the shaft power, , to the energy changes in the fluid. Equating the&Wshaftshaft powers from Eqns. (3) and (8)

h dm h dm UC dm UC dmw w02

01 2 1

& & & &∫ ∫ ∫ ∫− = − (9)

If we approximate the flow quantities by their mean values, then we can write

h h U C U Cw w02 01 2 2 1 1− = − (10)

For an incompressible-flow compression machine (from eqn. (5))

( )g H H H U C U CL w w2 1 2 2 2 2− + = −

and letting ΔH = H2 - H1 (the total head rise seen across the machine) and ΔHE = H2 - H1 + HL = ΔH + HL (the "Euler head") then

g H U C U CE w wΔ = −2 2 1 1 (11)

Eqns. 9-11 are versions of the famous Euler Pump and Turbine Equation (or Euler Equation). TheEuler equation is the fundamental equation of turbomachinery design. It relates the specification (for example,the head rise required) to the blade speed of the machine and the changes in flow velocity that it must produceto achieve the required performance. As described later, these changes in flow velocity are directly related tothe rotational speed and geometry (eg. blade shapes, etc.) of the machine.

Note that the Euler equation involves the full energy transfer between the machine and the fluid,including the energy that will be dissipated in overcoming friction. For a pump

ΔΔH H

Epump

=η

ΔH will be specified to the designer. But from eqn. (11), ΔHE is needed to determine the flow turning(change in UCw) which will achieve the required ΔH. Thus, to design the machine we need to know itsefficiency. As a result, the design process becomes iterative.

C W U= +

3.4 COMPONENTS OF ENERGY TRANSFER

We now examine in more detail the process of energy transfer within the rotor. Recall that

absolute velocity = relative velocity + velocity of moving reference frame

The drawing shows a hypothetical velocity diagram at outlet (station 2) for the generalized rotor (asimilar diagram could be drawn for station 1)

From the Euler Equation

&

&

Wm

g H h U C U CshaftE w w= = = −Δ Δ 0 2 2 1 1 (12)

We then rewrite the velocity terms on the RHS in terms of the velocity vectors in the drawing

C C C Ca w r22

22

22

22= + + (a)

and similarly for the relative velocity (the components are not labelled on the figure to avoid clutter)

( )W W W W

C U C Ca w r

a w r

22

22

22

22

22

2 22

22

= + +

= + − +(b)

Solve (a) and (b) for Ca22 + Cr2

2 and equate

C C W U U C Cw w w22

22

22

22

2 2 222− = − + −

Then

( )U C C U Ww2 2 22

22

221

2= + −

Similarly for the velocity triangles at the inlet, station1,

( )U C C U Ww1 1 12

12

121

2= + −

Substituting into (12)

( ) ( ) ( )( )&

&

( ) ( ) ( )

Wm

g H h C C U U W WshaftE= = = − + − + −Δ Δ 0 2

212

22

12

12

221

21 2 3

(13)

Note that (13) is another (and useful) version of the Euler Equation.

Now consider the physical interpretation of the three terms on the RHS of (13).

Term (1), is clearly the kinetic energy change of the fluid across the rotor. In a pump,( )12 2

212C C−

blower or compressor, the kinetic energy of the fluid normally increases across the rotor. Some of this kineticenergy can be converted to static pressure rise in a subsequent diffuser or set of stators.

To see the physical meaning of the other two terms, apply the SFEE between the inlet and outlet ofthe rotor again, assuming adiabatic flow and neglecting potential energy changes:

& & &mP C

u W mP C

ushaft1 1

2

12 2

2

22 2ρ ρ+ +

⎛

⎝⎜

⎞

⎠⎟ + = + +

⎛

⎝⎜

⎞

⎠⎟

Substitute for from the Euler Eqn., (13), and solve for the static pressure rise through the rotor passage&Wshaft

( ) ( ) ( )P P U U W W u u2 1 22

12

12

22

2 112

12

− = − + − − −ρ ρ ρ (14)

Equation (14) shows that there is some direct compression (or expansion) work done inside the rotor bladepassage and it is associated with the changes in U and W that the fluid experiences as it passes through therotor. Note that if there is friction present, u2 > u1, and this reduces the pressure rise that would be achieved bya compression machine, as one would expect.

Term (2), is then energy transfer to the fluid due to the centrifugal compression (or( )12 2

212U U−

expansion) of the fluid as it passes through the rotor ("centrifugal energy" change). The rotation of the fluidimposed by the rotor results in a radial pressure gradient to balance the centrifugal forces on the fluid particles.

For example, consider a centrifugal pump or compressor rotor for the limiting case where there is noflow (say that a valve has been closed in the discharge duct). The fluid particles trapped inside the rotor travel

2

1

F

(+)

(-)

ω

UW1

W2

in circular paths. The force required to give the corresponding acceleration towards the axis of rotation issupplied by the radial pressure gradient that is set up in the rotor.

For this case, W1 = W2 = 0, and from (14) then

( )P P U U2 1 22

121

2− = −ρ

Thus, a radial machine will produce a pressure rise even for no flow. The delivery pressure for this case issometimes known as the “shut-off head”.

When there is flow, the fluid particles that move through the radial pressure field will likewise becompressed (or expanded) and the corresponding work per unit mass is accounted for by term (2) in Eqn. (13).

Term (3), represents the change in pressure energy due to the change in fluid velocity( )12 1

222W W−

relative the rotor. Consider the flow in a the rotor-blade passage of an axial compressor. Neglecting friction(u2 = u1) and if the stream tube is at constant radius (so that U1 = U2) then from Eqn. (14)

( )P P W W2 1 12

221

2− = −ρ (15)

As shown in the sketch, a typical compressor rotor passage increases incross-sectional area as the relative flow is turned towards the axial(which is necessary in order to increase the Cw in the absolute frame). From continuity, W2 < W1 and from (15) there is a correspondingpressure rise. The passage is thus a diffuser. The forces exerted on thefluid by the blade surfaces cause the static pressure to rise between inletand outlet, and since there is also displacement associated with theseforces (since the rotor is moving) work is being done on the fluid.

Note that the pressure rise along the rotor blade passage can cause separation of the blade boundarylayers and therefore stalling of the airfoils. We therefore find it necessary to limit the change in W that wepermit in a given blade passage.

Summarizing:

(a) Term (1) in Eqn. (13) represents the change in kinetic energy (dynamic pressure) of the fluid dueto the work done on it in the rotor.(b) Terms (2) and (3) represent the direct static pressure changes (compression or expansion work)which occur inside the rotor.

In general, all three components of energy transfer will tend to be present in all rotors. However, foraxial rotors the centrifugal compression tends to be small (since U1 – U2 for every streamtube that passesthrough the rotor), whereas it is large in radial rotors.

3.5 VELOCITY DIAGRAMS AND STAGE PERFORMANCE PARAMETERS

3.5.1 Simple Velocity Diagrams for Axial Stages

A turbomachinery stage generally consists of two blade rows, a rotor and a set of stators:• A compressor stage normally has a rotor followed by a row of stators. As noted in 3.4, some

static pressure rise can occur inside the rotor. The stators can produce a further staticpressure rise by reducing the fluid velocity.

• A turbine stage normally has a row of stators ("inlet guide vanes" or "nozzles") followed by arotor. The nozzles impart swirl to the flow, accelerating it and thus causing a static pressuredrop. The rotor then extracts energy from the fluid by removing the swirl. This may beaccompanied by a further static pressure drop inside the rotor.

Consider a thin streamtube passing through an axial compressor stage (say near the mean radius):

We then draw a hypothetical set of velocity vectors as they might appear in the axial plane:

Note that the inlet flow has been assumed to have some swirl (α1 … 0.0). Therefore, there must beanother stage or a set of inlet guide vanes ahead of the present stage. The stators have also been shaped togive a stage outlet flow vector equal to the inlet vector (C3 = C1). This is sometimes referred to as a “normalstage”.

Even for an axial stage, as the flow passes through the stage, the streamtube may vary slightly inradius. Thus, in general U1 … U2. Also, due to the density changes and changes in the cross-sectional area ofthe annulus, the axial velocity at different locations may vary (Ca1 … Ca2). However, across a given axial rotorblade, the radial shift in any given streamline tends to be quite small. For reasons discussed later, it is alsoundesirable to have the axial velocity change significantly along the machine. The latter is the reason for thetapering of the annulus which is seen in most multistage compressors and turbines.

For discussion purposes only, we may therefore make the following simplifying assumptions for axialstages:

(i) Assume the streamline radius is constant through a rotor: U1 = U2.(ii) Assume constant axial velocity through a given stage: Ca1 = Ca2 = Ca3.

The resulting velocity diagrams are sometimes known as the “simple” velocity diagrams (or velocitytriangles). For actual design calculations, we would not make these simplifications: we would use the true,general velocity diagrams. But in practice most axial stages come close to satisfying the simplifyingassumptions and therefore the conclusions which we will draw about the stage behaviour, based on the simplevelocity triangles, will be quite realistic.

One convenient feature of the simple velocity triangles is that we can combine the inlet and outlettriangles because of the common blade speed vector U. We can therefore draw the velocity triangles for theaxial compressor stage as follows:

3.5.2 Degree of Reaction

If the pressure is rising in the direction of the flow (ie. if there is “diffusion”), then there is a danger ofthe boundary layers on the walls separating. When this happens on a turbomachinery blade, there is generallya large reduction in the efficiency of the machine and an impairment of its ability to transfer energy to or fromthe fluid. In the case of compressors, boundary layer separation can lead to the very serious phenomena ofstall and surge which will be discussed later.

Diffusion is present most obviously in compressors since they are specifically intended to raise thepressure of the fluid. While overall the pressure drops through a turbine stage, diffusion may still be presentlocally on the blade surfaces. Thus, the possibility of boundary layer separation is a concern in the design ofboth compressors and turbines.

As evident from the velocity triangles, pressure rise can occur in both blade rows of a compressorstage. Intuitively, it would seem beneficial to divide the diffusion fairly evenly between the blade rows. Similarly, in a turbine stage both blade rows can benefit from the expansion. The choice of the split inpressure rise or drop between the two blade rows is one of the considerations for the designer of aturbomachinery stage.

We define the degree of reaction, Λ

( ) ( )[ ]( )

Λ =

=− + −

−

rate of energy transfer by pressure change inside the rotortotal rate of energy transfer

U U W W

h h

12 2

212

12

22

02 01

(16)

which can also be written

Λ =−−

h hh h

2 1

02 01(17)

where h = static enthalpy, h0 = total enthalpy. Using the Steady Flow Energy Equation or Euler Equation,there are several alternative ways of expressing the denominator in (16) and (17).

If the flow is assumed incompressible and isentropic, and the stage inlet and outlet velocities are thesame (ie. if is a “normal stage”), (17) reduces to

ΛΔΔ

=PP

rotor

stage(18)

Thus, (16) and (17) are also approximate measures of the fraction of the static pressure change which occursacross the rotor.

A well-designed pump, fan or compressor will then have Λ > 0 in order to spread the diffusionbetween the blade rows. A value of Λ . 0.5 has often been used. In an open machine, such as a Pelton wheelturbine, P1 = P2 = Patm and Λ = 0. A machine with Λ = 0 is known as an impulse machine. Impulse wheels aresometimes used for axial turbines, particularly steam turbines.

The effect of the choice of Λ on the machine geometry can be seen by examining the velocitydiagrams for a few examples.

Axial-Flow Impulse Turbine (Λ = 0):

Consider the mean radius. Assume incompressible flow, constant annulus area and no radial shift inthe streamlines. Thus U1 = U2 = U and from continuity, Ca0 = Ca1 = Ca2 since . We therefore&m C Aa annulus= ρhave the conditions for simple velocity triangles. The turbine stage will look as follows:

The basis for the stage geometry is as follows:

Nozzles: We must accelerate the flow through the nozzles, since all expansion is to occur inhere (Λ = 0): ie. we want C1 > C0. This can be done by turning the flow since thiswill reduce the area of the flow passage from A0 to A1noz (for the constant height,A1noz = A0cosα1N). Bear in mind that Ca0 = Ca1 from continuity.

Rotor Blades: For Λ = 0, we need W1 = W2 (since U1 = U2). Thus we need A2rot = A1rot, which isobtained with β1N = β2N. Therefore, the impulse turbine will have equal inlet andoutlet metal angles.

What determines the value of α1N which is chosen? From the Euler Equation:

( )& & &W m U C U C mU Cw w w= − =2 2 1 1 Δ

Redraw the velocity triangles with the common blade speeds U superimposed. Note that ΔCw = Cw2 - Cw1 willbe negative, consistent with our sign convention that power in ispositive. The magnitude of ΔCw (for a given U) is clearly related toα1. Thus, the required plays a direct role in determining the&Wvelocity triangles, and ultimately the metal angles.

Note also that to sketch the blade shapes we assumed thatthe fluid leaves a blade row at the metal angle:

α α β β1 1 2 2= ′ = ′,

This is not strictly true, as will be discussed later, but is often areasonable first approximation. It is sometimes known as the "EulerApproximation".

Axial-Flow Turbine with Λ>0 (Reaction Turbine):

Again assume constant streamline radius, constant annulus area and incompressible flow. Then U1 =U2 and Ca1 = Ca2 as before. The nozzles will again impart swirl to obtain some expansion. To get expansion inthe rotor, need W2 > W1 and thus *β2N* > *β1N*. An example of the geometry of a reaction turbine is then asfollows:

U

Cw1 (+)

Cw2 (+)

ΔCwC1

Ca1 = Ca2

α1 (+)C2

W1

W2

Axial-Flow Compressor with Λ>0:

Again, assume U1 = U2 and Ca1 = Ca2. To get static pressure rise across the rotor we need W2 < W1. Examining the compressor used as an example in Section 3.5.1, it is evident that this compressor meets thisrequirement:

3.5.3 de Haller Number

The importance of diffusion in compressor blade rows was discussed in Section 3.5.2. By selecting adegree of reaction close to 50%, the diffusion is shared roughly equally between the rotor and the stators. However, this does not address the question of whether the blade rows will be able to sustain the level ofdiffusion which is being asked of them. We will later examine diffusion limits which are used in the detaileddesign of the blade rows. However, it is useful to have a simple approximate criterion for diffusion which canbe applied at the point in the design where we are taking basic decisions about the velocity triangles.

An axial compressor blade row in effect forms a rectangular diffusing duct. Based on variouscompressor designs of the time, de Haller in the mid 1950’s suggested that the maximum static pressure risewhich could be achieved in axial compressor blade passages is given by

C P

Vp,max .= =

Δ12

0 442ρ (a)

where ΔP = static pressure rise between inlet and outlet of the blade rowV = velocity at the inlet to the passage (relative velocity for rotors, absolute for stators).

Taking a rotor blade passage and assuming no change in radius of the streamlines (so that there is nocentrifugal compression) and neglecting friction, from Section 3.4 the static pressure rise is

P P W W2 1 12

221

212

− = −ρ ρ .

Substituting into (a) and simplifying,

WW

2

10 75

⎛

⎝⎜

⎞

⎠⎟ =

min

. .

The ratio W2/W1 (or Cout/Cin for a row of stators) is known as the de Haller number.

The de Haller limit should be used as a rough guide only. It does not take into account details of theblade passage design which can improve the diffusion capability of the passage. Successful moderncompressor designs have used values of the de Haller number as low as 0.65. The de Haller number should beused mainly to alert the designer to the fact that the level of diffusion in a particular compressor blade rowmay present a design challenge.

3.5.4 Work Coefficient

From the Euler Equation

( )∆

∆h U C U C

UCw w

w

0 2 2 1 1= −=

and for an axial machine with simple velocity triangles (so that U1 U2 = U)

∆ ∆h U Cw0 = .

From the velocity triangles, if we vary U, adjusting Ca to maintain geometrically similar triangles, then

∆C Uw ∝

and ∆h U02∝ .

Thus, the power transfer varies as U2. The head or enthalpy change "per unit U2" is a useful measure of thestage loading and is known as the work coefficient, , where

( )ψ = = =

∆ ∆ ∆hU

UC

Ug HU

w E02 2 2

For “high” , we are taking full advantage of the blade speed and we have “high stage loading”: we willspecify what constitutes “high” for different types of machines in Section 3.5.6.

For a centrifugal machine, tip speed, U2, would be used in .

For an axial machine with simple velocity triangles (so that U1 U2 = U)

ψ = =U C

UCU

w w∆ ∆2

Normally, is taken as positive. For our sign convention, h0 and Cw are negative for turbines. Therefore, we use absolute values in

3.5.5 Flow Coefficient

Consider two compressor rotors designed for the same service (same Q, ΔP0 and N):

The same mean radii have been used so that the rotors have the same blade speeds U. From the Eulerequation, , and to achieve the same , and thus the same pressure rise, they must thereforeΔ Δh U Cw0 = Δh0have the same change in swirl velocity, ΔCw. As a result, the rotors have the same work coefficient(ψ = ΔCw/U) and thus the same loading. However, rotor B has twice the axial velocity of rotor A: this isachieved by reducing the cross-sectional area of the machine. This change obviously has a significant effecton the rotor blade geometry. It also has aerodynamic consequences:

(i) For rotor B, both the absolute and relative velocities have been increased. Since losses generallyvary as 0.5ρV2 (where V = W for the rotor), rotor B will, all other things being equal, have poorerefficiency than rotor A.(ii) All other things are not equal. Note that the increase in Ca in rotor B has had the effect ofincreasing the de Haller number (W2/W1). Thus, the diffusion has been reduced in rotor B, which isaerodynamically favourable.

We can thus identify an additional important parameter which must be chosen by the designer, the flowcoefficient, φ:

φ =CU

a

For a centrifugal compressor, we would use Cr2/U2, where Cr2 is the radial component of velocity at the rotoroutlet.

Note that for the compressors shown, the change in flow coefficient did not in fact change the degreeof reaction. As you will show in Problem Set 3, the symmetry of the velocity triangles for both machinesimplies that they both have 50% reaction.

3.5.6 Choice of Stage Performance Parameters for Design

We have identified four useful performance parameters: the degree of reaction, the de Haller number,the work coefficient and the flow coefficient. Experience shows that to design a stage with good efficiency, φ,ψ and Λ, and for fans and compressors, the de Haller number, should be kept within certain ranges.

DesignParameter

Fans, Pumps, Compressors Axial Turbines

Axial Centrifugal

φ 0.2 6 0.7 . 1 (at outlet) 0.4 6 1.2

ψ 0.3 6 0.6 0.6 6 1.0 (see later) 0.3 6 3.0<0.5 - “Lightly Loaded”>1.5 - “Highly Loaded”

Λ 0.3 6 0.7 (Not much used) 0 6 1

de Haller >0.65 (well-designed machineswith clean inlet flow)>0.80 (simple design, poor inletflow uniformity)

See Section 6.4.3 N/A

For compressible-flow axial turbines, Smith ( S.F. Smith, "A Simple Correlation of TurbineEfficiency," J. Royal Aero. Soc., Vol. 49, July 1965, pp. 467-470.) developed a very useful figure (the “Smithchart”) which summarizes the influence of φ and ψ on the efficiency of the stage:

Variation of Stage Efficiency with φ and ψ (for Zero Clearance).

The "Smith Chart" or "Smith Diagram" presents the results for a large number of turbine tests (forboth model and full-scale machines) conducted at Rolls-Royce from 1945 to 1965. Over that period, the flowover the tip of the rotor blades ("tip leakage") was considerably reduced. The tip-leakage flow is an importantsource of losses and as a result there was significant improvement in efficiency. To isolate the influence of thestage loading and shape of the velocity triangles, the efficiencies were corrected back to their zero-clearanceequivalents. Thus, efficiencies for actual machines can be expected to be lower than those shown by a coupleof percentage points. Note that the degree of reaction is not mentioned on the Smith chart. The turbines usedto generate the chart had a range of degrees of reaction. However, the performance of turbines is not stronglydependent on the degree of reaction, provided reasonable values are used.

The Smith chart is well known and is widely used by axial turbine designers during the preliminarystages of design. The usefulness of the Smith chart makes it surprising that comparable charts are not morewidely used by axial and centrifugal compressor designers. Part of the reason lies in the important role playedby diffusion (expressed through both the degree of reaction and the de Haller number) in compressorperformance. Thus a single “Smith chart” for compressors is not feasible. However, it is possible to generatea small number of charts, each for a different value of degree of reaction say, and then use these in design. Inthe late 1980's Casey (M.V. Casey, “A Mean Line Prediction Method for Estimating the PerformanceCharacteristics of an Axial Compressor Stage,” Proceedings, I.Mech.Eng., C264/87, 1987, pp. 273-285.)calculated compressor stage performance for a wide range of conditions. In a recent textbook, Lewis (R.I.Lewis, “Turbomachinery Performance Analysis”, Arnold, London, 1996) took this data to generate “Smithcharts” for axial compressors for three values of degree of reaction: 50, 70 and 90%. Note the rapiddeterioration in efficiency when the de Haller number is less than about 0.7.

The use of the guidelines presented in this section will be illustrated in the next chapter.

“Smith” Charts for Axial Compressors: (a) Λ = 0.5, (b) Λ = 0.7, (c) Λ = 0.9.

3.6 EFFICIENCY OF TURBOMACHINES

3.6.1 Incompressible-Flow Machines

The definitions of efficiency used for incompressible-flow machines have been discussed briefly inearlier sections. The definitions are repeated here for completeness.

Fundamentally, the efficiency of a turbomachine is defined in terms of a comparison with a related“ideal” machine in which there are no losses. However, there are small conceptual differences between thedefinitions of efficiency used for incompressible- and compressible-flow machines. These will therefore beclarified now.

(a) Pumps, Fans and Blowers

From the steady flow energy equation,

& & &W mg H m hshaft E= =Δ Δ 0

where ΔHE = Euler head = head equivalent of the shaft power input to the machine= head rise that would be achieved in the ideal (no losses) machine with the

same shaft power input as the actual machine.

The fluid power is defined as the useful, mechanical power that actually appears in the fluid across themachine

& &W mg Hfluid = Δ

where ΔH = total head rise that is actually observed across the machine.

The Euler head and the actual total head are related by

Δ ΔH H HE L= −

where HL is the head loss due to friction inside the machine. Neglecting elevation changes, we can also write

Δ Δ

Δ Δ

P g HP g H

actual

ideal E

0

0

,

,

=

=

ρρ

We then define the efficiency for a pump, fan or blower as

ηρ

pumpshaft E

actual

ideal

Fluid powerShaft power

Qg HW

HH

PP

= = = =Δ Δ

Δ

Δ

Δ&,

,

0

0

To help visualize the significance of this definition, and for comparison with the definition of efficiency usedfor compressors, we represent the processes on the Δh0 versus s diagram.

The specification calls for the machine to raise thefluid head by ΔH, or the total pressure by

. With the same shaft powerΔ ΔP P P g Hactual0 02 01, = − = ρinput per unit mass flow (Δh0), the ideal machine would raisethe pressure by ΔP0,ideal = PN02 - P01. Thus, the efficiency forpumps, fans and blowers is defined by comparing the head ortotal pressure rises for the actual and an ideal machine thathave the same shaft power input. As described below, thedefinition of efficiency for compressors is slightly different.

(b) Turbines

For turbines, the head drop, ΔH, or pressure drop ΔP0,actual that is available is normally specified. However, some of the fluid power released by the fluid is used in overcoming friction inside the machine andis therefore not available to be extracted as shaft power output, . That is, & &W mg Hshaft E= Δ

Δ ΔH H HE L= +

The turbine efficiency is then defined as

ηρρturbine

E E ideal

actual

Shaft powerFluid power

Qg HQg H

HH

PP

= = = =ΔΔ

ΔΔ

Δ

Δ0

0

,

,

The physical interpretation can again be seen in terms of theh0 versus s diagram. The actual pressure drop is ΔP0,actual = P01 - P02and the shaft power extracted is . In an& & &W mg H m hshaft E= =Δ Δ 0ideal machine, a smaller pressure drop, ΔP0,ideal = P01 - PN02, would beneeded to produce the same shaft power output. Thus, the turbineefficiency is defined in terms of two machines that have same shaftpower output. The comparison is between the head or total pressuredrops required to obtain that shaft power output in the ideal andactual machines. The similarity with the definition used for pumps,fans and blowers is evident.

h0

s

P01

P02

Δh0

P02

IDEAL

ACTUAL

h0

s

P01

P02

Δh0P02

IDEAL

ACTUAL

3.6.2 Compressible-Flow Machines

The efficiency of compressible flow machines is defined slightly differently. The comparison is againbetween ideal and actual machines. However, instead of the shaft power input or output, the common basis isthe pressure rise or drop across the machines.

(a) Compressors

The h0-s diagram is again used to comparethe processes used to define the efficiency. Forcompressible-flow machines, the pressure rise ordrop across the machine is generally expressed interms of the total pressure ratio. The h0-s diagramshows the ideal and actual compression processesneeded to obtain the same pressure ratio, P02/P01. Forthe ideal machine, the shaft power required is

& & ,W m hideal ideal= Δ 0

while for the actual machine

& & .,W m hactual actual= Δ 0

The compressor efficiency is then defined as the ratio of the shaft powers required to produce the samepressure ratio in the ideal and actual machines:

ηcideal

actual

ideal

actual

ideal

actual

WW

m hm h

hh

= = =&

&

&

&

,

,

,

,

Δ

Δ

Δ

Δ0

0

0

0

If we assume that the working fluid is a perfect gas, then h0 = CpT0, and it is common to present theprocesses on a T0-s diagram, rather than the h0-s diagram. The efficiency can then be written

ηcp ideal

p actual

C TC T

T TT T

= =′ −−

Δ

Δ0

0

02 01

02 01

,

,

For any isentropic process involving a perfect gas,

P constρ γ = .

where γ = Cp/Cv, the specific heat ratio. Then using the perfect gaslaw, P = ρRT, we can write

′=⎛

⎝⎜

⎞

⎠⎟

−

TT

PP

02

01

02

01

1γγ

h0

s

P01

P02

IDEAL

ACTUAL

Δh0,actual

Δh0,ideal

s

P01

P02

IDEAL

ACTUAL

T02

T01

T0

T02

Then

( )Δh

C T T

C T PP

actualp

c

p

c

002 01

01 02

01

1

1

, =′ −

=⎛

⎝⎜

⎞

⎠⎟ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

η

η

γγ

and this expression allows the shaft power required to drive the actual machine, , to be& & ,W m hactual actual= Δ 0related to the specified pressure ratio.

(b) Turbines

The efficiency of compressible-flow turbines is similarly defined by comparing the shaft powerproduced by the expansion through the same pressure ratio for an ideal and the actual machine. Following thesame procedure as for the compressor, we obtain

ηtactual

ideal

actual

ideal

WW

hh

T TT T

= = =−

′ −

&

&,

,

Δ

Δ0

0

02 01

02 01

and

Δh C TPPactual p t0 01

02

01

1

1, =⎛

⎝⎜

⎞

⎠⎟ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

η

γγ

Note the expression for Δh0, actual will be negative, consistent withour sign convention that power into a machine is positive.

3.6.3 Polytropic Efficiency

Consider a multi-stage axial compressor consisting of a number of stages with equal stage pressureratios. If the stages are designed using the same technology, it is reasonable that they will each have the samestage isentropic efficiency. It is then possible to calculate the overall pressure ratio and isentropic efficiencyfor the machine as a whole.

Let PRs = stage total-pressure ratioηs = stage isentropic efficiency

It can then be shown that the actual temperature at the outlet of the Nth stage is

( )T TPR

Ns

s

N

0 1 01

1

11

+

−

= +−⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

γγ

η

The overall pressure ratio for the N stages is

s

P01

P02IDEAL

ACTUAL

T02

T01

T0

T02

( )PR PRc sN=

and the isentropic temperature rise for the whole compressor is then

( )

′ − = −⎛

⎝⎜⎜

⎞

⎠⎟⎟

= −⎛

⎝⎜⎜

⎞

⎠⎟⎟

+

−

−

T T T PR

T PR

N c

sN

0 1 01 01

1

01

1

1

1

γγ

γγ

and thus the overall isentropic efficiency is

( )

η

η

γγ

γγ

cN

N

s

N

s

s

N

T TT T

PR

PR

=′ −

−=

−

+−

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

+

+

−

−

0 1 01

0 1 01

1

1

1

1 1 1

For example, if PRs = 1.2 and ηs = 0.9, the resulting variation of the overall pressure ratio and overallisentropic efficiency with the numberof stages is shown in the figure. Asseen, the overall efficiency decreases asthe pressure ratio increases.

When cycles for gas turbineengines are being investigated, it isnormal to examine the effect of varyingpressure ratio. It is evident thatassuming a constant value of theoverall compressor isentropicefficiency is not valid for suchinvestigations. To account for theeffect of the pressure ratio on theisentropic efficiency, the concept of thesmall-stage or polytropic efficiency hasbeen introduced.

From the Second Law of Thermodynmaics, for a general infinitesimal process

dhdP

T ds00

00= +

ρ

Then for an isentropic process (ds = 0)

1 2 3 4 5 6 7 8 9 10 1112

Overall Compressor Pressure Ratio

Ove

rall

Com

pres

sorI

sent

ropi

cEf

ficie

ncy

1 2 3 4 5 6 7 8 9 100.8

0.82

0.84

0.86

0.88

0.9

Stage PR = 1.2Stage ηisen = 0.9

EFFECT OF PRESSURE RATIO ON OVERALL ISENTROPIC EFFICIENCY

Number of Stages

dhdP

′ =00

0ρ

Define the polytropic efficiency, ηp, as the isentropic efficiency for the infinitesimal process

dh dhp′ =0 0η

∴ =dP

dhp0

00ρ

η

Then assuming a perfect gas, h0 = CpT0 and P0 = ρ0RT0. Also Cp - Cv = R, or , and the fluidC Rp = −

γγ 1

properties are assumed constant through the process. Then

( )dPC T

d C T

C Tpp

p

p

0

0 0

0

0ρη=

dP

RT

dTTp

0

0 0

0

01

γγ

ρη

−

=

ordTT

dPPp

0

0

0

0

1=

−γη γ

Integrating this between the start and end of a finite process

ln lnTT

PPp

02

01

02

01

1⎛

⎝⎜

⎞

⎠⎟ =

− ⎛

⎝⎜

⎞

⎠⎟

γη γ

orTT

PP

p02

01

02

01

1

=⎛

⎝⎜

⎞

⎠⎟

−γη γ

For a compression process, the isentropic efficiency is defined as

ηchh

=′Δ

Δ0

0

where and where ηpc is the polytropic efficiency forΔ ′ = −⎛

⎝⎜⎜

⎞

⎠⎟⎟

−

h C T PRp0 01

1

1γγ Δh C T PRp

pc0 01

1

1= −⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

−γη γ

the compressor. Then

η

γγ

γη γ

cPR

PR pc

=−

−

−

−

1

11

1

For a turbine,

ηthh

=′

ΔΔ

0

0

and, with inlet at 3 and outlet at 4, it can then be shown that

( )

η

η γ

γ

γγ

t

PP

PP

pt

=

−⎛

⎝⎜

⎞

⎠⎟

−⎛

⎝⎜

⎞

⎠⎟

−

−

1

1

04

03

1

04

03

1

The following figure shows the resulting variation of isentropic efficiency with pressure ratio for an assumedpolytropic efficiency of 0.9 and γ = 1.4, for both a compressor and a turbine. Also shown are the earlierresults for the multistage compressor with stage pressure ratio of 1.2.

The concept of polytropic efficiency should be used with caution. It is only valid if the machine canbe considered to employ comparable technology and produce comparable performance as the pressure ratio isvaried. For this reason, it should be applied only to explore the influence of pressure ratio on performance formultistage machines. It is assumed that the pressure ratio is varied by adding or removing comparable stages. Polytropic efficiency should not be used to predict how the efficiency of a single stage will vary as its design

Pressure Ratio

Isen

tropi

cEf

ficie

ncy

1 2 3 4 5 6 7 8 9 10

0.86

0.88

0.9

0.92

0.94

VARIATION OF ISENTROPIC EFFICIENCY WITH PRESSURE RATIOPolytropic Efficiency, ηp = 0.9, γ = 1.4

1 2 34

56

78

910

1112

10

Turbine

Compressor

pressure ratio is changed. As will be shown later, stage performance is closely related to its tip speed. Forexample, to increase the design pressure ratio of a compressor stage, the tip speed must normally be increased. This in turn results in higher flow velocities generally. As these velocities reach and exceed the speed ofsound, shock waves will appear, providing a source of additional losses that is not present at lower speeds. Thus, as the stage pressure ratio is changed, the technology cannot be considered to remain unchanged.

4.2 CONTROL VOLUME ANALYSIS FOR AXIAL-COMPRESSOR BLADE SECTION

4.2.1 Force Components

Consider the control volume for the flow through one blade passage:

Take unit depth in the z direction. Also, make the following simplifying assumptions(i) Incompressible flow(ii) Constant axial velocity through the passage: Ca1 = Ca2 = Ca.

The blade exerts a force F on the flow thought the passage. This is divided into axial and tangentialcomponents X and Y. By definition, the lift generated by a turbomachinery blade L is the component of the bladeforce normal to the vector mean flow direction through the blade row. The drag D is the component of the bladeforce parallel to the vector mean flow direction.

Then apply the linear momentum equation to the control volume. In the x direction:

( )( ) ( ) ( )

ΣF m V V

X P s P s m C Cx x x

a a

= −

+ × − × = −

&

&

2 1

1 2 2 11 1

Note that the pressure forces along the left and right faces of the control volume exactly balance each other in boththe x and y directions. Then since we have assumed Ca1 = Ca2, the x-wise momentum equation reduces to

( )X P P s= −2 1 (1)

For the y direction:

( )ΣF m V Vy y y= −& 2 1

and since the pressure forces on the control volume cancel each other in the y-direction, the only force in the y-

C2 P2Ca2

α2

Cw2

s

Yαm

XF

DL

P1

Ca

Ca1

Cw1

C1 α1x

y

Cm

Cwm

αm

A B

CD

direction is that due to blade, Y

( ) ( ) ( )( )( )

Y C s C C

C s C Ca w w

a w w

= × − − −

= −

ρ

ρ

1 1 2

1 2 (2a)

Since

tan , tanα α11

22= =

CC

CC

w

a

w

awe can also write

( )Y C sa= −ρ α α21 2tan tan (2b)

From the definition total pressure for incompressible flow, the total pressure loss through the passage isgiven by

( ) ( )ΔP P P P P C C0 01 02 1 2 12

221

2= − = − + −ρ

From the velocity triangles,

( ) ( )( )( )

C C C C C C

C C C Cw a w a

w w w w

12

22

12

12

22

22

1 2 1 2

− = + − +

= + −since Ca1 = Ca2. Then

( ) ( )( )ΔP P P C C C Cw w w w0 1 2 1 2 1 212

= − + + −ρ

substituting for from (2a)C Cw w1 2−

( )ΔP Xs

C C YC sw w

a0 1 2

12

= − + +⎛

⎝⎜

⎞

⎠⎟ρ

ρ

and using the vector mean flow direction through the passage, , we can write( )tan tan tanα α αm = +12 1 2

( )ΔPs

X Y m01

= − + tanα (3)

From the force vector triangles, the drag D can be expressed in terms of X and Y as follows

( )D Y X

X Ym m

m m

= −

= − +

sin coscos tan

α αα α (4)

and substituting from (3)

D P s m= Δ 0 cosα (5)

Then from the definition of the drag coefficient

C D

C c

P s

C c

D

m

m

m

=×

=

12

1

12

2

0

2

ρ

α

ρ

Δ cos

and finally, since σ = c/s is the solidity

ΔP C CD mm

021

21

= ⎛⎝⎜

⎞⎠⎟

σ ραcos

(6a)

or alternatively, using Cm = Ca/cosαm,

ΔP C CD am

02

3

12

1= ⎛

⎝⎜⎞⎠⎟

σ ραcos

(6b)

As will be seen later, some axial fan and compressor prediction procedures use the airfoil drag coefficient to expressthe loss performance for the blade row. Equation (6a) or (6b) can then be used to express this as a total pressureloss.

Returning to the lift force, from the force triangles the lift L can be expressed as

L X Ym m= +sin cosα α (7)

Solving (4) for X

X Y Dm

m= −tan

cosα

α

and substituting into (7)

L Y D Y

Y D

mm

m m

mm

= −⎛

⎝⎜

⎞

⎠⎟ +

= −

tancos

sin cos

costan

αα

α α

αα

Then substituting for Y from (2b)

( )LC s

Da

mm= − −

ρα

α α α2

1 2costan tan tan (8)

By definition

C L

C c

L

C cL

m am

= =⎛

⎝⎜

⎞

⎠⎟

12

12

12 22

ρ ραcos

then

( )C

C s

C c

D

C cL

a

m

am

m

m=−

⎛

⎝⎜

⎞

⎠⎟

−

ρα

α α

ρα

ρα

2

1 2

22

212

1 12

costan tan

cos

tan

or

( )C sc

CL m D m= ⎛⎝⎜⎞⎠⎟

− −2 1 2cos tan tan tanα α α α (9a)

Since the drag force is normally much smaller than the lift, the drag term is often omitted from (9a),

( )C scL m= ⎛

⎝⎜⎞⎠⎟

−2 1 2cos tan tanα α α (9b)

4.2.2 Circulation

Any lifting surface has circulation. By definition, the circulation Γ is

Γ = ∫ V dSS (10)

where the integral is evaluated along any closed contour enclosing the lifting surface. VS is the tangentialcomponent of the flow velocity along the enclosing curve and S is arc length. For the axial-compressor airfoil, thecurve A-B-C-D-A shown on the control volume in the last section is a convenient curve for use in (10):

Γ = + + +∫ ∫ ∫ ∫V dS V dS V dS V dSSA

B

SB

C

SC

D

SD

A

Since B-C and D-A are periodic surfaces with identical lengths and velocity distributions,

V dS V dSSB

C

SD

A

∫ ∫= −

and their contributions to Γ cancel. Along A-B and C-D, VS is simply Cy (= Cw) along the respective segments. Thedirection of the integration changes so that the integrals will have opposite signs (since Cw1 and Cw2 have the samesign for the control volume shown). Thus, we can write

Γ = −C s C sy y1 2

We are assuming constant axial velocity, and since (and Cx = Ca), we can writetanα = C Cy x

( )Γ = −C sa tan tanα α1 2 (11)

From Eqn. (8), neglecting the drag term and substituting from (11) we can also write

L Cm= ρ Γ

which is the expression given by the Kutta-Joukowski Theorem for an isolated airfoil. Note that in the case of theblade row, the “undisturbed” velocity seen by the airfoil is in fact the vector mean velocity through the passage.

4.3 IDEALIZED STAGE GEOMETRY AND AERODYNAMIC PERFORMANCE

4.3.1 Meanline Analysis

For preliminary design, we typically consider just the flow at the mean radius and treat the flowthrough stage as one-dimensional. The mean radius is normally defined as the radius that divides the flowarea in half:

( )r r rm h t2 2 2 2= +

where rh = hub radius and rt = tip radius. This approach is known as meanline analysis.

The first step is to define the meanline velocity triangles, starting from the specification ( or&m P− Δ 0) and using the guidelines for φ, ψ etc. from Chapter 3. To illustrate the procedure, we will use aQ H− Δ

semi-quantitative example. Assuming an incompressible flow machine, we will define the velocity trianglesfor a stage, consisting of a rotor and a row of stators, that delivers a head rise ΔH at a volume flow rate Q. From the general guidelines, we choose the following values for the mean radius:

Flow coefficient: φ = =CU

a 05.

Work coefficient: ψ = =g HU

EΔ2 0 4.

Degree of reaction: Λ = 05.

Note that if we were using the Lewis charts from Section 3.5.6, we would probably choose a slightly highervalue of φ for this value of ψ:

To proceed, we need the value of the Euler head rise, ΔHE = ΔH/η. Therefore, we need to guess avalue for the stage efficiency η. This can be done from experience, or from the specific speed plots in Chapter2, or from the approximate correlations shown in Section 3.5.6. Later, we will see how to calculate theefficiency of the stage we have designed. If this efficiency is different from the one we have guessed here, wewill have designed the stage with an incorrect value of the ΔHE and it will not match the required performanceΔH. If this turns out to be the case, we will have to return to the beginning and revise the design. Thus, thedesign of a turbomachine inherently tends to be iterative: to design the machine we need its efficiency, but wedo not know its efficiency until we have designed it.

Having estimated ΔHE we can then calculate the absolute blade speed (at the mean radius) from ourchosen value of the work coefficient ψ:

1 2 3

rt

rh

rm

Ug HE=Δψ

With U determined, the axial velocity at mean radius follows from the chosen value of the flow coefficient φ:

C Ua = φ

The chosen value of φ also determines the relative magnitudes of Ca and U as they will appear in the velocitydiagram: in this case, Ca = 0.5U. Finally, having established Ca, the required annulus area for the stagefollows from one-dimensional continuity:

A mC

QCa a

= =&

ρ

We will assume “simple” velocity diagrams, as defined in Section 3.5.1. That is, we assume that theannulus is shaped such that Ca and U remain constant through the stage: U1 = U2 = U, Ca1 = Ca2 = Ca3 = Ca. Then from the Euler equation

g H U C U C U CE w w wΔ Δ= − =2 2 1 1

Knowing U, we now know ΔCw. Note also that for the simple velocity diagrams, ψ can be written

ψ =ΔCU

w

and we therefore also know the relative magnitudes of U and ΔCw in the velocity triangles: ΔCw = 0.4U.

Finally, we make use of the degree of reaction Λ to completely define the velocity triangles. Since wehave chosen 50% reaction, equal amounts diffusion areoccurring in the rotor and the stators. Thus the de Hallernumbers for the rotor and stators must be the same:

WW

CC

2

1

3

2=

On Problem Set #3, you will show that for simplevelocity diagrams, this is achieved by making thevelocity triangles for the inlet and outlet of the rotorsymmetrical, and by designing the stators so that C3 =C1. That is, we design the stators so that the flow at thestage outlet is identical to the flow that entered the stage. The rotor velocity triangles will then look as shown.

With the velocity triangles established, we candetermine the de Haller numbers:

WW

CC

2

1

3

20 678= = .

This is approaching the limit of about 0.65 that was

( )+2α ( )+1α ( )+1wC

( )+2wC wCΔU

1C

2C

( )−1β( )−2β

21 aa CC =

2W

1W

recommended in Section 3.5.6 and we will therefore have to monitor our design for the possibility of stall. Asnoted in Section 3.5.5, we could reduce the diffusion levels by increasing the flow coefficient φ.

Note that to achieve 50% reaction in this stage, theinlet flow must have a swirl angle α1. Thus, there must eitherbe a stage ahead of the present one, or a set of inlet guidevanes, that leave the required amount of swirl in the flow. Theflow from this stage will also leave with swirl α3 = α1, so thatC3 = C1.

Suppose instead that the inlet swirl was specified. Forexample, if this is the first stage in the machine then we willnormally have no swirl in the flow, α1 = 0. Using the samevalues of φ and ψ, the velocity triangles will then look asshown. We can then show that the resulting degree of reactionis Λ = 0.8. This means that the diffusion is much higher in therotor than in the stators and this might at first be a matter forconcern. However, consider the values of the de Hallernumbers (we will assume that the flow leaves the stage withno swirl, C3 = C1):

Rotor:WW

2

10 699= .

Stators:CC

3

20 781= .

As expected, the value is lower for the rotor than for the stators. However, the diffusion is actually less thanfor the 50% reaction machine. As a result, this stage may be just as feasible as the earlier stage, despite thehigh value of degree of reaction.

Having determined the velocity triangles, the next step is to define the blade geometries that willproduce the required velocities.

( )+2α

01 =α

01 =wC

( )+2wC wCΔ

U

1C

2C

( )−1β

( )−2β

21 aa CC =

2W1W

4.3.2 Blade Geometries Based on Euler Approximation

For the idealized analysis, we define the blade geometry using the assumption that the fluid leaves theblade row parallel to the metal angle at the trailing edge of the blades: this is known as the EulerApproximation. In a later section, we will develop the procedures for estimating the actual outlet flow angle,which will turn out to be slightly different. To bring the flow smoothly into the blade passage, we will alsomake the leading edge metal angle parallel to the inlet flow angle. We can then define the shapes of the bladesfor the 50% reaction stage as follows:

As indicated on the drawing:(i) to bring the flow smoothly onto the leading edge of the rotor blades.′ =β β1 1(ii) In the relative frame of reference, the flow must leave the rotor blade passage at to produce theβ2required turning. Based on the Euler approximation, the flow will leave the trailing edge at the metalangle and we therefore use .′ =β β2 2(iii) The stators see the flow in the absolute frame. To bring the flow smoothly onto the leading edgeof the stator blades we therefore make .′ =α α2 2(iv) Again, the flow is assumed to leave the stators at the metal angle, and we use .′ = =α α α3 3 1

Note that with the assumptions made, the rotor and stator blade geometries are identical for the 50% reactionstage.

4.3.3 Off-Design Performance of the Stage

The geometry of the idealized stage was defined to give the required performance at the design point:that is, at the design flow rate and rotational speed. However, any turbomachine will often be operated awayfrom its design point. The idealized analysis can also be used to give reasonable predictions of how the stagewill perform for off-design operating points.

(a) Effect of Varying Flow Rate

Consider first the effect of a reduction in flow rate at fixed blade speed U (i.e. at constant RPM). Theresulting velocity triangles will look as follows:

The new velocity triangles were arrived at as follows:

(i) Based on the Euler Approximation, the flow will still leave the blade rows at the metal angle. Therefore, α1, β2 (and α3) are unchanged. Recall that there must be a set of stators or inlet guide vanesahead of the rotor to account for the inlet swirl. (ii) From continuity, Ca is reduced and thus so is C1. In a quantitative calculation, the new value ofCa would just be obtained from , where Q is the new volume flow rate, and A is theC Q Aa =annulus area as established at the design point.(iii) The magnitude of W2 is also reduced, by continuity, but the direction is unchanged.

From the velocity triangles, Cw1 has decreased while Cw2 has increased. As a result, the change inswirl velocity ΔCw has increased. From the Euler equation

g H U CE wΔ Δ=

and the head rise produced by the machine will be increased. Equivalently, for a compressibleΔ ΔH HE= ηflow machine, , and the corresponding pressure ratio, , will be increased. Note that this isΔh0 P P02 01consistent with the increase in incidence (“angle of attack”) at the leading edge of the rotor blade. As a result

of this, the blade should develop greater lift, do more work on the fluid, and thus increase the head rise. Onthe other hand, increasing the incidence will eventually lead to stalling of the blade. Thus, reducing the flowrate through a compressor stage will move it towards stall. Note that the incidence was also increased for thestators, bringing them closer to stall as well.

Clearly, we can use the velocity triangles and the Euler equation to predict the quantitative stagecharacteristic for the idealized stage. It is convenient to express the characteristic in terms of the work andflow coefficients. The flow turning is

ΔC C Cw w w= −2 1

and from the velocity triangles (noting that Ww2 is negative for the conventional compressor velocity triangles)

C C C U W U Cw a w w a1 1 2 2 2= = + = +tan tanα β

Then( )ΔC U Cw a= + −tan tanβ α2 1

and dividing by U

( )ΔCU

CU

w a= + −1 2 1tan tanβ α

orψ φ= +1 m

Thus, the ψ versus φ curve (effectively, the head rise versus flow rate characteristic) is a straight line withslope

m = −tan tanβ α2 1

For the present case, the symmetry of the velocity triangles implies that and the slope is thenβ α2 1= −. For α1 > 0, as is the case here, this gives a negative slope and an inverse relationship betweenm = − 2 1tanα

head rise and flow rate, as inferred above.

Alternatively, since the characteristicpasses through the design point (say, φD and ψD),we can write

m D

D=

−ψφ

1

and the slope of the characteristic is seen to bedetermined by the choice of design point (notealso that in all cases ψ = 1 at φ = 0 for the idealcharacteristic). Interestingly, the characteristicwill be steeper for a more lightly-loaded stage(lower design work coefficient ψD) as illustrated inthe plot.

1.00.50.0 φ

0.5

1.0

ψ

ψD1

ψD2

ψD3

INCREASINGDESIGN-POINT

LOADINGφD

(b) Effect of Varying Blade Speed

It is also worth looking briefly at the effect of varying the blade speed at constant flow rate. Using theEuler Approximation again, it can be shown that the change in the velocity triangles will look as follows:

From the triangles, since U has decreased. For the work coefficient, ,φ φ= >C Ua D ψ = ΔC UwΔCw has clearly decreased, but so has U. However, ΔCw has decreased more rapidly than U; as can be seen, asmall further decrease in U would reduce ΔCw to zero. We therefore conclude that and φψ ψ= <ΔC Uw Dand ψ are again seen to vary inversely. In summary, any deviation from the design point will cause the a givencompressor to move along the same ψ versus φ characteristic.

It is also worth noting that the reduction in rotational speed has had a very strong effect on theabsolute work transfer:

g H U CE wΔ Δ=

Since ΔCw decreases directly with U (and in fact faster than U) the head rise varies approximately as

g H kUEΔ ≈ 2

and the head rise delivered by the stage, at a fixed value of flow rate, will change strongly with the rotationalspeed: for example, reducing the speed by a factor of 2 will reduce the head rise by about a factor of 4. Thus,high rotational speed is essential to obtain high pressure rise from a compressor stage. This will be illustratedfurther in later sections.

As seen, the Euler Approximation results in an idealized ψ versus φ characteristic for the stage that isa straight line with a negative slope.

We have already noted that some changes in operating point will result in positive values of theincidence at the leading edge of the airfoils. If this incidence becomes too large, we would expect the airfoilsto stall. Also, we would expect the efficiency of the stage to be best when the rotor and stator blades areoperating at the design point. We can therefore project what the actual stage characteristic is likely to bebased on the idealized characteristic:

The characteristic shown applies for all rotational speeds. As noted, there is a strong effect ofrotational speed on the absolute performance (say ΔH for a given Q). To emphasize this, the characteristicsare often plotted in absolute terms as variations of ΔH (or ΔP0) versus Q (or ) for constant values of&mrotational speed N. The corresponding curves are easily calculated from the non-dimensional characteristic. The resulting map will look as follows.

ψ

Dψ

η

Dφ φ

USING EULERAPPROXIMATION

STALL

LIKELYACTUAL

MAXIMUM η

HΔ

N

maxη

Q

η CONSTANT

CONSTANT

On each of the constant speed lines, there will be a point that corresponds to the design point valuesof φ and ψ on the non-dimensional characteristic. At each of those points, the velocity triangles will besimilar, as indicated in the drawing. In each case, the relative velocity vector at the rotor inlet is lined up withthe metal angle and the flow comes smoothly onto the leading edge. As shown, we would therefore expectthat the machine will operate at its maximum efficiency at each of those points, apart perhaps for some smalleffect of differing Reynolds numbers. Also, as we will see later, frictional losses vary as V2 and thus thehigher flow velocities with increasing rotational speed will result in higher frictional losses. This effect willbe partly offset by the fact that the Reynolds number is also increasing.

Later in the chapter, we will examine to what degree actual machines match the performancecharacteristics we have inferred from the velocity triangles in this section.

4.3.4 Spanwise Blade Geometry

Finally, we use the idealized stage analysis to give an example of how the blade shape will varyacross the span. For this example, we will take the case with no inlet swirl from Section 4.3.1. At the meanradius, φ = 0.5 and ψ = 0.4. For discussion purposes, we will also take the hub-to-tip ratio, HTR = rh/rt as 0.5. Note that since the cross-sectional area is determined by the flow rate and the choice of flow coefficient φ,once we choose the HTR, we can calculate the various required radii, rh, rm and rt. Finally, with the meanradius known and the mean blade speed Um fixed by the choice of work coefficient ψ, we have the rotationalspeed, ω = Um/rm.

To define the resulting spanwise geometry, we assume that the inlet axial velocity Ca is constantacross the span and that we want the same total head rise, , at every spanwise section. Thisg H U CE wΔ Δ=fixes the ΔCw as a function of radius and allows us to draw the velocity triangles for each spanwise section. The drawing shows the resulting velocity triangles and the blade geometry based on the Euler Approximation,for three spanwise sections. The table on the next page summarizes the corresponding values of theperformance parameters.

U

C1

W1

W2

rm

rt

rh

W1

W2

W1

W2

C2

C2

U W1

C2

U

C1

C1

Parameter TIP MEAN ROOT

Flow Coefficient, φ 0.395 0.5 0.791

Work Coefficient, ψ 0.25 0.4 1.0

Degree of Reaction, Λ 0.875 0.8 0.5

de Haller Number (Rotor) 0.788 0.699 0.62

de Haller Number (Stators) 0.845 0.781 0.62

Note:

(i) This blade design is clearly not acceptable. The work coefficient is far too high at the root and the deHaller numbers there also indicate too much diffusion. The blade will need to be redesigned. If the stage isstill to produce uniform pressure rise across the span, the mean line work coefficient will have to be reduced.(ii) The blade exhibits considerable twist across the span. Both this and the large variation in the designparameters is a function of the hub-to-tip ratio, HTR = rh/rt. Increasing the HTR will make the blade moreuniformly loaded across the span, but since the cross-sectional area is fixed (by the choice of φ), this hasconsequences for the tip diameter of the machine and the rotational speed. This is demonstrated in thefollowing sketch, which shows three different blades with the same annulus cross-sectional area but differentvalues of HTR. In multi-stage compressors, the HTR will normally increase along the machine since thecross-sectional area is decreased to keep the axial velocity high. This is illustrated by the cross-section of thecompressor from the GE LM2500+ gas turbine engine (17 stages, PR = 23.3).

HTR 0.3 0.5 0.8RPM Higher Lower

From: Wadia et al., ASME 99-GT-210

4.4 CHOICE OF SOLIDITY - BLADE LOADING LIMITS

The design parameters introduced in the last chapter apply to a stage or a blade row. Experience hasshown that it is possible to design a stage of good efficiency if the guidelines for those design parameters arefollowed. The parameters also fully define the velocity triangles and the corresponding airfoil geometries. However, the guidelines give no information about the number and the spacing of those airfoils: in otherwords, about the solidity σ = c/s of the blade rows.

For a blade row, the larger the spacing between the airfoils the larger the mass flow that each airfoil isrequired to turn. From the control volume analysis in Section 4.2, the resulting lift coefficient was given by

( )

( )

C scL m

m

= ⎛⎝⎜⎞⎠⎟

−

= ⎛⎝⎜

⎞⎠⎟

−

2

2 1

1 2

1 2

cos tan tan

cos tan tan

α α α

σα α α

and it is seen to vary directly with spacing, or inversely with the solidity. Just as for an isolated airfoil, there isan upper limit to the lift that a turbomachinery blade can develop before it stalls. For a given set of inlet andoutlet flow angles, it is possible to stay below the loading limit by making the solidity of the blade row largeenough. Thus, the solidity of the blade row is selected on the basis of a blade loading limit. This is in contrastto the work coefficient, ψ, which was a stage loading limit.

In the past, loading limits for compressor blades have sometimes been expressed in terms of the liftcoefficient (Horlock, 1958). In the early 1950s, Howell suggested that a well-designed compressor airfoil willstall at

CCCL

1

2

3

33⎛

⎝⎜

⎞

⎠⎟ ≈ .

and designers of low-solidity fans have sometimes used the criterion

C csL

⎛⎝⎜⎞⎠⎟≤ 11.

However, expressing the loading limit simply in terms of CL has been found to be unreliable. Recent practicehas therefore taken a somewhat different approach.

Howell (British Practice)

In the 1950s, Howell conducted an extensive series of cascade measurements on the compressorairfoils that were commonly used in British compressor design. The performance was measured for a widerange of the design parameters, including the flow turning angle and solidity. Howell varied the amount offlow turning up to the onset of stall. The corresponding total-pressure losses were also measured. Howellsuggested that a suitable design turning angle for a blade row was that which corresponded to about 80% ofthe turning that would result in stall. He also found that the losses were close to a minimum at this condition. He therefore presented a correlation that could be used to estimate the solidity that would result in the bladerow operating at 80% of the stalling turning angle. This correlation is shown in the next figure (taken fromSaravanamuttoo et al., 2001).

Knowing the design deflection and outlet flow angle from the velocity triangles, Fig. 5.14 can be used to select a suitable value of solidity (note that the plot is expressed in terms of s/c = 1/σ).

Lieblein (NASA Design Practice)

Like Howell in Britain, in the 1950s NACA (now NASA) conducted an extensive set of cascademeasurements to determine the performance of compressor airfoils for a wide range of geometric andaerodynamic parameters. As described later, these results became the basis for a compressor design systemwhich is now widely used, both in North America and in Europe (including Britain).

The drawing shows the hypothetical velocity distribution around a compressor blade.

C

C1

Cmax

x/c 1.00

C2

0

C2

C1

C1

SUCTION SURFACE

The performance of the blade is limited by the deceleration (that is, the diffusion or adverse pressuregradient) on the suction surface of the airfoil. If the diffusion is too great, the boundary layer separates, theblade stalls, and the losses increase significantly. Lieblein proposed a parameter to measure the severity of thediffusion:

DC C

C=

−max 2

1(1)

As usual, relative velocities W would be used for rotor blades.

Unfortunately, Cmax is a function of the detailed flow around the particular airfoil, which would not beknown early in design. However, the larger the lift (or circulation) being generated by the airfoil the largerCmax must be. From Section 4.2.2, the circulation is given by

( )Γ Δ= − =s C C s Cw w w1 2

and thus we can write( )( )

C C f

C f s Cw

max = +

= +1

1

Γ

Δ

Substituting into (1),

( )DCC C

f s Cw= − +1 12

1 1Δ

Experiments showed that the following form for D correlates the loss and stalling behaviour of a wide rangeof blade geometries:

DCC

CCw= − +1

22

1 1

Δσ (2)

This parameter is known as the diffusion factor. Note that (2) depends only on the upstream anddownstream velocities, which are known oncethe velocity triangles are established.

The figure (taken from NASA SP-36,1966) shows the variation of the total pressureloss coefficient, ω1, with D. As seen, the lossesrise sharply for D > 0.65, implying the onset ofstall. At the design point, the diffusion factorshould therefore be less than this. A suitablevalue might be D = 0.3 - 0.4. With D chosen,the only unknown in (2) is the solidity and it cantherefore be used to select the value of σ.

4.5 EMPIRICAL PERFORMANCE PREDICTIONS

4.5.1 Introduction

The idealized stage analysis used in Section 4.3 made a number of assumptions that are not fullysatisfied in practice. For example, the flow angle at the trailing edge does not precisely match the metal angle,as assumed in the Euler Approximation. Nor does matching the inlet flow angle to the inlet metal anglenecessarily result in the lowest losses. Finally, we need methods for estimating the losses generally, in orderpredict the efficiency of the stage and thus complete its design. To accomplish a more realistic stage analysis,we need to draw on correlations for the behaviour of actual blade geometries, as determined experimentally. Such empirical correlations were alluded to in the discussion of blade-loading limits in the last section.

Two systems for empirical performance predictions of axial compressors have been used fairlywidely. The British system, connected mainly with the name of Howell, will be discussed since it is relativelyeasy to apply in hand calculations. However, it omits the influence of a number of blade geometricparameters, does not directly apply to all the families of blade geometries that are in common use, and hassomewhat limited ability to predict the influence of factors such as compressibility.

A more comprehensive, but less easily applied, prediction system was developed by NASA during the1950s and 60s. This system is summarized in a famous document, NASA SP-36, “Aerodynamic Design ofAxial-Flow Compressors” published in 1965. SP-36 continues to form the basis for much practical axial-compressor design, both in North America and outside. The correlations presented in SP-36 have also beenre-evaluated and updated from time to time so that the system continues to be applicable.

It should be mentioned the largest gas turbine engine companies (eg. Pratt & Whitney, GeneralElectric and Rolls-Royce) have to some extent developed their own compressor design systems that reflecttheir in-house design philosophies and proprietary blade profile designs. However, these systems are oftenstructured in similar ways and strongly influenced by the design systems that are available in the openliterature.

4.5.2 Blade Design and Analysis Using Howell’s Correlations

The figure shows the nomenclature used by Howell:

Nomenclature:

s = blade spacingc = blade chord (solidity σ = c/s)ζ = stagger angleθ = α1' - α2' = camber anglea = distance of maximum camber aft of blade leading edget = maximum thickness of blade

i = incidence = α1 - α1'δ = deviation = α2 - α2' = difference between outlet flow angle and metal angleε = flow turning = α1 - α2

The nomenclature applies for a stationary blade row. For a rotor, replace α by β and use the relativecomponents of velocity.

Typical results obtained by Howell for a particular cascade geometry are shown in the followingfigure. The figure (taken from Horlock, 1958) shows the variation of flow turning, ε and the total pressureloss as a function of the incidence, i.

The cascade performance should depend on the blade and cascade geometry as well as the flowconditions. Howell suggested that:

( )( )

ε δ

θ α

, , , ,

, , , ,

losses f blade geometry cascade geometry flow conditions

f a c s c i

=

= 2

He also found that the results collapse well onto universal curves if they are normalized in terms of the resultsat the "nominal" (or "design"or "reference") flow condition for each cascade. The nominal condition isdefined, somewhat arbitrarily, as the condition at which the flow turning, ε, is 0.8 of the value at stall. Stall isthe appearance of boundary layer separation, towards the trailing edge, on the low pressure side of the blade.The appearance of stall manifests itself in a rise in the losses and an impairment of the ability of the blade toturn the flow. For convenience, Howell defined the stalling incidence as the positive incidence at which thelosses have increased to twice their minimum value. This definition is fairly easy to apply to experimentaldata. As the figure above indicates, it also seems to correspond fairly well to the point of maximum flowturning. The latter point could perhaps have been use as an alternative for identifying the “stalling” incidence.

The superscript * is used designate nominal values of the flow quantities. Thus

ε* = nominal deflection = 0.8 εstall

The corresponding values of i, δ and α2 are designated i*, δ* and α2*.

Howell’s correlations can be presented in a small number of formulae and graphs.

(a) Deviation at the trailing edge:

δ θσ

* = ⎛⎝⎜

⎞⎠⎟

mn1 (1)

where

m ac

= ⎛⎝⎜

⎞⎠⎟

+0 23 2500

22.*α

(2)

with all angles are measured in degrees.

For normal compressor rotor and stator blades n = 0.5. For the inlet guide vanes (IGVs) ahead of acompressor stage, Howell suggested using n = 1.0 and a constant value of m = 0.19. Unlike typicalcompressor rotor and stator blades, IGVs form an accelerating flow passage. They therefore behave more likea turbine blade row and this accounts for the difference in the behaviour of the deviation.

(b) Flow turning:

Howell found that the nominal flow turning, ε*, correlated quite well with just the flow outlet angle,α2*, and the solidity of the blade row, σ = c/s

ε α* * ,= ⎛⎝⎜

⎞⎠⎟

f sc2

The correlation is usually presented graphically ( Fig. 5.14 from Saravanamuttoo et al.) and was used inSection 4.4 to select the solidity.

The blade will often be used at other than the nominal (“design”) flow conditions. Howell was able tocorrelate fairly successfully the “off-design” behaviour of the cascades by plotting the results against the non-dimensional relative incidence, irel = (i - i*)/ε*. Figure 3.17 (taken from Dixon) shows the normalized flowturning, ε/ε* as a function of irel. The figure also shows the variation of the losses (expressed as a “dragcoefficient”) with relative incidence. As seen, the losses are close to a minimum at the nominal condition. Loss estimates will be discussed separately later.

(c) Reynolds number effects:

Howell obtained most of his cascade data for a Reynolds number of 300,000 (based on blade chordand upstream velocity). The resistance of the suction-surface boundary layer to separation is a function of thethickness of the boundary layer and whether it is laminar or turbulent. Thus, the flow turning behaviour of theblade row is a function of the Reynolds number, particularly at low values. Howell examined the dependenceof the flow turning on the Reynolds number. Figure 3.3 (taken from Horlock) shows the effect of Reynoldsnumber on the nominal turning.

The correlations presented to this point can be used to predict the flow turning capability of acompressor blade row. As mentioned, loss estimates will be considered later.

The correlations can be used in two ways: for analysis or for design.

Analysis: Predicting the performance of a blade row of specified geometry.Design: Determining the geometry of a blade row which produces a specified performance.

The approach is a little different for each case. Each will be described and the analysis mode will then beillustrated with an example.

Analysis Mode Calculations:

In this case, the inlet flow direction (α1 or β1) is specified and the blade row geometry is known (α1',α2', a/c, and σ = c/s). The goal is to predict the outlet flow angle, α2.

(i) The performance depends strongly on α2*. Since it is not known initially, it must be determined(by iteration). Guess a value of α2*. Use equations (1) and (2) to calculate δ*. Then

α α δ2 2* *= ′ +

Compare this value with the assumed α2*, revise as necessary and repeat until α2* and δ* areconsistent.(ii) Read the value of ε* from Fig. 5.14. Then

α α ε

α α

1 2

1 1

* * *

* *

= +

= − ′i

The nominal conditions are now known.(iii) If the actual i = α1* - α1' is different from i* then the blade row is operating "off-design". Fig.3.17 would then be used to determine the actual flow turning. The Reynolds number correctionwould be applied to the turning if appropriate.

Design Mode Calculations:

Again, the inlet flow direction (α1 or β1) would be specified. Typically, the shape of the camber line(ie. a/c) would also be selected. The goal is then to choose a blade row geometry (α1', α2', and σ = c/s) whichwill give the desired outlet flow angle, α2. This application of the correlations is a little more complicatedsince there is in fact a range of geometries which will satisfy the requirements.

One possible approach is to use the nominal values for the design point. This is reasonable sincenominal conditions give near-minimum losses and provide some stall margin. Then

and α α α α2 2 1 1* *= = ε α α* = −1 2

With α2* and ε* known, Fig. 5.14 is now used to choose the solidity, σ (this was the way that Fig 5.14 wasused in Section 4.4). Since the blade row is operating at the nominal conditions, the deviation will also be thatgiven by Eqns. (1) and (2). However, δ* is also a function of the camber, θ. From the drawing of the cascade,the flow turning is related to the camber by

(or in this case )ε θ δ= + −i ε θ δ* * *= + −i

Thus, the value of the camber will depend on the choice made for i*. Howell’s correlations indicate that thereis no unique choice for the design incidence, although he recommends that a value be chosen of a few degreesat most. Reductions in camber can be compensated for by increases in incidence, and vice versa. Note thatthese changes will also result in a change in the stagger of the blade row. In summary, according to theHowell’s correlations a variety of blade geometries can produce identical aerodynamic performance. Thisgives the designer some freedom to tailor the blade geometry to meet other possible requirements: eg. tosimplify the spanwise variation in the blade geometry, to alter a natural frequency, or to alter the stress level insome region.

The Howell cascade measurements were made for the British C family of compressor blade profiles. Therefore, a compressor designed according to the correlations is most likely to match the predictedperformance if the same blade profiles are used in the machine. The C4 profile, one of the most widely usedof the C-family profiles, is described in an appendix to these notes.

For use in computer programs or with analysis software (such as Mathcad or Matlab), the graphs forthe Howell’s correlations have been fitted by polynomials. These curve and surface fits are also given in anappendix.

4.5.3 Blade Design and Analysis Using NASA SP-36 Correlations

The NASA correlations are based on a large body of cascade data collected for blades using theNACA 65-series airfoil profile shape (Emery et al., "Systematic Two-Dimensional Cascade Tests of NACA65-Series Compressor Blades at Low Speeds," NACA Report 1368, 1958). The results are correlated anddesign procedures are summarized in NASA SP-36 ("Aerodynamic Design of Axial-Flow Compressors",1965). SP-36 also includes data for double circular-arc (DCA) blades which have been used to designtransonic compressors.

As noted, Howell’s correlations do not give clear guidance for the choice of design incidence. Whilethe nominal incidence, i*, is a reasonable choice for the design point, it is also clear from Fig. 3.17 that usingi* does not in general minimize the profile losses. Howell’s correlations also do not take into account somegeometric parameters which are known to affect the blade performance, such the ratio of maximum-thickness-to-chord, tmax/c. Finally, the Howell’s correlations are most suitable for analyzing the performance of a bladerow of specified geometry ("analysis mode") rather than determing a geometry which gives a desiredperformance ("design mode").

By comparison, the NASA correlations are intended particularly for use in design mode, althoughthey can also be used for analysis. They guide the designer to a choice of design incidence which nominallyminimizes the profile losses. The correlations also account for more aspects of the blade geometry. Thedrawback to using the NASA correlations is that reference must be made to more graphs than for the Howell’scorrelations.

For consistency with the SP-36 graphs, the procedures will be described in terms of the nomenclatureused by NASA.

As with the Howell correlations, the incidence and deviation are defined in terms of some referenceflow condition, although the definition of this condition is slightly different. Fig. 131 (from SP-36) shows thedefinition of the reference incidence, iref. It is theincidence half way between two off-design values ofincidence at which the losses are equal. SP-36usually refers to this as the “minimum-lossincidence” although the losses will only be aminimum if the loss “bucket” is symmetrical. Asevident from Fig. 3.17, this is not normally the case. Nevertheless, the reference condition will be nearminimum loss and thus would be a reasonable choicefor the design point. The deviation produced at thereference incidence is designated as δref.

For specified inlet and outlet flow angles, β1and β2, the required flow turning, Δβ = β1 - β2, is related to the camber, incidence and deviation by

Δβ θ δ= + −i

If we use the reference values of incidence and deviation then

Δβ θ δ= + −iref ref (1)

It was found that the deviation angle and the minimum-loss incidence vary linearly with the blade camber:

i i nm

ref

ref

= +

= +0

0

θ

δ δ θ

where i0 and δ0 are the values for the same blade when it has zero camber. Substituting into (1), the requiredcamber is given by

θβ δ

=+ −+ −

Δ 0 0

1i

n m(2)

The correlations are then used to find the values of the four unknowns on the right-hand side of (2).

The minimum-loss incidence at zero camber is written

( ) ( ) ( )i K K ii sh i t0 0 10= (3)

where

(i0)10 = minimum-loss incidence for a blade with zero camber and 10% thickness

(Ki)sh = shape correction to be applied when blades of other than the 65-series profile arebeing used

(Ki)t = thickness correction for blades with other than 10% thickness

For 65-A10 series blades, the correlations for the incidence related quantities are given on thefollowing graphs from NASA SP-36 (the graphs are reproduced at the end of the section):

(i0)10 = f1(β1,σ) Fig. 137

n = f2(β1,σ) Fig. 138

(Ki)t = f3(t/c) Fig. 142

For 65-series blades the shape correction, (Ki)sh, is simply 1.0. However, it has been suggested that the samecorrelations can be used to design C-series (C4 etc.) blades with circular-arc camber lines by setting (Ki)sh = 1.1, and to design DCA blading by setting (Ki)sh = 0.7.

The zero-camber deviation, δ0, is obtained in a similar way:

( ) ( ) ( )δ δδ δ0 0 10= K K

sh t (4)where

(δ0)10 = reference deviation for a blade with zero camber and 10% thickness

(Kδ)sh = shape correction to be applied when blades of other than the 65-series profile arebeing used

(Kδ)t = thickness correction for blades with other than 10% thickness

For 65-A10 series blades, the correlations are given on the following graphs:

(δ0)10 = f4(β1,σ) Fig. 161

(Kδ)t = f5(t/c) Fig. 172

As with the incidence, for 65-series blades the shape correction for deviation, (Kδ)sh, is simply 1.0. For C4 andDCA the same values of the shape correction as for incidence have been suggested: 1.1 and 0.7 respectively.

The deviation gradient, m, is also a function of β1 and σ. It is usually obtained using a deviation rulesimilar to that used in the Howell’s correlations:

mm

b= =σ

σ1 0. (5)

where

mσ=1.0 = value of m for a solidity σ = 1.0

= f6(β1) Fig. 163

b = f7(β1) Fig. 164.

Eqn. (2) defines the camber required for the blade if the reference conditions are chosen as the designpoint. However, there may be a variety of reasons to choose a different incidence at the design point, in thesame way that nominal conditions might not be used when designing a compressor using Howell’scorrelations. If i is different from iref then δ will also be different from δref. The resulting value of δ can bepredicted from

( )δ δ δ= + − ⎛

⎝⎜⎞⎠⎟ref ref

ref

i i ddi (6)

where (dδ/di)ref is given in Fig. 177 as a function of σ and β1.

The procedures just outlined can be used by the designer to obtain a blade row with a geometry whichwill result in the required performance: that is, they are suitable for use in design mode. Of course, somedecisions must already have been made concerning the type of blading (C-series, 65-series, DCA etc.), thecamber line shape, if other than 65-series blades are used, and the maximum thickness.

Eqn. (6) also allows the correlations to be used in analysis mode. For analysis mode calculations thefollowing approach would be used:

(i) For the specified geometry and design inlet-flow direction, β1, the reference conditions are firstdetermined.(ii) For an off-design inlet value of β1, Eqn. (6) would then be used to predict the deviation. Thisdefines the outlet flow direction, β2, and the off-design velocity triangle is then known.

As with the Howell’s correlations, curve and surface fits for the SP-36 correlations are given in anAppendix.

4.6 LOSS ESTIMATION FOR AXIAL-FLOW COMPRESSORS

4.6.1 Blade-Passage Flow and Loss Components

The drawing shows schematically the flow through the blade passage of a compressor rotor. Inaddition to the frictional effects in the boundary layers on the surfaces of the rotor blades, there are a numberof other flow features that can generate losses. The losses due to each of these features are normally estimatedindividually and then simply added to estimate the resultant losses through the blade passage.

For axial machines (both compressors and turbines), the losses are therefore subdivided into:

(i) Profile losses: These are the losses generated by friction in blade-surface boundary layers, by thesudden expansion in area at the trailing edge, and by the mixing out of the wakedownstream of the blade.

(ii) Secondary losses: The slower-moving flow in endwall boundary layers is "over turned" by the blade-to-blade pressure field, as shown in the drawing. The fluid swept towards the lowpressure (“suction”) side of the passage is blocked by the blade surface and rolls upinto a "passage vortex" that generates additional losses through high shear stresses atthe endwalls and as it mixes with the downstream flow. The boundary-layerseparation around the blade leading edge also results in a "horseshoe vortex".

(iii) Annulus losses: These are generated by friction on the endwalls, mainly upstream and downstream ofthe blade passage. The endwall losses inside the passage are normally assigned tothe secondary losses.

(iv) Tip-leakage losses: There must be some clearance between the rotor blade tips and the compressorcasing. The flow that is driven through the tip gap rolls up into a "tip-leakagevortex" as it interact with the main passage flow. There are viscous (frictional)losses inside the gap, but most of the tip-leakage losses are generated throughdownstream mixing with the surrounding fluid.

In transonic and supersonic compressors, there will be additional losses due to the presence of shockwaves.

4.6.2 Loss Estimation Using Howell’s Correlations

Howell gave simple correlations, expressed mostly in terms of drag coefficients, to estimate the losses:

(i) Profile Losses:

The profile losses were expressed as a function of both the incidence and the spacing-to-chord ratio,s/l (Howell used the symbol l for chord length), as shown earlier in Dixon Fig. 3.17 (repeated here).

(ii) Secondary Losses:

Howell concluded that the secondary losses at the endwalls depended primarily on the lift beinggenerated by the airfoils, since this determined the pressure difference that drives the flow across the passageto form the secondary flow. Thus

C CDS L= 0 018 2.

where from Section 4.2.1 the blade lift coefficient is given by

( )C scL m=

−2 1 2tan tan cosα α α

and

αα α

m =+

arctan

tan tan1 2

2

For the rotor flow, we would use the relative flow angles, 1 and 2, as usual.

(iii) Annulus Losses:

C sh

sc

ch ARDA = = =0 02 0 02 0 02. . .

σ

where h = blade height = rt - rh, AR = blade aspect ratio = h/c.

(iv) Tip-Clearance Losses:

The tip clearance loss is found to be a strong function of the height of the clearance gap comparedwith the blade span h. Howell suggested that a 1% increase in the rotor clearance gap would reduce the stageefficiency by 3%:

∆η τclearance h

= 3

With the “drag coefficients” corresponding to the losses determined, the corresponding total-pressurelosses can be calculated from Eqn. (6a) or (6b) from Section 4.2.1. Equation (6b) is usually the mostconvenient:

∆P C Closs D am

02

3

12

1, cos

=

σ ρ

α(6b)

Section 4.6.5 explains how to use the estimated total-pressure losses to obtain the stage efficiency.

The NASA system for axial compressor loss prediction, described next, uses direct correlations fortotal pressure loss coefficient, rather than for drag coefficient.

4.6.3 Loss Estimation Using NASA SP-36 Correlations

(i) Profile Losses

The profile loss system presented inNASA SP-36 is associated with the name ofLieblein, as were the blade loading limitspresented in Section 4.4. In that section, it wasseen that the profile losses correlated quitewell with the diffusion factor defined byLieblein, which was defined as

DC C

C=

−max 2

1(1)

However, Lieblein subsequently argued that the profile losses should depend primarily on the amountof diffusion on the suction side of the blade. He therefore introduced an alternative parameter, known as theequivalent diffusion ratio:

DCCeq = max

2(2)

Note that Deq resembles the deHaller number. Whereas the deHaller number defines the net diffusion betweenthe inlet and outlet of the blade row, Deq defines the local diffusion on the suction side of the airfoil. Liebleinthen correlated the profile losses with Deq and this approach has since been widely adopted.

As with the diffusion factor D, the exact value of the Deq is only known if the detailed flow around theairfoil is known. For use in the early stages of design, an approximate value of Deq, estimated from thecirculation, is therefore used. The following correlation appears to be widely accepted:

( ) ( )Deq = + −

coscos

. .cos

tan tanαα

ασ

α α2

1

12

1 2112 0 61 (3)

The profile losses are reflected in a momentum deficitin the wake, as measured by the momentum thickness downstream of the airfoil:

θ = −

CC

CC

dyref ref

s2

2

2

20

1, ,

where C2,ref is the velocity outside the wake. Thecorresponding total-pressure loss coefficient is then given by

ω θ σα

αα

=

2

2

1

2

2

c coscoscos

(4)

C

C1

Cmax

x/c 1.00

C2

0

C2

C1

C1

SUCTION SURFACE

yrefC ,2

( )yC2

0

s

where

ωρ

=−P P

C

01 02

121

2(5)

The loss correlation is then expressed in terms of the variation of the momentum thickness ratio, /c,with equivalent diffusion ratio, Deq:

( )θc

f Deq= (6)

The figure shows the original data set, obtained for NACA 65-series compressor airfoils, that wasused by Lieblein. Also shown are various curve fits for the function in (6) that have been proposed over theyears. Note that losses begin to rise sharply at Deq 2.0 and this would be interpreted as the onset of stall. Forthe original diffusion factor, Eqn (1), the corresponding value was D 0.6 (see Section 4.4)

Recently, Konig et al. (W.M. Konig, D.K. Hennecke & L. Fottner, “Improved Blade Profile Loss andDeviation Models for Advanced Transonic Compressor Bladings: Part I - A Model for Subsonic Flow,”ASME Journal of Turbomachinery, Vol. 118, January 1996, pp. 73-80.) investigated whether the Liebleincorrelation approach worked equally well for more recent compressor airfoil shapes. Their data are shown inthe next figure, along with the same curve fits.

Equivalent Diffusion Ratio, Deq

Wak

eM

omen

tum

Thic

knes

sR

atio

,θ//c

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.02

0.04

0.06

0.08

0.1

0.12

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.02

0.04

0.06

0.08

0.1

0.12

AungierWilson & KorakianitisKoch & SmithCasey/StarkeKonig et al

Lieblein Data

AXIAL COMPRESSOR PROFILE LOSSES AT DESIGN INCIDENCEComparison of Correlations with Lieblein Data

Although there is some evidence that more recent blade designs can tolerate somewhat higher valuesof Deq before stalling, the curve fit suggested by Aungier (R.H. Aungier, Axial-Flow Compressors, ASMEPress, 2003) seems as reasonable as any, for both data sets:

( ) ( )θc

D Deq eq= + − + −

0 004 10 31 1 0 4 12 8

. . . . (7)

Summarizing the procedure for estimating the profile losses:

(1) Deq is estimated from the velocity triangles and the blade row solidity using (3).(2) From Deq obtain the momentum thickness ratio, /c, using (7).(3) The total-pressure loss coefficient is then calculated from (4).

The method outlined here assumes that the blade is operating at its minimum-loss incidence, i* (seeSection 4.5.3). If i > i* then Lieblein suggested that (3) should be replaced by

( ) ( ) ( )D a i ieq = + − + −

coscos

. .cos

tan tan * .αα

ασ

α α2

1

12

1 2

1 43112 0 61

where a = 0.0117 for NACA 65-series blades and 0.007 for C4-series circular-arc blades.

+ ++

+ +

Equivalent Diffusion Ratio, Deq

Wak

eM

omen

tum

Thic

knes

sR

atio

,θ/c

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.02

0.04

0.06

0.08

0.1

0.12

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.02

0.04

0.06

0.08

0.1

0.12

AungierWilson & KorakianitisKoch & SmithCasey/StarkeKonig et al

AXIAL COMPRESSOR PROFILE LOSSES AT DESIGN INCIDENCEComparison of Correlations with Konig et al. Data

Konig et al. Data

(ii) Endwall Losses

NASA SP-36 does not provide clear guidance for estimating either the secondary losses or tipclearance losses for the purposes of meanline analysis.

Instead, most recent text books (eg. Japikse & Baines) and papers seem to recommend a methoddeveloped by Koch & Smith at General Electric (Koch, C.C. and Smith, L.H., “Loss Sources and Magnitudesin Axial-Flow Compressors,” ASME J. Eng. for Power, Vol. 98, 1976, pp. 411-424). The method providescombined estimates for the effects on stage efficiency of both secondary flows and tip leakage. This isphysically reasonable since, where both are present, the secondary and tip-leakage flows are in close proximity and tend to interact significantly. Unfortunately, the method is somewhat difficult to apply since it requires afairly detailed knowledge of the stage geometry. It is also necessary to specify how close the stage is to stallat the operating point for which the loss estimates are being made. Nevertheless, because of the importance ofendwall losses and the apparent widespread acceptance of the Koch & Smith method, it is worth examining.

The final output of the method is a correction to the stage efficiency, expressed in the form

ηη

δ

νP

h

h

=−

−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

1 2

1 2

*

(1)

where ηP = stage efficiency as calculated from the profile losses onlyδ* = average displacement thickness of the two endwall boundary layersν = average tangential force-deficit thickness for the two endwall boundary

layers

The tangential force-deficit thickness is a measure of the reduction in blade force near the endwalls due to thelower fluid velocity present in the endwall boundary layers.

Koch & Smith provide correlations, derived from very wide-ranging tests conducted on a large, low-speed compressor test rig, for estimating the values of δ* and ν. The drawing defines some of the geometricparameters that appear in the correlations.

s = spacingλ = stagger angleg = staggered spacing

= scosλ In addition, the following are used

ε = tip clearanceh = blade spanξ = axial gap between rotor and stators

In the correlations, average values of the parameters are used. Forexample, the staggered spacing used is the average value for the rotor andstator blade passages. Similarly, the tip clearance would be the average ofthe values for the rotors and the stators. Normally, this would result in theclearance value being half of that for the rotor blades, since the statorclearance is usually zero. However, stators are sometimes cantilevered

s

g

λ

λ

c

from the casing wall and have a clearance at the hub wall. If the stators are variable pitch, they will also needclearance.

The Koch & Smith correlation is embodied in three graphs.

(a) Displacement thickness. The first graph is used to estimate the displacement thickness as a function ofthe clearance and the pressure rise ratio:

2δ ε*

,max

,g

fC

C gP

P

=⎛

⎝⎜

⎞

⎠⎟

ΔΔ

where

ΔΔC PqP =

with the static pressure rise across the stage and the average of the inlet dynamic pressures for the rotorΔP qand stator rows. is the maximum value of the static pressure rise coefficient for the same stage,ΔCP ,maxcorresponding to the stalling of the stage.

The pressure rise ratio is probably the most difficult input to obtain. However, for preliminary design it may be sufficient to choose a value that seems generally consistent with the stage and blade loading that hasbeen chosen. For example, if the deHaller numbers are low and the solidities have been selected to giverelatively high values of the diffusion factors, the pressure rise ratio would be expected to be towards thehigher end of the scale.

ΔCP/ΔCP,max

2δ*/g

0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

ε/g = 0.0

0.025

0.050

0.075

ε/g = 0.10

(b) Effect of Axial Spacing. Koch & Smith concluded that the average displacement thickness of the endwallboundary layer would vary with the axial spacing ξ between the rotor blade and the stators. If that spacing isdifferent from 0.35s, then the following correction is applied to the displacement thickness given by theprevious figure.

(c) Force Deficit Thickness. Finally the force-deficit thickness is correlated against the displacementthickness as given in the following figure.

Axial Gap/Blade Spacing, ξ/s

2δ*/(

2δ*)

ref

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.8

0.85

0.9

0.95

1

1.05

1.1

ΔCP/ΔCP,max

2ν/2δ*

0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

The staggered gap, g, is not a commonly-occurring variable in performance correlations. However,the parameters in which it appears can be related to more familiar ones. For example,

ε ε

ε σλ

g hhc

cs

sg

hAR

=

=cos

(2)

Values of the tip clearance ε are often specified as a fraction of the blade span h. Therefore, reasonable valuesof ε/h would be known early in design. Eqn. (2) also implies that the Koch & Smith correlation can be used toconduct parametric studies to investigate the influence on the endwall losses of common design parameterssuch as the solidity σ and the blade aspect ratio AR = h/c.

For use in Eqn. (1), note that

2 2

2

δ δ

δ λσ

* *

* cosh g

gs

sc

ch

g AR

=

=

4.6.4 Effects of Incidence and Compressibility

The Howell correlation for the profile losses for C-series airfoils presented in Section 4.6.2 includedthe influence of incidence. As seen, the losses rose more rapidly with positive incidence than with negative. However, the precise behaviour of the losses with incidence is strongly influenced by the geometry of theblade section.

In addition, the Howell results apply only for low subsonic values of the inlet Mach number. The lossbehaviour of the airfoil is also strongly influenced by the inlet Mach number.

We do not have time in this course to go into these issues in detail. Therefore, only somerepresentative results are presented to illustrate the complexities.

The figure (taken from SP-36) shows the variation of profile losses with both incidence and inletMach number for four different airfoil and cascade geometries.

Note that the two examples of the British C4-series airfoils differ mainly in the shape of the camberlines and yet their sensitivity to both the inlet Mach number and the incidence are significantly different.

The double circular arc (DCA) profiles were specifically developed by NACA for use in transoniccompressors. It is seen that their sensitivity to Mach number is delayed to a higher inlet Mach number thansome of the other shapes.

The strong influence of the detailed airfoil geometry on the behaviour at both off-design incidenceand with increasing inlet Mach number obviously makes it more difficult to devise simple correlations for thelosses, liked those presented in Sections 4.6.2 and 4.6.3. For a recent attempt, see W.M. Konig, D.K.Hennecke & L. Fottner, “Improved Blade Profile Loss and Deviation Models for Advanced TransonicCompressor Bladings: Part II - A Model for Supersonicc Flow,” ASME Journal of Turbomachinery, Vol. 118,January 1996, pp. 81-87.

4.6.5 Relationship Between Losses and Efficiency

With the total-pressure losses across the stage determined from the correlations described inΣP loss0,

Section 4.6.2 or 4.6.3, the total-pressure rise across the stage is then:

∆ ∆ Σ∆P P Pideal loss0 0 0= −, ,

where P0,ideal is the pressure rise that would have been obtained in an ideal machine having the same workinput.

(i) For compressible flow of a perfect gas, the ideal pressure rise is

∆P P P where PP

TTideal0 03 01

03

01

03

01

1

, = ′ − ′ =

−γ

γ

and T03 is the (actual) final T0 corresponding to the work input: h0 = Cp(T03 - T01)

Then the actual P03 is obtained from

P P P loss03 03 0= ′ − Σ ,

where is as predicted for the rotor and stators together.ΣP loss0,

The corresponding efficiency can be calculated by determining the power required to compressthrough the same pressure ratio, P03/P01, in an isentropic machine:

′ =

−

TT

PP

03

01

03

01

1γγ

03P ′

03P

0T

03T ′01P

03T

01T

s

Then

η = ′ −−

T TT T

03 01

03 01

For hand calculations, the following approximation results in only a small error

η ≅∆

∆P

P ideal

0

0,

If any of the loss components is expressed in terms of an efficiency decrement, as is sometimes the case withtip-leakage loss, its contribution to the total pressure losses can be estimated from

∆ ∆ ∆P PTL clearance ideal0 0, ,≅ η

and this would be included in the earlier summation of losses.

(ii) For incompressible flow, we can use

( )∆ ∆ ∆P g H U C U C U Cideal E w w w0 2 2 1 1, = = − ≅ρ ρ ρ

and

η =∆

∆P

P ideal

0

0,

where, as before ∆ ∆ Σ∆P P Pideal loss0 0 0= −, ,

4.7 COMPRESSOR STALL AND SURGE

4.7.1 Blade Stall and Rotating Stall

As the flow rate through a compressor or fan is reduced at constant rotational speed, the velocitytriangles show that the incidence at the leading edge of the rotor blades is increased. If the incidence becomestoo large, the blades may stall.

The disorganized flow in the stalled region partially blocks the blade passage. As a result some of thefluid that was previously passing through the stalled passage is diverted to the adjacent passages, as indicatedin the middle figure. This has the effect of increasing the incidence at the airfoil lying next to the stalledregion while reducing the incidence at the other adjacent airfoil. The increase in incidence on the one adjacentairfoil may cause it to stall in turn. On the other hand, the airfoil which has its incidence reduced will movefurther away from stall, or may unstall if it was previously stalled. Thus, there is a tendency for the stall cellto migrate from blade passage to blade passage in the opposite direction to the rotation. This phenomenon isknown as rotating stall. The stall cells move in the opposite direction to the rotation at a relative speed whichis about half the rotational speed.

The rotating stall can take a number of patterns. It may involve only one blade passage, or a largenumber of adjacent blade passages around the annulus. If the rotor is at the front of a multi-stage compressor,it will have a relatively low hub-to-tip ratio. As seen in Section 4.3.4, the loading will then vary considerablyacross the span and it will be the hub region that will have the highest loading and therefore be the most likelyto stall. In that case, the may stall cells may only involve part of the span of the blade. If the blades have highhub-to-tip ratio, the stall is more likely to extend across the full span.

NORMAL ATTACHEDFLOW

SEPARATION OFBLADE BOUNDARY

LAYER - BLADE STALLROTATING STALL

MIGRATION OFSTALL CELL

Particularly if the rotating stall occurs at low speed and only involves part of the span, it may not be adanger to the machine. It is nevertheless undesirable since:

(i) The stalled passages, and therefore the stage, produce less pressure rise.(ii) The stage losses will be higher, leading to lower efficiency.(iii) The fluctuating forces on the blades as they successively stall and unstall will be a source ofnoise.

For reasons discussed in Section 4.8, it is fairly common for the early stages of multi-stage axialcompressors to experience some rotating stall at low rotational speeds. If it is present only during start-up andshut-down of the machine, this may be acceptable.

4.7.2 Surge

If the stall is very extensive, the pressure rise may be affected to the point that the slope of the ΔP0

versus characteristic becomes positive. As will be shown in Chapter 7, if this occurs the system of which&mthe compressor is a part can become dynamically unstable. If this instability is triggered, the result is knownas surge. The peak point of the compressor characteristic is therefore often identified as the surge point.

LOW HTR -ROTATING STALL

HIGH HTR -FULL SPAN STALL, ALL PASSAGES

.constN

m&

0PΔ

ROTATINGSTALL

SURGE

In Chapter 7, an approximate, unsteady-flow analysis is developed for a compression system. Thisanalysis identifies the various factors that will make the system more or less prone to surge. However, asimple physical argument can illustrate the sequence of events that might occur during a surge event.

Consider a gas turbine engine that is operating at high speed and high power or thrust output.

At steady state, the combustor is being “filled” with gas by the compressor at the same rate as it isbeing “drained” through the turbine. At any instant in time, there is a fairly large mass of gas present in thevolume of the combustor. Now suppose that the last stage of the compressor suddenly stalls. This might bedue to some disturbance that causes a drop in the mass flow rate through the machine. Since the last stagenormally has a high hub-to-tip ratio, the stall may involve the full span of the blades and all of the passages, asdescribed in the last section.

If the stall is very extensive, there will be an abrupt drop in the pressure of the gas delivered by thecompressor. The flow area through the turbine is relatively small and this limits the outflow through theturbine. Consequently, the pressure in the combustor will drop somewhat gradually. It the rate of pressuredrop is too slow, the situation can occur that the pressure in the combustor is higher than the pressure at thecompressor outlet. Since fluid tends to flow from a region of high pressure to one of low pressure, it istherefore possible for the high pressure gases from the combustor to flow back upstream into the compressor. The pressure in the combustor will eventually drop to below the compressor discharge pressure, at which pointthe flow in the compressor may re-establish itself. However, if the conditions that led to the initial stall are stillpresent, the whole process can repeat. The cyclic flow reversal in the compressor can result in very largefluctuating forces on the blades which can destroy the machine. In the gas turbine engine, the abrupt drop incompressed air supplied to the combustor can also lead to over-temperatures and resultant serious damage tothe turbines.

4.8 MULTI-STAGE COMPRESSORS

We now examine the aerodynamic behaviour of multi-stage compressors.

For arguments sake, we will consider a hypothetical four-stage compressor made up of stages withidentical aerodynamic characteristics and thus identical stage design points. Therefore, at design point for themachine as a whole, each of the stages will be running at their individual design points, which occur for thesame value of the flow coefficient φ= Ca/U for all of the stages. Assume also that the mean radius, and thusthe blade speed U, is the same for all four stages. Since the density of the gas increases across each successivestage, to maintain the constant axial velocity Ca needed to keep φ constant it is necessary to reduce the annulusarea along the machine. This variation in the cross-sectional area would be determined at the design-pointflow conditions.

Now consider what happens if the compressor is run at a rotational speed that is lower than the designvalue. To see the effect, we will just consider the first two stages:

TARGET CaVARIATION

AREA VARIATIONTO ACHIEVETARGET Ca

ρ VARIATIONDUE TO

COMPRESSION

Ca

ρ

A

U

Ca

Ca

U

ACm aρ=&

DESIGN POINT

AmCa ρ&

=OR

φDφ

ψ

ψD

FOUR IDENTICAL STAGES

DESIGN N

1

2

3

4

STAGE I STAGE II

21 3

Since the annulus area has been adjusted such that at the design point the two stages have the sameflow coefficient:

φ φ φI II DD D= =

For stage I

φ IaI

DD

DCU

=

and the corresponding pressure rise is

and the outlet density ΔP P PID= −3 1 ρ3

1

3D

D

D

P PRT

I=+ Δ

Then for stage II

Cm

AaD

D

D

33 3

=&

ρφ II

a

DD

DCU

= 3

where, as mentioned, A3 was adjusted to give .φ φI IID D=

Now consider the effect of halving N (ie. halving U) while also halving (ie. halving Ca) to keep&mStage I operating at its design φ:

φ φI

a

D

I

C

U

D

D= =

1212

1

Thus, Stage I will also be operating at its design ψ. However, the absolute Δh0 (and thus ΔP0 varies as U2 andthe pressure rise is therefore reduced to

and Δ ΔP PD≅14

ρ3

1

3

14=

+P P

RT

DΔ

Neglecting the changes in T3, which will be relatively much smaller than the changes in P3, then

ρρ

3

3

14 10

D

P P

P P

D

D

≅+

+<

Δ

Δ.

The corresponding change in Ca3 is then

( )CC

mA

mA

mm

ka

a D DD

D

D3

3

3 3

3 3

3

3

10 12

= = = > ⎛⎝⎜

⎞⎠⎟=

&

&

&

&.

ρ

ρ

ρρ

where k > 1/2. Then the flow coefficient for Stage II

φ φIIa a

D

IICU

kC

U

kD

D= = =3 3

12

12

and since k > 1/2, . Thus, the non-dimensional operating point for Stage II shifts to a lower value ofφ φII II D>

ψ than the design value. Stage I undercompresses the fluid due to the reduction in U2. But stage II undercompresses the fluid even more than Stage I due to the reduction in both U2 and ψ. This effect onlyincreases in the subsequent stages. For the 4-stage compressor with four identical stages we would thereforeexpect to see the following pattern of operating points:

If we now reduce the mass flow rate at the low speed operatingpoint, keeping N constant, the flow coefficient for Stage I will belowered. Stage I will then be producing slightly higher pressure rise. However, the effect of the low U2 is much greater than the smallincrease in ψ and Stage I will still be producing much lower pressurerise than at the design N. Consequently, Stage I is still under-compressing the fluid and the downstream stages will again be atsuccessively higher values of φ. We therefore conclude that if wethrottle the flow further, Stage I will be the first to reach its stallingvalue of φ.

DESIGN CaVARIATION

AREA VARIATION(FIXED)

ρ VARIATION(UNDER-

COMPRESSION)

Ca

ρ

A

U

Ca

Ca

U

DESa mACm && <= ρ

REDUCED RPM, REDUCED m&

NEW CaVARIATION

AmCa ρ&

=

φDφ

ψ

ψD

LOW N, STAGE 1 AT DESIGN

12

3

4

φDφ

ψ

ψD

LOW N, THROTTLED

1

2

3

4

If we then consider operating points above the design N, similararguments will lead to the conclusion that each stage is over-compressing thefluid. Consequently, the Ca into successive stages decreases and so does theflow coefficient φ. We would therefore expect to see the approximate patternof operating points shown. Note that if we throttle the flow further, it will nowbe the last stage which stalls first.

Combining these arguments, we can plot the expected map for the compressor as a whole.

Note that:(i) Over most of the map, we assume that the stalling of any stage results in compressor surge. As aresult, when the onset of the stall switches from the front to the back of the machine (near the designN), there is a discontinuity in the slope (or “knee”) in the surge line.

At low values of N, stall is expected to occur first in the first stage of the compressor. However, since the early stages of the compressor have lower hub-to-tip ratios, the stall there is morelikely to be part-span, rotating stall (as discussed in Section 4.7). This, combined with the fact thatthe absolute forces on the blades will be low at low N, means that some degree of rotating stall isacceptable at low N. As a result, at the low end of the map the surge line has a “kink”, indicating thatsome early-stage stall is allowed.(ii) At the design point of the compressor, all of the individual stages are operating at their designpoints and therefore have their maximum efficiencies. From the earlier discussion, it is evident that atany other operating point at most one of the stages will be operating at best efficiency. Therefore, theefficiency of the overall compressor will be less than its value at design. For this reason, the lines ofconstant efficiency are shown as closed contours surrounding the design point.

φDφ

ψ

ψD

HIGH N, STAGE 1 AT DESIGN

1

2

3

4

01

01

PTm&

01TN

01

02

PP

CONSTANTSPEED LINE

HYPOTHETICALSTEADY-STATE

OPERATING LINE

HIGH-SPEEDOPERATING POINT

STAGE STALL LINE

DESIGNPOINT

COMPRESSORSURGE LINE

1 2 3

4

4

3 2 1

η

ηmax

The compressor map shown is a hypothetical one. In practice, the individual stages in a multi-stagemachine will not all have identical characteristics. Nor are the stall lines for the individual stages likely tocross at exactly the same point on the map, and as a result the knee in the surge line will probably not be aswell defined. Nevertheless, many of the features are reproduced by actual compressor maps, as shown on thefollowing:

NASA 8-Stage Research Compressor

Pratt & Whitney TF30 LP Compressor

4.9 ANALYSIS AND DESIGN OF LOW-SOLIDITY STAGES - BLADE-ELEMENT METHODS

For solidities, , less than about 0.4 each blade can be treated as an isolated airfoil. Note that = 0.4was the lowest value of solidity that appeared on the NASA SP-36 correlations (Section 4.5.3). Usually, theblade is divided into a series of spanwise segments or blade elements. Three-dimensional flow effects in theform of spanwise flows are usually neglected, although the downwash induced by the trailing vortex system issometimes taken into account. This approach, known as the "blade-element method", is commonly used todesign propellers and low-performance axial fans.

Consider the flow relative to a blade element. The element behaves like an isolated airfoil in a streamin the direction of the vector mean of the inlet and outlet flows:

ZLL = zero-lift line of blade elementWm = vector mean velocity relative to blade element

WC

C QAm

a

mm a= =

+

=

cos; arctan

tan;

ββ

β β1 tan 2

2

= angle of attack of blade element = angle between ZLL and WmL = lift force on blade element (perpendicular to Wm)

= 1/2Wm2crCL (1)

where c = chord length of blade elementr = radial width of blade elementCL = lift coefficient of blade element (as obtained from airfoilcharacteristics and )

D = drag force on blade element (parallel to Wm)(normally D L)

X = axial component of force on blade elementX L sinm (see note at end of section)

Y = tangential component of force on blade element

CL, CD = fns [, section shape, Re] (2)- as obtained from airfoil data

The axial force is obtained from the momentum equation (with Ca = const.):

( )F N X A P r r Px B= = =∆ ∆ ∆ ∆2π (3a)

where NB = no. of bladesP = static pressure difference across the blade row

Substituting for X in terms of the lift coefficient

N W c rC r r P

N W cC r P

B m L m

B m L m

12

2

12

2

2

2

ρ β π

ρ β π

∆ ∆ ∆

∆

≅

≅

sin

sin (3b)

For the input power, from the energy equation

( )∆ ∆∆ ∆

∆ ∆ ∆ ∆W TQ P

Q h Q U CinR

w= = ≅ ≅ωη

ρ ρ00 (4a)

assuming P0 is small so that P0/R h0, and where

T = torque applied to flow through annulus width rQ = volume flow rate through annulus width rR = rotor efficiency (R =1 if CD = 0)P0 = total pressure difference across rotor (usually P0 P since C1 C2)

Substituting for the torque in terms of the components of the lift and drag forces (T = NBrY)

( ) ( )∆ ∆ ∆ ∆ ∆ ∆ cos sinW N W c r C C r Q P Q U Cin B m L m D mR

w= +

= =1

22 0ρ β β ω

ηρ (4b)

Rotor and stator blade rows can then be designed using Eqns. (1) - (4). Iteration will generally benecessary since W2 is a function of L, which is a function Wm, which in turn is a function of W2. Theanalysis would be performed at enough spamwise sections to define the full blade geometry.

Propeller analysis usually takes into account the "downwash" induced along the blade span by thetrailing tip vortices from the blades. The downwash would slightly alter the effective flow incidence seen bythe blade and thus the lift it develops.

To make the velocity triangle diagram clearer, the blade was sketched with somewhat lower staggerangle than would normally be found in practice. The diagram shows the force triangles for a more realisticvalue of the stagger angle:

Note that D makes a noticeable contribution to the magnitude of Y but has a much smaller influence on themagnitude of X. This is the reason that D can be neglected when determining X, but needs to includedwhen determining Y.

∆L

∆X

∆Y

∆D

x

y

Wm∆F

5.2 IDEALIZED STAGE GEOMETRY AND AERODYNAMIC PERFORMANCE

The geometry of an axial-flow turbine blade is similar that of an axial-flow compressor blade, exceptthat camber is usually much larger. The stage consists of a set of stators ("nozzles") followed by a rotor. Thenozzles control the swirl in the flow entering the rotor and the rotor then extracts work from the fluid byremoving swirl. This arrangement of components results in stage aerodynamic characteristics that are verydifferent from those obtained for an axial compressor.

We begin again by estimating the stage performance based on an idealized stage:(i) Simple velocity triangles are assumed: constant axial velocity through the stage and constant meanradius, resulting in constant blade speed where the mean streamline enters and leaves the rotor.(ii) Approximate blade geometries are obtained using the Euler Approximation.

Consider again the reaction turbine sketched in Section 3.5. The drawing shows the velocitytriangles:

Now reduce the mass flow at constant N, using the Euler Approximation to determine the outlet flowangles. From the drawing shown over, the flow coefficient is reduced

φ φ= <CU

aD

Clearly, ΔCw is smaller than at design. This is also consistent with the reduction in rotor blade incidence. Thus

ψ ψ= = <Δ ΔhU

CU

wD

02

Therefore, ψ varies directly with φ. Compare this with the case of compressors where they varied inversely.

Next, consider reducing U while holding Ca constant:

From the velocity triangles

φ φ ψ ψ= > = >CU

CU

aD

wD

and the same trend is found as when the mass flow rate was changed.

Now consider the absolute output. From the Euler equation

Δ Δh U Cw0 =and from the velocity triangles, ΔCw increased as U decreased. It is not entirely clear whether the productUΔCw has increased or decreased. However, it is clear that it, and therefore Δh0, has not changed very much. Compare this with the compressor case, where a reduction in U resulted in a large reduction in UΔCw:

Summarizing, based on the velocity triangles, the aerodynamic performance characteristics of axialcompressors and turbines differ in two main ways:

(i) ψ versus φ, and therefore ΔP0 versus , is negative for compressors, positive for turbines.&m(ii) The energy transfer Δh0 is a strong function of U for compressors, but only a weak function forturbines.

The following figures show the actual characteristics of the gas-generator turbine of the Orenda OT-2gas turbine engine. Note that it is conventional to use the pressure ratio as the independent variable forplotting turbine aerodynamic characteristics.

The characteristics confirm that the mass flow-pressure ratio characteristic is only a weak function ofthe rotational speed. However, this does not mean that the rotational speed is not important in order to have ahigh output of useful work. As seen from the velocity triangles, if the rotational speed is reduced below thedesign value, the energy released by the fluid, Δh0 = UΔCw, may not be changed very much, but this is alsoaccompanied by high incidence on the rotor. This will lead to higher losses and therefore poor efficiency. This is confirmed by the OT-2 efficiency curves. Thus, to have high energy release by the fluid and to recovermost of that energy as useful shaft power output, it is necessary to have high rotational speed.

5.3 EMPIRICAL PERFORMANCE PREDICTIONS

Cascade results are used for meanline analysis of turbines in much the same way as for axialcompressors. Again, primarily British results will be presented, but these are also widely used in NorthAmerica.

5.3.1 Flow Outlet Angle

Turbine blade rows, for gas turbine engines in particular, often operate at choked conditions or withmildly supersonic outlet flow conditions. The correlations for outlet flow angles for such blade rows aregenerally divided into two sections: one for low speeds (usually taken as M2 0.5-0.7) and one for the soniccondition (M2 = 1.0). For intermediate values of M2 the outlet angle is usually assumed to vary linearlybetween the low-speed and the sonic values.

(i) Low Speed (M2 0.5)

As mentioned earlier, the Carter & Hughes correlation for deviation (used by Howell for compressors)has also been used for turbines:

δ θ=

m s

c

n

where = camber angle and the value of m is obtained from Fig. 3.6 (from Horlock).

For turbines, n is generally taken as 1.0, as used for compressor inlet guide vanes (as opposed to the value of1/2 used for compressor rotor and stator blades). However, the Carter & Hughes correlation tends to over-estimate the deviation for most modern turbine blades.

A more satisfactory (but less convenient) correlation is that due to Ainley & Mathieson (A-M). Theircorrelation uses the so-called gauge angle g as a reference angle to which the actual outlet angle is related:

θ gos

=

−cos 1

where o = throat opening and s = blade spacing. For an infinitesimallythin blade which is straight from the throat to the trailing edge, the gaugeangle would define the direction normal to the throat line.

For low speed flow, A-M correlated the outlet flow angle 2with the gauge angle. Fig. 7.13 (from Saravanamuttoo et al.) shows thevariation for a “straight-backed” blade: that is, a blade for which thesuction side is straight from the throat point to the trailing edge. Thecurve in the figure can be approximated by

α 2111625 12=

−−. cos o

s

However, most turbine blades are not straightbacked. Instead they have a certain amount of “unguidedturning” as defined by the angle u. In A-M’s day, if thesuction surface was not straight from the throat to thetrailing edge, it was usually defined by a circular arc. A-M therefore corrected the outlet angle as follows:

α 2111625 12 4=

− +

−. cos os

se

where e is the suction side radius of curvature. Unfortunately, modern turbine blades usually do not usecircular arcs to define their surface shapes. As a result, eis not constant and generally not known. To use the A-M correlation it is therefore necessary to obtain an“equivalent” value of e. An approximate value can becalculated from the unguided turning angle as follows:

se o

s

u=

−

πθ

180 12

for u measured in degrees.

θgo

s

θg

(ii) Sonic Condition

For M2 = 1.0 and a straight-backed blade, A-M indicated that the outlflow angle would be equal to thegauge angle:

α 21=

−cos os

For a curved-back blade, this was again corrected for the suction side radius of curvature. The results werepresented graphically but can be approximated by the following curve fit:

α 21

1 787 4 1281=

−

−+

−cos sin. .o

sse

os

se

As mentioned, for 0.5 M2 1.0 the value of 2 is obtained by linear interpolation:

( )( )α α α α2 2 2 2 22 0 5 2 0 5 2 1.02 1= − − −

= = =M M MM

. .

5.3.2 Choice of Solidity - Blade Loading

5.3.2.1 Zweifel Coefficient

In 1945, Zweifel introduced a tangential force coefficient to measure the loading of turbine blades. Consider the control volume enclosing a single airfoil in a row of turbine blades. The CV extends unit depthin the z direction.

Apply the linear momentum equation in the y direction:

( )ΣF m V Vy y y= −& 2 1 (1)

Because the top and bottom faces of the CV are periodic boundaries, the pressure forces on them exactlybalance each other in both the x and y directions. Thus, the only contribution to ΣFy is the blade force Y. Then

( )Y m C Cw w= +& 2 1 (2)

From the velocity triangles,

C Cw a1 1 1= tanα C Cw a2 2 2= tanα

and for unit span, . Note that we are using here a common convention in turbine design&m C sa= ×ρ2 2 1practice that α1, α2, Cw1, and Cw2 are all taken to be positive: that is, we are not rigidly following the signconventions introduced earlier. Then (2) can be written

α1

α1

x

y

X

Y

Ca1

Ca2

cx

s

Cw1

Cw2

C2

P1

P2

C1

Y sCCCa

a

a

= +⎛

⎝⎜

⎞

⎠⎟ρ α α2 2

22

1

22tan tan

or, since ,C Ca2 2 2= cosα

( )Y C sCC

a

a

= +⎛

⎝⎜

⎞

⎠⎟

12

22 22 2

2 21

22ρ α α αcos tan tan (3)

The tangential force in (3) is just the integrated effect of the pressure distribution around the airfoil:

( )Y P P dxPS SS

cx= −∫0

Zweifel then defined a reference, “ideal” loading distribution. This corresponds to the maximum loading thatcould be achieved with the same inlet and outlet conditions while avoiding adverse pressure gradients on thesuction surface. This distribution, which is not physically realizable, corresponds to a pressure on the pressureside of P0 and a pressure on the suction side equal to the discharge pressure P2. The resulting “ideal”tangential force is then

( )Y P P c C cideal x x= − × =0 2 2 221 1

2ρ (4)

The Zweifel coefficient is then obtained by taking the ratio of the actual to the ideal tangential forces

Z YYideal

=

Substituting from (3) and (4) then

PS

P

P0

P1

P2

0 cxx

SS

"IDEAL" DISTRIBUTION

ACTUALDISTRIBUTION

2220 2

1 CPP ρ=−

Z sc

CCx

a

a

=⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟2 2

2 21

21cos tan tanα α α (5)

Note that this definition neglects the sign convention for angles. For a typical turbine blade, α1 and α2 haveopposite signs. If the signs of α1 and α2 are taken into account then the coefficient becomes:

Z sc

CCx

a

a

=⎛

⎝⎜

⎞

⎠⎟ −

⎛

⎝⎜

⎞

⎠⎟2 2

2 21

21cos tan tanα α α

As usual, for rotor blades β replaces α. The normal definition of the solidity is σ = c/s. The way the Zweifelcoefficient is defined results in the “solidity” being expressed in terms in term of the axial chord length, cx,rather than the true chord, c. The relationship between the true chord and axial chord can be seen from thedrawing, where ζ is the stagger angle:

Zweifel (1945) concluded, based on European cascade data from the 1930s and 1940s, that Z .

0.8 gave minimum profile losses. Thus, for given velocity triangles, the “optimum” s/cx is that which gives thevalue of Z which results in minimum profile losses:

• If s/cx is too high (which corresponds to low solidity), losses will be high due to separation,• If s/cx is too low, profile losses are high because of excessive wetted area.

Using the Zweifel coefficient to choose s/cx is analogous to the use of the diffusion factor to select the solidityfor axial compressors. Since Zweifel’s time, profile design has improved and today turbines are oftendesigned with considerably higher values of Z ( Z = 1.00-1.05 is common).

CX

S

C

ζ

Z = 1.37

5.3.2.2 Ainley & Mathieson Correlation

Ainley & Mathieson developed a widely used loss system (see next section), based on British turbinecascade data from the 1940s and 1950s. They likewise identified the geometries that gave minimum profilelosses for different combinations of inlet and outlet flow angles. These optimum geometries, expressed asoptimum s/c (“spacing-to-chord” ratio or “pitch-to-chord” ratio) were presented graphically as shown in Fig.7.14 (from Saravanamuttoo et al.). This figure can therefore be used to choose solidity (as an alternative to theZweifel criterion).

Unfortunately, Zweifel and Ainley & Mathieson expressed“solidity” differently: cx/s versus c/s. This makes it difficult to comparethe geometries that would be obtained using each approach, for the sameset of velocity triangles. The two ratios are related through the staggerangle, ζ, of the blade row (see the figure on the previous page),since . However, the value of the stagger angle is not fixedcosζ = c cxby the inlet and outlet flow (or metal) angles. This is illustrated in thefigure at the right, which shows two actual, very highly-loaded (Z = 1.37)low pressure turbine blade rows that were designed for identical inlet andoutlet flow angles (α1 = 35o, α2 = 60o). The two blades clearly have verydifferent stagger angles. This is the result of different decisions regardingthe detailed pressure distributions around the blades. The blade with thehigh stagger angle was designed to be “forward-loaded”: that is, todevelop most of its lift on the forward part of the airfoil. The one withthe lower stagger angle is much more “aft-loaded”. The two airfoils haveidentical values of cx/s, and thus have the same values of Z. However,they clearly have different values of s/c and therefore cannot both havethe “optimum” geometry according to Fig. 7.14.

Despite these difficulties, it is possible to make an approximatecomparison between the results obtained by the two different approachesto choosing the blade spacing. Kacker & Okapuu (KO; see Appendix E)provided a correlation that gives the typical values of stagger angle that would be seen for differentcombinations of α1 and α2:

Z = 1.028

Z = 1.56

Z = 1.034

Z = 1.26

Z = 0.911

Z = 0.909Z = 0.990

Using values of stagger angle obtained from K-O Fig. 5, the following figure shows the values of theZweifel coefficient for selected combinations of inlet and outlet flow angles. It is evident that the optimumgeometries based on the Ainley & Mathieson correlations lead to higher values of Z than Zweifel originallyrecommended. Very high values of Z are obtained for impulse blades (α1 = α2). Since the Ainley &Mathieson loss system was specifically based on loss measurements made for impulse blades, these resultssuggest that relatively higher values of Zweifel coefficient can be tolerated in the rotor blades for stages withlow values of degree of reaction, especially if the total flow turning is low.

5.3.3 Losses

In both North America and Europe, most loss estimates for axial-flow turbines are based on a losssystem developed by Ainley & Mathieson (AM) in the UK in the early 1950s (ARC R&M 2974, 1957; seealso Saravanamuttoo et al.). The AM system has been updated a couple of times to reflect improvements inblade design: for the design-point conditions, this was done most recently by Kacker & Okapuu (KO) of Pratt& Whitney Canada (Kacker, S.C. and Okapuu, U., “A Mean Line Prediction Method for Axial Flow TurbineEfficiency”, ASME J. Eng. for Power, Vol. 104, January 1982, pp. 111-119).

The KO system will be summarized here. The figures from the paper have also been fitted to curvesor surfaces and these fits are given in Appendix E.

For turbines, the total-pressure loss coefficient Y is defined as

YP

P Ploss=−

Δ 0

02 2

, (1)

Note that in this case, the loss is non-dimensionalized by the outlet dynamic pressure, whereas the inlet valueis used in the loss coefficients for axial compressors.

As for compressors, losses are again divided into components and these are then added linearly toobtain the total losses:

( )Y Y f Y Y YTotal P S TET TC= + + +Re (2)

where the subscripts designate the components as follows: P = profile, S = secondary, TET = trailing-edgethickness, TC = tip clearance. f(Re) represents a correction for the effects of Reynolds number on the profilelosses. The effect of Reynolds number on the other loss components is not well documented but it is believedto be small.

The following figure shows the blade nomenclature used in the KO system. Note that they do notfollow the sign convention we defined earlier. Using that convention, the inlet and outlet flow and metalangles will often have opposite signs because of the high turning that is normally present in turbine bladerows. It becomes a nuisance to keep track of the signs and therefore it is common practice by turbinedesigners to take both the inlet and outlet angles as positive, as shown in KO Fig. 3.

Profile Losses:

The profile loss is obtained as the weighted average of the losses for two extreme cases with the sameoutlet flow angle: a nozzle blade (maximum blade-passage acceleration) and an impulse blade (zeroacceleration). In the original AM system, the expression took the form:

( )Y Y Y Yt c

P AM P nozzle P impulse P nozzle, , , ,max

.= +

⎛

⎝⎜

⎞

⎠⎟ −

⎡

⎣⎢⎢

⎤

⎦⎥⎥⎛⎝⎜

⎞⎠⎟

βα

βα

1

2

2

0 2

1

2 (3)

where tmax is the maximum thickness of the blade. Note that KO use α for "air" angles, β for "blade" angles. The two reference loss coefficients were presented graphically by AM, as shown in Fig. 1 (for nozzles) andFig. 2 (for impulse blades). Note also that for a given value of the outlet angle α2 there is a value of solidityσ = c/s that minimizes the profile losses. This was the origin of the "optimum σ" that is plotted on Fig. 7.14 inSection 5.3.2.2.

Kacker & Okapuu compared the AM predictions of profile losses with those obtained from turbineairfoils of more recent design. They concluded that the AM loss systems significantly over-estimates thelosses for modern turbine blades. The KO profile loss correlation therefore takes the form

( )Y YP KO P AM, ,.= 23

0 914 (4)

where the factor of 0.914 was introduced to correct the AM loss estimate to that for zero trailing-edgethickness (since KO handle trailing-edge losses separately) and the factor of 2/3 reflects the improvements inprofile design since Ainley & Mathiesons’ time.

As seen, Eqn. (3) includes a correction for the maximum thickness of the airfoil: the data in Figs. 1and 2 apply for a maximum thickness of 20% of the chord length. Decisions about the maximum airfoilthicknesses would not normally be made at the stage of a meanline analysis for the blade row. However, KOexamined the range of maximum thicknesses observed for a number of recent actual designs and provided thecorrelation shown in Fig. 4. Knowing the flow turning from the velocity triangles, this figure can then be usedto obtain a reasonable value for the thickness, ahead of the detailed design of the blade.

The estimates obtained from the correlations described above apply for low speed flows. The turbinesin gas turbine engines normally operate under compressible flow conditions. The Mach number levelsencountered depend to some degree on where the turbine is located in the engine:

High Pressure Turbine (HPT). The HPT is located immediately downstream of the combustor anddrive the high pressure compressor. To minimize the number of stages, HPTs are typically designedto operate at transonic outlet flow conditions.Low Pressure Turbine (LPT). The LPT drives the low pressure compressor, and the fan stage in aturbofan engine. The fan has a large tip diameter and to keep the tip Mach numbers acceptable, thefan shaft must rotate at a much lower speed than the high-pressure spool. The tip diameter of the LPTis much smaller than that of the fan and as a result it runs at a relatively low blade speed. This in turnresults in lower flow velocities generally. It is therefore normal for the flow around LPT airfoils to besubsonic everywhere.

As a result of these differences, the strongest effects of compressiblity are normally seen in HPTs. Thefollowing Schlieren photos (taken from E. Detemple-Laake, “Measurement of the Flow Field in the BladePassage and Side Wall Region of a Plane Turbine Cascade,” AGARD-CP_469, 1989) show the flow throughan HPT blade passage with exit Mach numbers of 0.9 (left) and 1.25 (right):

The profile losses can be affected by compressibility effects in at least two ways:

(i) Inlet Shock Losses. The high levels of curvature around the leading edges of turbine bladesresult in high local velocities in this region. For inlet relative Mach numbers as low as 0.6, patches ofsupersonic flow, terminating in a shock, can appear on the suction side of the airfoil.(ii) Channel Acceleration and Outlet Shocks. A turbine blade passage is normally an acceleratingflow channel. As the outlet Mach number increases, there is a tendency for the blade surfaceboundary layers to be thinned and their contribution to the losses actually decreases slightly. As theoutlet Mach number approaches 1.0, patches of supersonic flow, terminating in shocks, may begin toappear on the aft suction surface. Finally, as the outlet Mach number becomes supersonic, expansionwaves and shocks appear in the trailing edge region. In addition to directly contributing additionaltotal pressure losses, it is common for one or more of the shocks to impinge on the surface of theadjacent blade. This can cause boundary layer separation, which would further increase the losses. This effect can be seen from the following figure, which shows Detemple-Laake’s cascade operatingat an outlet Mach number of 1.30.

The following figure shows the relative profile losses as a function of exit Mach number for anotherHPT cascade (from Mee et al., “An Examination of the Contributions to Loss on a Transonic Turbine Blade inCascade,” ASME J. Turbomachinery, Vol. 114, January 1992, pp. 155-162).

The complexity of the compressibility effects makes it difficult to predict their influence on thelosses. Kacker & Okapuu provide procedures for estimating the contributions to the profile losses; seethe paper for details.

Finally, KO give the following Reynolds number corrections for profile losses:

( )f for

for

for

cc

c

cc

ReRe

Re

. Re

ReRe

.

.

=×

⎛⎝⎜

⎞⎠⎟

≤ ×

= × < <

= ⎛⎝⎜

⎞⎠⎟

>

−

−

2 102 10

10 2 10 10

1010

5

0 45

5 6

6

0 26

where the Reynolds number is based on the chord length and exit velocity.

Secondary Losses:

As in Howell’s correlations for compressors, the AM/KO loss systems indicate that the secondarylosses in axial turbines are a function of CL

2:

( )Y f ARCs cS

L

m

=⎛

⎝⎜

⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟0 04 2

1

2 22

3.coscos

coscos

αβ

αα

(5)

where

( )Cs c

Lm= +2 1 2tan tan cosα α α

( )α α αm = −⎛⎝⎜

⎞⎠⎟

−tan tan tan12 1

12

and as before, all angles are taken as positive.

The loss coefficients give the total-pressure losses as averaged over the total mass flow ratethrough the blade passage. As the aspect ratio of the blade becomes larger, a smaller fraction of the spanis occupied by the secondary flow and the loss associated with it becomes averaged over an increasinglylarger mass flow rate. Consequently, the mass-averaged loss coefficient varies inversely with the aspectratio. This effect is embodied in the aspect ratio correction, f(AR) in Eqn. (5). Kacker & Okapuu foundthat the AM loss system tended to over-estimate the effect of aspect ratio on blades of very low aspectratio (which are often used in modern HPTs). In the KO loss system, the correction for blade aspect ratiotherefore takes the following form:

( )f ARh c

h cfor h c

h cfor h c

=− −

≤

= >

1 0 25 22

1 2

.

(6)

Kacker & Okapuu also provide a compressibility correction for the secondary losses (see thepaper).

Trailing-Edge Losses:

Due to the finite thickness of the trailing edge, the streamtube experiences a sudden increase inarea as it leaves the blade passage. The resulting sudden-expansion loss is correlated in terms of analternative form of loss coefficient, known as an energy loss coefficient, Δφ2, as a function of the ratio ofthe trailing-edge thickess to the throat opening. KO correlated the values for nozzle blades and impulseblades separately, as shown in Fig. 14.

In the same way as for the profile losses, thetrailing-edge loss for an arbitrary blade is expressed as the weighted average of the values for nozzle andimpulse blades:

( ) ( ) ( )( )Δ Δ Δ Δφ φβα

φ φβ β α βTET TET TET TET2

02 1

2

22

02

1 1 2 1= +

⎛

⎝⎜

⎞

⎠⎟ −= = = (7)

Aspect Ratio, h/c

Asp

ectR

atio

Cor

rect

ion,

f(AR

)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

Ainley & Mathieson

Kacker & Okapuu (Eqn. (6))

The energy loss coefficient is then converted to the usual total pressure loss coefficient as follows:

Y

M

M

TETTET

=

−−

−−

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

− +−⎛

⎝⎜⎞⎠⎟

−−

−−

1 12

11

1 1

1 1 12

22

2

1

22 1

γφ

γ

γγ

γγ

Δ

and for incompressible flow this reduces to

YTETTET

= −1 12Δφ

Tip-Clearance Loss:

For unshrouded blades, KO express the effects of tip-clearance losses as a correction to theefficiency:

Δηη α0 2

0 93= .cos

kh

RR

Tip

Mean

(8)

where η0 is the efficiency for zero tip clearance and k is the tip clearance. Note that Eqn. (8) indicatesthat a 1% increase in tip clearance, relative to blade span, will result in a 1% reduction in efficiency. Thisis considerably lower sensitivity than the 3% reduction that is predicted by Howell’s correlation for axialcompressors. As seen, KO also found the loss to be a function of the hub-to-tip ratio of the blade, since

, where HTR = RHub/RTip. ( )R R HTRMean Tip= +12

1 2

Low-pressure turbine blades are often shrouded to reduce the tip-leakage flow and losses. KOrecommend the following expression to estimate the tip-leakage losses for a shrouded rotor blade row:

Y ch

kc

Cs cTC

L

m

=′⎛

⎝⎜⎞⎠⎟

⎛

⎝⎜

⎞

⎠⎟0 37

0 78 2 22

3.coscos

. αα

(9)

where kN is the effective tip clearance and

( )′ =k k

Number of seals 0 42.

To illustrate the relative magnitudes of the various components of loss, the predicted losscomponents for two different turbine stages, one subsonic and one transonic, will be quoted (taken fromMoustapha et. al., Axial and Radial Turbines, Concepts NREC, 2003, pp. 89-90). The table summarizesthe design parameters for the two stages:

Subsonic Turbine Transonic Turbine

Pressure Ratio 1.97 3.76

Work Coefficient, ψ 1.31 2.47

Flow Coefficient, φ 0.47 0.64

Reaction, Λ (%) 50 30

Stage Efficiency (%) 88 83.5

Stators Rotor Stators Rotor

Exit Mach Number 0.67 0.82 1.1 1.14

Total Flow Turning (o) 60 78 76 124

Blade Aspect Ratio, h/c 0.71 1.25 0.70 1.44

Tip Clearance, k/h (%) 1.5 1.5

Zweifel Coefficient 0.74 0.88 0.84 0.76

The figures show the resulting values of the loss coefficients:

Subsonic Turbine

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Profile TrailingEdge

Secondary TipClearance

Total

Loss Component

Loss

Coe

ffici

ent,

Y

StatorsRotor

Transonic Turbine

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Profile TrailingEdge

Secondary TipClearance

Total

Loss Component

Loss

Coe

ffici

ent,

Y

StatorsRotor

PW100 Turboprop

Compressor Pressure Ratio

η,Ef

ficie

ncy

(%)

Tip

Mac

hN

umbe

r

0 5 10 1550

55

60

65

70

75

80

85

90

95

100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η, (Rotor Only)

η (Rotor + Diffuser)

DiffuserLoss

Data for rotor only: Senoo, Y., Hayami, H., Kinoshita, Y. and Yamasaki, H., "Experimental Study on Flow in a Supersonic CentrifugalImpeller," ASME J. Eng. for Power, Vol. 101, Jan. 1979, pp. 32-41.Data for rotor with PWC pipe diffuser: Kenny, D.P., "A Comparison of the Predicted and Measured Performance of High Pressure RatioCentrifugal Compressor Diffusers," ASME Paper 72-GT-54, 1972.

Influence of diffuser design and diffuser pinch on pressure ratio, surge line and choking mass flow rate: Japikse, D. “Decisive Factors inAdvanced Centrifugal Compressor Design and Development,” I.MechE, Orlando, FL, November 2000.

6.2 IDEALIZED STAGE CHARACTERISTICS

Consider the outlet flow from a centrifugal rotor with backswept vanes. Assume that there is no swirlin the rotor inlet flow and that the fluid is incompressible. Assume also the Euler Approximation so that theflow leaves the rotor parallel to the metal angle at the vane trailing edge.

From the Euler equation

∆ ∆h g H U C U CU C

E w w

w

0 2 2 1 1

2 2

= = −=

and tan β22 2

2

=−C U

Cw

r

orC U C

U QA

w r2 2 2 2

22

2

= +

= +

tan

tan

β

β

Then

g H U U QAE∆ = +2

22

22tan β

or, dividing by N2D2

g HN D

K K QND

E∆2 2 1 2 3= + (1)

where KU

N Dconst1

22

2 2

2

60= =

=π .

U2

Cr2

Cw2

ω

W2

C2 β2 (+) β'2 (+)

and

KU DNA2

2

22= tan β

Equation (1) is the equation for the idealized head rise versus flow rate characteristic. Within the EulerApproximation (2 = 2), the slope of the characteristic, K2, is constant and has the same sign as 2, as shownin the sketch:

Note that whereas the slope of the H vs Q characteristic for axial machines was always negative(assuming < 1.0, which experience has shown is necessary), radial machines can have charateristics witheither positive of negative slopes, depending on the geometry of the vanes at the outlet:

Forward-swept vanes:• Highest head rise.• dHE/dQ > 0 is destabilizing, but losses can provide some stable operating range (see later

section).• Very high C2: puts heavy demands on diffuser to recover pressure.• Suitable where want to maximize head rise, efficiency is not a serious concern and surge is

not a problem.

Radial vanes:• Simplest to manufacture.• No bending stresses in vanes due to centrifugal effects (were therefore favoured in early gas

turbine engine applications of centrifugal compressors).

Backward-swept vanes:• Lower head rise.• Wide stable operating range (because dHE/dQ < 0).• Lower C2: reduces diffuser losses.

Recall from Section 3.4 that there will be a pressure rise through a radial rotor due to centrifugalcompression, even with no flow. From Eqn. (1), for Q = 0, HE = U2

2/g (or h0 = U22). The head rise at zero

flow is known as the "shut-off" head.

6.3 EMPIRICAL PERFORMANCE PREDICTIONS

6.3.1 Rotor Speed and Tip Diameter

The rotor speed and size can be estimated from correlations using two different approaches: fromspecific speed and specific diameter; or from the flow coefficient and work coefficient.

Specific Speed:

In Chapter 2, we have already used specific speed as a basis for selecting the type of turbomachinethat is suitable for a particular application. The following figures (from C. Rodgers, “Specific Speed andEfficiency of Centrifugal Impellers” in "Performance Prediction of Centrifugal Pumps and Compressors" ed.S. Gopalakrishnan et al., ASME International Gas Turbine Conference, New Orleans, March 1980, pp.191-200.) show more specific data for unshrouded centrifugal rotors, including the effect of several geometricand aerodynamic parameters.

The definition of specific speed used here is based average density (or average volume flow rate):

( )N

Q Q

g HS =

+

ω 1 2

12

34

2

∆

Compare this with the usual non-dimensional specific speed:

( )Ω

∆=

ωQ

g H

1

12

34

Note that for a typical case, will be slightly larger than .NS

If there are no constraints on the rotational speed, then one would normally choose the value ofthat gives the highest ( ). With chosen and estimated, the Cordier diagram can beN S NS ≅ −0 6 08. . N S

used to choose the diameter.

Work Coefficient and Flow Coefficient:

Aungier (R.H. Aungier, Centrifugal Compressors - A Strategy for Aerodynamic Design and Analysis,ASME Press, New York, 2000) presents a convenient correlation of work coefficient versus flow coefficientfor industrial compressors of several configurations: with shrouded and unshrouded impellers; and withvaneless and vaned diffusers. The correlations are based on results for compressors with pressure ratios up toabout 3.5, but can probably be extrapolated to somewhat higher values. Aungier defines the non-dimensionalparameters as follows:

Flow Coefficient: Work Coefficient:φρ π

=mr U1 2

22

µ Prefh

U=

∆ 0

22

where h0ref is the total enthalpy rise for the reversible process with the same pressure ratio. P is the stagepolytropic efficiency. Aungier’s correlations are presented in the following two figures:

The figures are applied as follows:

(a) Select to give the best stage polytropic efficiency, P, and read the corresponding workcoefficient, P. From

µ Prefh

U=

∆ 0

22

calculate U2.

(b) Then from the chosen

φρ π

=mr U1 2

22

calculate r2.

With the tip radius and tip blade speed defined, the rotational speed is known. If the rotational speedis constrained (eg. driving motors are only available for certain speeds) then Fig. 6-1 or Fig. 6-2 can be used toselect a compromise size and rotational speed that minimizes the impact on the stage efficiency.

6.3.2 Rotor Inlet Geometry

From the Euler equation

Δ Δh g H U C U CE w w0 2 2 1 1= = −

If there are no IGVs, Cw1 = 0 and the work transfer depends entirely on the rotor tip or outlet conditions. Forgood efficiency, the impeller inlet must nevertheless be well designed (eg. the inducer inlet metal angle mustbe matched to the inlet relative flow vector) and correctly sized.

Consider three rotors designed for the same , U2 and with the same outlet geometry so that all three&mgive the same Δh0. The critical region for frictional losses (which vary as V2), cavitation and compressibilityeffects is at the vane tip at the inlet, since that is where the relative velocity is the highest and static pressurethe lowest. The drawing shows the resulting inlet tip velocity triangles for three different inlet sizes:

To allow room for a shaft, or for a nut to hold the rotor to the end of the shaft, typically r1h = 0.2r2 to 0.35r2. For a given r1h, it is evident from the inlet velocity triangles that there is an optimum r1t that minimizes theinlet relative velocity and Mach number:

r1t

r1h

U1t

W1t

r2

U1t

U1t

C1tC1t C1t

LARGE EYEHIGH U1tLOW C1t

SMALL EYELOW U1tHIGH C1t

W1tW1t

W1t

M1t

r1t

OPTIMUM

6.3.3 Rotor Outlet Width

Consider the effect on the outlet velocity triangles of varying the rotor outlet width (or outlet vaneheight) b2. The outlet metal angle is adjusted to maintain constant Cw2 and thus give the same pressure rise. From continuity

( )&m C A C r br r= =ρ ρ π2 2 2 2 2 2 22

and thus for a fixed , the choice of b2 determines the radial component of velocity at the rotor outlet:&m

Summarizing the effect of different choices of b2:

LARGE b2 SMALL b2

C2

Lower(Good)

Higher(Bad - Larger diffusionrequired downstream)

W2 Lower Higher

W2/W1

Lower(Bad - Larger diffusion

required in rotor passage)

Higher(Good)

β2 Higher Lower

The value of b2 would thus be chosen to obtain a compromise between high diffusion inside the rotorpassage and high diffusion in the downstream diffuser (which serves the same function as the stators in anaxial compressor stage).

Note that W2/W1 is again the de Haller number. Various papers and textbooks provide guidelines for

2β

2bSMALL

2bLARGE

2wC

2b

2C

2rC

2W

2U

2W2C

2rω

choosing the de Haller number for centrifugal fan, compressor, and pump rotor passages:

(1) Aungier (2000) Recommended: W2/W1 > 0.75Never exceed: W2/W1 < 0.65

(2) Wilson & Korakianitis (1998) Recommended: W2/W1 > 0.8

(3) Rodgers (1978) Recommended: W2/W1 > 0.71

(4) Yoshinaga (PWC document, 1982) Low PR compressors and fans: W2/W1 > 0.8High PR compressors (up to 8.0) W2/W1 > 0.6

where W1 = value of relative velocity at inlet mean radius.

6.3.4 Rotor Outlet Metal Angle - Slip

From Section 6.3.3, the required outlet flow angle 2 was seen to be related to the choice of b2. Thecorresponding metal angle 2depends on the deviation, which is called “slip” in centrifugal machines. Theslip in turn depends on the rotor “solidity”: that is, the number of vanes, Z. Thus, the choices for 2 and Z areinter-related.

Consider a backswept rotor:

Because of slip, the rotor imparts less swirl to the flow than for the “ideal” case, for which 2 =2 (that is, theEuler approximation is taken to hold in the ideal case). Since Cw2 < Cw2, the h0 is reduced by this effect. Wethen define the slip factor as

σ =′

C

Cw

w

2

2

where 1.0.

A number of correlations have been proposed for . The one due to Stodola has been widely used:

σ

π β

φ β= −

′

− ′

11

2

2 2

Zcos

tan

where 2 = Cr2/U2 and 2 is the backsweep or forwardsweep angle (taken as positive in both cases). Stanitzsuggested a slightly simpler form:

σ

π

φ β= −

− ′1

0 63

1 2 2

.

tanZ

Wiesner (F.J. Wiesner, “A Review of Slip Factors for Centrifugal Impellers,” ASME Trans., J. Eng.for Power, October 1967, pp. 558-572) reviewed the available slip factor correlations and pointed out that theStodola, Stanitz and similar correlations are only valid for impellers with long blades. Wiesner recommendedthe Busemann correlation which takes into account the influence of r1/r2 and provided the following curve fit:

Letting = r1/r2, and identifying a limiting value of given by

εβ

lim. cos

it

Ze

=′

18 16 2

then for limit (ie. longer vanes)

σβ

= −′

1 20 7

sin.Z

and for > limit (ie. shorter vanes)

σβ ε ε

ε= −

′

−−−

1 1

12

0 7

3cos.

lim

limZit

it

The figure shows the predicted variation with Z and for an example backsweep angle of 45o (taken fromAungier, 2000), who provides an equivalent butslightly different curve fit:

CENTRIFUGAL COMPRESSOR - NUMBER OF VANES

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

45

Rodgers, Ns = 0.6Rodgers, Ns = 0.7Rodgers, Ns = 0.8Wilson ZmaxWilson Zmin

Backsweep Angle (Deg.)

Num

ber o

f Van

es, Z

6.3.5 Choice of Number of Vanes - Vane Loading

Wilson & Korakianitis (Design of High-Efficiency Turbomachinery and Gas Turbines, 2nd ed., Prentice-Hall, 1998) provide a broad guideline for selecting thenumber of blades, as function of the vane angle at the tip, asshown in the figure at right.

More recently, Rodgers (2000) presented acorrelation for the number of vanes which, according to hisloss estimates, gives the best rotor efficiency:

Z =′25 2cosβ

Ω

where Ω is the usual non-dimensional specific speed. Comparison with the Wilson & Korakianitis figure suggeststhe Rodgers’ correlation is very conservative, leading to verylarge numbers of vanes.

Aungier (2000) outlines a method of selecting thenumber of vanes based directly on the vane loading. Hesuggests the following limit:

20 9

2 1

ΔWW W+

≤ .

where ΔW is the maximum relative velocity difference acrossthe vane. ΔW can be estimated from

ΔWD UZ LB

=2 2 2π ψ

where ψ = work coefficient = Δh0/U22 and LB is the length of the

vane along the mean camber line. A reasonable initial estimateof LB can be obtained from

L zb D D

B I= −⎛⎝⎜

⎞⎠⎟+

−′

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Δ 2 2 1

22

12 cosβ

where ΔzI is the axial length of the rotor.

6.3.6 Losses

The actual stage characteristics are different from ideal due to slip and losses. Slip reduces output butdoes not affect efficiency since the required input power is reduced along with the output.

Sources of losses:

(1) Disc friction: - friction on outer surface of impeller- since this torque is not exerted on the through-flowing fluid, it does not appear inthe Euler work, gQHE

(2) Leakage: - fluid leaks through the tip gap leading to losses as in axial machines- if the rotor is shrouded, compressed fluid can leak through the clearance back to theinlet, to be recompressed over and over again- thus, more fluid is compressed than is delivered by the machine, increasing thepower required and showing up as an apparent loss

(3) Inlet: - at other than design Q, flow angle and metal angle will be mismatched at theleading edge, resulting in separation and additional losses- a simple, inexpensive machine with no inducer will have significant inlet losses atall operating conditions

(4) Impeller: - frictional and separation losses inside the impeller channels- roughly Q2

(5) Diffuser/Volute: - frictional and separation losses roughly Q2

- for vaned or pipe diffuser, additional leading-edge losses when Q Qdesign (like (3))- for volute, sudden expansion losses due abrupt change in area

The figure shows the approximate trend of the loss components with flow rate:

The next figures show the resulting stage characteristics, taking into account slip and losses, forbackward-swept and forward-swept vanes:

(i) Backward-swept vanes: (ii) Forward-swept vanes:

Note that due to the effects of the losses the machine with forward-swept vanes also has some stableoperating range (dHE/dQ < 0.0), although it tends to be narrower and does not include the design point.

Taking into account the losses, the required shaft power is

&( ) ( )W gQ H gQ H Disc Bearing Friction PowerE th l E th= + +ρ ρ∆ ∆

where HE(th) = theoretical Euler head (Euler head with slip but no losses)Ql = leakage flow (volume flow which leaks from outlet back to inlet, to be

recompressed) for a shrouded rotor

The actual head delivered is

∆ ∆ ∆H H HE th L= −( )

where HL = sum of losses (3) + (4) + (5)

and the corresponding efficiency is

η ρoverall

shaft

gQ HW

=∆

As usual, for compressible flow substitute for Q and h0 for gH.m

1

Fig. 7.1 Compressor operating points.Fig. 7.2 Four-component compression system.

CHAPTER 7 Static and Dynamic Stability of Compression Systems

7.1 INTRODUCTION

It was mentioned in Chapter 4 that surge is verydangerous to axial compressors. While centrifugalcompressors are more rugged than axial machines, surge isstill dangerous and should be avoided.

It was also noted that surge is a dynamic instability whichdepends on not just the characteristics of the compressor butalso on the aerodynamic characteristics of the othercomponents to which it is connected. It is possible to developa simple lumped-parameter analysis for a compressionsystem. Such an analysis can provide useful insights intowhich characteristics of the system encourage or delay theonset of surge. For further information see Stenning (1980),Greitzer (1980, 1981) and Cumpsty (1989).

7.2 STATIC STABILITY

A system is statically stable if, when it is disturbed by asmall amount from its equilibrium operating point, a reactionarises which tends to restore it to the equilibrium condition.Static stability is normally a necessary, but not sufficientcondition for dynamic stability

Consider the compressor characteristic shown in Fig. 7.1.Points A - D are all equilibrium operating points ()P0,load =)P0,machine at the given ). Consider point A and suppose&mthat a small disturbance causes an increase in :&m(i) The machine delivers less )P0 than required by the loadat this .&m(ii) The flow rate in the load must therefore decrease,causing the system to move back towards A.

The same argument can be made for points B and D.Thus, operating points A, B and D are statically stableoperating points.

Point C is different. If is disturbed to a larger value,&mthe machine delivers more )P0 than the load requires at thenew . The flow in the load will therefore increase even&mfurther and the operating point moves further from theequilibrium point. Thus C is a statically unstable operatingpoint.

Static stability does not guarantee that the system willfinally settle at the original equilibrium operating point, onlythat tend to move back towards the equilibrium point. Thesystem may overshoot and oscillate about the operating point.If it eventually settles at the original operating point, thesystem is dynamically stable.

7.3 DYNAMIC STABILITY - SURGE

A simple analysis can be developed to predictapproximately the dynamic stability characteristics of acompression system. A compressible flow system will beexamined. Only minor modifications are needed to make itapply to an incompressible flow system.

Fig. 7.2 shows schematically a simple system consistingof four components:

(1) A compressor(2) A duct(3) A plenum, in which mass can be stored.(4) A throttle, represented by a valve, which provides

the main pressure loss in the system. To a firstapproximation, the throttle could also represent theturbine in a gas turbine engine.

2

Fig. 7.3 Compressor characteristics.

m m m P P P etc2 2 2 3 3 3= + ′ = + ′ .

( )P P C m2 01 1− = (1)

( )d P Pdm

dCdm

c2 01

1 1

−= = (2)

dPdm

dPdm

dPdm

c2

1

01

1

2

1− = = (3)

dP cdm

P

P

m

m

2 1

2

2

1

1

∫ ∫=

( )P P c m m2 2 1 1− = −

′ = ′P cm2 1 (4)

′ = ′P cm2 2 (5)

The flow through the components is treated as one-dimensional. Thus, the flow at any point is characterized bya single value of P, T, C, etc. (if necessary, these would&mbe interpreted as the local average values). The analysis willconsider perturbations about an equilibrium operating pointand the perturbations will be assumed to be small.

The instantaneous value of any flow quantity isrepresented by the sum of the mean value plus theinstantaneous (small) perturbation:

where m2 is the mass flow rate at plane 2 (the dot is omittedfor convenience). The goal of the analysis is to determine thebehaviour of the perturbations over time after some initialdisturbance has occurred. If m2N, P3N etc. eventually decreaseto zero, the system is dynamically stable at the operatingpoint in question.

The approach used is known as the lumped-parametermethod: equations for the behaviour of each component aredeveloped separately and they are then linked by the flowconditions at the interfaces between the components.Consider each component in turn:

(1) Compressor

The pressure rise across the across the compressor,represented by P2 - P01, is a function of the inlet mass flow:

where C is the function which defines the compressorcharacteristic (see Fig. 7.3). The gradient at any operatingpoint along the characteristic is

and since we are assuming that the perturbations from theoperating point are small, we can assume that c is constant inour analysis. That is, we linearize the compressorcharacteristic at the operating point of interest.

If we assume that P01 is constant (ie. that the compressordraws fluid from a large, constant pressure reservoir) thenfrom (2)

and integrating (3) for a small deviation away from theequilibrium point

From the definition of the perturbations, this can be written

This is then the perturbation equation for the compressor.

If we assume that the internal volume is small, so thatessentially no mass can be stored in the compressor, then m1= m2 at all times and an alternative to (4) is

(2) Duct

We assume that the losses in the system occur primarilyin the throttle so that we can neglect the frictional losses inthe duct. We also neglect the volume of the duct relative tothe volume of the plenum. Therefore, the duct introducesonly inertia: a pressure difference is present between stations2 and 3 only when the fluid in the duct is being accelerated ordecelerated.

The equation governing the behaviour of the duct can beobtained either by performing a force balance on the freebody consisting of the cylinder of fluid in the duct or by

3

( )

( )

( )

ΣF ddt

mu

P A P A ddt

A Lu

L ddt

Au

x =

− =

=

2 3 ρ

ρ

P P LA

dmdt2 3

2− = (6)

( ) ( )ΣF ddt

u dV mu muxV

out in= + −∫ ρ

ΣF ddt

u Adx ddt

m dx L dmdtx

L L= =

⎛

⎝⎜⎜

⎞

⎠⎟⎟ =∫ ∫ρ

02

0

2

( ) ( ) ( )P P P P L

A

d m m

dt2 2 3 32 2

+ ′ − + ′ =+ ′

′ − ′ =′P P L

Admdt2 3

2 (7)

m m V ddt2 3

3− =ρ (8)

Pργ = const.

ddt P

dPdt

RTdPdt

adPdt

ρ ργ

γ

3 3

3

3

3

3

32

3

1

1

=

=

=

m m Va

dPdt2 3

32

3− =

′ − ′ =′m m V

adPdt2 3

32

3 (9)

applying the unsteady momentum equation to a controlvolume occupying the duct. For both analyses, we willneglect density changes along the duct.

(i) Force balance:

and DAu = m2 is the instantaneous mass flow rate at all pointsin the duct (since density changes are neglected), so that

(ii) Control volume analysis:

For the control volume in the duct

Since the density is constant along the duct, the instantaneousinflows and outflows of momentum must be identical, andonly the first term, the momentum accumulation term,remains on the right-hand side:

After substituting for EFx in terms of the inlet and outletpressures, (6) is again obtained.

We then substitute into (6) in terms of the perturbations

and since there are no losses in the duct, the mean inlet andoutlet pressures must be the same. Thus, the perturbationequation for the duct becomes

(3) Plenum

The plenum can be a mass storage component. Applyingconservation of mass to the plenum:

where m3 = mass flow rate through the valve. Changes in themass in the plenum will be reflected in the density of thestored gas. In a pump system, a reservoir with a free surfaceor a surge tank would similarly act as a mass storagecomponent.

If the compression or expansion process is isentropic,then

Differentiating with respect to time and assuming a perfectgas

where a3 = speed of sound at the plenum conditions. Thenfrom (8)

Substituting in terms of the perturbation quantities, theperturbation equation for the plenum is obtained:

(4) Throttle

The throttle is handled in exactly the same way as thecompressor: the load line is linearized at the equilibriumoperating point. If the valve is choked, the mass flow rate

4

′ = ′P f m3 3 (10)

′ = ′

′ − ′ =′

′ − ′ =′

′ = ′

P cm

P P LA

dmdt

m m Va

dPdt

P f m

2 2

2 32

2 332

3

3 3

(11)

(12)

(13)

(14)

cm f m LA

dmdt

′ − ′ =′

2 32 (15)

f m f m f Va

dPdt

′ − ′ =′

2 332

3 (16)

f m cm f Va

dPdt

LA

dmdt

′ − ′ =′−

′2 2

32

3 2 (17)

dPdt

dPdt

LA

d mdt

′=

′−

′3 22

22

dPdt

c dmdt

′=

′2 2

dPdt

c dmdt

LA

d mdt

′=

′−

′3 22

22

f Va

dPdt

c f Va

dmdt

f Va

d mdt3

23

32

2

32

22

2′=

′−

′(18)

( )f Va

LA

d mdt

LA

c f Va

dmdt

f c m32

22

232

22 0′

+ −⎛

⎝⎜⎜

⎞

⎠⎟⎟

′+ − ′ = (19)

m d xdt

s dxdt

kx2

2 0+ + = (20)

through it is a function of only the upstream pressure, P3. Ifit is not choked, the pressure downstream is assumed to beconstant. Then the perturbation equation for the throttlebecomes:

where f is the local slope of the load line (note that f willalways be positive).

Characteristic Equation for the System

Summarizing, there are four perturbation equations forthe components in the system:

These are four equations in the four unknowns m2N, m3N, P2Nand P3N. Solving for any one of the unknowns from (11) -(14) leads to a second-order ordinary differential equation forthe variation in time for that unknown.

For example, solving for m2N, substitute (13) and (14)into (11):

Multiply (13) by f (noting that f is non-zero and alwayspositive),

Then subtract (15) from (16) to eliminate m3N

Differentiate (12) with respect to time and rearrange to obtainan expression for dP3N/dt:

and from (11)

Thus

or

Substituting (18) into (17) and rearranging

This is seen to be a second-order ordinary differentialequation in m2N.

It can be shown that the corresponding equation for anyof the other three perturbations would have the savecoefficients as (19).

Within the assumptions of the analysis, the coefficientsare constant and, given initial conditions for m2N and dm2N/dt,(19) can readily be solved to determine the response of thesystem. As noted earlier, if m2N tends to 0 with increasingtime, the system is dynamically stable.

A useful analogy can be drawn between the presentsystem and a mass-spring-damper system for which thegoverning equation is (for free vibrations)

where s = damping coefficient, k = spring constant. Twoconditions must be met for the system governed by (20) to bestable:

(i) k > 0 - that is, the spring constant must be positive

The equivalent condition in (19) is that f > c, which is

5

s k mc = 2

x Ae ss

km

tss

km

t

c

c= −⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−sin 1

2

φ

LA

c f Va

or c LaAV f

− > >32

32

0

precisely the requirement for static stability which wearrived at with qualitative arguments in Section 7.2.

(ii) s > 0 - that is, the damping must be positive

This is evident from the solution to the equation: thesystem has a critical value of the damping coefficient, sc,given by

If s < sc, the system is under-damped and the solutiontakes the form

Thus, the system oscillates sinusoidally in time, with themagnitude of the peak displacement being controlled bythe exponential term.

Since m, k, and sc are all positive, if s > 0 theexponential term decreases in time, the magnitude of thefluctuations decays, and the system is seen to bedynamically stable. If s > sc, the system is over-dampedand the solution is no longer oscillatory but it againincludes exponential terms which are a function of s.Again, if s is negative the exponential terms grow in timeand the system moves away from the equilibrium point inan unstable way.

Applying these ideas to the compressor system, it is seenthat there are two contributions to the system damping:

(a) positive (stabilizing) damping is supplied by theinertia of the fluid in the duct (the L/A term), and(b) potentially negative damping is supplied by the terminvolving the slopes of the compressor and throttlecharacteristics.

Since f is always positive, the sign of the damping term iscontrolled by the sign of the slope of the compressorcharacteristic, c. If c < 0 (as it normally is at higher flowrates) strong positive damping will be present and the systemwill be stable. The condition for instability is then

Thus, the system will become unstable for some positivevalue of the slope of the compressor characteristic, theprecise magnitude being a function of a number of systemparameters.

It is not immediately clear whether it requires a largepositive value of c (large in comparison to f, for example) todestabilize the system, but note that as L tends to zero c alsotends to zero. Therefore, a compressor which is connected toa plenum by a very short length of duct will become unstableessentially at the peak of the compressor characteristic. Thatis why the latter is often used as a criterion for predictingsurge. In general, we would expect to encounter thecondition for dynamic instability near the peak of thecharacteristic and probably long before we reach thecondition for static instability (operating point C on theoriginal )P0 versus diagram) .&m

The relationship between stall and surge now is a littleclearer. For a typical compressor characteristic, as the flowrate through the machine is reduced the output peaks andeventually begins to reduce. This is generally the result ofincreasingly extensive stall: perhaps an increasing number ofrotating stall cells and/or cells of increasing spanwise extentas the flow rate is reduced. Stall thus prepares the conditionsfor surge. Note that the appearance of stall is a phenomenonof the compressor itself, not the system. On the other hand,surge is an unstable condition in compression system inwhich flow quantities, including the compressor mass flowand delivery pressure, undergo oscillatory fluctuations whichgrow over time. In systems such as gas turbine engines, thesefluctuations can reach destructive magnitudes in a very smallnumber of cycles.

References

Cumpsty, N.A., 1989, Compressor Aerodynamics, Longman,Harlow.

Greitzer, E.M., 1980, “Review - Axial Compressor StallPhenomena,” ASME J. Fluids Engineering, Vol. 102, June1980, pp. 134-151.

Greitzer, E.M., 1981, “The Stability of Pumping Systems,”ASME J. Fluids Engineering, Vol. 103, June 1981, pp. 193-242.

Stenning, A.H., 1980, “Rotating Stall and Surge,” ASME J.Fluids Engineering, Vol. 102, March 1980, pp. 14-20.

1

APPENDIX A:

Curve and Surface Fits for Howell’s Correlations for Axial Compressor Blades

(a) Design-Point flow Deflection, * (C,R & S, Fig. 5.14)

* is a function of inlet flow angle, 2 and s/c (=1/):

With: A = 33.5293 B = -0.530812 C = -15.2599D = 0.00209610 E = -0.677212 F = 0.187148

*(2,s/c) = A + B2 + C ln(s/c) + D22 + E(ln(s/c))2 + F2ln(s/c)

Applies for: 0 < 2 < 70o, 0.5 < s/c < 1.5 (or 0.666 < < 2.0).

(b) Reynolds Number Correction for Design-Point Deflection (Horlock Fig. 3.3)

With A = 0.664154 B = 22.1578 C = 1.03819 D = 4.71864

where Re is the Reynolds number based on inlet velocity and blade chord divided by 105.

(c) Off-Design Deflection (Dixon Fig. 3.17)

where

The curve fit is applicable for -0.8 < irel < 0.8.

(d) Profile Drag Coefficient, CDp (Dixon Fig. 3.17)

For values of irel from -0.7 to 0.3 the profile drag coefficient, CDp, is a function of solidity and irel:

2

CDp1A(irel,s/c) = -0.02842irel(s/c)2 + 0.004381(s/c)3 - 0.00788(s/c)2 - 0.003979(s/c) + 0.07753irel(s/c)

CDp1B(irel,s/c) = -0.01542irel2 (s/c) + 0.02277 - 0.04429irel + 0.05002irel

2 + 0.009207irel3

CDp1(irel,s/c) = CDp1A(irel,s/c) + CDp1B(irel,s/c)

This curve fit is applicable for 0.5 < s/c < 1.5 (or 0.666 < < 2.0).

For values of irel greater than 0.3, CDp is a function of the relative incidence only:

CDp2(irel) = 0.01665 - 0.004181irel - 0.01908irel2 + 0.06477irel

3 + 0.3949irel4 + 0.3426irel

5

1

(1)

APPENDIX B:

C4 Compressor Blade Profiles

Like NACA 4-digit airfoils, the C-series compressor blades are defined by a symmetricalthickness distribution which is superimposed on a specified mean, or camber, line. As indicated in theHowell correlations, both circular arc and “parabolic” arc camber lines have been used with C-seriesblades.

For the blade with a parabolic arc camber line, the point of maximum camber lies at other thanmid-chord. Typically, the point of maximum camber lies towards to leading edge; that is, a/c < 0.5.

The relationship between the camber angle 2 (= 21 + 22), a/c and b/c is:

and

The term parabolic arc camber line is somewhat misleading. The mean line is not defined by asingle parabola, or even by two joined parabolas. For example, to define a polynomial which passes

2

(2)

(3)

(4)

(5)

through (0,0) with slope tan21 and through (a,b) with zero slope requires at least a cubic. The followingdiscussion will consider mainly the circular arc camber line.

Setting a/c = 0.5 in Eqn (1),

The equations of the camber line and its inclination, Nc, are then

and

The co-ordinates of the upper and lower sides of the blade are then

where yt is the local thickness of the blade. For the C4 profile, the blade thickness distribution is given by

where t is the maximum thickness of the blade as a fraction of the chord length.

The geometry of C-series blade is designated using a shorthand notation. For example, a bladedesignated 10C4/30C50 refers to a blade with a C4 profile and: 10% maximum thickness, circular arccamber, camber angle 30o and maximum camber at 50% chord (the last piece of information is redundant

3

since circular arc camber has already been specified). The resultant geometry is shown:

1

APPENDIX C:

Curve and Surface Fits for NASA Correlations for Axial Compressor Blades

(a) Minimum-Loss Incidence (SP-36 Fig. 137)

The surface fit gives the minimum loss incidence for a blade of zero camber and 10% thickness asa function of inlet flow angle, $1, and solidity, F:

With: A00 = -0.13571 A01 = 0.075795 A02 = 9.1315x10-4

A10 = 0.015986 A11 = 0.074959 A20 = -2.4954x10-4

i0(10)($1,F) = A00 + A01F + A02F2 + A10$1 + A11$1F + A20$1

2

Valid for: 0.4 < F < 2.0, 0.0 < $1 < 70.0.

(b) Slope Factor, n, for Minimum-Loss Incidence (SP-36 Fig. 138)

With: A00 = -0.066879 A01 = 0.05897 A02 = -0.054019A03 = 0.033568 A04 = -7.1706x10-3 A10 = -6.0476x10-3

A11 = 7.402x10-3 A12 = -2.5749x10-3 A13 = 2.6067x10-4

A20 = -3.3001x10-5 A21 = -3.084x10-5 A22 = 1.3955x10-5

A30 = 8.0286x10-7 A31 = -1.2016x10-7 A40 = -9.1961x10-9

n1($1,F) = A00 + A01F + A02F2 + A03F

3 + A04F4 + A10$1 + A11$1F + A12$1F

2 + A13$1F3

n2($1,F) = A20$12 + A21$1

2F + A22$12F2 + A30$1

3 + A31$13F + A40$1

4

n($1,F) = n1($1,F) + n2($1,F)

Valid for: 0.4 < F < 2.0, 0.0 < $1 < 70.0.

(c) Thickness Correction, (Ki)t, for Minimum-Loss Incidence (SP-36 Fig. 142)

Valid for: 0.0 < t/c < 0.12, probably usable up to t/c = 0.15.

2

(d) Zero Camber Deviation Angle, *0 (SP-36 Fig. 161)

With: A00 = 0.053535 A01 = -0.29275 A02 = 0.71879A03 = -0.75902 A04 = 0.3706 A05 = -0.067233A10 = -3.838x10-3 A11 = 0.02838 A12 = -0.02068A13 = 3.4149x10-3 A14 = 5.8448x10-4 A20 = 3.5333x10-4

A21 = 2.0917x10-4 A22 = 3.0519x10-4 A23 = -1.2273x10-4

A30 = -1.3124x10-5 A31 = -1.0755x10-5 A32 = 1.7229x10-6

A40 = 2.3356x10-7 A41 = 1.1718x10-7 A50 = -1.4651x10-9

*o1($1,F) = A00 + A01F + A02F2 + A03F

3 + A04F4 +A05F

5 + A10$1 + A11$1F + A12$1F2 + A13$1F

3 + A14$1F4

*o2($1,F) = A20$12 + A21$1

2F + A22$12F2 + A23$1

2F3 + A30$13 + A31$1

3F + A32$13F2

+ A40$14 + A41$1

4F + A50$15

*o($1,F) = *o1($1,F) + *o2($1,F)

Valid for: 0.4 < F < 2.0, 0.0 < $1 < 70.0.

(e) Parameters for Deviation Rule (SP-36 Figs. 163,164)

The slope factor for the deviation rule is given by

where mF=1($1) = 0.170 + 6.2698x10-5$1 + 1.4096x10-5$12 + 1.9823x10-7$1

3

b($1) = 0.965 - 2.5464x10-3$1 + 4.2695x10-5$12 - 1.3182x10-6$1

3

Valid for: 0.0 < $1 < 70.0.

(f) Thickness Correction, (K*)t, for Deviation (SP-36 Fig. 172)

Valid for: 0.0 < t/c < 0.12, probably usable up to t/c = 0.15.

(g) Gradient of Deviation Angle with Incidence, d*o/di (SP-36 Fig. 177)

3

Valid for: 0.4 < F < 1.8, 0.0 < $1 < 70.0.

1

APPENDIX D:

NACA 65-Series Compressor Blade Profiles

The 65-series blade geometry is not represented by closed-form analytical expressions. Instead, itis necessary to work with tabulated values:

x/c Thickness(for t = 0.10c)

yt/c

Camber Line (for CL = 1.0)

yc/c dyc/dx

0.0 0.0 0.0 ---

0.005 0.00752 0.00250 0.42120

0.0075 0.00890 0.00350 0.38875

0.0125 0.01124 0.00535 0.34770

0.025 0.01571 0.00930 0.29155

0.050 0.02222 0.01580 0.23430

0.075 0.02709 0.02120 0.19995

0.10 0.03111 0.02585 0.17485

0.15 0.03746 0.03365 0.13805

0.20 0.04218 0.03980 0.11030

0.25 0.04570 0.04475 0.08745

0.30 0.04824 0.04860 0.06745

0.35 0.04982 0.05150 0.04925

0.40 0.05057 0.05355 0.03225

0.45 0.05029 0.05475 0.01595

0.50 0.04870 0.05515 0.0

0.55 0.04570 0.05475 -0.01595

0.60 0.04151 0.05355 -0.03225

0.65 0.03627 0.05150 -0.04925

0.70 0.03038 0.04860 -0.06745

0.75 0.02451 0.04475 -0.08745

0.80 0.01847 0.03980 -0.11030

0.85 0.01251 0.03365 -0.13805

0.90 0.00749 0.02585 -0.17485

0.95 0.00354 0.01580 -0.23430

1.00 0.00150 0.0 (-0.23430)

2

(1)

The thickness distribution is given for a NACA 65-010 blade which has been modified to give afinite trailing-edge thickness of 0.3% of the chord length. The baseline thickness distribution has zerothickness at the trailing edge and therefore cannot be manufactured. The nominal maximum thickness is10% of chord. For blades with other values of maximum thickness, the tabulated distribution is simplyscaled accordingly.

The table indicates that maximum camber is at 50% of chord. However, the camber line is not asimple circular arc. In fact, the slope of the camber line tends to infinity at the leading and trailing edges. At the leading edge, this gives a "droop" to the nose of the blade which is believed to reduce its sensitivityto incidence.

Because of the camber line shape, there is no simple relationship between the camber angle, asdefined earlier, and the magnitude of the maximum camber. Instead, the camber line shape is related tothe nominal maximum lift coefficient which the blade shape would achieve as an isolated airfoil. Thecamber line shape quoted applies for a nominal lift coefficient CL = 1.0. To generate compressor bladeswith a desired camber angle, the following can be used to relate an equivalent circular arc camber angle tothe nominal CL:

for 2 in degrees.

To generate the geometry for a 65-series compressor blade with a particular camber angle, 2:(i) From (1), determine the nominal CL.(ii) Scale the camber line co-ordinates and slope values by (CL/1.0).(iii) Calculate the blade-surface co-ordinates by superimposing the tabulated thicknessdistribution (scaled as necessary if the maximum thickness is to be different from 10% of chord)on the camber line using Eqns. (5) from Appendix B.

The drawing compares the 10C4/30C50 blade with the 65-series which has the same maximum thicknessand the equivalent camber:

1

APPENDIX E:

Curve and Surface Fits for Kacker & Okapuu Loss Systemfor Axial Turbines

Kacker & Okapuu ("A Mean Line Prediction Method for Axial Flow Turbine Efficiency," ASMETrans., J. Eng. for Power, Vol. 104, January 1982, pp. 111-119) presented an updated version of theAinley & Mathieson loss system for axial turbines. The Kacker & Okapuu (KO) system presents a basisfor estimating the complete losses, and thus the efficiency, of an axial turbine at its design point. For acomplete outline of the loss system see the paper.

Some aspects of the loss system are presented only in graphical form in the paper. Therefore anumber of figures have been digitized and curves or surfaces fitted to the data. This appendix documentsthe curve fits and, in some cases, demonstrates the quality of the fits graphically. The figure numbersrefer to the figures in the Kacker & Okapuu paper.

(a) Ainley & Mathieson (AMDC) Profile-Loss Coefficients (Figs. 1, 2)

KO use the AMDC correlation for profile loss coefficient, with corrections for Reynolds number,exit Mach number, channel acceleration, and improvements in design. The AMDC loss coefficient isobtained as a weighted average of the values for a nozzle blade ($1 = 0) and an impulse blade. Thesevalues are obtained from the plots shown in Figures 1 and 2. The data in these figures have been fitted topolynomial surfaces of the form:

The values of the coefficients follow:

(i) Nozzle Blade, (Fig. 1)

a0,0 = 0.358716a0,1 = -1.43508a0,2 = 1.57161a0,3 = -0.496917a1,0 = -0.0112815a1,1 = 0.0548594a1,2 = -0.0555387a1,3 = 0.014165a2,0 = 0.000175083a2,1 = -0.000824937a2,2 = 0.000652287a2,3 = -7.30141E-05a3,0 = -8.61323E-07a3,1 = 3.95998E-06a3,2 = -1.89698E-06a3,3 = -4.9954E-07

2

(ii) Impulse Blade, (Fig. 2)

a0,0 = 0.0995503a0,1 = 0.182837a0,2 = 0.01603a1,0 = 0.00621508a1,1 = -0.0283658a1,2 = 0.011249a2,0 = -7.10628E-05a2,1 = 0.000327648a2,2 = -0.000122645

(b) Stagger Angle (Fig. 5)

In the early stages of design, axial chord rather than true chord of the blades is often specified. However, the profile loss correlations require the solidity of the blade row, which is based on the truechord. KO present an approximate correlation for the stagger angle as a function of the inlet and outletangles. The true chord can then be calculated from the axial chord. The graphical data are again fitted toa surface, using a polynomial of the form:

with coefficients,

a0,0 = -2.90463 a0,1 = 0.307036 a0,2 = 0.370176E-02a1,0 = 0.412797 a1,1 = -0.355369E-01 a1,2 = -0.194938E-03a2,0 = 0.593956E-02 a2,1 = 0.389157E-03 a2,2 = 1.74147E-06

The surface fit and the digitized values are compared over.

3

(c) Inlet Mach Number Ratio (Fig. 6)

A correction is made for shock losses at the leading edge of the blade. Since the Mach numbertends to be higher at the hub than at midspan, KO present a correlation for the hub Mach number as afunction of the midspan value and the hub-to-tip ratio. The shock loss is then calculated from theestimated hub Mach number. The following polynomials were fitted to the curves of Figure 6:

(i) Rotors

(ii) Nozzles

4

(d) Trailing-Edge Energy Coefficient (Fig. 14)

The trailing-edge losses are expressed in terms of the energy coefficient. This was correlatedwith the ratio of the trailing-edge thickness to the throat opening. Curves were presented for nozzle andimpulse blades. The values from these curves are then averaged in a weighted way to give the coefficientfor a blade of arbitrary inlet and outlet flow angles.

(i) Impulse (Rotor) Blade

(ii) Nozzle

5

1

APPENDIX F:

Centrifugal Stresses in Axial Turbomachinery Blades

1.0 Introduction

As briefly mentioned in lectures, the design of a turbomachine involves a trade-off between oftenconflicting considerations: aerodynamics, heat transfer, materials, stresses, and vibrations (not to mentioncost). While our focus is on the aerodynamics, it is obviously wasteful to develop even a preliminaryaerodynamic design for a turbomachine which cannot be built for stress reasons.

Turbomachinery blades experience significant unsteady forces which lead to vibratory stresses,and both low cycle and high cycle fatigue are important considerations. However, the level of the steadystress determines the margin which is available for these unsteady stresses. In turbines, creep distortion isan important consideration and the steady centrifugal stress is also the starting point for a creep analysis. Thus, if the steady centrifugal stresses are kept within established limits, the design is likely to bemechanically feasible. Fortunately, the steady centrifugal stresses in the rotor blades can be estimatedfairly easily in the early stages of the aerodynamic design.

A later section gives some criteria for judging whether the centrifugal stresses are acceptable. These criteria apply primarily to the high-performance machines used in gas turbine engines. The stressesare particularly high in low hub-to-tip ratio fan blades and in turbine blades; they are much lower innormal compressor blades. A survey of typical, industrial axial-flow fans from several manufacturersshows that peak tip speeds are consistently below 120 m/s. It is believed that this limit is related to thestresses which can be sustained by the rather simple blade attachments, rather than stresses in the actualblades. Higher tips speeds can be used but these require a switch to a considerably more expensivemethod of attachment.

2.0 Steady Centrifugal Stresses

2

(1)

(2)

(3)

Consider the forces on the small blade element shown.

Then

and this can then be integrated from radius R to the tip, RT, (with a specified blade area variation) toobtain the centrifugal stress at R.

Constant Section Blade:

With dA = 0, integrating (1):

and the maximum stress occurs at the root:

Tapered Blades:

The cross-sectional area of turbomachinery blades often varies from hub to tip. If the areadecreases, the root stress will be reduced from the value given by (2). Taking into account the taper, thehub stress can be written

where K depends on the nature of the taper in cross-sectional area:

(a) Blade with constant cross-section.

(b) Blade with linear taper.

3

(4)

where

The cross-sectional area of the blade is roughly proportional to the product of the chord length (c) and themaximum thickness (tmax). Thus, the area ratio can be approximated by

If both the chord length and the maximum thickness are tapered linearly from the hub to the tip, tomaintain constant maximum thickness-to-chord ratio, the cross-sectional area will in fact varyparabolically. It can be shown that the resultant centrifugal stresses will be lower than for linear taper. However, for HTR > 0.5 the stresses are very similar and the assumption of linear taper gives a good,slightly conservative, estimate of the hub stress.

3.0 Allowable Stress Levels

From Eqn. (3)

where N = RPM and A = annulus area of the stage. Rearranging,

From the density and stress limits for currently available blade materials, values of the right-hand side of(4) can defined by the structural engineer. The aerodynamicist can then use these to verify that theproposed design is feasible mechanically. The following table gives values of KAN2 which are

4

reasonably representative of the current stress limits for axial turbomachines:

MACHINE TYPE KAN2

(A in inches2,N in RPM)

KAN2

(A in m2,N in RPM)

Compressor 8-10 x 1010 5.2-6.5 x 107

High-Pressure Turbine (HPT) 4-5 x 1010 2.5-3.2 x 107

Shrouded Low-Pressure Turbine (LPT) 6-8 x 1010 3.8-5.2 x 107

Unshrouded LPT 8-10 x 1010 5.2-6.5 x 107