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Department of Mechanical and Aerospace Engineering

CARLETON UNIVERSITY

MECH 5401

TURBOMACHINERY

SUPPLEMENTARY COURSE NOTES

S.A. Sjolander

January 2010

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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. Euler pump 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. Sjolander January 2010

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CARLETON UNIVERSITY

MECH 4305 - Fluid Machinery

TABLE OF CONTENTS

Page

1.0 INTRODUCTION

1.1 Course Objectives

1.2 Positive-Displacement Machines vs Turbomachines

1.3 Types of Turbomachines

2.0 NON-DIMENSIONAL PARAMETERS AND SIMILARITY

2.1 Dimensional Analysis - Review

2.2 Application to Turbomachinery

2.2.1 Non-Dimensional Parameters for Incompressible-Flow Machines

2.2.2 Effect of Reynolds Number

2.2.3 Performance Curves for Incompressible-Flow Turbomachines

2.2.4 Non-Dimensional Parameters for Compressible Flow Machines

2.2.5 Performance Curves for Compressible-Flow Turbomachines

2.3 Load Line and Operating Point

2.4 Classification of Turbomachines - Specific Speed

2.5 Selection of Machine for a Given Application - Specific Size

2.6 Cavitation

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS AND

THERMODYNAMICS

3.1 Steady-Flow Energy Equation

3.2 Angular Momentum Equation

3.3 Euler Pump and Turbine Equation

3.4 Components of Energy Transfer

3.5 Velocity Diagrams and Stage Performance Parameters

3.5.1 Simple Velocity Diagrams for Axial Stages

3.5.2 Degree of Reaction3.5.3 de Haller Number

3.5.4 Work Coefficient

3.5.5 Flow Coefficient

3.5.6 Choice of Stage Performance Parameters for Design

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3.6 Efficiency of Turbomachines

3.6.1 Incompressible-Flow Machines

3.6.2 Compressible-Flow Machines

4.0 AXIAL-FLOW COMPRESSORS, FANS AND PUMPS

4.1 Introduction

4.2 Control Volume Analysis for Axial-Compressor Blade Section

4.2.1 Force Components

4.2.2 Circulation

4.3 Idealized Stage Geometry and Aerodynamic Performance

4.3.1 Meanline Analysis

4.3.2 Blade Geometries Based on Euler Approximation

4.3.3 Off-Design Performance of the Stage

4.3.4 Spanwise Blade Geometry

4.4 Choice of Solidity - Blade Loading Limits

4.5 Empirical Performance Predictions

4.5.1 Introduction

4.5.2 Blade Design and Analysis Using Howell’s Correlations

4.5.3 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 Components

4.6.2 Loss Estimation Using Howell’s Correlations4.6.3 Loss Estimation Using NASA SP-36 Correlations

4.6.4 Effects of Incidence and Compressibility

4.6.5 Relationship Between Losses and Efficiency

4.7 Compressor Stall and Surge

4.7.1 Blade Stall and Rotating Stall

4.7.2 Surge

4.8 Aerodynamic Behaviour of Multi-Stage Axial Compressors

4.9 Analysis and Design of Low-Solidity Stages - Blade-Element Methods

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5.0 AXIAL-FLOW TURBINES

5.1 Introduction

5.2 Idealized Stage Geometry and Aerodynamic Performance


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

5.3.2.1 Zweifel Coefficient

5.3.2.2 Ainley & Mathieson Correlation

5.3.3 Losses

6.0 CENTRIFUGAL COMPRESSORS, FANS AND PUMPS

6.1 Introduction

6.2 Idealized Stage Characteristics


6.3.1 Rotor Speed and Tip Diameter 6.3.2 Rotor Inlet Geometry

6.3.3 Rotor Outlet Width

6.3.4 Rotor Outlet Metal Angle - Slip

6.3.5 Choice of Number of Vanes - Vane Loading

6.3.6 Losses

7.0 STATIC AND DYNAMIC STABILITY OF COMPRESSION SYSTEMS

7.1 Introduction

7.2 Static Stability

7.3 Dynamic Stability - Surge

Appendix A: Curve and Surface Fits for Howell’s Correlations for Axial Compressor Blades

Appendix B: C4 Compressor Blade Profiles

Appendix C: Curve and Surface Fits for NASA SP-36 Correlations for Axial Compressor Blades

Appendix D: NACA 65-Series Compressor Blade Profiles

Appendix E: Curve and Surface Fits for Kacker & Okapuu Loss System for Axial Turbines

Appendix F: Centrifugal Stresses in Axial Turbomachinery Blades

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CARLETON 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 of

turbomachines which are used most often in these engines: axial and centrifugal compressors and

axial 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 University

Press, 1994.

A recent book published for use with a short course offered by Concepts-NREC, a company in

Vermont which develops courses on various turbomachinery topics for industry. Reasonably

good. One of the few books on turbomachinery fluid mechanics which also addresses mechanical

design 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 Penn

State University. The emphasis is on more advanced topics, particularly computational

techniques. Brief and somewhat weak on fundamentals and the concepts used in preliminary

design. For these reasons, not well suited as a companion to this course. However, someone

continuing in turbomachinery aerodynamic design will probably want to have a copy of the book

in his/her personal library.

Additional Readings

The Library has a number of older textbooks on turbomachinery in which you may find material

of interest: see for example the books by Vavra, Csanady and Balje. The following books are ones I have

found 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 it

covers.

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The following two, relatively short books were written by the man who subsequently helped to

found the Whittle Turbomachinery Laboratory at Cambridge University. He spent a number of

years 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 engineering

information. They remain in-print thanks to an American publisher who specializes in reprinting

classic 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, (reprinted

by Krieger).

The next book is by a more recent Director of the Whittle Laboratory. In the Preface he explicitly

disclaims any intention to present design information. However, it presents a detailed, relatively

up-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 published

published by Longman, like Cumpsty and Cohen, Rodgers & Saravanamuttoo. It is the least

satisfactory of the three, and is apparently going out of print. Nevertheless, worth being aware of

since 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 reflect

largely British practice. The North American approach to axial compressor design was developed

by 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 courses

on many aerospace engineering topics, including turbomachinery aerodynamics. The following

are 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.

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NOMENCLATURE FOR TURBOMACHINES

ENERGY TRANSFER TO THE FLUID

ENERGY TRANSFER FROM THE FLUID

Fans

Blowers

Turbines

Turbo-expanders

Wind mills/Wind turbines

Gases

Liquids

Incompressible flow

Compressible flow

Both

Compressors

Propellers

Pumps

Gases

Both

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lift force

drag force

L

D

⎛

⎝ ⎜

⎞

⎠⎟ =

⎛

⎝ ⎜

⎞

⎠⎟

model prototype

P R T 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 also

be 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 for

the two flows. This includes geometric ratios, which are needed for “geometric similarity”. For example, if

the 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 ratios

will 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 flow

is compressible and the working fluid is a perfect gas.

For a particular airfoil shape, the flow is completely determined by:

c - chord

α - angle of attack U - freestream velocity

ρ - fluid density

μ - fluid viscosity

R - gas constant

a - speed of sound

γ - 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 our

particular 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

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ρ μ

ρ

μ × × × =

× × ×

1

3

U c U c

M

L

LT

M

L

T L

U

a Mach number M = ,

( ) DU c

DU c

or D

U c

C

M L

T

L

M

T

L L L

D× × ××

= ≡

× × ×

1 1 11 1

2

1

2 22

2

3 2

2

ρ ρ ρ

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

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

If n = no. of independent physical variables

r = no. of basic dimensions (eg. Mass, Length, Time, Temp. (θ), etc.)

Then (n - r) criteria of similarity exist

eg. for the airfoil n = 8

r = 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 of

similarity

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 between

two 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 M D = Re, , ,α γ

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C

C

L

D

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., to

form other valid non-dimensional ratios. This does not provide any new information, simply a rearrangement

of known information. However, the resulting ratios may be useful alternative ways of looking at the

information. 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 density

μ - fluid viscosity

N (or ω) - machine speed; revs or rads per unit time

Q - volume flow rate through the machine

Note that the choice of independent variables is somewhat arbitrary. One way to visualize what are

possible independent variables and what are dependent variables is to imagine a test being conducted on the

machine in the laboratory. The variables which, when set, fully determine the operating point of the machine

is 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, the

head or pressure rise produced or power absorbed are then dependent functions of the characteristics of the

machine. Alternatively, if the throttling valve is adjusted to produce a particular pressure rise, then we lose

control over the flow rate and it becomes a dependent variable. The independent variables listed above are the

most common choices for incompressible flow machines that raise the pressure of the fluid. All other

variables are then dependent. For example

W&

Q

Q

D

N

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ΔH - total head rise across machine (or sometimes, total pressure rise)

- shaft power absorbed by the machine&W T - torque absorbed by the machine

η - 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 number ρ

μ

ρ

μ

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 & ,W

N D f

Q

N D

N D

ρ

ρ

μ 3 5 3

2

= ⎛ ⎝ ⎜ ⎞

⎠⎟

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

( )

( )

Δ H f D N Q

W f D N Q etc

=

=

1

2

, , , ,

& , , , , .

ρ μ

ρ μ

Q N D

Q

N D

L

T

T

L

× × =

× ×

1 1

1

1

3 3

3

3

& &

W N D

W

N D

M L

T

T

T L

M L

× × × =

⎛

⎝ ⎜

⎞

⎠⎟

× × ×

1 1 1

1

1

3 5 3 5

2

2 3 3

5

ρ ρ

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

H P

g

V

g z

statichead dynamic head elevation head

= + +

= + +

ρ

2

2

and H can be interpreted physically as the mechanical energy content per unit weight. However, the energy

content is more commonly expressed on a per unit mass basis:

g H 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 a

dependent 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 take

different 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 and blowers, the efficiency is usually defined as:

η pump

useful power transferred to fluid

input power

fluid power

shaft power = =

g H N D

g H

N D

LT

L T L

Δ Δ

× × =

× ×

1 1

11

2 2 2 2

2

2

2

g H

N Dor

g H

N D f

Q

N D

N DΔ2 2 2 2 3

2⎛

⎝ ⎜

⎞

⎠⎟ =

⎛

⎝ ⎜

⎞

⎠⎟,

ρ

μ

g H

N D

g H

N D

P

N D

Δ Δ Δ2 2 2 2

0

2 2= =

ρ

ρ ρ

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and

fluid power mass flow rate mechanical energy change per unit mass

m g H

Qg H

= ×

= ×

=

& Δ

Δ ρ

Thus

Similarly, for turbines:

η ρ

ρ

pump

Qg H

W

Q

N D

g H

N D

W

N D

Flow Coefficient Head Coefficient

Power Coefficient

=

=

⎛

⎝ ⎜

⎞

⎠⎟ ⎛

⎝ ⎜

⎞

⎠⎟

⎛

⎝ ⎜

⎞

⎠⎟

= ×

Δ

Δ

&

&

3 2 2

3 5

η ρ

turb

shaft power

fluid power

W

Qg H

Power Coefficient

Flow Coefficient Head Coefficient

= =

=×

&

Δ

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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 to

turbulent mixing. Viscosity has a minor direct effect and losses tend to vary slowly with Re: recall from the

Moody chart that in pipe flow the friction factor varies much more slowly with Re for turbulent flow than for

laminar flow. Thus, if the Reynolds numbers are high and the differences in Re are not too large between the

machines being compared, Re is often neglected as a criterion of similarity. We can then use, as an

approximation

Where Re variations can not be neglected, a number of empirical relations have been proposed for

correcting 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 and

Reynolds number level. For example, the ASME Power Test Code (PTC-10, 1965) suggests the following

values:

n = 0.1 for centrifugal compressors

n = 0.2 for axial compressors

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

with increasing Re.

g H

N D

W

N Detc fns

Q

N D

N D

fns Q

N D

∆

2 2 3 5 3

2

3

,

, , . ,

, Re

ρ η

ρ

µ =

=

g H

N D

W

N Detc fns

Q

N Donly

∆

2 2 3 5 3,

, , . ρ

η =

1

1

−

−=

η

η

P

M

M

P

n

Re

Re(1)

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Reynolds Number Based on Blade Chord

Taken from: AGARD-LS-167

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2.2.3 Performance Curves for Incompressible-Flow Turbomachines

Relationships such as

(neglecting Re) g H

N D

f Q

N D

∆

2 2 3=

imply that if we test a family of geometrically-similar, incompressible-flow machines (different sizes, different

speeds etc.), the resulting data will fall on a single line if expressed in non-dimensional form. For example, the

non-dimensional coefficients for a pump of fan might appear as follows (we will discuss later why the curves

will have the particular trends shown):

The thick curves are used to suggest variations which could be due to the neglected Re effects, and perhaps

some secondary effects which were not included in the original list of independent parameters (e.g. mild

compressibility effects for a fan or blower). The dashed line indicates the likely "design point": the preferred

operating point, since the efficiency is best there.

Because of the universality of the performance curves, the tests could be conducted for a single

machine 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. #1

Q 1 for the form of non-dimensional parameters which are often used for hydraulic turbines.

C o e f f i c i e n t s

Likely "Design Point"

3D N

Q

22D N

H g

53D N

W

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2.2.4 Non-Dimensional Parameters for Compressible-Flow Turbomachines

We now develop the criteria of similarity for compressible-flow turbomachines. Assuming the

working fluid is a perfect gas, a suitable list of independent variables which control performance is as follows:

N D m

a

or

T

P

or R, , & , , , , ,

01

01

01

01

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ρ

μ γ

where

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

(stagnation speed of sound)a RT 01 01= γ

ˆ could use T01 rather than a01

(perfect gas)P RT 01 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) ND

a01

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

D a ρ 01

2

01

(3) or we could use againμ D

m& Re=

1 ρ

μ

01

2 N D

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

C

p

v

All other performance coefficients are then functions of these four coefficients (as always, geometrical

similarity is assumed).

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Dependent performance coefficients:

The main change from incompressible-flow machines is in the form of the pressure change

coefficient. Instead of the head or total pressure coefficient, we conventionally use the pressure ratio:

P

PP machine outlet total pressure02

0102

=

Then

P

P

W

N Detc fns

ND

a

m

a D

02

01 01

3 5

01 01 01

2,

&

, , . , &

, Re, ρ

η ρ

γ = ⎛

⎝ ⎜

⎞

⎠⎟ (1)

The form of the independent coefficients used here is very general. The main assumption that has

been made is that the working fluid is a perfect gas. We can make use of some of the perfect gas expressions

to rewrite the independent parameters in a somewhat more convenient form:

(1) Speed coefficient:

ND

a

ND

RT

N

T

D

R01 01 01

= =γ γ

(2) Flow coefficient:

& & &m

a D

m

P

RT RT D

m T

P

R

D ρ γ

γ 01 01

201

01

01

2

01

01

2

1= =

Then (1) can be written

P

P

W

N Detc fns

N

T

D

R

m T

P

R

D

02

01 01

3 5

01

01

01

2

1,

&

, , . ,&

, Re, ρ

η

γ γ γ =

⎛

⎝ ⎜⎜

⎞

⎠⎟⎟ (2)

This is the form of the parameters that is appropriate for the most general case, where we are relating the

performance of geometrically-similar, compressible-flow turbomachines of different sizes and operating with

different 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 the

calculation of the coefficients. If we then omit the known, constant fluid properties we can write:

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P

P

W

N Detc fns

N D

T

m T

P D

02

01 01

3 5

01

01

01

2,

&

, , . ,&

, Re ρ

η = ⎛

⎝ ⎜⎜

⎞

⎠⎟⎟ (3)

This form of the coefficients is suitable for relating geometrically-similar machines with different sizes but

with the same working fluid. Note that by assuming the same working fluid, we have reduced the number of criteria of similarity by one. The main disadvantage to this form of the coefficients is that the speed and flow

coefficients 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 operating

at different inlet conditions, then D is a known constant and is often omitted:

P

P

W

N Detc fns

N

T

m T

P

02

01 01

3 5

01

01

01

,&

, , . ,&

, Re ρ

η = ⎛

⎝ ⎜⎜

⎞

⎠⎟⎟ (4)

This is the form of the independent coefficients typically used to present the performance characteristics of the

compressors and turbines for gas turbine engines.

As with incompressible-flow machines, it is sometimes possible to neglect Re as a criterion of

similarity (by the same arguments used in Section 2.2.2). Note that the speed and flow coefficients are again

dimensional.

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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 curves

will take the form:

P

P f

ND

T

m T

P Detc02

01

1

01

01

01

2=

,

.

Thus, whereas our performance tests for the incompressible-flow machines led to a single curve for each

dependent 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 look

as 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 be

neglected, then we would have to generate a series of such graphs, each one containing the performance data

for a different value of the reference Re.

01

02

P

P

01T

D N

01T

D N

2

01

01

D P

T m

INCREASING

CHOKING

SURGE LINE

(UPPER LIMIT OF

STABLE OPERATION)

LINE OF CONSTANT

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(b) Turbine Characteristic:

In a gas turbine engine, the pressure ratio developed by the compressor is applied across the turbine at

the hot end of the engine. The mass flow rate swallowed by the turbine and its power output are then

dependent functions of the turbine characteristics. That is, as far as the turbine is concerned the pressure ratio

is imposed and is effectively an independent parameter. When presenting performance data, we generally plot

independent parameters on the “x axis” and dependent parameters on the “y axis”, as was done on the

compressor 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

D P

T m

01

02

P

P

STATORS CHOKED

LINES OF CONSTANT

01T

D N

01T

D N CONSTANT

2

01

01

D P

T m

01

02

P

P

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NASA 8-Stage Research Axial Compressor

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2.3 LOAD LINE AND OPERATING POINT

The performance diagrams discussed in the earlier sections present a wide range of conditions at

which the machine can operate. For example, the compressor in the last section can operate stably at any point

to the right of the surge line. The precise point at which a turbomachine actually operates depends on the load

to 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 Δ H H machine load = Δ ΔP Pmachine load 0 0, ,=

Thus, the operating point is where the machine and load , or , characteristicsΔ H 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 other

components 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 by

compressor/turbine matching conditions (a propulsion nozzle will also influence operating points - see

Saravanamuttoo et al., Ch. 8 & 9).

H

Q

N1

N2

N3

LOAD LINE

PUMP CHARACTERISTICS

AT CONSTANT SPEED

A

B

C

COMPRESSOR

COMBUSTOR

TURBINE

fuelm&

Cm&

Tm&

outW&

CW&

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

& & &

& & &

m m m

N 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 elevation

between reservoir and tailwater pond (minus any losses in the penstock). Since

&W gQ H T T = η ρ Δ

to produce varying power (according to electrical demand), it is necessary to vary the equilibrium Q, at fixed

ΔH. Furthermore, since the electricity must be generated at fixed frequency, we do not have the option of

varying 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.

H

Q

1

2

3

CONSTANT SPEED LINES -

SAME SPEED,

DIFFERENT BLADE SETTINGS

LOAD LINE

NEGLECTING FRICTION

LOAD LINE

INCLUDING FRICTION

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(with N in revs/s in the coeffcients)

g ∆H⋅

N2

D2

Head coefficient:

Q

N D3

⋅

Flow coefficient:

RPM N 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

H e a d C o e f f i c i e n t

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

pump:

m2/s ν 10 6−

:=Viscosity (water):

mL 125:=Pipe length:

(smooth)mmd pipe 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):

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2.4 CLASSIFICATION OF TURBOMACHINES - SPECIFIC SPEED

Neglecting Reynolds number effects, for a given family of geometrically-similar incompressible-flow

turbomachines the efficiency is a function of one criterion of similarity only. Normally we use the flow

coefficient as the independent parameter. That is

η = ⎛

⎝ ⎜

⎞

⎠⎟ f

Q

N Donly

3

Thus, the maximum0 will occur for this family (say family A) at some particular value of Q/ND3. For another family of machines, the maximum0 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 we

knew the value of Q/ND3 that we required in a given application, we would choose the machine that gives the

best value of efficiency at that value of Q/ND3. Unfortunately, this idea presupposes that we know the

diameter of the machine. In general, this will not be the case. We therefore look for an alternative parameter

to Q/ND3 that does not involve the size of the machine to use as a basis for classifying families of

turbomachines.

We can always form valid new non-dimensional parameters by combining existing ones. Combine

the flow and head coefficients to eliminate D:

( )

Q

N D

g H

N D

NQ

g H

3

1

2

2 2

3

4

1

2

3

4

⎛

⎝ ⎜

⎞

⎠⎟

⎛

⎝ ⎜

⎞

⎠⎟

=

∆ ∆

Following convention, we then define

( )

Ω

∆

= ω Q

g H

1

2

3

4

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 of

turbomachines:

3DN

Q

FAMILY A

FAMILY B

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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

FANS, BLOWERS ANDCOMPRESSORS (BRITISH UNITS)

N RPM cfs

ft

S 1 3

4

= Ω = N

S 1

129

PUMPS (AMERICAN

MANUFACTURERS)

N RPM USgpm

ft

S 2 3

4

= Ω = N

S 2

2730

HYDRAULIC TURBINES(BRITISH UNITS)

N RPM HP

ft

S 3 5

4

= Ω = N S 3

42

(IF WORKING FLUID IS WATER)

HYDRAULIC TURBINES(METRIC UNITS)

N RPM metric HP

m

S 4 5

4

= Ω = N

S 4

187

(IF WORKING FLUID IS WATER)

FANS, BLOWERS ANDCOMPRESSORS (METRIC UNITS) N

RPM m s

m

S 5

3

3

4

= Ω = N

S 5

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 the

efficiencies that can be expected. These efficiencies apply for machines that are well-designed, correctly sized

for their applications, and operating at their design points.

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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 H t t = η ρ ∆

we can rewrite the specific speed as

( ) ( )Ω

∆ ∆

= =ω ω

η ρ

Q

g H

W

g H 3

4

5

4

&

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

N N W

H

S =

&

∆5

4

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

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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 design

tools such as computational fluid dynamics. This is illustrated in the following figure which shows the

variation 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 from

Shepherd and concluded that efficiencies had improved noticeably since Shepherd’s time. They also projected

that there would be further improvements by 2000, as shown in the figure.

Specific Speed, NS

E f f i c i e n c y ,

η

101

102

1030.4

0.5

0.6

0.7

0.8

0.9

1

Shepherd (1956): 1948 Data

Japikse & Baines (1994): 1990 Data

Japikse & Baines (1994): 2000 Projected

Positive-DisplacementMachines

CentrifugalMachines

Axial-F lowMachines

10 20 40 60 80 100 200 400 600 1000

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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 fact

that, 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 stress

considerations). Often the selection process will involve varying the speed to get a specific speed which

results in good efficiency.

From the duty, one can work out the specific speed and then use the figures in Sec. 2.4 to select an

appropriate type of machine. However, the efficiencies shown on the figures will be achieved only if the

machine 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 and

may 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 using

Q/ND3, we define a new parameter, the "specific size":

( )∆

∆=

D g H

Q

1

4

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 H duty

3

4

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

( ) D g H

Qduty

∆∆

1

4

=

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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 fact

that, 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 stress

considerations). Often the selection process will involve varying the speed to get a specific speed which

results in good efficiency.

From the duty, one can work out the specific speed and then use the figures in Section 2.4 to select an

appropriate type of machine. However, the efficiencies shown on the figures will be achieved only if the

machine 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 layers

will 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 using

Q/ND3, we define a new parameter, the "specific size" ):

( )∆

∆=

D g H

Q

1

4

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 H duty

3

4

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

( ) D g H

Q

duty

∆∆

1

4

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

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Example (Section 2.5):

A small hydraulic turbine is to deliver a power of 1000 kW. The total head available is 6 m. and theturbine 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.

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2.6 CAVITATION

If the local absolute static pressure falls below the vapour pressure of a liquid, it will boil, forming

vapour cavities or bubbles. This is known as cavitation. When the bubbles collapse, brief, very high forces

are created which can cause rapid erosion of metal surfaces. Cavitation will also cause significant

performance deterioration. Thus, cavitation should be avoided.

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

outlet 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

H P

g

V

gabs

abs

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 or

suction side of the machine. hv is the head corresponding to the vapour pressure of the liquid,

hP

gv

vap=

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 the

total head, Habs, but directly related to it. Thus, the onset of cavitation will occur for some critical, positive

value of Hsv.

1

2

01P

Tf

1P

SVgH

2P

2

12

1V

vP

2

22

1V

P

01P

1P

2P

vP

2

221 V

crit icalSVgH

o

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

( )S

Q

gH sv

= ω

3

4

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) while

reducing 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 valve

can 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 avoid

cavitation it has been found that the values of Si are fairly similar:

For pumps: Si . 2.5 - 3.5 N.B.: near the design point

For 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

H

sv crit

Δ

SSi

H

3DN

Q

DATA FOR CONSTANT

INCEPTION

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where is the critical value of : that is, the value at cavitation inception. However, the value of H sv crit H svσ will vary with the details of the design of the machine. This can be illustrated by considering two pump

impellers 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

gH sv

= ω

3

4

it follows that the two machines have the same critical value of S: Si1 = Si2. However, the two rotors do not

have the same value of ΔH. In fact, the larger rotor will produce a significantly larger ΔH because of its

higher tip speed (ΔH varies as (ND)2, as implied by the form of the head coefficient; see also later sections).Thus, at cavitation

σ σ 1

1

1

2

2

2

= > = H

H

H

H

sv crit sv crit , ,

Δ Δ

since ΔH1 < ΔH2. Consequently, the Thoma parameter should be used only within a geometrically-similar

family of machines. For example, a critical value of σ determined from model tests can be used to predict the

conditions 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 normal part of selecting a hydraulic machine for a particular duty.

D1

D2

1

2

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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 flowrate of 18.9 m3/sec and running at 180 RPM. What is the maximum height above the tailwater level that thiturbine can be installed if cavitation is to be avoided? The draft tube is a length of diffusing duct at the exitof 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 largecompared with the draft tube outlet so that the flow is effectively being dumped into a very large reservoir at

the draft tube outlet.

TAIL POND

6m

h3m

DRAFT TUBE OUTLET

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1 2

m& m&

Q&

shaftW&

E dm d Q W E dmshaft 1 2∫ ∫ ∫+ + =& & & &

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

E u P C

gz

thermal mechanical

h C

gz

= + + +⎛

⎝ ⎜

⎞

⎠⎟

+

= + +

ρ

2

2

2

2

(2)

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS

AND 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 be

written

where = mass flow rate of fluid&m

E = energy per unit mass for fluid

= rate of heat transfer to the machine&

Q

= shaft power into the machine&W shaft

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 mass

C = absolute velocity of fluid

C2/2 = kinetic energy per unit mass

gz = potential energy per unit mass

h = P/ρ + u = enthalpy per unit mass

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( )

& &

&

W m h C

h C

m h h

shaft = +⎛

⎝ ⎜

⎞

⎠⎟ − +

⎛

⎝ ⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −

2

2

2

1

1

2

02 01

2 2

(3a)

&

&

W

m

or

Q

u P C

gz u P C

gzshaft =

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ + +⎛

⎝ ⎜

⎞

⎠⎟ − + + +

⎛

⎝ ⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ρ ρ ρ

2

2 2

2

2 1

1 1

2

12 2

(4)

u u

g H total head loss due to friction inside the machine L

2 1− = = " "

( )&W Q g H H H

Q g H Q g H

shaft L

L

= − +

= +

ρ

ρ ρ

2 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 frictional

effects 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 P

g

C

g 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 is

the 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

0

1

(3b)

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We defined the efficiency of a pump or blower as

η pump

fluid power

shaft power =

thenη

ρ

ρ ρ pump

L

L

Qg H

Qg H QgH

H

H

=+

=+

ΔΔ

Δ

1

1

(6)

As shown later, we have ways to estimate the various contributions to HL (eg. frictional losses at the walls vary

as 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 H shaft L= − ρ ρ Δ

since the friction inside the machine now reduces the shaft power output compared with the fluid power

released by the fluid, as given by ρQgΔH. We then define turbine efficiency

η turbine

shaft power out

fluid power =

Efficiency is discussed further in Section 3.6.

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T r C dm r C dmout in

0 = × − ×

T rC dm rC dmw

out

w

in

= − (7)

( ) ( )T m rC m rC w out w in= −

3.2 ANGULAR-MOMENTUM EQUATION

The energy transfer between the fluid and the machine occurs by tangential forces exerted on the fluid

as 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):

T orque 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 axis

Cw = tangential component of absolute velocity

Or using mean values

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3.3 EULER PUMP AND TURBINE EQUATION

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

C = absolute velocity

W = 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 component

w = "whirl" (circumferential or tangential) component (subscripts t and θ also used)

Angles:

α = absolute velocity

αN = stator blade metal angles

β = relative velocity

βN = rotor blade metal angles

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

machines.

Sign conventions: The question of signs only arises with reference to velocity components and

angles 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 −)

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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 the

fluid in the form of a tangential force. The corresponding shaft power is

&W T shaft = 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 ω 12

12

(8)

where U = r ω is the blade speed.

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

h dm h dm UC dm UC dmw w02

0

1 2 1

& & & &∫ ∫ ∫ ∫− = − (9)

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

h h U C U C w w02 0 1 2 2 1 1− = − (10)

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

( )g H H H U C U C L 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 C E w wΔ = −2 2 1 1 (11)

Eqns. 9-11 are versions of the famous Euler Pump and Turbine Equation (or Euler Equation). The

Euler 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 produce

to achieve the required performance. As described later, these changes in flow velocity are directly related to

the 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

Δ Δ

H H

E

pump

=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 its

efficiency. As a result, the design process becomes iterative.

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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 (a

similar diagram could be drawn for station 1)

From the Euler Equation

&

&

W

m g H h U C U C

shaft

E 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 C a w r 22

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 C

a w r

a w r

22

22

22

22

22

2 2

2

22

= + +

= + − +(b)

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

2 and equate

C C W U U C C w w w22

22

22

22

2 2 222− = − + −

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Then

( )U C C U W w2 2 22 22 221

2= + −

Similarly for the velocity triangles at the inlet, station1,

( )U C C U W w1 1 12 12 121

2= + −

Substituting into (12)

( ) ( ) ( )( )&

&

( ) ( ) ( )

W

mg H h C C U U W W

shaft

E = = = − + − + −Δ Δ 0 22

12

22

12

12

221

2

1 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

22

12C C −

blower or compressor, the kinetic energy of the fluid normally increases across the rotor. Some of this kinetic

energy 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 of

the rotor again, assuming adiabatic flow and neglecting potential energy changes:

&

&

&m

P C

u W m

P C

ushaft 1 1

2

12 2

2

22 2 ρ ρ + +

⎛

⎝ ⎜

⎞

⎠⎟ + = + +

⎛

⎝ ⎜

⎞

⎠⎟

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

( ) ( ) ( )P P U U W W u u2 1 22 12 12 22 2 11

2

1

2− = − + − − − ρ ρ ρ (14)

Equation (14) shows that there is some direct compression (or expansion) work done inside the rotor blade

passage and it is associated with the changes in U and W that the fluid experiences as it passes through the

rotor. Note that if there is friction present, u2 > u1, and this reduces the pressure rise that would be achieved by

a compression machine, as one would expect.

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

22

12

U U −

expansion) of the fluid as it passes through the rotor ("centrifugal energy" change). The rotation of the fluid

imposed 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 no

flow (say that a valve has been closed in the discharge duct). The fluid particles trapped inside the rotor travel

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2

1

F

(+)

(-)

U

W1

W2

in circular paths. The force required to give the corresponding acceleration towards the axis of rotation is

supplied 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 U 2 1 22 121

2− = − ρ

Thus, a radial machine will produce a pressure rise even for no flow. The delivery pressure for this case is

sometimes known as the “shut-off head”.

When there is flow, the fluid particles that move through the radial pressure field will likewise be

compressed (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

12

22W 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 W 2 1 12 221

2− = − ρ (15)

As shown in the sketch, a typical compressor rotor passage increases in

cross-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 corresponding

pressure rise. The passage is thus a diffuser. The forces exerted on the

fluid by the blade surfaces cause the static pressure to rise between inlet

and outlet, and since there is also displacement associated with theseforces (since the rotor is moving) work is being done on the fluid.

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Note that the pressure rise along the rotor blade passage can cause separation of the blade boundary

layers and therefore stalling of the airfoils. We therefore find it necessary to limit the change in W that we

permit 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, for

axial rotors the centrifugal compression tends to be small (since U1 – U2 for every streamtube that passes

through the rotor), whereas it is large in radial rotors.

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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 static pressure rise by reducing the fluid velocity.

• A turbine stage normally has a row of stators ("inlet guide vanes" or "nozzles") followed by a

rotor. The nozzles impart swirl to the flow, accelerating it and thus causing a static pressure

drop. The rotor then extracts energy from the fluid by removing the swirl. This may be

accompanied 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:

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Note that the inlet flow has been assumed to have some swirl (α1 … 0.0). Therefore, there must be

another stage or a set of inlet guide vanes ahead of the present stage. The stators have also been shaped to

give a stage outlet flow vector equal to the inlet vector (C3 = C1). This is sometimes referred to as a “normal

stage”.

Even for an axial stage, as the flow passes through the stage, the streamtube may vary slightly in

radius. Thus, in general U1 … U2. Also, due to the density changes and changes in the cross-sectional area of the annulus, the axial velocity at different locations may vary (Ca1 … Ca2). However, across a given axial rotor

blade, the radial shift in any given streamline tends to be quite small. For reasons discussed later, it is also

undesirable to have the axial velocity change significantly along the machine. The latter is the reason for the

tapering 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 axial

stages:

(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 velocity

triangles). 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 simplifying

assumptions and therefore the conclusions which we will draw about the stage behaviour, based on the simple

velocity triangles, will be quite realistic.

One convenient feature of the simple velocity triangles is that we can combine the inlet and outlet

triangles because of the common blade speed vector U. We can therefore draw the velocity triangles for the

axial compressor stage as follows:

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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 of

the boundary layers on the walls separating. When this happens on a turbomachinery blade, there is generally

a large reduction in the efficiency of the machine and an impairment of its ability to transfer energy to or from

the fluid. In the case of compressors, boundary layer separation can lead to the very serious phenomena of

stall and surge which will be discussed later.

Diffusion is present most obviously in compressors since they are specifically intended to raise the

pressure of the fluid. While overall the pressure drops through a turbine stage, diffusion may still be present

locally on the blade surfaces. Thus, the possibility of boundary layer separation is a concern in the design of

both compressors and turbines.

As evident from the velocity triangles, pressure rise can occur in both blade rows of a compressor

stage. 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 in

pressure rise or drop between the two blade rows is one of the considerations for the designer of a

turbomachinery stage.

We define the degree of reaction,Λ

( ) ( )[ ]( )

Λ =

=

− + −

−

rate of energy transfer by pressure change inside the rotor

total rate of energy transfer

U U W W

h h

1

2 2

212

12

22

02 01

(16)

which can also be written

Λ =−

−

h h

h 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 the

same (ie. if is a “normal stage”), (17) reduces to

Λ Δ

Δ=

P

P

rotor

stage

(18)

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

across the rotor.

A well-designed pump, fan or compressor will then have Λ > 0 in order to spread the diffusion

between 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 withΛ = 0 is known as an impulse machine. Impulse wheels are

sometimes used for axial turbines, particularly steam turbines.

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The effect of the choice of Λ on the machine geometry can be seen by examining the velocity

diagrams for a few examples.

Axial-Flow Impulse Turbine (Λ = 0):

Consider the mean radius. Assume incompressible flow, constant annulus area and no radial shift in

the streamlines. Thus U1 = U2 = U and from continuity, Ca0 = Ca1 = Ca2 since . We therefore&m C Aa annulus= ρ

have 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 in

here (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 is

obtained with β1N = β2N. Therefore, the impulse turbine will have equal inlet andoutlet metal angles.

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

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

Redraw the velocity triangles with the common blade speeds U superimposed. Note that ΔCw = Cw2 - Cw1 will

be negative, consistent with our sign convention that power in is

positive. The magnitude ofΔCw (for a given U) is clearly related to

α1. Thus, the required plays a direct role in determining the&W

velocity triangles, and ultimately the metal angles.

Note also that to sketch the blade shapes we assumed that

the fluid leaves a blade row at the metal angle:

α α β β 1 1 2 2= ′ = ′,

This is not strictly true, as will be discussed later, but is often a

reasonable first approximation. It is sometimes known as the "Euler

Approximation".

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 in

the rotor, need W2 > W1 and thus *β2N* > *β1N*. An example of the geometry of a reaction turbine is then asfollows:

U

Cw1

(+)

Cw2

(+)

CwC

1

Ca1

= Ca2

1 (+)

C2

W1

W2

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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 this

requirement:

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3.5.3 de Haller Number

The importance of diffusion in compressor blade rows was discussed in Section 3.5.2. By selecting a

degree 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 of

diffusion 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 can

be 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 various

compressor designs of the time, de Haller in the mid 1950’s suggested that the maximum static pressure rise

which could be achieved in axial compressor blade passages is given by

C P

V p,max .= =

Δ

1

2

0 442

ρ (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 no

centrifugal compression) and neglecting friction, from Section 3.4 the static pressure rise is

P P W W 2 1 12

221

2

1

2− = − ρ ρ .

Substituting into (a) and simplifying,

W

W

2

1

0 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 the

blade passage design which can improve the diffusion capability of the passage. Successful modern

compressor designs have used values of the de Haller number as low as 0.65. The de Haller number should be

used mainly to alert the designer to the fact that the level of diffusion in a particular compressor blade row

may present a design challenge.

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3.5.4 Work Coefficient

From the Euler Equation

( )

∆

∆

h U C U C

UC

w w

w

0 2 2 1 1= −

=

and for an axial machine with simple velocity triangles (so that U 1 U2 = U)

∆ ∆h U C w0 = .

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

∆C U w ∝

and ∆h U 0

2∝ .

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 = = =

∆ ∆ ∆h

U

UC

U

g H

U

w E 0

2 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 U 1 U2 = U)

ψ = =U C

U

C

U

w w∆ ∆

2

Normally,

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