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Department of Mechanical and Aerospace Engineering CARLETON UNIVERSITY

MECH 5401 TURBOMACHINERY

SUPPLEMENTARY COURSE NOTES

S.A. Sjolander January 2010

CARLETON UNIVERSITY Department of Mechanical & Aerospace Engineering MECH 5401 - Turbomachinery COURSE 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. Blade loading and choice of solidity. Degree of reaction.

4

Correlations for performance estimation at the design point for: axial compressors, axial turbines and centrifugal compressors. Approximate off-design performance: compressor maps and turbine characteristics.

5

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

6

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

7

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

8

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

9

Compressible flow effects: choking in turbomachinery blade rows; shock waves in transonic compressors 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 wakeblade interaction. Approximate analysis of unsteady behaviour of compression systems: dynamic system instability (surge); factors affecting compressor surge.

11

Current issues in turbomachinery aerodynamics. Very high loading for weight and blade-count reduction. 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. Multidisciplinary optimization.

S.A. Sjolander January 2010

Department of Mechanical and Aerospace Engineering CARLETON UNIVERSITY MECH 4305 - Fluid Machinery TABLE OF CONTENTS Page 1.0

INTRODUCTION 1.1 1.2 1.3

2.0

Course Objectives Positive-Displacement Machines vs Turbomachines Types of Turbomachines

NON-DIMENSIONAL PARAMETERS AND SIMILARITY 2.1 2.2

Dimensional Analysis - Review Application to Turbomachinery 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5

2.3 2.4 2.5 2.6

3.0

Non-Dimensional Parameters for Incompressible-Flow Machines Effect of Reynolds Number Performance Curves for Incompressible-Flow Turbomachines Non-Dimensional Parameters for Compressible Flow Machines Performance Curves for Compressible-Flow Turbomachines

Load Line and Operating Point Classification of Turbomachines - Specific Speed Selection of Machine for a Given Application - Specific Size Cavitation

FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS AND THERMODYNAMICS 3.1 3.2 3.3 3.4 3.5

Steady-Flow Energy Equation Angular Momentum Equation Euler Pump and Turbine Equation Components of Energy Transfer Velocity Diagrams and Stage Performance Parameters 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6

Simple Velocity Diagrams for Axial Stages Degree of Reaction de Haller Number Work Coefficient Flow Coefficient Choice of Stage Performance Parameters for Design

3.6

Efficiency of Turbomachines 3.6.1 3.6.2

4.0

Incompressible-Flow Machines Compressible-Flow Machines

AXIAL-FLOW COMPRESSORS, FANS AND PUMPS 4.1 4.2

Introduction Control Volume Analysis for Axial-Compressor Blade Section 4.2.1 4.2.2

4.3

Idealized Stage Geometry and Aerodynamic Performance 4.3.1 4.3.2 4.3.3 4.3.4

4.4 4.5

Blade Passage Flow and Loss Components Loss Estimation Using Howell’s Correlations Loss Estimation Using NASA SP-36 Correlations Effects of Incidence and Compressibility Relationship Between Losses and Efficiency

Compressor Stall and Surge 4.7.1 4.7.2

4.8 4.9

Introduction Blade Design and Analysis Using Howell’s Correlations Blade Design and Analysis Using NASA SP-36 Correlations

Loss Estimation for Axial-Flow Compressors 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5

4.7

Meanline Analysis Blade Geometries Based on Euler Approximation Off-Design Performance of the Stage Spanwise Blade Geometry

Choice of Solidity - Blade Loading Limits Empirical Performance Predictions 4.5.1 4.5.2 4.5.3

4.6

Force Components Circulation

Blade Stall and Rotating Stall Surge

Aerodynamic Behaviour of Multi-Stage Axial Compressors Analysis and Design of Low-Solidity Stages - Blade-Element Methods

5.0

AXIAL-FLOW TURBINES 5.1 5.2 5.3

Introduction Idealized Stage Geometry and Aerodynamic Performance Empirical Performance Predictions 5.3.1 5.3.2

5.3.3 6.0

CENTRIFUGAL COMPRESSORS, FANS AND PUMPS 6.1 6.2 6.3

Introduction Idealized Stage Characteristics Empirical Performance Predictions 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6

7.0

Flow Outlet Angle Choice of Solidity - Blade Loading 5.3.2.1 Zweifel Coefficient 5.3.2.2 Ainley & Mathieson Correlation Losses

Rotor Speed and Tip Diameter Rotor Inlet Geometry Rotor Outlet Width Rotor Outlet Metal Angle - Slip Choice of Number of Vanes - Vane Loading Losses

STATIC AND DYNAMIC STABILITY OF COMPRESSION SYSTEMS 7.1 7.2 7.3

Appendix A: Appendix B: Appendix C: Appendix D: Appendix E: Appendix F:

Introduction Static Stability Dynamic Stability - Surge

Curve and Surface Fits for Howell’s Correlations for Axial Compressor Blades C4 Compressor Blade Profiles Curve and Surface Fits for NASA SP-36 Correlations for Axial Compressor Blades NACA 65-Series Compressor Blade Profiles Curve and Surface Fits for Kacker & Okapuu Loss System for Axial Turbines Centrifugal Stresses in Axial Turbomachinery Blades

Department of Mechanical and Aerospace Engineering CARLETON UNIVERSITY MECH 4305 - Fluid Machinery Recommended Texts S.L. Dixon, Fluid Mechanics, Thermodynamics of Turbomachinery, 5th ed., Elsevier ButterworthHeineman, 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 preliminary aerodynamic 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 while others 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.

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 the famous 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.

NOMENCLATURE FOR TURBOMACHINES ENERGY TRANSFER TO THE FLUID Fans Blowers

Gases Incompressible flow

Pumps

Liquids

Compressors

Gases

Compressible flow

Propellers

Both

Both

ENERGY TRANSFER FROM THE FLUID Turbines Turbo-expanders Wind mills/Wind turbines

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 that around 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

⎛ lift force ⎞ ⎛ L⎞ =⎜ ⎟ ⎟ ⎜ ⎝ drag force ⎠ model ⎝ D ⎠ prototype 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 α γ - specific heat ratio Note that the pressure and temperature are not quoted. U For a perfect gas,

P = ρ RT

a = γ RT

L

D M c

Thus, by specifying a, γ and R, we have implicitly specified T. Similarly, with ρ and R specified, and T 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.

(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 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:

ρ ×

1

× U

× c

μ LT M L × × × L 3 M T L

=

ρU c μ

which is clearly the Reynolds number, Re α

is already non-dimensional and can be used directly as a criterion of similarity is also already non-dimensional

γ

U a

=

Mach number , M

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

×

1 U2

×

1 = c ×1

L L3 × × T2 M

T2 L2

×

1 LL

D ×

M

1

ρ

then C D = f ( Re, M , α , γ )

D ρU 2 c

or

D 1 ρU 2 c 2

(≡ CD )

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

CL CD

=

L D

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 in Section 2.4. D Q

N

W& Q

For a given geometry, the independent variables that determine performance are usually taken as . D ρ μ N (or ω) Q-

machine size (usually rotor outside diameter) fluid density fluid viscosity machine speed; revs or rads per unit time 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 (by controlling 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

ΔH W& T η -

total head rise across machine (or sometimes, total pressure rise) shaft power absorbed by the machine torque absorbed by the machine efficiency of the machine

ΔH = f 1 ( D, N , Q, ρ , μ ) W& = f 2 ( D, N , Q, ρ , μ ) etc. 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:

Q ×

1 N

×

L3 × T

T 1

×

1 D

Q

=

3

N D3

1 L3

This is known as the flow coefficient, capacity coefficient or flow number (2)

Fluid properties (specifically, viscous effects):

ρ N DD = μ

ρ N D2 μ

ie. the Reynolds number

All other non-dimensional ratios or coefficients then depend on these two criteria of similarity. For power coefficient (non-dimensional work per unit time)

W&

×

1 N3

⎛ M L2 ⎞ ⎜ 2 ⎟ ⎝ T ⎠ × T then

W&

ρ N 3 D5

1

×

T3 1 =

ρ

×

L3 M

1 D5

×

×

=

W& ρ N 3 D5

1 L5

⎛ Q ρ N D2 ⎞ ⎟ , f⎜ 3 μ ⎠ ⎝ ND

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

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

P V2 + +z ρ g 2g = static head + dynamic head + elevation head

H=

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:

g ΔH L T

2

1

×

N

2

T2 1

L ×

× ×

1 D

g ΔH

=

2

N 2 D2

1 L2

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:

⎛ gH ⎞ ⎜ or 2 2 ⎟ ⎝ N D ⎠

g ΔH N 2 D2

=

⎛ Q ρ N D2 ⎞ ⎟ , f⎜ 3 μ ⎠ ⎝ ND

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

g ΔH 2

N D

2

=

ρ g ΔH ρ N 2 D2

=

Δ P0

ρ N 2 D2

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

and

fluid power = mass flow rate × mechanical energy change per unit mass

Thus

=

m& ×

=

ρ Qg ΔH

η pump

=

gΔH

ρ Qg ΔH W&

⎛ Q ⎞ ⎛ g ΔH ⎞ ⎜ ⎟⎜ ⎟ ⎝ N D3 ⎠ ⎝ N 2 D2 ⎠ = ⎛ W& ⎞ ⎜ ⎟ ⎝ ρ N 3 D5 ⎠ =

Flow Coefficient × Head Coefficient Power Coefficient

Similarly, for turbines:

η turb

= =

shaft power W& = fluid power ρ Qg ΔH Power Coefficient Flow Coefficient × Head Coefficient

2.2.2

Effect of Reynolds Number We have shown that in general for incompressible flow:

g ∆H W& , , η , etc. = N 2 D2 ρ N 3 D5

æ Q ρ N D2 ö fnsç , ÷ µ ø è N D3 æ Q ö fnsç , Re÷ 3 è ND ø

=

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

g ∆H W& , , η , etc. = N 2 D 2 ρ N 3 D5

ö æ Q fnsç only÷ 3 ø è ND

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

1 − η P æ Re M ö =ç ÷ 1 − η M è Re P ø

n

(1)

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.

Reynolds Number Based on Blade Chord

Taken from: AGARD-LS-167

2.2.3

Performance Curves for Incompressible-Flow Turbomachines Relationships such as g ∆H N 2 D2

=

æ Q ö fç ÷ (neglecting Re) 3 è ND ø

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):

Coefficients

Likely "Design Point"

W& ρN 3 D 5

η g∆H N 2 D2

Q N D3 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.

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:

⎛ a 01 ⎞ ⎛ P01 ⎞ ⎜ ⎟ ⎜ ⎟ N , D, m& , ⎜ or ⎟ , ⎜ or ⎟ , μ , R, γ ⎜ ⎟ ⎜ ⎟ ⎝ T01 ⎠ ⎝ ρ 01 ⎠ where m& = mass flow rate (rather than Q as measure of flow rate)

a 01 = γ RT01

(stagnation speed of sound)

ˆ could use T01 rather than a01 P01 = ρ 01 RT01

(perfect gas)

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)

ND a 01

speed parameter (effectively the tip Mach number)

(2)

m& ρ 01 D 2 a 01

flow parameter (effectively the axial Mach number)

(3)

μD 1 ρ N D2 = or we could use 01 again μ m& Re

(4) γ =

Cp Cv

specific heat ratio (which is already non-dimensional)

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

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:

P02 P01

P02 = machine outlet total pressure

Then

⎛ ND ⎞ m& , , Re, γ ⎟ (1) fns⎜ 2 ⎝ a 01 ρ 01 a 01 D ⎠

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

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 01

=

ND

γ RT01

N

D T01 γ R

=

(2) Flow coefficient:

m& ρ 01 a 01 D 2

=

m& P01 γ RT01 D 2 RT01

=

m& T01 P01

R 1 γ D2

Then (1) can be written

P02 W& , η , etc. = , P01 ρ 01 N 3 D 5

⎛ N fns⎜⎜ ⎝ T01

D

γR

,

m& T01 P01

⎞ R 1 , Re, γ ⎟⎟ 2 γ D ⎠

(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:

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

⎛ N D m& T01 ⎞ fns⎜⎜ , , Re⎟⎟ 2 ⎝ T01 P01 D ⎠

(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:

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

⎛ N m& T01 ⎞ fns⎜⎜ , , Re⎟⎟ P01 ⎝ T01 ⎠

(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.

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: P02 P01

=

æ ND m & T01 ö ÷ , f 1 çç 2 ÷ è T01 P01 D ø

etc.

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")

P02 P01

LINE OF CONSTANT

ND SURGE LINE (UPPER LIMIT OF STABLE OPERATION)

T01

CHOKING

ND INCREASING

T01

& T01 m P01D 2

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.

(b) Turbine Characteristic: P02 P01

STATORS CHOKED

LINES OF CONSTANT

ND T01

& T01 m P01D 2

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:

& T01 m P01D 2

CONSTANT

ND T01

P02 P01

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

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 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)

Qmachine = Qload (or, for compressible flow, m& machine = m& load )

(2)

ΔH machine = ΔH load (or ΔP0,machine = ΔP0,load )

& , characteristics Thus, the operating point is where the machine and load ΔH vs Q , or ΔP0 vs m 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 the system. 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. ΔH

LOAD LINE

C

PUMP CHARACTERISTICS AT CONSTANT SPEED

B N3

A N2 N1

Q

(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). & fuel m COMBUSTOR

W& C COMPRESSOR

&C m

W& out TURBINE

&T m

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

m& T

= m& C + m& fuel

NC W&

=

T

NT = W& C + W& 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& T = η T ρ gQ ΔH 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.

CONSTANT SPEED LINES SAME SPEED, DIFFERENT BLADE SETTINGS

ΔH β1

β2

β3

LOAD LINE NEGLECTING FRICTION

LOAD LINE INCLUDING FRICTION

Q

EXAMPLE (Section 2.3): A pump is connected to the piping system shown. What flow rate of water will be pumped for the two valve settings?

VALVE K = 1, K=10 K = 0.9

K = 0.9 K = 0.9

K = 1 (EXIT LO SS) K = 0.9 6 m.

WATER

K = 0.9 PUMP

K = 0.5 (ENTRY LOSS)

Pipe diameter:

dpipe := 50

Pipe length:

L := 125

Viscosity (water):

ν := 10

−6

mm

(smooth)

m m2/s

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

Pump speed:

Flow coefficient:

4

N := 1750 RPM Q

3 3

N⋅ D Head coefficient:

Pump Characteristics

D := 30 cm

g⋅ ∆H 2 2

N D

Head Coefficient

Pump diameter:

2

(with N in revs/s in the coeffcients) 1

0

0

0.002

0.004 0.006 Flow Coefficient

0.008

0.01

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

FAMILY B FAMILY A

η =

⎛ Q ⎞ only⎟ f⎜ 3 ⎠ ⎝ ND

Q ND3

Thus, the maximum 0 will occur for this family (say family A) at some particular value of Q/ND3. For another family of machines, the maximum 0 might occur at a different value of Q/ND3. We could therefore classify 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: 1

⎛ Q ⎞2 ⎜ ⎟ 3 ⎝ ND ⎠ ⎛ g ∆H ⎞ ⎜ 2 2⎟ ⎝N D ⎠

3 4

1

=

NQ 2 3

( g ∆H ) 4

Following convention, we then define 1

Ω =

ω Q2

( g ∆H )

3 4

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

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 following table can be used to convert between the various definitions used:

AREA OF APPLICATION FANS, BLOWERS AND COMPRESSORS (BRITISH UNITS)

SPECIFIC SPEED

N S1 =

RPM cfs

EQUIVALENT S

Ω=

N S1 129

Ω=

N S2 2730

Ω=

N S3 42

3

ft 4 PUMPS (AMERICAN MANUFACTURERS)

HYDRAULIC TURBINES (BRITISH UNITS)

N S2 =

RPM USgpm ft

N S3 =

RPM HP ft

HYDRAULIC TURBINES (METRIC UNITS)

N S4 =

3 4

5 4

RPM metric HP

N S5 =

Ω=

5

m4 FANS, BLOWERS AND COMPRESSORS (METRIC UNITS)

(IF WORKING FLUID IS WATER)

RPM m 3 s 3

N S4 187

(IF WORKING FLUID IS WATER)

Ω=

N S5 53

m4

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.

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& t = η t ρ Qg ∆H

we can rewrite the specific speed as

Ω=

ω Q

( g ∆H )

3 4

=

ω W& 5

η ρ ( g ∆H ) 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 NS =

N W& 5

∆H 4 The following figure (from Shepherd, 1956) shows the variation of the power specific speed for hydraulic turbines of different geometries.

The plots shown above were based on data that is as much as 50 years old. One might expect that over 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.

1

0.9

Axial-Flow Machines

Efficiency, η

0.8

0.7

0.6

Centrifugal Machines

Positive-Displacement Machines

0.5

0.4 1 10 10

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

20

40

60

2 80 10 100

200

Specific Speed, N S

400

600

1000 103

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 For turbines

N, Q and )H (or )P0) are typically specified.

& and )H (or )P0) are typically specified. N, 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 0.5DV2 (and with gases, 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 0 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" ): 1

∆=

D( g ∆H ) 4 Q

The specific size for a given machine is then the value of ) at which it achieves its best efficiency. The value of ) depends on the machine type (i.e. S) and to some degree on its detailed design. However, in the early 1950s 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

é ω Q Ω = êê 3 êë ( g ∆H ) 4

ù ú ú úû duty

(2) From S, read ) from the Cordier diagram and size the machine such that 1 é 4 D g ∆ H ( ) ê ê Q êë

ù ú =∆ ú úû duty

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 For turbines

N, Q and )H (or )P0) are typically specified.

& and )H (or )P0) are typically specified. N, 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 as 0.5DV2 (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. 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" ): 1

∆=

D( g ∆H ) 4 Q

The specific size for a given machine is then the value of ) at which it achieves its best efficiency. The value of ) depends on the machine type (i.e. S) and to some degree on its detailed design. However, in the early 1950s 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 Q Ω=⎢ 3 ⎢⎣ ( g ∆H ) 4

⎤ ⎥ ⎥ ⎥⎦ duty

(2) From S, read ) from the Cordier diagram and size the machine such that 1 ⎡ ⎢ D( g ∆H ) 4 ⎢ Q ⎢⎣

⎤ ⎥ =∆ ⎥ ⎥⎦ duty

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, 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 sv = H abs − hv

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

⎡P V2⎤ H abs = ⎢ abs + ⎥ 2g ⎦ suction side ⎣ ρg 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,

hv =

Pvap

ρg

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

P01 1 ρ V 12 2

P

P1

P01 1 ρ V 22 2

ρ gH SV

P2

P1

Pv

Pv f (T )

o

1 ρ V 22 2

P2

ρ gH SV

critical

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

S=

ω Q 3

( gH sv ) 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:

ΔH η DATA FOR CONSTANT

Q ND 3

INCEPTION

S

Si

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 lowpressure 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: For turbines:

Si . 2.5 - 3.5 Si . 3.5 - 5.0

N.B.: near the design point

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 sv

crit

ΔH

where H sv crit is the critical value of H sv : that is, the value at cavitation inception. However, the value of σ 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:

2 1 D2 D1

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=

ω Q 3

( gH sv ) 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

σ1 =

H sv

crit ,1

Δ H1

> σ2 =

H sv

crit ,2

ΔH 2

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.

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 this turbine 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 at the draft tube outlet.

6m

3m

h

TAIL POND

DRAFT TUBE OUTLET

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS AND THERMODYNAMICS 3.1 STEADY-FLOW ENERGY EQUATION Consider a control volume containing a turbomachine: 1

2

& m

& m

W& shaft

Q& 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

∫E

1

∫

dm& + d Q& + W& shaft =

∫E

2

dm&

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

& 1 + Q& + W& shaft = mE & 2 mE where

(1)

m& E Q&

= mass flow rate of fluid = energy per unit mass for fluid = rate of heat transfer to the machine

W& shaft

= shaft power into the machine

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

E

⎛ P C2 ⎞ = u + ⎜ + + gz⎟ 2 ⎝ρ ⎠ thermal + mechanical =h+

where

u P/ρ C C2/2 gz h

= = = = = =

C2 + gz 2

internal thermal energy per unit mass (= CvT) flow work (“pressure energy”) per unit mass absolute velocity of fluid kinetic energy per unit mass potential energy per unit mass P/ρ + u = enthalpy per unit mass

(2)

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

⎡⎛ C2 ⎞ ⎛ C2 ⎞⎤ = m& ⎢⎜ h2 + 2 ⎟ − ⎜ h1 + 1 ⎟ ⎥ 2 ⎠ ⎝ 2 ⎠ ⎥⎦ ⎢⎣⎝

W& shaft

= m& (h02 − h01 )

where

(3a)

C2 = stagnation enthalpy 2 = C P T0 for perfect gases

= h+

h0

For general non-uniform flows, we would write

∫

∫

W& shaft = h0 dm& − h0 dm& 2

1

(3b)

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:

W& shaft

⎛ m& ⎞ ⎞ ⎛ ⎞⎤ P C2 P C2 ⎜ ⎟ ⎡⎛ = ⎜ or ⎟ ⎢⎜ u2 + 2 + 2 + gz 2 ⎟ − ⎜ u1 + 1 + 1 + gz1 ⎟ ⎥ ρ ρ 2 2 ⎠ ⎝ ⎠ ⎦⎥ ⎜ ⎟ ⎢⎝ ⎝ ρ Q⎠ ⎣

(4)

It is also common to write

u2 − u1 = H L = " total head loss" due to friction inside the machine g The total head is a measure of the total mechanical energy content of the fluid

H

= total head

=

P C2 + +z ρ g 2g

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

W& shaft

=

ρ Q g ( H 2 − H1 + H L )

=

ρ Q g ΔH + ρ Q g H L

(5)

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”.

We defined the efficiency of a pump or blower as

η pump = then

fluid power shaft power

ρQgΔH ρQgΔH + ρQgH L 1 = H 1+ L ΔH

η pump =

(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& shaft

= ρ Q g ΔH − ρ Q g H L

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.

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):

Torque applied to fluid in CV = outflow of angular momentum - inflow of angular momentum T0

=

ò

out

&− r × C dm

ò

in

& r × C dm

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

T

=

ò rC

out

where

r = Cw =

w

ò

& − rC w dm & dm in

radial distance from the axis tangential component of absolute velocity

Or using mean values T

& (rC w ) out − m & (rC w ) in = m

(7)

3.3

EULER PUMP AND TURBINE EQUATION We will use the following nomenclature in this and the subsequent sections: β (−)

ROTOR U α (+)

β (−) α (+)

C U W

STATORS

β (−)

C W U

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

Subscripts: a r w

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

Angles: α αN β βN

= absolute velocity = stator blade metal angles = relative velocity = 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 the turbomachinery 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.

Consider again the general turbomachinery rotor

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

=

T

∫ rC

w

∫

dm& − rC w dm&

2

(7)

1

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& shaft = T ω and multiplying through by ω in (7)

∫ = ∫ UC

∫ rωC

W& shaft = rω C w dm& − 2

2

& w dm

−

1

∫

& w dm

(8)

UC w dm&

1

where U = rω is the blade speed. But the SFEE also relates the shaft power, W& shaft , to the energy changes in the fluid. Equating the shaft powers from Eqns. (3) and (8)

∫ h dm& − ∫ h dm& = ∫ UC 0

2

0

1

2

&− w dm

∫ UC

& w dm

(9)

1

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

h02 − h01

= U 2 C w2 − U 1C w1

(10)

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

g ( H 2 − H1 + H L ) = U 2 C w 2 − U 2 C w 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 ΔH E = U 2 C w2 − U 1 Cw1

(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

ΔH E =

ΔH

η pump

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.

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

C =W +U 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& shaft m&

= g ΔH E = Δh0 = U 2 C w2 − U 1C w1

(12)

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

C22 = Ca22 + C w2 2 + Cr22

(a)

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

W22 = Wa22 + Ww22 + Wr22 = Ca22 + (U 2 − C w2 ) + Cr22 2

Solve (a) and (b) for Ca22 + Cr22 and equate

C22 − C w2 2 = W22 − U 22 + 2U 2 C w2 − C w2 2

(b)

Then

U 2 C w2 =

(

1 2 C2 + U 22 − W22 2

)

Similarly for the velocity triangles at the inlet, station1,

U 1 C w1 =

(

1 2 C1 + U 12 − W12 2

)

Substituting into (12)

W& shaft m&

= g ΔH E = Δh0 =

((

) (

) (

1 C22 − C12 + U 22 − U 12 + W12 − W22 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).

(

)

1 2 C2 − C12 is clearly the kinetic energy change of the fluid across the rotor. In a pump, 2 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. Term (1),

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:

⎛ P C2 ⎞ m& ⎜ 1 + 1 + u1 ⎟ + W& shaft 2 ⎝ ρ ⎠

⎛P ⎞ C2 = m& ⎜ 2 + 2 + u2 ⎟ 2 ⎝ ρ ⎠

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

P2 − P1

=

(

)

(

)

1 1 ρ U 22 − U 12 + ρ W12 − W22 − ρ (u2 − u1 ) 2 2

(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.

(

)

1 2 U 2 − U 12 is then energy transfer to the fluid due to the centrifugal compression (or 2 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. Term (2),

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

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. (+) F 2

(-)

ω

1

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

P2 − P1

=

(

1 ρ U 22 − U 12 2

)

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).

(

)

1 2 W1 − W22 represents the change in pressure energy due to the change in fluid velocity 2 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) Term (3),

(

1 P2 − P1 = ρ W12 − W22 2

)

(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 these forces (since the rotor is moving) work is being done on the fluid.

W2

W1 U

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 due to 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.

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:

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:

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 1 U 22 − U 12 + W12 − W22 2 (16) (h02 − h01 )

[(

)]

) (

which can also be written

Λ =

h2 − h1 h02 − h01

(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 =

ΔProtor ΔPstage

(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 wheel turbine, P1 = P2 = Patm and Λ = 0. A machine with Λ = 0 is known as an impulse machine. Impulse wheels are sometimes used for axial turbines, particularly steam turbines.

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 m& = ρCa Aannulus . We therefore 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 this will 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 and outlet metal angles.

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

& ΔC w W& = m& (U 2 C w2 − U 1C w1 ) = mU 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 of ΔCw (for a given U) is clearly related to Ca1 = Ca2 α1. Thus, the required W& plays a direct role in determining the velocity triangles, and ultimately the metal angles. C2

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

α 1 = α 1′ , β 2 = β 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".

Cw2 (+)

α1 (+) U

W2

C1

Cw1 (+)

ΔCw

W1

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 as follows:

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:

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 detailed design 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,max =

where ΔP V

ΔP = 0.44 1 2 ρV 2

(a)

= static pressure rise between inlet and outlet of the blade row = 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

1 1 P2 − P1 = ρW12 − ρW22 . 2 2 Substituting into (a) and simplifying,

⎛ W2 ⎞ = 0.75 . ⎜ ⎟ ⎝ W1 ⎠ 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.

3.5.4

Work Coefficient

From the Euler Equation ∆h0 = U 2 C w2 − U 1C w1 = ∆(UCw )

and for an axial machine with simple velocity triangles (so that U1. U2 = U) ∆h0 = U∆C w .

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

and

∆C w

∝ U

∆h0

∝ U2 .

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

ψ =

∆h0 U

2

=

∆(UCw ) U

2

=

g∆H E U2

For “high” R, we are taking full advantage of the blade speed and we have “high stage loading”: we will specify what constitutes “high” R for different types of machines in Section 3.5.6. For a centrifugal machine, tip speed, U2, would be used in R. For an axial machine with simple velocity triangles (so that U1. U2 = U)

ψ =

U∆C w U

2

=

∆Cw U

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

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 Euler equation, Δh0 = UΔCw , and to achieve the same Δh0 , and thus the same pressure rise, they must therefore have 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 is achieved by reducing the cross-sectional area of the machine. This change obviously has a significant effect on the rotor blade geometry. It also has aerodynamic consequences: (i) For rotor B, both the absolute and relative velocities have been increased. Since losses generally vary as 0.5ρV2 (where V = W for the rotor), rotor B will, all other things being equal, have poorer efficiency than rotor A. (ii) All other things are not equal. Note that the increase in Ca in rotor B has had the effect of increasing the de Haller number (W2/W1). Thus, the diffusion has been reduced in rotor B, which is aerodynamically favourable. We can thus identify an additional important parameter which must be chosen by the designer, the flow coefficient, φ:

φ=

Ca U

For a centrifugal compressor, we would use Cr2/U2, where Cr2 is the radial component of velocity at the rotor outlet. Note that for the compressors shown, the change in flow coefficient did not in fact change the degree of reaction. As you will show in Problem Set 3, the symmetry of the velocity triangles for both machines implies 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. Design Parameter

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 1.5 - “Highly Loaded”

Λ

0.3 6 0.7

(Not much used)

061

de Haller

>0.65 (well-designed machines with clean inlet flow) >0.80 (simple design, poor inlet flow uniformity)

See Section 6.4.3

N/A

For compressible-flow axial turbines, Smith ( S.F. Smith, "A Simple Correlation of Turbine Efficiency," J. Royal Aero. Soc., Vol. 49, July 1965, pp. 467-470.) developed a very useful figure (the “Smith chart”) 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 (for both model and full-scale machines) conducted at Rolls-Royce from 1945 to 1965. Over that period, the flow over the tip of the rotor blades ("tip leakage") was considerably reduced. The tip-leakage flow is an important source of losses and as a result there was significant improvement in efficiency. To isolate the influence of the stage loading and shape of the velocity triangles, the efficiencies were corrected back to their zero-clearance equivalents. Thus, efficiencies for actual machines can be expected to be lower than those shown by a couple of percentage points. Note that the degree of reaction is not mentioned on the Smith chart. The turbines used to generate the chart had a range of degrees of reaction. However, the performance of turbines is not strongly dependent 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 preliminary stages of design. The usefulness of the Smith chart makes it surprising that comparable charts are not more widely used by axial and centrifugal compressor designers. Part of the reason lies in the important role played by diffusion (expressed through both the degree of reaction and the de Haller number) in compressor performance. Thus a single “Smith chart” for compressors is not feasible. However, it is possible to generate a small number of charts, each for a different value of degree of reaction say, and then use these in design. In the late 1980's Casey (M.V. Casey, “A Mean Line Prediction Method for Estimating the Performance Characteristics 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 “Smith charts” for axial compressors for three values of degree of reaction: 50, 70 and 90%. Note the rapid deterioration in efficiency when the de Haller number is less than about 0.7.

“Smith” Charts for Axial Compressors: (a) Λ = 0.5, (b) Λ = 0.7, (c) Λ = 0.9. The use of the guidelines presented in this section will be illustrated in the next chapter.

3.6

EFFICIENCY OF TURBOMACHINES 3.6.1 Incompressible-Flow Machines

The definitions of efficiency used for incompressible-flow machines have been discussed briefly in earlier 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 the definitions of efficiency used for incompressible- and compressible-flow machines. These will therefore be clarified now. (a) Pumps, Fans and Blowers

From the steady flow energy equation,

& ΔH E = m& Δh0 W& shaft = mg 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 the machine

& ΔH W& fluid = mg where

ΔH

=

total head rise that is actually observed across the machine.

The Euler head and the actual total head are related by ΔH = ΔH E − H L

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

ΔP0,actual = ρ gΔH ΔP0,ideal = ρ gΔH E We then define the efficiency for a pump, fan or blower as

η pump =

ΔP0,actual Fluid power ρQgΔH ΔH = = = Shaft power ΔH E ΔP0,ideal W& shaft

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

The specification calls for the machine to raise the fluid head by ΔH, or the total pressure by ΔP0,actual = P02 − P01 = ρgΔH . With the same shaft power input per unit mass flow (Δh0), the ideal machine would raise the pressure by ΔP0,ideal = PN02 - P01. Thus, the efficiency for pumps, fans and blowers is defined by comparing the head or total pressure rises for the actual and an ideal machine that have the same shaft power input. As described below, the definition of efficiency for compressors is slightly different.

P02 h0

P02

Δh0

P01

ACTUAL IDEAL

s

(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 and & ΔH E . That is, is therefore not available to be extracted as shaft power output, W& shaft = mg ΔH = ΔH E + H L

The turbine efficiency is then defined as

η turbine =

ΔP0,ideal Shaft power ρQgΔH E ΔH E = = = ρQgΔH ΔH ΔP0,actual Fluid power

The physical interpretation can again be seen in terms of the h0 versus s diagram. The actual pressure drop is ΔP0,actual = P01 - P02 & ΔH E = m& Δh0 . In an and the shaft power extracted is W& shaft = mg ideal machine, a smaller pressure drop, ΔP0,ideal = P01 - PN02, would be needed to produce the same shaft power output. Thus, the turbine efficiency is defined in terms of two machines that have same shaft power output. The comparison is between the head or total pressure drops required to obtain that shaft power output in the ideal and actual machines. The similarity with the definition used for pumps, fans and blowers is evident.

h0 P01

ACTUAL

Δh0

IDEAL

P02 P02

s

3.6.2 Compressible-Flow Machines The efficiency of compressible flow machines is defined slightly differently. The comparison is again between ideal and actual machines. However, instead of the shaft power input or output, the common basis is the pressure rise or drop across the machines. (a) Compressors The h0-s diagram is again used to compare the processes used to define the efficiency. For compressible-flow machines, the pressure rise or drop across the machine is generally expressed in terms of the total pressure ratio. The h0-s diagram shows the ideal and actual compression processes needed to obtain the same pressure ratio, P02/P01. For the ideal machine, the shaft power required is

h0

P02

Δh0,actual P01

ACTUAL

Δh0,ideal IDEAL

W&ideal = m& Δh0, ideal while for the actual machine

s

W& actual = m& Δh0, actual . The compressor efficiency is then defined as the ratio of the shaft powers required to produce the same pressure ratio in the ideal and actual machines:

ηc =

m& Δh0, ideal Δh0, ideal W&ideal = = & Wactual m& Δh0, actual Δh0, actual

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

ηc =

C p ΔT0, ideal C p ΔT0, actual

=

T02′ − T01 T02 − T01

For any isentropic process involving a perfect gas,

P

ργ

T0

P02

T02 T02

= const .

P01

ACTUAL IDEAL

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

T02′ ⎛ P02 ⎞ =⎜ ⎟ T01 ⎝ P01 ⎠

γ −1 γ

T01

s

Then

Δh0, actual =

C p (T02′ − T01 )

ηc

γ −1 ⎛ ⎞ C p T01 ⎜ ⎛ P02 ⎞ γ ⎟ − 1⎟ = ⎟ ⎜⎜ η c ⎜ ⎝ P01 ⎠ ⎟ ⎝ ⎠

& Δh0, actual , to be and this expression allows the shaft power required to drive the actual machine, W& actual = m related to the specified pressure ratio. (b) Turbines The efficiency of compressible-flow turbines is similarly defined by comparing the shaft power produced by the expansion through the same pressure ratio for an ideal and the actual machine. Following the same procedure as for the compressor, we obtain

ηt =

W& actual Δh0, actual T02 − T01 = = T02′ − T01 Δh0, ideal W&ideal

T0

and

P01

T01 ACTUAL

Δh0, actual

γ −1 ⎛ ⎞ ⎜ ⎛ P02 ⎞ γ ⎟ = C p T01η t ⎜ ⎜ − 1⎟ ⎟ ⎜ ⎝ P01 ⎠ ⎟ ⎝ ⎠

IDEAL

P02

T02 T02

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

s

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

PRs = stage total-pressure ratio ηs = stage isentropic efficiency

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

T0 N +1

γ −1 ⎛ ⎞ γ −1 PR ( ) ⎜ ⎟ s = T01 ⎜ 1 + ⎟⎟ ηs ⎜ ⎝ ⎠

The overall pressure ratio for the N stages is

N

PRc = ( PRs )

N

and the isentropic temperature rise for the whole compressor is then γ −1 ⎛ ⎞ ⎜ T0′N +1 − T01 = T01 PRc γ − 1⎟ ⎜ ⎟ ⎝ ⎠

⎛ = T01 ⎜⎜ PRsN ⎝

(

)

γ −1 γ

⎞ − 1⎟⎟ ⎠

and thus the overall isentropic efficiency is

ηc =

T0′N +1 − T01 T0 N +1 − T01

N ( γ −1)

=

PRs

γ

−1

γ −1 γ

N

⎞ ⎛ ⎜ PRs − 1⎟ ⎟ ⎜1 + ηs ⎟ ⎜ ⎠ ⎝

−1

For example, if PRs = 1.2 and ηs = 0.9, the resulting variation of the overall pressure ratio and overall isentropic efficiency with the number of stages is shown in the figure. As EFFECT OF PRESSURE RATIO ON OVERALL ISENTROPIC EFFICIENCY seen, the overall efficiency decreases as the pressure ratio increases. When cycles for gas turbine engines are being investigated, it is normal to examine the effect of varying pressure ratio. It is evident that assuming a constant value of the overall compressor isentropic efficiency is not valid for such investigations. To account for the effect of the pressure ratio on the isentropic efficiency, the concept of the small-stage or polytropic efficiency has been introduced.

Overall Compressor Isentropic Efficiency

0.9 1

2

3

4

5

6

0.88

Number of Stages

7

8

9

10

11

0.84

Stage PR = 1.2 Stage ηisen = 0.9

0.82

0.8

1

2

3

4

5

6

7

Overall Compressor Pressure Ratio

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

dh0 = Then for an isentropic process (ds = 0)

12

0.86

dP0

ρ0

+ T0 ds

8

9

10

dh0′ =

dP0

ρ0

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

dh0′ = η p dh0

∴

dP0

ρ0

= η p dh0

Then assuming a perfect gas, h0 = CpT0 and P0 = ρ0RT0. Also Cp - Cv = R, or C p =

Rγ , and the fluid γ −1

properties are assumed constant through the process. Then

(

d C p T0 dP0 = ηp C p T0 ρ 0 C p T0 dP0

γ ρ 0 RT0 γ −1 or

= ηp

)

dT0 T0

dT0 γ − 1 dP0 = T0 η p γ P0

Integrating this between the start and end of a finite process

⎛ T ⎞ γ − 1 ⎛ P02 ⎞ ln⎜ 02 ⎟ = ln⎜ ⎟ ⎝ T01 ⎠ η p γ ⎝ P01 ⎠ γ −1

or

T02 ⎛ P02 ⎞ η pγ =⎜ ⎟ T01 ⎝ P01 ⎠

For a compression process, the isentropic efficiency is defined as

ηc =

Δh0′ Δh0

γ −1 γ −1 ⎛ ⎞ ⎞ ⎛ η γ ⎜ pcγ ⎟ ⎜ Δ h = C T PR − 1 where and Δh0 = C p T01 PR − 1⎟ where ηpc is the polytropic efficiency for 0′ p 01 ⎜ ⎟ ⎟ ⎜ ⎠ ⎝ ⎝ ⎠ the compressor. Then

ηc =

PR

γ −1 γ γ −1 η pcγ

PR

−1 −1

For a turbine,

ηt =

Δh0 Δh0′

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

ηt =

⎛P ⎞ 1 − ⎜ 04 ⎟ ⎝ P03 ⎠

η pt (γ − 1)

⎛P ⎞ 1 − ⎜ 04 ⎟ ⎝ P03 ⎠

γ

γ −1 γ

The following figure shows the resulting variation of isentropic efficiency with pressure ratio for an assumed polytropic efficiency of 0.9 and γ = 1.4, for both a compressor and a turbine. Also shown are the earlier results for the multistage compressor with stage pressure ratio of 1.2. VARIATION OF ISENTROPIC EFFICIENCY WITH PRESSURE RATIO Polytropic Efficiency, ηp = 0.9, γ = 1.4

0.94

Isentropic Efficiency

Turbine

0.92

0.9 1

2

3

4

5

6

7

8

0.88

9

10

11

12

Compressor

0.86 1

2

3

4

5

6

7

8

9

10

Pressure Ratio

The concept of polytropic efficiency should be used with caution. It is only valid if the machine can be considered to employ comparable technology and produce comparable performance as the pressure ratio is varied. For this reason, it should be applied only to explore the influence of pressure ratio on performance for multistage 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 is changed. As will be shown later, stage performance is closely related to its tip speed. For example, 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 of sound, 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: P1

y A

B C1

x

α1

Ca1

Cw1

Cm

αm

Y αm

Ca

Cwm

X

F

L

D

s α2

D C2

C

Ca2

P2

Cw2

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 tangential components X and Y. By definition, the lift generated by a turbomachinery blade L is the component of the blade force normal to the vector mean flow direction through the blade row. The drag D is the component of the blade force parallel to the vector mean flow direction. Then apply the linear momentum equation to the control volume. In the x direction:

ΣFx = m& (V x 2 − V x1 )

X + P1 ( s × 1) − P2 ( s × 1) = m& (Ca 2 − Ca1 ) Note that the pressure forces along the left and right faces of the control volume exactly balance each other in both the x and y directions. Then since we have assumed Ca1 = Ca2, the x-wise momentum equation reduces to

X = ( P2 − P1 ) s For the y direction:

(

ΣFy = m& V y 2 − V y1

(1)

)

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

direction is that due to blade, Y

Y = ρCa ( s × 1)(( − C w1 ) − ( − C w2 )) = ρCa s(C w1 − C w2 )

(2a)

Since

C w1 , Ca

tan α 1 =

tan α 2 =

C w2 Ca

we can also write

Y = ρCa2 s( tan α 1 − tan α 2 )

(2b)

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

ΔP0 = P01 − P02 = ( P1 − P2 ) +

(

1 ρ C12 − C22 2

)

From the velocity triangles,

(

) (

C12 − C22 = C w21 + Ca21 − C w2 2 + Ca22 = (C w1 + C w2 )(C w1 − C w2 )

)

since Ca1 = Ca2. Then

ΔP0 = ( P1 − P2 ) +

1 ρ (Cw1 + Cw 2 )(Cw1 − Cw 2 ) 2

substituting for Cw1 − Cw 2 from (2a)

ΔP0 = −

X 1 ⎛ Y ⎞ + ρ ⎜ ( C w1 + C w 2 ) ⎟ s 2 ⎝ ρ Ca s ⎠

and using the vector mean flow direction through the passage, tan α m =

ΔP0 =

1 (− X + Y tan α m ) s

1 ( tan α 1 + tan α 2 ) , we can write 2

(3)

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

D = Y sin α m − X cosα m

= cosα m ( − X + Y tan α m )

(4)

and substituting from (3)

D = ΔP0 s cos α m

(5)

Then from the definition of the drag coefficient

CD =

D

1 ρCm2 c × 1 2 ΔP s cos α m = 0 1 ρCm2 c 2

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

⎛1 ⎞ 1 ΔP0 = C D σ ⎜ ρCm2 ⎟ ⎝2 ⎠ cos α m

(6a)

or alternatively, using Cm = Ca/cosαm,

1 ⎞ ⎛1 ΔP0 = CDσ ⎜ ρCa2 ⎟ ⎠ cos3 α m ⎝2

(6b)

As will be seen later, some axial fan and compressor prediction procedures use the airfoil drag coefficient to express the loss performance for the blade row. Equation (6a) or (6b) can then be used to express this as a total pressure loss. Returning to the lift force, from the force triangles the lift L can be expressed as

L = X sin α m + Y cos α m

(7)

Solving (4) for X

X = Y tan α m −

D cos α m

and substituting into (7)

⎛ D ⎞ L = ⎜ Y tan α m − ⎟ sin α m + Y cos α m cos α m ⎠ ⎝ =

Y − D tan α m cos α m

Then substituting for Y from (2b)

L=

ρCa2 s ( tan α 1 − tan α 2 ) − D tan α m cos α m

By definition

CL =

L L = 1 1 ⎞ ρCm2 c 1 ρC 2 c⎛⎜ ⎟ a 2 2 2 ⎝ cos α m ⎠

(8)

then

ρCa2 s ( tan α 1 − tan α 2 ) cos α m D tan α m CL = − 1 ⎛ ⎞ 2 1 1 ρ Cm c ⎟ ρCa2 c⎜ 2 2 2 ⎝ cos α m ⎠ or

⎛ s⎞ C L = 2⎜ ⎟ cos α m ( tan α 1 − tan α 2 ) − C D tan α m ⎝ c⎠

(9a)

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

⎛ s⎞ C L = 2⎜ ⎟ cos α m ( tan α 1 − tan α 2 ) ⎝ c⎠

(9b)

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

∫

Γ = VS dS

(10)

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

Γ=

∫

B A

VS dS +

∫

C B

VS dS +

∫

D

C

VS dS +

∫

A D

VS dS

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

∫

C B

VS dS = −

∫

A D

VS dS

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

Γ = C y1 s − C y 2 s We are assuming constant axial velocity, and since tan α = C y C x (and Cx = Ca), we can write

Γ = Ca ( tan α 1 − tan α 2 ) s

(11)

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

L = ρ Cm Γ which is the expression given by the Kutta-Joukowski Theorem for an isolated airfoil. Note that in the case of the blade 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

rt rm 1

2

3

rh

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

(

)

rm2 = rh2 + rt2 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 ( m& − ΔP0 or Q − ΔH ) and using the guidelines for φ, ψ etc. from Chapter 3. To illustrate the procedure, we will use a semi-quantitative example. Assuming an incompressible flow machine, we will define the velocity triangles for 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: C φ = a = 0.5 Flow coefficient: U

gΔH E

Work coefficient:

ψ =

Degree of reaction:

Λ = 0.5

U2

= 0.4

Note that if we were using the Lewis charts from Section 3.5.6, we would probably choose a slightly higher value 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 a value for the stage efficiency η. This can be done from experience, or from the specific speed plots in Chapter 2, or from the approximate correlations shown in Section 3.5.6. Later, we will see how to calculate the efficiency of the stage we have designed. If this efficiency is different from the one we have guessed here, we will have designed the stage with an incorrect value of the ΔHE and it will not match the required performance ΔH. If this turns out to be the case, we will have to return to the beginning and revise the design. Thus, the design of a turbomachine inherently tends to be iterative: to design the machine we need its efficiency, but we do 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 our chosen value of the work coefficient ψ:

U=

gΔH E

ψ

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

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

A=

Q m& = ρ Ca Ca

We will assume “simple” velocity diagrams, as defined in Section 3.5.1. That is, we assume that the annulus is shaped such that Ca and U remain constant through the stage: U1 = U2 = U, Ca1 = Ca2 = Ca3 = Ca. Then from the Euler equation gΔH E = U 2 C w2 − U 1C w1 = UΔC w

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

ψ =

ΔC w U

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 we have chosen 50% reaction, equal amounts diffusion are occurring in the rotor and the stators. Thus the de Haller Ca1 = C a 2 numbers for the rotor and stators must be the same:

W2 C3 = W1 C2

α 2 (+ )

α 1 (+ )

On Problem Set #3, you will show that for simple velocity diagrams, this is achieved by making the velocity triangles for the inlet and outlet of the rotor symmetrical, and by designing the stators so that C3 = C1. That is, we design the stators so that the flow at the U stage 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 can determine the de Haller numbers:

W2 C3 = = 0.678 W1 C2 This is approaching the limit of about 0.65 that was

Cw 1 (+ )

C1

Cw 2 (+ ) C2 W1 W2

β 1 (− )

β 2 (− )

ΔCw

recommended in Section 3.5.6 and we will therefore have to monitor our design for the possibility of stall. As noted 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, the inlet flow must have a swirl angle α1. Thus, there must either be a stage ahead of the present one, or a set of inlet guide vanes, that leave the required amount of swirl in the flow. The flow from this stage will also leave with swirl α3 = α1, so that C3 = C1 . Suppose instead that the inlet swirl was specified. For example, if this is the first stage in the machine then we will normally have no swirl in the flow, α1 = 0. Using the same values of φ and ψ, the velocity triangles will then look as shown. We can then show that the resulting degree of reaction is Λ = 0.8. This means that the diffusion is much higher in the rotor than in the stators and this might at first be a matter for concern. However, consider the values of the de Haller numbers (we will assume that the flow leaves the stage with no swirl, C3 = C1): Rotor:

W2 = 0.699 W1

Stators:

C3 = 0.781 C2

C a1 = Ca 2 C1

α 2 (+ ) Cw 2 (+ )

ΔCw

C2

U W1

α1 = 0 W2

Cw 1 = 0

β 2 (− ) β 1 (− )

As expected, the value is lower for the rotor than for the stators. However, the diffusion is actually less than for the 50% reaction machine. As a result, this stage may be just as feasible as the earlier stage, despite the high value of degree of reaction. Having determined the velocity triangles, the next step is to define the blade geometries that will produce the required velocities.

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 the blade row parallel to the metal angle at the trailing edge of the blades: this is known as the Euler Approximation. 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 also make the leading edge metal angle parallel to the inlet flow angle. We can then define the shapes of the blades for the 50% reaction stage as follows:

As indicated on the drawing: (i) β 1′ = β 1 to bring the flow smoothly onto the leading edge of the rotor blades. (ii) In the relative frame of reference, the flow must leave the rotor blade passage at β 2 to produce the required turning. Based on the Euler approximation, the flow will leave the trailing edge at the metal angle and we therefore use β 2′ = β 2 . (iii) The stators see the flow in the absolute frame. To bring the flow smoothly onto the leading edge of the stator blades we therefore make α 2′ = α 2 . (iv) Again, the flow is assumed to leave the stators at the metal angle, and we use α 3′ = α 3 = α 1 . Note that with the assumptions made, the rotor and stator blade geometries are identical for the 50% reaction stage.

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 away from its design point. The idealized analysis can also be used to give reasonable predictions of how the stage will 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). The resulting 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 vanes ahead 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 of Ca would just be obtained from Ca = Q A , where Q is the new volume flow rate, and A is the 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 in swirl velocity ΔCw has increased. From the Euler equation gΔH E = UΔC w

and the head rise produced by the machine ΔH = ηΔH E will be increased. Equivalently, for a compressible flow machine, Δh0 , and the corresponding pressure ratio, P02 P01 , will be increased. Note that this is consistent 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. On the other hand, increasing the incidence will eventually lead to stalling of the blade. Thus, reducing the flow rate through a compressor stage will move it towards stall. Note that the incidence was also increased for the stators, bringing them closer to stall as well. Clearly, we can use the velocity triangles and the Euler equation to predict the quantitative stage characteristic for the idealized stage. It is convenient to express the characteristic in terms of the work and flow coefficients. The flow turning is ΔC w = C w2 − C w1

and from the velocity triangles (noting that Ww2 is negative for the conventional compressor velocity triangles) C w1 = Ca tan α 1

Then

C w2 = U + Ww2 = U + Ca tan β 2

ΔC w = U + Ca ( tan β 2 − tan α 1 )

and dividing by U

ΔC w C = 1 + a ( tan β 2 − tan α 1 ) U U

or

ψ = 1 + mφ

Thus, the ψ versus φ curve (effectively, the head rise versus flow rate characteristic) is a straight line with slope m = tan β 2 − tan α 1 For the present case, the symmetry of the velocity triangles implies that β 2 = − α 1 and the slope is then m = − 2 tan α 1 . For α1 > 0, as is the case here, this gives a negative slope and an inverse relationship between head rise and flow rate, as inferred above. Alternatively, since the characteristic passes through the design point (say, φD and ψD), we can write

m=

1.0

ψ

ψ D −1 φD

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

ψD3

0.5

ψD2 ψD1 INCREASING DESIGN-POINT LOADING

φD 0.0

0.5

φ

1.0

(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 the Euler Approximation again, it can be shown that the change in the velocity triangles will look as follows:

From the triangles, φ = Ca U > φ D since U has decreased. For the work coefficient, ψ = ΔC w U , ΔCw has clearly decreased, but so has U. However, ΔCw has decreased more rapidly than U; as can be seen, a small further decrease in U would reduce ΔCw to zero. We therefore conclude that ψ = ΔC w U < ψ D and φ and ψ are again seen to vary inversely. In summary, any deviation from the design point will cause the a given compressor 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 the absolute work transfer: gΔH E = UΔC w

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

gΔH E ≈ kU 2 and the head rise delivered by the stage, at a fixed value of flow rate, will change strongly with the rotational speed: 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 illustrated further in later sections.

As seen, the Euler Approximation results in an idealized ψ versus φ characteristic for the stage that is a straight line with a negative slope. We have already noted that some changes in operating point will result in positive values of the incidence at the leading edge of the airfoils. If this incidence becomes too large, we would expect the airfoils to stall. Also, we would expect the efficiency of the stage to be best when the rotor and stator blades are operating at the design point. We can therefore project what the actual stage characteristic is likely to be based on the idealized characteristic: MAXIMUM

ψ

ψD

η

η

STALL

USING EULER APPROXIMATION

LIKELY ACTUAL

φD

φ

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

η max

ΔH

η

CONSTANT

N

Q

CONSTANT

On each of the constant speed lines, there will be a point that corresponds to the design point values of φ and ψ on the non-dimensional characteristic. At each of those points, the velocity triangles will be similar, as indicated in the drawing. In each case, the relative velocity vector at the rotor inlet is lined up with the metal angle and the flow comes smoothly onto the leading edge. As shown, we would therefore expect that the machine will operate at its maximum efficiency at each of those points, apart perhaps for some small effect of differing Reynolds numbers. Also, as we will see later, frictional losses vary as V2 and thus the higher flow velocities with increasing rotational speed will result in higher frictional losses. This effect will be 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 performance characteristics 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 vary across the span. For this example, we will take the case with no inlet swirl from Section 4.3.1. At the mean radius, 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 mean radius known and the mean blade speed Um fixed by the choice of work coefficient ψ, we have the rotational speed, ω = Um/rm. To define the resulting spanwise geometry, we assume that the inlet axial velocity Ca is constant across the span and that we want the same total head rise, gΔH E = UΔCw , at every spanwise section. This 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 the performance parameters.

C1 rt W1

C2

U

W1

W2

C1

rm

C2 W1 U

W2

C1 rh W1 U C2

W2

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 de Haller numbers there also indicate too much diffusion. The blade will need to be redesigned. If the stage is still 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 design parameters is a function of the hub-to-tip ratio, HTR = rh/rt. Increasing the HTR will make the blade more uniformly loaded across the span, but since the cross-sectional area is fixed (by the choice of φ), this has consequences for the tip diameter of the machine and the rotational speed. This is demonstrated in the following sketch, which shows three different blades with the same annulus cross-sectional area but different values of HTR. In multi-stage compressors, the HTR will normally increase along the machine since the cross-sectional area is decreased to keep the axial velocity high. This is illustrated by the cross-section of the compressor from the GE LM2500+ gas turbine engine (17 stages, PR = 23.3).

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

0.3

RPM

Higher

0.5

0.8 Lower

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 has shown that it is possible to design a stage of good efficiency if the guidelines for those design parameters are followed. 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 other words, 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 is required to turn. From the control volume analysis in Section 4.2, the resulting lift coefficient was given by

⎛ s⎞ C L = 2⎜ ⎟ cosα m ( tan α 1 − tan α 2 ) ⎝ c⎠ ⎛ 1⎞ = 2⎜ ⎟ cosα m ( tan α 1 − tan α 2 ) ⎝σ ⎠ and it is seen to vary directly with spacing, or inversely with the solidity. Just as for an isolated airfoil, there is an upper limit to the lift that a turbomachinery blade can develop before it stalls. For a given set of inlet and outlet flow angles, it is possible to stay below the loading limit by making the solidity of the blade row large enough. Thus, the solidity of the blade row is selected on the basis of a blade loading limit. This is in contrast to 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 lift coefficient (Horlock, 1958). In the early 1950s, Howell suggested that a well-designed compressor airfoil will stall at 3

⎛C ⎞ C L ⎜ 1 ⎟ ≈ 3.3 ⎝ C2 ⎠ and designers of low-solidity fans have sometimes used the criterion

⎛ c⎞ C L ⎜ ⎟ ≤ 11 . ⎝ s⎠ However, expressing the loading limit simply in terms of CL has been found to be unreliable. Recent practice has therefore taken a somewhat different approach. Howell (British Practice) In the 1950s, Howell conducted an extensive series of cascade measurements on the compressor airfoils that were commonly used in British compressor design. The performance was measured for a wide range of the design parameters, including the flow turning angle and solidity. Howell varied the amount of flow turning up to the onset of stall. The corresponding total-pressure losses were also measured. Howell suggested that a suitable design turning angle for a blade row was that which corresponded to about 80% of the 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 blade row operating at 80% of the stalling turning angle. This correlation is shown in the next figure (taken from Saravanamuttoo 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 cascade measurements to determine the performance of compressor airfoils for a wide range of geometric and aerodynamic parameters. As described later, these results became the basis for a compressor design system which 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

C2 SUCTION SURFACE

Cmax

C1

C1

C2

0 0

x/c

1.0

The performance of the blade is limited by the deceleration (that is, the diffusion or adverse pressure gradient) on the suction surface of the airfoil. If the diffusion is too great, the boundary layer separates, the blade stalls, and the losses increase significantly. Lieblein proposed a parameter to measure the severity of the diffusion: − C2 C D = max (1) C1 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 be known early in design. However, the larger the lift (or circulation) being generated by the airfoil the larger Cmax must be. From Section 4.2.2, the circulation is given by

Γ = s(C w1 − Cw2 ) = sΔC w and thus we can write

C max = C1 + f ( Γ )

= C1 + f ( sΔC w )

Substituting into (1),

D =1−

C2 1 + f ( sΔC w ) C1 C1

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

D =1−

ΔC w C2 + C1 2σ C1

(2)

This parameter is known as the diffusion factor. Note that (2) depends only on the upstream and downstream velocities, which are known once the velocity triangles are established. The figure (taken from NASA SP-36, 1966) shows the variation of the total pressure loss coefficient, ω1, with D. As seen, the losses rise sharply for D > 0.65, implying the onset of stall. At the design point, the diffusion factor should therefore be less than this. A suitable value might be D = 0.3 - 0.4. With D chosen, the only unknown in (2) is the solidity and it can therefore 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 fully satisfied 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 angle necessarily result in the lowest losses. Finally, we need methods for estimating the losses generally, in order predict 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 fairly widely. The British system, connected mainly with the name of Howell, will be discussed since it is relatively easy to apply in hand calculations. However, it omits the influence of a number of blade geometric parameters, does not directly apply to all the families of blade geometries that are in common use, and has somewhat 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 the 1950s and 60s. This system is summarized in a famous document, NASA SP-36, “Aerodynamic Design of Axial-Flow Compressors” published in 1965. SP-36 continues to form the basis for much practical axialcompressor design, both in North America and outside. The correlations presented in SP-36 have also been re-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, General Electric and Rolls-Royce) have to some extent developed their own compressor design systems that reflect their in-house design philosophies and proprietary blade profile designs. However, these systems are often structured in similar ways and strongly influenced by the design systems that are available in the open literature.

4.5.2 Blade Design and Analysis Using Howell’s Correlations The figure shows the nomenclature used by Howell:

Nomenclature: s c ζ θ a t

= = = = = =

i = δ = ε =

blade spacing blade chord (solidity σ = c/s) stagger angle α1' - α2' = camber angle distance of maximum camber aft of blade leading edge maximum thickness of blade incidence = α1 - α1' deviation = α2 - α2' = difference between outlet flow angle and metal angle flow turning = α1 - α2

The nomenclature applies for a stationary blade row. For a rotor, replace α by β and use the relative components of velocity. Typical results obtained by Howell for a particular cascade geometry are shown in the following figure. The figure (taken from Horlock, 1958) shows the variation of flow turning, ε and the total pressure loss as a function of the incidence, i.

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

ε , δ , losses = f (blade geometry , cascade geometry , = f(

a c, θ ,

s c,

flow conditions) i, α 2

)

He also found that the results collapse well onto universal curves if they are normalized in terms of the results at the "nominal" (or "design"or "reference") flow condition for each cascade. The nominal condition is defined, somewhat arbitrarily, as the condition at which the flow turning, ε, is 0.8 of the value at stall. Stall is the 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 to turn the flow. For convenience, Howell defined the stalling incidence as the positive incidence at which the losses have increased to twice their minimum value. This definition is fairly easy to apply to experimental data. As the figure above indicates, it also seems to correspond fairly well to the point of maximum flow turning. 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:

⎛ 1⎞ δ * = mθ ⎜ ⎟ ⎝σ ⎠

n

(1)

where 2

α 2* ⎛ a⎞ m = 0.23⎜ 2 ⎟ + ⎝ c⎠ 500

(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 a compressor stage, Howell suggested using n = 1.0 and a constant value of m = 0.19. Unlike typical compressor rotor and stator blades, IGVs form an accelerating flow passage. They therefore behave more like a 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

s⎞ ⎛ ε * = f ⎜ α 2* , ⎟ ⎝ c⎠ The correlation is usually presented graphically ( Fig. 5.14 from Saravanamuttoo et al.) and was used in Section 4.4 to select the solidity. The blade will often be used at other than the nominal (“design”) flow conditions. Howell was able to correlate fairly successfully the “off-design” behaviour of the cascades by plotting the results against the nondimensional relative incidence, irel = (i - i*)/ε*. Figure 3.17 (taken from Dixon) shows the normalized flow turning, ε/ε* as a function of irel. The figure also shows the variation of the losses (expressed as a “drag coefficient”) 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 chord and upstream velocity). The resistance of the suction-surface boundary layer to separation is a function of the thickness of the boundary layer and whether it is laminar or turbulent. Thus, the flow turning behaviour of the blade row is a function of the Reynolds number, particularly at low values. Howell examined the dependence of the flow turning on the Reynolds number. Figure 3.3 (taken from Horlock) shows the effect of Reynolds number on the nominal turning.

The correlations presented to this point can be used to predict the flow turning capability of a compressor 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 be illustrated 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 2* = α 2 ′ + δ * Compare this value with the assumed α2*, revise as necessary and repeat until α2* and δ* are consistent. (ii) Read the value of ε* from Fig. 5.14. Then

α 1* = α 2* + ε * i * = α 1* − α 1′ 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 correction would 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) which will give the desired outlet flow angle, α2. This application of the correlations is a little more complicated since 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 since nominal conditions give near-minimum losses and provide some stall margin. Then

α 2* = α 2 α 1* = α 1

and ε * = α 1 − α 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 was used in Section 4.4). Since the blade row is operating at the nominal conditions, the deviation will also be that given 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

ε =i +θ −δ

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

Thus, the value of the camber will depend on the choice made for i*. Howell’s correlations indicate that there is no unique choice for the design incidence, although he recommends that a value be chosen of a few degrees at most. Reductions in camber can be compensated for by increases in incidence, and vice versa. Note that these changes will also result in a change in the stagger of the blade row. In summary, according to the Howell’s correlations a variety of blade geometries can produce identical aerodynamic performance. This gives the designer some freedom to tailor the blade geometry to meet other possible requirements: eg. to simplify the spanwise variation in the blade geometry, to alter a natural frequency, or to alter the stress level in some 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 predicted performance if the same blade profiles are used in the machine. The C4 profile, one of the most widely used of 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 for the Howell’s correlations have been fitted by polynomials. These curve and surface fits are also given in an appendix.

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 the NACA 65-series airfoil profile shape (Emery et al., "Systematic Two-Dimensional Cascade Tests of NACA 65-Series Compressor Blades at Low Speeds," NACA Report 1368, 1958). The results are correlated and design 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 design transonic compressors. As noted, Howell’s correlations do not give clear guidance for the choice of design incidence. While the nominal incidence, i*, is a reasonable choice for the design point, it is also clear from Fig. 3.17 that using i* does not in general minimize the profile losses. Howell’s correlations also do not take into account some geometric parameters which are known to affect the blade performance, such the ratio of maximum-thicknessto-chord, tmax/c. Finally, the Howell’s correlations are most suitable for analyzing the performance of a blade row of specified geometry ("analysis mode") rather than determing a geometry which gives a desired performance ("design mode"). By comparison, the NASA correlations are intended particularly for use in design mode, although they can also be used for analysis. They guide the designer to a choice of design incidence which nominally minimizes the profile losses. The correlations also account for more aspects of the blade geometry. The drawback to using the NASA correlations is that reference must be made to more graphs than for the Howell’s correlations. For consistency with the SP-36 graphs, the procedures will be described in terms of the nomenclature used by NASA. As with the Howell correlations, the incidence and deviation are defined in terms of some reference flow condition, although the definition of this condition is slightly different. Fig. 131 (from SP-36) shows the definition of the reference incidence, iref. It is the incidence half way between two off-design values of incidence at which the losses are equal. SP-36 usually refers to this as the “minimum-loss incidence” although the losses will only be a minimum if the loss “bucket” is symmetrical. As evident from Fig. 3.17, this is not normally the case. Nevertheless, the reference condition will be near minimum loss and thus would be a reasonable choice for the design point. The deviation produced at the reference incidence is designated as δref. For specified inlet and outlet flow angles, β1 and β2, the required flow turning, Δβ = β1 - β2, is related to the camber, incidence and deviation by

Δβ = θ + i − δ If we use the reference values of incidence and deviation then

Δβ = θ + i ref − δ ref

(1)

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

i ref = i 0 + nθ

δ ref = δ 0 + mθ where i0 and δ0 are the values for the same blade when it has zero camber. Substituting into (1), the required camber is given by

θ=

Δβ + δ 0 − i 0 1+ 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 0 = ( K i ) sh ( Ki ) t (i 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 are being 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 the following 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 same correlations 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 0 = ( Kδ ) sh ( Kδ ) t (δ 0 ) 10

(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 are being 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 and DCA 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 rule similar to that used in the Howell’s correlations:

m=

mσ =1.0

σb

(5)

where mσ=1.0

b

=

value of m for a solidity σ = 1.0

=

f6(β1)

Fig. 163

=

f7(β1)

Fig. 164.

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

⎛ dδ ⎞ δ = δ ref + (i − i ref )⎜ ⎟ ⎝ di ⎠ ref

(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 which will result in the required performance: that is, they are suitable for use in design mode. Of course, some decisions must already have been made concerning the type of blading (C-series, 65-series, DCA etc.), the camber 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 the following approach would be used: (i) For the specified geometry and design inlet-flow direction, β1, the reference conditions are first determined. (ii) For an off-design inlet value of β1, Eqn. (6) would then be used to predict the deviation. This defines 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 an Appendix.

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. In addition to the frictional effects in the boundary layers on the surfaces of the rotor blades, there are a number of other flow features that can generate losses. The losses due to each of these features are normally estimated individually 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 the sudden expansion in area at the trailing edge, and by the mixing out of the wake downstream of the blade. (ii) Secondary losses: The slower-moving flow in endwall boundary layers is "over turned" by the blade-toblade pressure field, as shown in the drawing. The fluid swept towards the low pressure (“suction”) side of the passage is blocked by the blade surface and rolls up into a "passage vortex" that generates additional losses through high shear stresses at the endwalls and as it mixes with the downstream flow. The boundary-layer separation 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 of the blade passage. The endwall losses inside the passage are normally assigned to the secondary losses. (iv) Tip-leakage losses: There must be some clearance between the rotor blade tips and the compressor casing. The flow that is driven through the tip gap rolls up into a "tip-leakage vortex" 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 through downstream mixing with the surrounding fluid. In transonic and supersonic compressors, there will be additional losses due to the presence of shock waves.

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 being generated by the airfoils, since this determined the pressure difference that drives the flow across the passage to form the secondary flow. Thus C DS = 0.018 C L2

where from Section 4.2.1 the blade lift coefficient is given by

æ sö C L = 2ç ÷ ( tan α 1 − tan α 2 ) cos α m è cø

and

æ tan α 1 + tan α 2 ö α m = arctanç ÷ è ø 2 For the rotor flow, we would use the relative flow angles, $1 and $2, as usual. (iii) Annulus Losses: C DA = 0.02

s sc 0.02 = 0.02 = h c h σ AR

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 J compared with the blade span h. Howell suggested that a 1% increase in the rotor clearance gap would reduce the stage efficiency by 3%: ∆η clearance = 3

τ h

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

1 ö öæ æ1 ∆P0, loss = C D σ ç ρ Ca2 ÷ ç ÷ ø è cos 3 α m ø è2

(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 for total pressure loss coefficient, rather than for drag coefficient.

4.6.3 Loss Estimation Using NASA SP-36 Correlations

(i) Profile Losses C

The profile loss system presented in NASA SP-36 is associated with the name of Lieblein, as were the blade loading limits presented in Section 4.4. In that section, it was seen that the profile losses correlated quite well with the diffusion factor defined by Lieblein, which was defined as C − C2 D = max C1

C2 SUCTION SURFACE

Cmax

C1

C1

C2

0 0

(1)

x/c

1.0

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

C max C2

(2)

Note that Deq resembles the deHaller number. Whereas the deHaller number defines the net diffusion between the inlet and outlet of the blade row, Deq defines the local diffusion on the suction side of the airfoil. Lieblein then 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 the airfoil is known. For use in the early stages of design, an approximate value of Deq, estimated from the circulation, is therefore used. The following correlation appears to be widely accepted: Deq =

(cosα 1 ) 2 tan α − tan α ö÷ cosα 2 æç . + 0.61 112 ( 1 2 )÷ cosα 1 çè σ ø

(3) y

The profile losses are reflected in a momentum deficit in the wake, as measured by the momentum thickness 2 downstream of the airfoil:

C2,ref s C 2 (y )

θ =ò

s 0

C2 C2 ,ref

æ C çç 1 − 2 C2 ,ref è

ö ÷÷ dy ø 0

where C2,ref is the velocity outside the wake. The corresponding total-pressure loss coefficient is then given by 2

æ θ ö σ æ cosα 1 ö ω = 2ç ÷ ç ÷ (4) è c ø cosα 2 è cosα 2 ø

where

ω=

P01 − P02 1 ρC12 2

(5)

The loss correlation is then expressed in terms of the variation of the momentum thickness ratio, 2/c, with equivalent diffusion ratio, Deq:

θ = f ( Deq ) c

(6)

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

AXIAL COMPRESSOR PROFILE LOSSES AT DESIGN INCIDENCE Comparison of Correlations with Lieblein Data

Wake Momentum Thickness Ratio, θ//c

0.12

Aungier Wilson & Korakianitis Koch & Smith Casey/Starke Konig et al

0.1

0.08

0.06 Lieblein Data

0.04

0.02

0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Equivalent Diffusion Ratio, D eq

Recently, Konig et al. (W.M. Konig, D.K. Hennecke & L. Fottner, “Improved Blade Profile Loss and Deviation 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 Lieblein correlation approach worked equally well for more recent compressor airfoil shapes. Their data are shown in the next figure, along with the same curve fits.

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

Wake Momentum Thickness Ratio, θ/c

0.12

Aungier Wilson & Korakianitis Koch & Smith Casey/Starke Konig et al

0.1

0.08

0.06

0.04 Konig et al. Data

0.02 +

+

0

1

1.2

1.4

1.6

+

+

1.8

+

2

2.2

2.4

2.6

Equivalent Diffusion Ratio, D eq

Although there is some evidence that more recent blade designs can tolerate somewhat higher values of Deq before stalling, the curve fit suggested by Aungier (R.H. Aungier, Axial-Flow Compressors, ASME Press, 2003) seems as reasonable as any, for both data sets: 2 8 θ = 0.004 éê10 . + 31 . ( Deq − 1) + 0.4( Deq − 1) ùú ë û c

(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, 2/c, using (7). (3) The total-pressure loss coefficient T is then calculated from (4).

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

(cosα 1 ) 2 tan α − tan α + a i − i * cosα 2 æç . + 0.61 112 ( 1 2) cosα 1 çè σ

(

)

1.43

ö ÷ ÷ ø

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

(ii) Endwall Losses NASA SP-36 does not provide clear guidance for estimating either the secondary losses or tip clearance losses for the purposes of meanline analysis. Instead, most recent text books (eg. Japikse & Baines) and papers seem to recommend a method developed by Koch & Smith at General Electric (Koch, C.C. and Smith, L.H., “Loss Sources and Magnitudes in Axial-Flow Compressors,” ASME J. Eng. for Power, Vol. 98, 1976, pp. 411-424). The method provides combined estimates for the effects on stage efficiency of both secondary flows and tip leakage. This is physically 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 a fairly detailed knowledge of the stage geometry. It is also necessary to specify how close the stage is to stall at the operating point for which the loss estimates are being made. Nevertheless, because of the importance of endwall 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

⎛ 2δ * ⎞ 1− ⎜ ⎟ η ⎜ h ⎟ = η P ⎜ 1 − 2ν ⎟ ⎜ ⎟ ⎝ h ⎠ where

ηP δ* ν

= = =

(1)

stage efficiency as calculated from the profile losses only average displacement thickness of the two endwall boundary layers 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 the lower fluid velocity present in the endwall boundary layers. Koch & Smith provide correlations, derived from very wide-ranging tests conducted on a large, lowspeed compressor test rig, for estimating the values of δ* and ν. The drawing defines some of the geometric parameters that appear in the correlations. s λ g

= = = =

spacing stagger angle staggered spacing

λ

s cos λ

In addition, the following are used ε h ξ

= = =

tip clearance blade span axial gap between rotor and stators

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

s g

c

λ

from the casing wall and have a clearance at the hub wall. If the stators are variable pitch, they will also need clearance. 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 of the clearance and the pressure rise ratio:

⎛ ΔC P 2δ * ε⎞ = f⎜ , ⎟ g ⎝ ΔC P ,max g ⎠ where

ΔC P =

ΔP q

with ΔP the static pressure rise across the stage and q the average of the inlet dynamic pressures for the rotor and stator rows. ΔC P,max is the maximum value of the static pressure rise coefficient for the same stage, corresponding 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 has been chosen. For example, if the deHaller numbers are low and the solidities have been selected to give relatively high values of the diffusion factors, the pressure rise ratio would be expected to be towards the higher end of the scale.

0.55 0.5 0.45 0.4

2δ*/g

0.35 0.3

ε/g = 0.10 0.075

0.25

0.050

0.2

0.025

0.15

ε/g = 0.0

0.1 0.05 0

0.7

0.75

0.8

0.85

ΔC P/ΔCP,max

0.9

0.95

1

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

1.1

1.05

2δ*/(2δ*)ref

1

0.95

0.9

0.85

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Axial Gap/Blade Spacing, ξ/s

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

0.9 0.8 0.7

2ν/2δ*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.75

0.8

0.85

ΔCP/ΔCP,max

0.9

0.95

1

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 hc s = g hcsg ε AR σ = h cos λ

(2)

Values of the tip clearance ε are often specified as a fraction of the blade span h. Therefore, reasonable values of ε/h would be known early in design. Eqn. (2) also implies that the Koch & Smith correlation can be used to conduct parametric studies to investigate the influence on the endwall losses of common design parameters such as the solidity σ and the blade aspect ratio AR = h/c. For use in Eqn. (1), note that

2δ * 2δ * g s c = h g s ch =

2δ * cos λ g σ 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 included the 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 the blade section. In addition, the Howell results apply only for low subsonic values of the inlet Mach number. The loss behaviour 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 some representative results are presented to illustrate the complexities. The figure (taken from SP-36) shows the variation of profile losses with both incidence and inlet Mach 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 camber lines 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 transonic compressors. It is seen that their sensitivity to Mach number is delayed to a higher inlet Mach number than some of the other shapes.

The strong influence of the detailed airfoil geometry on the behaviour at both off-design incidence and with increasing inlet Mach number obviously makes it more difficult to devise simple correlations for the losses, 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 Transonic Compressor 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 ΣP0,loss determined from the correlations described in Section 4.6.2 or 4.6.3, the total-pressure rise across the stage is then: ∆P0 = ∆P0, ideal − Σ∆P0 , loss

where )P0,ideal is the pressure rise that would have been obtained in an ideal machine having the same work input. (i) For compressible flow of a perfect gas, the ideal pressure rise is γ

∆P0, ideal = P03′ − P01

where

P03′ æ T03 ö γ − 1 =ç ÷ P01 è T01 ø

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

T0

P′03

P03 T03 T03′

P01

T01

s

Then the actual P03 is obtained from P03 = P03′ − ΣP0, loss

where ΣP0,loss is as predicted for the rotor and stators together. The corresponding efficiency can be calculated by determining the power required to compress through the same pressure ratio, P03/P01, in an isentropic machine: T03′ æ P03 ö =ç ÷ T01 è P01 ø

γ −1 γ

Then

η=

T03′ − T01 T03 − T01

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

η≅

∆P0 ∆P0, ideal

If any of the loss components is expressed in terms of an efficiency decrement, as is sometimes the case with tip-leakage loss, its contribution to the total pressure losses can be estimated from ∆P0, TL ≅ ∆η clearance ∆P0, ideal

and this would be included in the earlier summation of losses. (ii) For incompressible flow, we can use ∆P0 , ideal = ρ g ∆H E = ρ (U 2 C w 2 − U 1 C w1 ) ≅ ρ U ∆C w

and

η= where, as before

∆P0 ∆P0, ideal

∆P0 = ∆P0,ideal − Σ∆P0,loss

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 velocity triangles show that the incidence at the leading edge of the rotor blades is increased. If the incidence becomes too large, the blades may stall. The disorganized flow in the stalled region partially blocks the blade passage. As a result some of the fluid that was previously passing through the stalled passage is diverted to the adjacent passages, as indicated in the middle figure. This has the effect of increasing the incidence at the airfoil lying next to the stalled region while reducing the incidence at the other adjacent airfoil. The increase in incidence on the one adjacent airfoil may cause it to stall in turn. On the other hand, the airfoil which has its incidence reduced will move further away from stall, or may unstall if it was previously stalled. Thus, there is a tendency for the stall cell to migrate from blade passage to blade passage in the opposite direction to the rotation. This phenomenon is known as rotating stall. The stall cells move in the opposite direction to the rotation at a relative speed which is about half the rotational speed.

MIGRATION OF STALL CELL

NORMAL ATTACHED FLOW

SEPARATION OF BLADE BOUNDARY LAYER - BLADE STALL

ROTATING STALL

The rotating stall can take a number of patterns. It may involve only one blade passage, or a large number 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 considerably across the span and it will be the hub region that will have the highest loading and therefore be the most likely to stall. In that case, the may stall cells may only involve part of the span of the blade. If the blades have high hub-to-tip ratio, the stall is more likely to extend across the full span.

LOW HTR ROTATING STALL

HIGH HTR FULL SPAN STALL, ALL PASSAGES

Particularly if the rotating stall occurs at low speed and only involves part of the span, it may not be a danger 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 of noise. For reasons discussed in Section 4.8, it is fairly common for the early stages of multi-stage axial compressors to experience some rotating stall at low rotational speeds. If it is present only during start-up and shut-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 & characteristic becomes positive. As will be shown in Chapter 7, if this occurs the system of which versus m the compressor is a part can become dynamically unstable. If this instability is triggered, the result is known as surge. The peak point of the compressor characteristic is therefore often identified as the surge point.

ΔP0 SURGE

ROTATING STALL

N const .

& m

In Chapter 7, an approximate, unsteady-flow analysis is developed for a compression system. This analysis identifies the various factors that will make the system more or less prone to surge. However, a simple 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 is being “drained” through the turbine. At any instant in time, there is a fairly large mass of gas present in the volume of the combustor. Now suppose that the last stage of the compressor suddenly stalls. This might be due to some disturbance that causes a drop in the mass flow rate through the machine. Since the last stage normally has a high hub-to-tip ratio, the stall may involve the full span of the blades and all of the passages, as described in the last section.

If the stall is very extensive, there will be an abrupt drop in the pressure of the gas delivered by the compressor. The flow area through the turbine is relatively small and this limits the outflow through the turbine. Consequently, the pressure in the combustor will drop somewhat gradually. It the rate of pressure drop is too slow, the situation can occur that the pressure in the combustor is higher than the pressure at the compressor outlet. Since fluid tends to flow from a region of high pressure to one of low pressure, it is therefore 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 point the flow in the compressor may re-establish itself. However, if the conditions that led to the initial stall are still present, the whole process can repeat. The cyclic flow reversal in the compressor can result in very large fluctuating forces on the blades which can destroy the machine. In the gas turbine engine, the abrupt drop in compressed air supplied to the combustor can also lead to over-temperatures and resultant serious damage to the 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 with identical aerodynamic characteristics and thus identical stage design points. Therefore, at design point for the machine as a whole, each of the stages will be running at their individual design points, which occur for the same value of the flow coefficient φ= Ca/U for all of the stages. Assume also that the mean radius, and thus the blade speed U, is the same for all four stages. Since the density of the gas increases across each successive stage, to maintain the constant axial velocity Ca needed to keep φ constant it is necessary to reduce the annulus area along the machine. This variation in the cross-sectional area would be determined at the design-point flow conditions. Ca U U

Ca

ψ

FOUR IDENTICAL STAGES

DESIGN N 1

m& = ρ Ca A

2

ψD

3 OR

4

Ca =

m&

ρ

ρ VARIATION DUE TO COMPRESSION

ρA

Ca

TARGET Ca VARIATION

φ

φD

A AREA VARIATION TO ACHIEVE TARGET Ca

DESIGN POINT

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

STAGE I

1

2

STAGE II

3

Since the annulus area has been adjusted such that at the design point the two stages have the same flow coefficient: φ I D = φ II D = φ D For stage I

φI = D

CaI D UD

and the corresponding pressure rise is

ΔPI D = P3 − P1 and the outlet density ρ 3D =

P1 + ΔPI D RT3D

Then for stage II

Ca 3 D =

m& D ρ 3D A3

φ II = D

Ca 3 D UD

where, as mentioned, A3 was adjusted to give φ I D = φ II D . Now consider the effect of halving N (ie. halving U) while also halving m& (ie. halving Ca) to keep Stage I operating at its design φ:

1 C a 1D 2 φI = = φ ID 1 UD 2 Thus, Stage I will also be operating at its design ψ. However, the absolute Δh0 (and thus ΔP0} varies as U2 and the pressure rise is therefore reduced to 1 P1 + ΔPD 1 4 ΔP ≅ ΔPD and ρ 3 = RT3 4 Neglecting the changes in T3, which will be relatively much smaller than the changes in P3, then

1 P + ΔPD ρ3 4 ≅ . < 10 ρ3 P + ΔPD D

The corresponding change in Ca3 is then

Ca 3 Ca 3 D

m& ρ 3 m& ρ A ⎛ 1⎞ = 3 3 = D = ( > 10 . )⎜ ⎟ = k m& D ⎝ 2⎠ ρ 3 m& D ρ 3D A3

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

φ II =

Ca 3 kCa 3D k = = φ II D 1 1 U UD 2 2

and since k > 1/2, φ II > φ II D . Thus, the non-dimensional operating point for Stage II shifts to a lower value of 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 only increases in the subsequent stages. For the 4-stage compressor with four identical stages we would therefore expect to see the following pattern of operating points:

Ca U U

ψ

Ca LOW N, STAGE 1 AT DESIGN 1 2

ψD

3

m& = ρ C a A < m& DES

4

Ca =

φD

m& ρA

ρ

ρ VARIATION (UNDERCOMPRESSION)

φ Ca

DESIGN Ca VARIATION NEW Ca VARIATION

A AREA VARIATION (FIXED)

REDUCED RPM, REDUCED

If we now reduce the mass flow rate at the low speed operating point, keeping N constant, the flow coefficient for Stage I will be lowered. Stage I will then be producing slightly higher pressure rise. However, the effect of the low U2 is much greater than the small increase in ψ and Stage I will still be producing much lower pressure rise than at the design N. Consequently, Stage I is still undercompressing the fluid and the downstream stages will again be at successively higher values of φ. We therefore conclude that if we throttle the flow further, Stage I will be the first to reach its stalling value of φ.

m&

ψ LOW N, THROTTLED 1 2 3

ψD

4

φD

φ

ψ

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

HIGH N, STAGE 1 AT DESIGN 4 3 2

ψD

1

φD

φ

Combining these arguments, we can plot the expected map for the compressor as a whole. 1

P02 P01

2

3 4

STAGE STALL LINE

HIGH-SPEED OPERATING POINT

ηmax

HYPOTHETICAL STEADY-STATE OPERATING LINE

DESIGN POINT

COMPRESSOR SURGE LINE

η CONSTANT SPEED LINE

N

4

T01 3

2

1

m& T01 P01

Note that: (i) Over most of the map, we assume that the stalling of any stage results in compressor surge. As a result, when the onset of the stall switches from the front to the back of the machine (near the design N), 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 more likely to be part-span, rotating stall (as discussed in Section 4.7). This, combined with the fact that the absolute forces on the blades will be low at low N, means that some degree of rotating stall is acceptable at low N. As a result, at the low end of the map the surge line has a “kink”, indicating that some early-stage stall is allowed. (ii) At the design point of the compressor, all of the individual stages are operating at their design points and therefore have their maximum efficiencies. From the earlier discussion, it is evident that at any other operating point at most one of the stages will be operating at best efficiency. Therefore, the efficiency of the overall compressor will be less than its value at design. For this reason, the lines of constant efficiency are shown as closed contours surrounding the design point.

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

NASA 8-Stage Research Compressor

Pratt & Whitney TF30 LP Compressor

4.9

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

For solidities, F, less than about 0.4 each blade can be treated as an isolated airfoil. Note that F = 0.4 was the lowest value of solidity that appeared on the NASA SP-36 correlations (Section 4.5.3). Usually, the blade is divided into a series of spanwise segments or blade elements. Three-dimensional flow effects in the form of spanwise flows are usually neglected, although the downwash induced by the trailing vortex system is sometimes taken into account. This approach, known as the "blade-element method", is commonly used to design propellers and low-performance axial fans. Consider the flow relative to a blade element. The element behaves like an isolated airfoil in a stream in the direction of the vector mean of the inlet and outlet flows:

ZLL Wm

= =

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

Wm =

P )L

= = =

)D

=

)X

=

)Y

=

CL, CD =

Ca Q æ tan β 1 + tanβ 2 ö ; β m = arctanç ÷ ; Ca = è ø A cos β m 2

angle of attack of blade element = angle between ZLL and Wm lift force on blade element (perpendicular to Wm) 1/2DWm2c)rCL (1) where c = chord length of blade element )r = radial width of blade element CL = lift coefficient of blade element (as obtained from airfoil characteristics and P) drag force on blade element (parallel to Wm) (normally )D )L) axial component of force on blade element )X . )L sin$m (see note at end of section) tangential component of force on blade element fns [P, section shape, Re] - as obtained from airfoil data

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

(2)

Fx = N B ∆X = A∆P = (2π r∆r ) ∆P where NB )P

= =

(3a)

no. of blades static pressure difference across the blade row

Substituting for )X in terms of the lift coefficient

ö æ1 N B ç ρWm2 c∆rC L ÷ sin β m ≅ 2πr∆r∆P ø è2 ö æ1 N B ç ρWm2 cC L ÷ sin β m ≅ 2πr∆P ø è2

(3b)

For the input power, from the energy equation

∆Q∆P0 ∆W&in = ∆Tω = ≅ ρ∆Q∆h0 ≅ ρ∆Q(U∆C w ) ηR

(4a)

assuming )P0 is small so that )P0/D0R . )h0, and where

)T )Q 0R )P0

= = = =

torque applied to flow through annulus width )r volume flow rate through annulus width )r rotor efficiency (0R =1 if CD = 0) 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 = NBr)Y)

∆Q∆P0 é1 ù ∆W&in = N B ê ρWm2 c∆r ( CL cos β m + CD sin β m )r úω = = ρ ∆Q(U∆Cw ) (4b) ηR ë2 û Rotor and stator blade rows can then be designed using Eqns. (1) - (4). Iteration will generally be necessary since W2 is a function of )L, which is a function Wm, which in turn is a function of W2. The analysis 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 the trailing tip vortices from the blades. The downwash would slightly alter the effective flow incidence seen by the blade and thus the lift it develops. To make the velocity triangle diagram clearer, the blade was sketched with somewhat lower stagger angle than would normally be found in practice. The diagram shows the force triangles for a more realistic value of the stagger angle:

y

∆D

x

∆L ∆X ∆F

Wm

∆Y

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

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, except that camber is usually much larger. The stage consists of a set of stators ("nozzles") followed by a rotor. The nozzles control the swirl in the flow entering the rotor and the rotor then extracts work from the fluid by removing swirl. This arrangement of components results in stage aerodynamic characteristics that are very different 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 mean radius, 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 velocity triangles:

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

φ=

Ca < φD U

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

ψ =

Δh0 ΔC w = U U2

φD

ψ =

Cw U

>ψ D

and the same trend is found as when the mass flow rate was changed. Now consider the absolute output. From the Euler equation Δh0 = UΔC w and from the velocity triangles, ΔCw increased as U decreased. It is not entirely clear whether the product UΔCw 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 axial compressors and turbines differ in two main ways: (i) ψ versus φ, and therefore ΔP0 versus m& , is negative for compressors, positive for turbines. (ii) The energy transfer Δh0 is a strong function of U for compressors, but only a weak function for turbines.

The following figures show the actual characteristics of the gas-generator turbine of the Orenda OT-2 gas turbine engine. Note that it is conventional to use the pressure ratio as the independent variable for plotting turbine aerodynamic characteristics. The characteristics confirm that the mass flow-pressure ratio characteristic is only a weak function of the rotational speed. However, this does not mean that the rotational speed is not important in order to have a high output of useful work. As seen from the velocity triangles, if the rotational speed is reduced below the design value, the energy released by the fluid, Δh0 = UΔCw, may not be changed very much, but this is also accompanied 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 recover most 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 axial compressors. Again, primarily British results will be presented, but these are also widely used in North America.

5.3.1

Flow Outlet Angle

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

æ sö δ = mθ ç ÷ è cø

n

where 2 = 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 of 1/2 used for compressor rotor and stator blades). However, the Carter & Hughes correlation tends to overestimate the deviation for most modern turbine blades. A more satisfactory (but less convenient) correlation is that due to Ainley & Mathieson (A-M). Their correlation uses the so-called gauge angle 2g as a reference angle to which the actual outlet angle is related:

æ oö θ g = cos −1 ç ÷ è sø

where o = throat opening and s = blade spacing. For an infinitesimally thin blade which is straight from the throat to the trailing edge, the gauge angle would define the direction normal to the throat line. For low speed flow, A-M correlated the outlet flow angle "2 with the gauge angle. Fig. 7.13 (from Saravanamuttoo et al.) shows the variation for a “straight-backed” blade: that is, a blade for which the suction side is straight from the throat point to the trailing edge. The curve in the figure can be approximated by

o

θg s

æ oö α 2 = 11625 . cos −1 ç ÷ − 12 è sø

However, most turbine blades are not straight backed. Instead they have a certain amount of “unguided turning” as defined by the angle 2u. In A-M’s day, if the suction surface was not straight from the throat to the trailing edge, it was usually defined by a circular arc. AM therefore corrected the outlet angle as follows:

æ oö æ sö α 2 = 11625 . cos −1 ç ÷ − 12 + 4ç ÷ è sø è eø where e is the suction side radius of curvature. Unfortunately, modern turbine blades usually do not use circular arcs to define their surface shapes. As a result, e is not constant and generally not known. To use the AM correlation it is therefore necessary to obtain an “equivalent” value of e. An approximate value can be calculated from the unguided turning angle as follows:

s = e

πθ u æ oö 180 1 − ç ÷ è sø

for 2u measured in degrees.

2

θg

(ii) Sonic Condition For M2 = 1.0 and a straight-backed blade, A-M indicated that the outlflow angle would be equal to the gauge angle: æ oö α 2 = cos −1 ç ÷ è sø

For a curved-back blade, this was again corrected for the suction side radius of curvature. The results were presented graphically but can be approximated by the following curve fit: s 1.787 + 4 .128 ö æ e æ oö çæ sö ÷ sin −1 æ o ö α 2 = cos ç ÷ − ç ç ÷ ç ÷ ÷ è sø è eø è sø è ø −1

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

(

α 2 = α 2 M 2 = 0.5 − (2 M 2 − 1) α 2 M 2 = 0.5 − α 2 M 2 =1.0

)

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 depth in the z direction.

Y y

x X

P1

C1

α1 Cw1

s

Ca1 P2 Ca2 α1 cx

Cw2 C2

Apply the linear momentum equation in the y direction:

(

ΣFy = m& V y 2 − V y1

)

(1)

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

Y = m& (C w 2 + C w1 )

(2)

From the velocity triangles, C w1 = Ca1 tanα 1

C w 2 = Ca 2 tanα 2

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

⎛ ⎞ C Y = ρ 2 sCa22 ⎜ tanα 2 + a1 tan α 2 ⎟ Ca 2 ⎝ ⎠ or, since Ca 2 = C2 cosα 2 ,

⎛ ⎞ C 1 Y = ρ 2 C22 (2 s)cos 2 α 2 ⎜ tanα 2 + a1 tanα 2 ⎟ Ca 2 2 ⎝ ⎠

(3)

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

P

"IDEAL" DISTRIBUTION

P0

P1

1 P0 − P2 = ρC22 2

PS

ACTUAL DISTRIBUTION

P2 SS

x

0

Y=∫

cx

0

cx

( PPS − PSS )dx

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

1 Yideal = ( P0 − P2 )c x ×1 = ρ 2 C22 c x 2

(4)

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

Z= Substituting from (3) and (4) then

Y Yideal

⎛ s⎞ ⎛ ⎞ C Z = 2⎜ ⎟ cos 2 α 2 ⎜ tanα 2 + a1 tanα 1 ⎟ Ca 2 ⎝ cx ⎠ ⎝ ⎠

(5)

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

⎛ s⎞ ⎛ ⎞ C Z = 2⎜ ⎟ cos 2 α 2 ⎜ tanα 2 − a1 tanα 1 ⎟ Ca 2 ⎝ cx ⎠ ⎝ ⎠ As usual, for rotor blades β replaces α. The normal definition of the solidity is σ = c/s. The way the Zweifel coefficient 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 the drawing, where ζ is the stagger angle: C X

ζ

S

C

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 the value 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 solidity for axial compressors. Since Zweifel’s time, profile design has improved and today turbines are often designed with considerably higher values of Z ( Z = 1.00-1.05 is common).

5.3.2.2 Ainley & Mathieson Correlation Ainley & Mathieson developed a widely used loss system (see next section), based on British turbine cascade data from the 1940s and 1950s. They likewise identified the geometries that gave minimum profile losses for different combinations of inlet and outlet flow angles. These optimum geometries, expressed as optimum 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 the Zweifel criterion).

Unfortunately, Zweifel and Ainley & Mathieson expressed “solidity” differently: cx/s versus c/s. This makes it difficult to compare the geometries that would be obtained using each approach, for the same set of velocity triangles. The two ratios are related through the stagger angle, ζ, of the blade row (see the figure on the previous page), since cosζ = cx c . However, the value of the stagger angle is not fixed by the inlet and outlet flow (or metal) angles. This is illustrated in the figure at the right, which shows two actual, very highly-loaded (Z = 1.37) low pressure turbine blade rows that were designed for identical inlet and outlet flow angles (α1 = 35o, α2 = 60o). The two blades clearly have very different stagger angles. This is the result of different decisions regarding the detailed pressure distributions around the blades. The blade with the high stagger angle was designed to be “forward-loaded”: that is, to develop most of its lift on the forward part of the airfoil. The one with the lower stagger angle is much more “aft-loaded”. The two airfoils have identical 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 have the “optimum” geometry according to Fig. 7.14. Despite these difficulties, it is possible to make an approximate Z = 1.37 comparison between the results obtained by the two different approaches to 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 different combinations of α1 and α2:

Using values of stagger angle obtained from K-O Fig. 5, the following figure shows the values of the Zweifel coefficient for selected combinations of inlet and outlet flow angles. It is evident that the optimum geometries based on the Ainley & Mathieson correlations lead to higher values of Z than Zweifel originally recommended. 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 results suggest that relatively higher values of Zweifel coefficient can be tolerated in the rotor blades for stages with low values of degree of reaction, especially if the total flow turning is low.

Z = 1.028 Z = 1.034

Z = 0.911

Z = 1.56 Z = 1.26

Z = 0.990 Z = 0.909

5.3.3

Losses

In both North America and Europe, most loss estimates for axial-flow turbines are based on a loss system developed by Ainley & Mathieson (AM) in the UK in the early 1950s (ARC R&M 2974, 1957; see also Saravanamuttoo et al.). The AM system has been updated a couple of times to reflect improvements in blade 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 Turbine Efficiency”, 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 curves or surfaces and these fits are given in Appendix E. For turbines, the total-pressure loss coefficient Y is defined as

Y=

ΔP0,loss P02 − P2

(1)

Note that in this case, the loss is non-dimensionalized by the outlet dynamic pressure, whereas the inlet value is used in the loss coefficients for axial compressors. As for compressors, losses are again divided into components and these are then added linearly to obtain the total losses:

YTotal = YP f ( Re) + YS + YTET + YTC

(2)

where the subscripts designate the components as follows: P = profile, S = secondary, TET = trailing-edge thickness, TC = tip clearance. f(Re) represents a correction for the effects of Reynolds number on the profile losses. The effect of Reynolds number on the other loss components is not well documented but it is believed to be small. The following figure shows the blade nomenclature used in the KO system. Note that they do not follow the sign convention we defined earlier. Using that convention, the inlet and outlet flow and metal angles will often have opposite signs because of the high turning that is normally present in turbine blade rows. It becomes a nuisance to keep track of the signs and therefore it is common practice by turbine designers 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 same outlet flow angle: a nozzle blade (maximum blade-passage acceleration) and an impulse blade (zero acceleration). In the original AM system, the expression took the form:

YP , AM

2 ⎡ ⎛ β1 ⎞ = ⎢YP ,nozzle + ⎜ ⎟ YP ,impulse − YP ,nozzle ⎝α2 ⎠ ⎢⎣

(

)

β1

⎤⎛ t c ⎞ α2 ⎥⎜ max ⎟ ⎥⎦⎝ 0.2 ⎠

(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) and Fig. 2 (for impulse blades). Note also that for a given value of the outlet angle α2 there is a value of solidity σ = c/s that minimizes the profile losses. This was the origin of the "optimum σ" that is plotted on Fig. 7.14 in Section 5.3.2.2.

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

YP , KO =

2 (0.914YP , AM ) 3

(4)

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

As seen, Eqn. (3) includes a correction for the maximum thickness of the airfoil: the data in Figs. 1 and 2 apply for a maximum thickness of 20% of the chord length. Decisions about the maximum airfoil thicknesses would not normally be made at the stage of a meanline analysis for the blade row. However, KO examined the range of maximum thicknesses observed for a number of recent actual designs and provided the correlation shown in Fig. 4. Knowing the flow turning from the velocity triangles, this figure can then be used to 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 turbines in gas turbine engines normally operate under compressible flow conditions. The Mach number levels encountered 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 and drive the high pressure compressor. To minimize the number of stages, HPTs are typically designed to operate at transonic outlet flow conditions. Low Pressure Turbine (LPT). The LPT drives the low pressure compressor, and the fan stage in a turbofan engine. The fan has a large tip diameter and to keep the tip Mach numbers acceptable, the fan shaft must rotate at a much lower speed than the high-pressure spool. The tip diameter of the LPT is much smaller than that of the fan and as a result it runs at a relatively low blade speed. This in turn results in lower flow velocities generally. It is therefore normal for the flow around LPT airfoils to be subsonic everywhere. As a result of these differences, the strongest effects of compressiblity are normally seen in HPTs. The following Schlieren photos (taken from E. Detemple-Laake, “Measurement of the Flow Field in the Blade Passage and Side Wall Region of a Plane Turbine Cascade,” AGARD-CP_469, 1989) show the flow through an 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 blades result in high local velocities in this region. For inlet relative Mach numbers as low as 0.6, patches of supersonic 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 accelerating flow channel. As the outlet Mach number increases, there is a tendency for the blade surface boundary layers to be thinned and their contribution to the losses actually decreases slightly. As the outlet Mach number approaches 1.0, patches of supersonic flow, terminating in shocks, may begin to appear on the aft suction surface. Finally, as the outlet Mach number becomes supersonic, expansion waves and shocks appear in the trailing edge region. In addition to directly contributing additional total pressure losses, it is common for one or more of the shocks to impinge on the surface of the adjacent 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 operating at an outlet Mach number of 1.30.

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

The complexity of the compressibility effects makes it difficult to predict their influence on the losses. Kacker & Okapuu provide procedures for estimating the contributions to the profile losses; see the paper for details. Finally, KO give the following Reynolds number corrections for profile losses: ⎛ Re c ⎞ f ( Re) = ⎜ ⎟ ⎝ 2 × 10 5 ⎠

− 0. 4

for

= 10 .

Re c ≤ 2 × 10 5

for 2 × 10 5 < Re c < 10 6

⎛ Re ⎞ = ⎜ 6c ⎟ ⎝ 10 ⎠

− 0.2

for

Re c > 10 6

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 secondary losses in axial turbines are a function of CL2: 2

⎛ cos α 2 ⎞ ⎛ C L ⎞ cos 2 α 2 YS = 0.04 f ( AR)⎜ ⎟⎜ ⎟ ⎝ cos β 1 ⎠ ⎝ s c ⎠ cos 3 α m

(5)

where CL = 2( tan α 1 + tan α 2 ) cos α m sc

1 α m = tan −1 ⎛⎜ ( tan α 2 − tan α 1 )⎞⎟ ⎝2 ⎠ and as before, all angles are taken as positive. The loss coefficients give the total-pressure losses as averaged over the total mass flow rate through the blade passage. As the aspect ratio of the blade becomes larger, a smaller fraction of the span is occupied by the secondary flow and the loss associated with it becomes averaged over an increasingly larger mass flow rate. Consequently, the mass-averaged loss coefficient varies inversely with the aspect ratio. This effect is embodied in the aspect ratio correction, f(AR) in Eqn. (5). Kacker & Okapuu found that the AM loss system tended to over-estimate the effect of aspect ratio on blades of very low aspect ratio (which are often used in modern HPTs). In the KO loss system, the correction for blade aspect ratio therefore takes the following form:

f ( AR) =

1 − 0.25 2 − h c

1 = hc

hc

for h c ≤ 2 (6)

for h c > 2

Aspect Ratio Correction, f(AR)

2.5

2 Ainley & Mathieson

1.5

1

Kacker & Okapuu (Eqn. (6))

0.5

0

0

0.5

1

1.5

2

2.5

3

Aspect Ratio, h/c

Kacker & Okapuu also provide a compressibility correction for the secondary losses (see the paper). Trailing-Edge Losses: Due to the finite thickness of the trailing edge, the streamtube experiences a sudden increase in area as it leaves the blade passage. The resulting sudden-expansion loss is correlated in terms of an alternative form of loss coefficient, known as an energy loss coefficient, Δφ2, as a function of the ratio of the trailing-edge thickess to the throat opening. KO correlated the values for nozzle blades and impulse blades separately, as shown in Fig. 14.

In the same way as for the profile losses, the trailing-edge loss for an arbitrary blade is expressed as the weighted average of the values for nozzle and impulse blades: 2

Δφ

2 TET

= Δφ

2 TET ( β1 = 0 )

(

⎛β ⎞ 2 2 + ⎜ 1 ⎟ Δφ TET ( β1 =α 2 ) − Δφ TET ( β1 = 0) ⎝α2 ⎠

)

(7)

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

YTET =

⎡ γ −1 2⎛ ⎞⎤ 1 M2 ⎜ − 1 ⎟⎥ ⎢1 − 2 2 ⎝ 1 − Δφ TET ⎠ ⎥⎦ ⎢⎣

γ −1 2⎞ ⎛ M2 ⎟ 1 − ⎜1 + ⎝ ⎠ 2

−

−

γ γ −1

−1

γ γ −1

and for incompressible flow this reduces to YTET =

1 2 Δφ TET

−1

Tip-Clearance Loss: For unshrouded blades, KO express the effects of tip-clearance losses as a correction to the efficiency:

Δη

η0

= 0.93

RTip k h cos α 2 R Mean

(8)

where η0 is the efficiency for zero tip clearance and k is the tip clearance. Note that Eqn. (8) indicates that a 1% increase in tip clearance, relative to blade span, will result in a 1% reduction in efficiency. This is considerably lower sensitivity than the 3% reduction that is predicted by Howell’s correlation for axial compressors. As seen, KO also found the loss to be a function of the hub-to-tip ratio of the blade, since 1 R Mean = RTip 1 + HTR 2 , where HTR = RHub/RTip. 2

(

)

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

YTC

c ⎛ k ′⎞ = 0.37 ⎜ ⎟ h⎝ c ⎠

0.78

2

⎛ C L ⎞ cos 2 α 2 ⎜ ⎟ ⎝ s c ⎠ cos 3 α m

(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 loss components for two different turbine stages, one subsonic and one transonic, will be quoted (taken from Moustapha et. al., Axial and Radial Turbines, Concepts NREC, 2003, pp. 89-90). The table summarizes the 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

Exit Mach Number

Stators

Rotor

Stators

Rotor

0.67

0.82

1.1

1.14

60

78

76

124

0.71

1.25

0.70

1.44

Total Flow Turning (o) Blade Aspect Ratio, h/c Tip Clearance, k/h (%)

1.5

Zweifel Coefficient

0.74

1.5

0.88

0.84

0.76

The figures show the resulting values of the loss coefficients:

Transonic Turbine

Subsonic Turbine 0.4

0.3

0.4 Stators

0.35

Rotor

Loss Coefficient, Y

Loss Coefficient, Y

0.35

0.25 0.2 0.15 0.1 0.05

0.3

Stators Rotor

0.25 0.2 0.15 0.1 0.05

0 Profile

Trailing Edge

Secondary

Tip Clearance

Loss Component

Total

0 Profile

Trailing Edge

Secondary

Tip Clearance

Loss Component

Total

PW100 Turboprop

2

95

1.8

90

1.6 η, (Rotor Only)

η, Efficiency (%)

85

1.4 Diffuser Loss

80 75

1.2 1

η (Rotor + Diffuser)

70

0.8

65

0.6

60

0.4

55

0.2

50

0

5

10

Tip Mach Number

100

0 15

Compressor Pressure Ratio

Data for rotor only: Senoo, Y., Hayami, H., Kinoshita, Y. and Yamasaki, H., "Experimental Study on Flow in a Supersonic Centrifugal Impeller," 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 Ratio Centrifugal 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 in Advanced 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 swirl in the rotor inlet flow and that the fluid is incompressible. Assume also the Euler Approximation so that the flow leaves the rotor parallel to the metal angle at the vane trailing edge. U2 W2 β2 (+)

C2

Cr2

β'2 (+)

Cw2

ω

From the Euler equation ∆h0 = g∆H E = U 2 C w 2 − U 1 C w1 = U 2 Cw2

and

tan β 2 =

Cw2 − U 2 Cr 2

or C w 2 = U 2 + Cr 2 tan β 2 Q = U2 + tan β 2 A2

Then g∆H E = U 22 + U 2

Q tan β 2 A2

or, dividing by N2D2

g∆H E Q = K1 + K 2 2 2 N D ND 3 2

where

U2 æπ ö K1 = 2 2 2 = ç ÷ = const . è 60 ø N D

(1)

and K2 =

U2D tan β 2 NA2

Equation (1) is the equation for the idealized head rise versus flow rate characteristic. Within the Euler Approximation ($N2 = $2), the slope of the characteristic, K2, is constant and has the same sign as $2, as shown in the sketch:

Note that whereas the slope of the )H vs Q characteristic for axial machines was always negative (assuming R < 1.0, which experience has shown is necessary), radial machines can have charateristics with either 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 centrifugal compression, even with no flow. From Eqn. (1), for Q = 0, )HE = U22/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: from specific 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 turbomachine that is suitable for a particular application. The following figures (from C. Rodgers, “Specific Speed and Efficiency 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 geometric and aerodynamic parameters.

The definition of specific speed used here is based average density (or average volume flow rate): 1

NS =

æ Q + Q2 ö 2 ωç 1 ÷ è 2 ø 3

( g∆H ) 4

Compare this with the usual non-dimensional specific speed:

Ω=

1 2 1

ωQ

3

( g∆H ) 4

Note that for a typical case, S will be slightly larger than N S . If there are no constraints on the rotational speed, then one would normally choose the value of N S that gives the highest 0 ( N S ≅ 0.6 − 0.8 ). With N S chosen and S estimated, the Cordier diagram can be 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 coefficient for industrial compressors of several configurations: with shrouded and unshrouded impellers; and with vaneless and vaned diffusers. The correlations are based on results for compressors with pressure ratios up to about 3.5, but can probably be extrapolated to somewhat higher values. Aungier defines the non-dimensional parameters as follows: Flow Coefficient:

φ=

& m Work Coefficient: ρ 1πr22 U 2

µP =

∆h0ref U 22

where )h0ref is the total enthalpy rise for the reversible process with the same pressure ratio. 0P is the stage polytropic efficiency. Aungier’s correlations are presented in the following two figures:

The figures are applied as follows: (a) Select N to give the best stage polytropic efficiency, 0P, and read the corresponding work coefficient, :P. From

µP =

∆h0ref U 22

calculate U2. (b) Then from the chosen N

φ=

& m ρ 1πr22U 2

calculate r2. With the tip radius and tip blade speed defined, the rotational speed is known. If the rotational speed is constrained (eg. driving motors are only available for certain speeds) then Fig. 6-1 or Fig. 6-2 can be used to select a compromise size and rotational speed that minimizes the impact on the stage efficiency.

6.3.2 Rotor Inlet Geometry From the Euler equation Δh0 = gΔH E = U 2 C w 2 − U 1 C w1

If there are no IGVs, Cw1 = 0 and the work transfer depends entirely on the rotor tip or outlet conditions. For good efficiency, the impeller inlet must nevertheless be well designed (eg. the inducer inlet metal angle must be matched to the inlet relative flow vector) and correctly sized. Consider three rotors designed for the same m& , U2 and with the same outlet geometry so that all three give the same Δh0. The critical region for frictional losses (which vary as V2), cavitation and compressibility effects is at the vane tip at the inlet, since that is where the relative velocity is the highest and static pressure the lowest. The drawing shows the resulting inlet tip velocity triangles for three different inlet sizes:

SMALL EYE LOW U1t HIGH C1t

LARGE EYE HIGH U1t LOW C1t

r2

r1t

r1h

C1t

W1t

C1t

U1t

W1t

C1t U1t

W1t

U1t

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 the inlet relative velocity and Mach number:

W1t

OPTIMUM

M1t

r1t

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

m& = ρ 2 Cr 2 A2 = ρ 2 Cr 2 (2πr2 b2 ) and thus for a fixed m& , the choice of b2 determines the radial component of velocity at the rotor outlet:

Cw 2 U2 C2

W2

Cr 2

SMALL b2

β2 LARGE b2 C2

W2

b2 r2

ω

Summarizing the effect of different choices of b2:

LARGE b2

SMALL b2

C2

Lower (Good)

Higher (Bad - Larger diffusion required 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 rotor passage and high diffusion in the downstream diffuser (which serves the same function as the stators in an axial compressor stage). Note that W2/W1 is again the de Haller number. Various papers and textbooks provide guidelines for

choosing the de Haller number for centrifugal fan, compressor, and pump rotor passages: (1) Aungier (2000)

Recommended: Never exceed:

W2/W1 > 0.75 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 > 0.8 High 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. The corresponding metal angle $2Ndepends on the deviation, which is called “slip” in centrifugal machines. The slip in turn depends on the rotor “solidity”: that is, the number of vanes, Z. Thus, the choices for $2N and Z are inter-related. Consider a backswept rotor:

Because of slip, the rotor imparts less swirl to the flow than for the “ideal” case, for which $2 =$2N (that is, the Euler approximation is taken to hold in the ideal case). Since Cw2 < Cw2N, the )h0 is reduced by this effect. We then define the slip factor F as

σ=

Cw 2 C ′ w2

where F # 1.0. A number of correlations have been proposed for F. The one due to Stodola has been widely used:

π ′ cosæç β 2 ö÷ ø è Z σ = 1− ′ 1 − φ 2 tanæç β 2 ö÷ ø è

where N2 = Cr2/U2 and $2N is the backsweep or forwardsweep angle (taken as positive in both cases). Stanitz suggested a slightly simpler form:

0.63

σ = 1−

π Z

1 − φ 2 tan β 2

′

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 the Stodola, Stanitz and similar correlations are only valid for impellers with long blades. Wiesner recommended the 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 1

ε lim it = e

æ 8.16 cos β ′ ö 2 ÷ ç ç ÷ Z è ø

then for , # ,limit (ie. longer vanes)

σ = 1−

sin β 2

′

Z 0.7

and for , > ,limit (ie. shorter vanes)

æ ç σ = ç 1− ç è

3 ö cos β 2 ′ ÷ æ æ ε − ε lim it ö ö ç 1− ç ÷ ÷ Z 0.7 ÷÷ çè è 1 − ε lim it ø ÷ø ø

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

6.3.5 Choice of Number of Vanes - Vane Loading Wilson & Korakianitis (Design of High-Efficiency Turbomachinery and Gas Turbines, 2nd ed., PrenticeHall, 1998) provide a broad guideline for selecting the number of blades, as function of the vane angle at the tip, as shown in the figure at right. More recently, Rodgers (2000) presented a correlation for the number of vanes which, according to his loss estimates, gives the best rotor efficiency:

Z=

25cos β 2 Ω

′

where Ω is the usual non-dimensional specific speed. Comparison with the Wilson & Korakianitis figure suggests the Rodgers’ correlation is very conservative, leading to very large numbers of vanes. Aungier (2000) outlines a method of selecting the number of vanes based directly on the vane loading. He suggests the following limit:

CENTRIFUGAL COMPRESSOR - NUMBER OF VANES

2 ΔW ≤ 0.9 W2 + W1

45

40

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

2π D2 U 2 ψ Z LB

30

where ψ = work coefficient = Δh0/U22 and LB is the length of the vane along the mean camber line. A reasonable initial estimate of LB can be obtained from

b ⎞ 1 ⎛ D − D1 ⎞⎟ ⎛ L B = ⎜ Δz I − 2 ⎟ + ⎜ 2 ⎝ 2 ⎠ 2 ⎜⎝ cos β ′ ⎟⎠

Number of Vanes, Z

ΔW =

35

25

20

15

10

2

5

where ΔzI is the axial length of the rotor. 0

0

10

20

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

30 40 Backsweep Angle (Deg.)

50

60

70

6.3.6 Losses

The actual stage characteristics are different from ideal due to slip and losses. Slip reduces output but does not affect efficiency since the required input power is reduced along with the output. Sources of losses: (1) Disc friction:

(2) Leakage:

(3) Inlet:

(4) Impeller: (5) Diffuser/Volute:

- friction on outer surface of impeller - since this torque is not exerted on the through-flowing fluid, it does not appear in the Euler work, DgQ)HE - 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 the inlet, to be recompressed over and over again - thus, more fluid is compressed than is delivered by the machine, increasing the power required and showing up as an apparent loss - at other than design Q, flow angle and metal angle will be mismatched at the leading edge, resulting in separation and additional losses - a simple, inexpensive machine with no inducer will have significant inlet losses at all operating conditions - frictional and separation losses inside the impeller channels - roughly " Q2 - 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, for backward-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 stable operating 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& = ρgQ∆H E ( th ) + ρ gQl ∆H E ( th ) + Disc & Bearing Friction Power

where

)HE(th) = Ql =

theoretical Euler head (Euler head with slip but no losses) 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 E ( th ) − ∆H L

where

)HL

=

sum of losses (3) + (4) + (5)

and the corresponding efficiency is

η overall =

ρgQ∆H W& shaft

& for DQ and )h0 for g)H. As usual, for compressible flow substitute m

CHAPTER 7 Static and Dynamic Stability of Compression Systems

7.1 INTRODUCTION

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

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

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

It was also noted that surge is a dynamic instability which depends on not just the characteristics of the compressor but also on the aerodynamic characteristics of the other components to which it is connected. It is possible to develop a simple lumped-parameter analysis for a compression system. Such an analysis can provide useful insights into which characteristics of the system encourage or delay the onset of surge. For further information see Stenning (1980), Greitzer (1980, 1981) and Cumpsty (1989).

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

7.2 STATIC STABILITY 7.3 DYNAMIC STABILITY - SURGE A system is statically stable if, when it is disturbed by a small amount from its equilibrium operating point, a reaction arises which tends to restore it to the equilibrium condition. Static stability is normally a necessary, but not sufficient condition for dynamic stability

A simple analysis can be developed to predict approximately the dynamic stability characteristics of a compression system. A compressible flow system will be examined. Only minor modifications are needed to make it apply to an incompressible flow system.

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

Fig. 7.2 shows schematically a simple system consisting of 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 first approximation, the throttle could also represent the turbine in a gas turbine engine.

Fig. 7.2 Four-component compression system.

Fig. 7.1 Compressor operating points.

1

The flow through the components is treated as onedimensional. Thus, the flow at any point is characterized by & etc. (if necessary, these would a single value of P, T, C, m be interpreted as the local average values). The analysis will consider perturbations about an equilibrium operating point and the perturbations will be assumed to be small. The instantaneous value of any flow quantity is represented by the sum of the mean value plus the instantaneous (small) perturbation:

m2 = m2 + m2′

d ( P2 − P01 ) dm1

=

dC =c dm1

and since we are assuming that the perturbations from the operating point are small, we can assume that c is constant in our analysis. That is, we linearize the compressor characteristic at the operating point of interest.

P3 = P3 + P3′ etc.

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

where m2 is the mass flow rate at plane 2 (the dot is omitted for convenience). The goal of the analysis is to determine the behaviour of the perturbations over time after some initial disturbance has occurred. If m2N, P3N etc. eventually decrease to zero, the system is dynamically stable at the operating point in question.

dP2 dP01 dP2 − = =c dm1 dm1 dm1

(3)

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

The approach used is known as the lumped-parameter method: equations for the behaviour of each component are developed separately and they are then linked by the flow conditions at the interfaces between the components. Consider each component in turn:

P2

m1

∫ dP = ∫ cdm 2

P2

1

m1

(

P2 − P2 = c m1 − m1

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

P2 − P01 = C(m1 )

(2)

)

From the definition of the perturbations, this can be written

P2′ = cm1′

(1)

(4)

This is then the perturbation equation for the compressor.

where C is the function which defines the compressor characteristic (see Fig. 7.3). The gradient at any operating point along the characteristic is

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

P2′ = cm2′

(5)

(2) Duct We assume that the losses in the system occur primarily in the throttle so that we can neglect the frictional losses in the duct. We also neglect the volume of the duct relative to the volume of the plenum. Therefore, the duct introduces only inertia: a pressure difference is present between stations 2 and 3 only when the fluid in the duct is being accelerated or decelerated. The equation governing the behaviour of the duct can be obtained either by performing a force balance on the free body consisting of the cylinder of fluid in the duct or by

Fig. 7.3 Compressor characteristics.

2

applying the unsteady momentum equation to a control volume occupying the duct. For both analyses, we will neglect density changes along the duct.

P2′ − P3′ =

L dm2′ A dt

(7)

(i) Force balance: (3) Plenum

d (mu) dt d P2 A − P3 A = ( ρ A Lu) dt d = L ( ρ Au) dt Σ Fx

=

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

m2 − m3 = V

L dm2 A dt

(8)

where m3 = mass flow rate through the valve. Changes in the mass in the plenum will be reflected in the density of the stored gas. In a pump system, a reservoir with a free surface or a surge tank would similarly act as a mass storage component.

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

P2 − P3 =

dρ3 dt

(6) If the compression or expansion process is isentropic, then

(ii) Control volume analysis:

P

ργ

For the control volume in the duct

ΣFx =

∫ u ρ dV + (mu)

d dt

V

out

− (mu)in

Differentiating with respect to time and assuming a perfect gas

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

ΣFx =

d dt

∫

L

u ρ Adx =

0

d⎛ ⎜ m2 dt ⎜⎝

dρ3 ρ dP3 = 3 dt P3 γ dt 1 dP3 = γ RT3 dt

⎞ dm dx⎟ = L 2 ⎟ dt 0 ⎠

∫

= const.

L

=

1 dP3 a32 dt

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

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

m2 − m3 = We then substitute into (6) in terms of the perturbations

(

) (

)

P2 + P2′ − P3 + P3′ =

(

L d m2 + m2′ A dt

)

V dP3 a32 dt

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

m2′ − m3′ = and since there are no losses in the duct, the mean inlet and outlet pressures must be the same. Thus, the perturbation equation for the duct becomes

V dP3′ a32 dt

(9)

(4) Throttle The throttle is handled in exactly the same way as the compressor: the load line is linearized at the equilibrium operating point. If the valve is choked, the mass flow rate

3

through it is a function of only the upstream pressure, P3. If it is not choked, the pressure downstream is assumed to be constant. Then the perturbation equation for the throttle becomes:

P3′ = f m3′

dP3′ dP2′ L d 2 m2′ = − dt dt A dt 2

(10)

and from (11)

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

dm′ dP2′ =c 2 dt dt

Characteristic Equation for the System

Thus

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

P2′ = cm2′

dP3′ dm′ L d 2 m2′ =c 2 − dt dt A dt 2

(11)

L dm2′ P2′ − P3′ = A dt V dP3′ m2′ − m3′ = 2 a3 dt

or

(12)

f V dP3′ c f V dm2′ f V d 2 m2′ = − a32 dt a32 dt a32 dt 2

(13)

P3′ = f m3′

(14)

Substituting (18) into (17) and rearranging

These are four equations in the four unknowns m2N, m3N, P2N and P3N. Solving for any one of the unknowns from (11) (14) leads to a second-order ordinary differential equation for the variation in time for that unknown.

f V L d 2 m2′ ⎛ L c f V ⎞ dm2′ + ( f − c) m2′ = 0 (19) + ⎜⎜ − 2 ⎟⎟ a32 A dt 2 a3 ⎠ dt ⎝A

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

cm2′ − f m3′ =

L dm2′ A dt

This is seen to be a second-order ordinary differential equation in m2N. It can be shown that the corresponding equation for any of the other three perturbations would have the save coefficients as (19).

(15)

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

f m2′ − f m3′ = f

V dP3′ a32 dt

Within the assumptions of the analysis, the coefficients are constant and, given initial conditions for m2N and dm2N/dt, (19) can readily be solved to determine the response of the system. As noted earlier, if m2N tends to 0 with increasing time, the system is dynamically stable.

(16)

A useful analogy can be drawn between the present system and a mass-spring-damper system for which the governing equation is (for free vibrations)

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

V dP ′ L dm2′ f m2′ − cm2′ = f 2 3 − A dt a3 dt

(18)

m

d2x dt

(17)

2

+s

dx + kx = 0 dt

(20)

where s = damping coefficient, k = spring constant. Two conditions must be met for the system governed by (20) to be stable:

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

(i) k > 0 - that is, the spring constant must be positive The equivalent condition in (19) is that f > c, which is

4

It is not immediately clear whether it requires a large positive value of c (large in comparison to f, for example) to destabilize the system, but note that as L tends to zero c also tends to zero. Therefore, a compressor which is connected to a plenum by a very short length of duct will become unstable essentially at the peak of the compressor characteristic. That is why the latter is often used as a criterion for predicting surge. In general, we would expect to encounter the condition for dynamic instability near the peak of the characteristic and probably long before we reach the condition for static instability (operating point C on the & diagram) . original )P0 versus m

precisely the requirement for static stability which we arrived 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: the system has a critical value of the damping coefficient, sc, given by

sc = 2 k m

If s < sc, the system is under-damped and the solution takes the form

x = Ae

−

s sc

k t m

⎛ 2 ⎛ s⎞ ⎜ sin⎜ 1 − ⎜ ⎟ ⎝ sc ⎠ ⎜ ⎝

The relationship between stall and surge now is a little clearer. For a typical compressor characteristic, as the flow rate through the machine is reduced the output peaks and eventually begins to reduce. This is generally the result of increasingly extensive stall: perhaps an increasing number of rotating stall cells and/or cells of increasing spanwise extent as the flow rate is reduced. Stall thus prepares the conditions for surge. Note that the appearance of stall is a phenomenon of the compressor itself, not the system. On the other hand, surge is an unstable condition in compression system in which flow quantities, including the compressor mass flow and delivery pressure, undergo oscillatory fluctuations which grow over time. In systems such as gas turbine engines, these fluctuations can reach destructive magnitudes in a very small number of cycles.

⎞ ⎟ k t + φ⎟ m ⎟ ⎠

Thus, the system oscillates sinusoidally in time, with the magnitude of the peak displacement being controlled by the exponential term. Since m, k, and sc are all positive, if s > 0 the exponential term decreases in time, the magnitude of the fluctuations decays, and the system is seen to be dynamically stable. If s > sc, the system is over-damped and the solution is no longer oscillatory but it again includes exponential terms which are a function of s. Again, if s is negative the exponential terms grow in time and the system moves away from the equilibrium point in an unstable way.

References Cumpsty, N.A., 1989, Compressor Aerodynamics, Longman, Harlow. Greitzer, E.M., 1980, “Review - Axial Compressor Stall Phenomena,” ASME J. Fluids Engineering, Vol. 102, June 1980, pp. 134-151.

Applying these ideas to the compressor system, it is seen that there are two contributions to the system damping:

Greitzer, E.M., 1981, “The Stability of Pumping Systems,” ASME J. Fluids Engineering, Vol. 103, June 1981, pp. 193242.

(a) positive (stabilizing) damping is supplied by the inertia of the fluid in the duct (the L/A term), and (b) potentially negative damping is supplied by the term involving the slopes of the compressor and throttle characteristics.

Stenning, A.H., 1980, “Rotating Stall and Surge,” ASME J. Fluids Engineering, Vol. 102, March 1980, pp. 14-20.

Since f is always positive, the sign of the damping term is controlled by the sign of the slope of the compressor characteristic, c. If c < 0 (as it normally is at higher flow rates) strong positive damping will be present and the system will be stable. The condition for instability is then

L cfV − 2 >0 A a3

or

c>

La32 AV f

Thus, the system will become unstable for some positive value of the slope of the compressor characteristic, the precise magnitude being a function of a number of system parameters.

5

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/F): With: A = 33.5293 D = 0.00209610

B = -0.530812 E = -0.677212

C = -15.2599 F = 0.187148

,*("2,s/c) = A + B"2 + C ln(s/c) + D"22 + E(ln(s/c))2 + F"2ln(s/c) Applies for: 0 < "2 < 70o, 0.5 < s/c < 1.5 (or 0.666 < F < 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: 1

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 + 0.05002irel2 + 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 < F < 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 + 0.06477irel3 + 0.3949irel4 + 0.3426irel5

2

APPENDIX B: C4 Compressor Blade Profiles

Like NACA 4-digit airfoils, the C-series compressor blades are defined by a symmetrical thickness distribution which is superimposed on a specified mean, or camber, line. As indicated in the Howell correlations, both circular arc and “parabolic” arc camber lines have been used with C-series blades. For the blade with a parabolic arc camber line, the point of maximum camber lies at other than mid-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:

(1)

and

The term parabolic arc camber line is somewhat misleading. The mean line is not defined by a single parabola, or even by two joined parabolas. For example, to define a polynomial which passes 1

through (0,0) with slope tan21 and through (a,b) with zero slope requires at least a cubic. The following discussion will consider mainly the circular arc camber line. Setting a/c = 0.5 in Eqn (1),

(2)

The equations of the camber line and its inclination, Nc, are then

(3)

and (4)

The co-ordinates of the upper and lower sides of the blade are then

(5)

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 blade designated 10C4/30C50 refers to a blade with a C4 profile and: 10% maximum thickness, circular arc camber, camber angle 30o and maximum camber at 50% chord (the last piece of information is redundant 2

since circular arc camber has already been specified). The resultant geometry is shown:

3

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 as a function of inlet flow angle, $1, and solidity, F: With:

A00 = -0.13571 A10 = 0.015986

A01 = 0.075795 A11 = 0.074959

A02 = 9.1315x10-4 A20 = -2.4954x10-4

i0(10)($1,F) = A00 + A01F + A02F2 + A10$1 + A11$1F + A20$12 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 A03 = 0.033568 A11 = 7.402x10-3 A20 = -3.3001x10-5 A30 = 8.0286x10-7

A01 = 0.05897 A04 = -7.1706x10-3 A12 = -2.5749x10-3 A21 = -3.084x10-5 A31 = -1.2016x10-7

A02 = -0.054019 A10 = -6.0476x10-3 A13 = 2.6067x10-4 A22 = 1.3955x10-5 A40 = -9.1961x10-9

n1($1,F) = A00 + A01F + A02F2 + A03F3 + A04F4 + A10$1 + A11$1F + A12$1F2 + A13$1F3 n2($1,F) = A20$12 + A21$12F + A22$12F2 + A30$13 + A31$13F + A40$14 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.

1

(d) Zero Camber Deviation Angle, *0 (SP-36 Fig. 161) With:

A00 = 0.053535 A03 = -0.75902 A10 = -3.838x10-3 A13 = 3.4149x10-3 A21 = 2.0917x10-4 A30 = -1.3124x10-5 A40 = 2.3356x10-7

A01 = -0.29275 A04 = 0.3706 A11 = 0.02838 A14 = 5.8448x10-4 A22 = 3.0519x10-4 A31 = -1.0755x10-5 A41 = 1.1718x10-7

A02 = 0.71879 A05 = -0.067233 A12 = -0.02068 A20 = 3.5333x10-4 A23 = -1.2273x10-4 A32 = 1.7229x10-6 A50 = -1.4651x10-9

*o1($1,F) = A00 + A01F + A02F2 + A03F3 + A04F4 +A05F5 + A10$1 + A11$1F + A12$1F2 + A13$1F3 + A14$1F4 *o2($1,F) = A20$12 + A21$12F + A22$12F2 + A23$12F3 + A30$13 + A31$13F + A32$13F2 + A40$14 + A41$14F + 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$13 b($1) = 0.965 - 2.5464x10-3$1 + 4.2695x10-5$12 - 1.3182x10-6$13

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) 2

Valid for: 0.4 < F < 1.8, 0.0 < $1 < 70.0.

3

APPENDIX D: NACA 65-Series Compressor Blade Profiles The 65-series blade geometry is not represented by closed-form analytical expressions. Instead, it is 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)

1

The thickness distribution is given for a NACA 65-010 blade which has been modified to give a finite trailing-edge thickness of 0.3% of the chord length. The baseline thickness distribution has zero thickness at the trailing edge and therefore cannot be manufactured. The nominal maximum thickness is 10% of chord. For blades with other values of maximum thickness, the tabulated distribution is simply scaled accordingly. The table indicates that maximum camber is at 50% of chord. However, the camber line is not a simple 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 sensitivity to incidence. Because of the camber line shape, there is no simple relationship between the camber angle, as defined earlier, and the magnitude of the maximum camber. Instead, the camber line shape is related to the nominal maximum lift coefficient which the blade shape would achieve as an isolated airfoil. The camber line shape quoted applies for a nominal lift coefficient CL = 1.0. To generate compressor blades with a desired camber angle, the following can be used to relate an equivalent circular arc camber angle to the nominal CL:

(1)

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 thickness distribution (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 thickness and the equivalent camber:

2

APPENDIX E: Curve and Surface Fits for Kacker & Okapuu Loss System for Axial Turbines Kacker & Okapuu ("A Mean Line Prediction Method for Axial Flow Turbine Efficiency," ASME Trans., J. Eng. for Power, Vol. 104, January 1982, pp. 111-119) presented an updated version of the Ainley & Mathieson loss system for axial turbines. The Kacker & Okapuu (KO) system presents a basis for estimating the complete losses, and thus the efficiency, of an axial turbine at its design point. For a complete outline of the loss system see the paper. Some aspects of the loss system are presented only in graphical form in the paper. Therefore a number of figures have been digitized and curves or surfaces fitted to the data. This appendix documents the curve fits and, in some cases, demonstrates the quality of the fits graphically. The figure numbers refer 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 is obtained as a weighted average of the values for a nozzle blade ($1 = 0) and an impulse blade. These values are obtained from the plots shown in Figures 1 and 2. The data in these figures have been fitted to polynomial surfaces of the form:

The values of the coefficients follow: (i)

Nozzle Blade,

(Fig. 1)

a0,0 = 0.358716 a0,1 = -1.43508 a0,2 = 1.57161 a0,3 = -0.496917 a1,0 = -0.0112815 a1,1 = 0.0548594 a1,2 = -0.0555387 a1,3 = 0.014165 a2,0 = 0.000175083 a2,1 = -0.000824937 a2,2 = 0.000652287 a2,3 = -7.30141E-05 a3,0 = -8.61323E-07 a3,1 = 3.95998E-06 a3,2 = -1.89698E-06 a3,3 = -4.9954E-07

1

(ii)

Impulse Blade,

(Fig. 2)

a0,0 = 0.0995503 a0,1 = 0.182837 a0,2 = 0.01603 a1,0 = 0.00621508 a1,1 = -0.0283658 a1,2 = 0.011249 a2,0 = -7.10628E-05 a2,1 = 0.000327648 a2,2 = -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 true chord. KO present an approximate correlation for the stagger angle as a function of the inlet and outlet angles. The true chord can then be calculated from the axial chord. The graphical data are again fitted to a surface, using a polynomial of the form:

with coefficients, a0,0 = -2.90463 a1,0 = 0.412797 a2,0 = 0.593956E-02

a0,1 = 0.307036 a1,1 = -0.355369E-01 a2,1 = 0.389157E-03

The surface fit and the digitized values are compared over.

2

a0,2 = 0.370176E-02 a1,2 = -0.194938E-03 a2,2 = 1.74147E-06

(c) Inlet Mach Number Ratio (Fig. 6) A correction is made for shock losses at the leading edge of the blade. Since the Mach number tends to be higher at the hub than at midspan, KO present a correlation for the hub Mach number as a function of the midspan value and the hub-to-tip ratio. The shock loss is then calculated from the estimated hub Mach number. The following polynomials were fitted to the curves of Figure 6: (i)

Rotors

(ii)

Nozzles

3

(d) Trailing-Edge Energy Coefficient (Fig. 14) The trailing-edge losses are expressed in terms of the energy coefficient. This was correlated with the ratio of the trailing-edge thickness to the throat opening. Curves were presented for nozzle and impulse blades. The values from these curves are then averaged in a weighted way to give the coefficient for a blade of arbitrary inlet and outlet flow angles. (i)

Impulse (Rotor) Blade

(ii)

Nozzle

4

5

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 often conflicting considerations: aerodynamics, heat transfer, materials, stresses, and vibrations (not to mention cost). While our focus is on the aerodynamics, it is obviously wasteful to develop even a preliminary aerodynamic 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 steady stress determines the margin which is available for these unsteady stresses. In turbines, creep distortion is an 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 be mechanically feasible. Fortunately, the steady centrifugal stresses in the rotor blades can be estimated fairly 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 stresses are particularly high in low hub-to-tip ratio fan blades and in turbine blades; they are much lower in normal compressor blades. A survey of typical, industrial axial-flow fans from several manufacturers shows that peak tip speeds are consistently below 120 m/s. It is believed that this limit is related to the stresses which can be sustained by the rather simple blade attachments, rather than stresses in the actual blades. Higher tips speeds can be used but these require a switch to a considerably more expensive method of attachment. 2.0

Steady Centrifugal Stresses

1

Consider the forces on the small blade element shown.

Then (1)

and this can then be integrated from radius R to the tip, RT, (with a specified blade area variation) to obtain the centrifugal stress at R. Constant Section Blade: With dA = 0, integrating (1):

and the maximum stress occurs at the root: (2)

Tapered Blades: The cross-sectional area of turbomachinery blades often varies from hub to tip. If the area decreases, the root stress will be reduced from the value given by (2). Taking into account the taper, the hub stress can be written

(3)

where K depends on the nature of the taper in cross-sectional area: (a)

Blade with constant cross-section.

(b)

Blade with linear taper.

2

where

The cross-sectional area of the blade is roughly proportional to the product of the chord length (c) and the maximum 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, to maintain constant maximum thickness-to-chord ratio, the cross-sectional area will in fact vary parabolically. 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, (4)

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 the proposed design is feasible mechanically. The following table gives values of KAN2 which are 3

reasonably representative of the current stress limits for axial turbomachines:

KAN2

KAN2

(A in inches2, N in RPM)

(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

MACHINE TYPE

4

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MECH 5401 TURBOMACHINERY

SUPPLEMENTARY COURSE NOTES

S.A. Sjolander January 2010

CARLETON UNIVERSITY Department of Mechanical & Aerospace Engineering MECH 5401 - Turbomachinery COURSE 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. Blade loading and choice of solidity. Degree of reaction.

4

Correlations for performance estimation at the design point for: axial compressors, axial turbines and centrifugal compressors. Approximate off-design performance: compressor maps and turbine characteristics.

5

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

6

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

7

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

8

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

9

Compressible flow effects: choking in turbomachinery blade rows; shock waves in transonic compressors 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 wakeblade interaction. Approximate analysis of unsteady behaviour of compression systems: dynamic system instability (surge); factors affecting compressor surge.

11

Current issues in turbomachinery aerodynamics. Very high loading for weight and blade-count reduction. 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. Multidisciplinary optimization.

S.A. Sjolander January 2010

Department of Mechanical and Aerospace Engineering CARLETON UNIVERSITY MECH 4305 - Fluid Machinery TABLE OF CONTENTS Page 1.0

INTRODUCTION 1.1 1.2 1.3

2.0

Course Objectives Positive-Displacement Machines vs Turbomachines Types of Turbomachines

NON-DIMENSIONAL PARAMETERS AND SIMILARITY 2.1 2.2

Dimensional Analysis - Review Application to Turbomachinery 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5

2.3 2.4 2.5 2.6

3.0

Non-Dimensional Parameters for Incompressible-Flow Machines Effect of Reynolds Number Performance Curves for Incompressible-Flow Turbomachines Non-Dimensional Parameters for Compressible Flow Machines Performance Curves for Compressible-Flow Turbomachines

Load Line and Operating Point Classification of Turbomachines - Specific Speed Selection of Machine for a Given Application - Specific Size Cavitation

FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS AND THERMODYNAMICS 3.1 3.2 3.3 3.4 3.5

Steady-Flow Energy Equation Angular Momentum Equation Euler Pump and Turbine Equation Components of Energy Transfer Velocity Diagrams and Stage Performance Parameters 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6

Simple Velocity Diagrams for Axial Stages Degree of Reaction de Haller Number Work Coefficient Flow Coefficient Choice of Stage Performance Parameters for Design

3.6

Efficiency of Turbomachines 3.6.1 3.6.2

4.0

Incompressible-Flow Machines Compressible-Flow Machines

AXIAL-FLOW COMPRESSORS, FANS AND PUMPS 4.1 4.2

Introduction Control Volume Analysis for Axial-Compressor Blade Section 4.2.1 4.2.2

4.3

Idealized Stage Geometry and Aerodynamic Performance 4.3.1 4.3.2 4.3.3 4.3.4

4.4 4.5

Blade Passage Flow and Loss Components Loss Estimation Using Howell’s Correlations Loss Estimation Using NASA SP-36 Correlations Effects of Incidence and Compressibility Relationship Between Losses and Efficiency

Compressor Stall and Surge 4.7.1 4.7.2

4.8 4.9

Introduction Blade Design and Analysis Using Howell’s Correlations Blade Design and Analysis Using NASA SP-36 Correlations

Loss Estimation for Axial-Flow Compressors 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5

4.7

Meanline Analysis Blade Geometries Based on Euler Approximation Off-Design Performance of the Stage Spanwise Blade Geometry

Choice of Solidity - Blade Loading Limits Empirical Performance Predictions 4.5.1 4.5.2 4.5.3

4.6

Force Components Circulation

Blade Stall and Rotating Stall Surge

Aerodynamic Behaviour of Multi-Stage Axial Compressors Analysis and Design of Low-Solidity Stages - Blade-Element Methods

5.0

AXIAL-FLOW TURBINES 5.1 5.2 5.3

Introduction Idealized Stage Geometry and Aerodynamic Performance Empirical Performance Predictions 5.3.1 5.3.2

5.3.3 6.0

CENTRIFUGAL COMPRESSORS, FANS AND PUMPS 6.1 6.2 6.3

Introduction Idealized Stage Characteristics Empirical Performance Predictions 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6

7.0

Flow Outlet Angle Choice of Solidity - Blade Loading 5.3.2.1 Zweifel Coefficient 5.3.2.2 Ainley & Mathieson Correlation Losses

Rotor Speed and Tip Diameter Rotor Inlet Geometry Rotor Outlet Width Rotor Outlet Metal Angle - Slip Choice of Number of Vanes - Vane Loading Losses

STATIC AND DYNAMIC STABILITY OF COMPRESSION SYSTEMS 7.1 7.2 7.3

Appendix A: Appendix B: Appendix C: Appendix D: Appendix E: Appendix F:

Introduction Static Stability Dynamic Stability - Surge

Curve and Surface Fits for Howell’s Correlations for Axial Compressor Blades C4 Compressor Blade Profiles Curve and Surface Fits for NASA SP-36 Correlations for Axial Compressor Blades NACA 65-Series Compressor Blade Profiles Curve and Surface Fits for Kacker & Okapuu Loss System for Axial Turbines Centrifugal Stresses in Axial Turbomachinery Blades

Department of Mechanical and Aerospace Engineering CARLETON UNIVERSITY MECH 4305 - Fluid Machinery Recommended Texts S.L. Dixon, Fluid Mechanics, Thermodynamics of Turbomachinery, 5th ed., Elsevier ButterworthHeineman, 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 preliminary aerodynamic 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 while others 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.

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 the famous 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.

NOMENCLATURE FOR TURBOMACHINES ENERGY TRANSFER TO THE FLUID Fans Blowers

Gases Incompressible flow

Pumps

Liquids

Compressors

Gases

Compressible flow

Propellers

Both

Both

ENERGY TRANSFER FROM THE FLUID Turbines Turbo-expanders Wind mills/Wind turbines

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 that around 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

⎛ lift force ⎞ ⎛ L⎞ =⎜ ⎟ ⎟ ⎜ ⎝ drag force ⎠ model ⎝ D ⎠ prototype 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 α γ - specific heat ratio Note that the pressure and temperature are not quoted. U For a perfect gas,

P = ρ RT

a = γ RT

L

D M c

Thus, by specifying a, γ and R, we have implicitly specified T. Similarly, with ρ and R specified, and T 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.

(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 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:

ρ ×

1

× U

× c

μ LT M L × × × L 3 M T L

=

ρU c μ

which is clearly the Reynolds number, Re α

is already non-dimensional and can be used directly as a criterion of similarity is also already non-dimensional

γ

U a

=

Mach number , M

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

×

1 U2

×

1 = c ×1

L L3 × × T2 M

T2 L2

×

1 LL

D ×

M

1

ρ

then C D = f ( Re, M , α , γ )

D ρU 2 c

or

D 1 ρU 2 c 2

(≡ CD )

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

CL CD

=

L D

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 in Section 2.4. D Q

N

W& Q

For a given geometry, the independent variables that determine performance are usually taken as . D ρ μ N (or ω) Q-

machine size (usually rotor outside diameter) fluid density fluid viscosity machine speed; revs or rads per unit time 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 (by controlling 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

ΔH W& T η -

total head rise across machine (or sometimes, total pressure rise) shaft power absorbed by the machine torque absorbed by the machine efficiency of the machine

ΔH = f 1 ( D, N , Q, ρ , μ ) W& = f 2 ( D, N , Q, ρ , μ ) etc. 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:

Q ×

1 N

×

L3 × T

T 1

×

1 D

Q

=

3

N D3

1 L3

This is known as the flow coefficient, capacity coefficient or flow number (2)

Fluid properties (specifically, viscous effects):

ρ N DD = μ

ρ N D2 μ

ie. the Reynolds number

All other non-dimensional ratios or coefficients then depend on these two criteria of similarity. For power coefficient (non-dimensional work per unit time)

W&

×

1 N3

⎛ M L2 ⎞ ⎜ 2 ⎟ ⎝ T ⎠ × T then

W&

ρ N 3 D5

1

×

T3 1 =

ρ

×

L3 M

1 D5

×

×

=

W& ρ N 3 D5

1 L5

⎛ Q ρ N D2 ⎞ ⎟ , f⎜ 3 μ ⎠ ⎝ ND

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

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

P V2 + +z ρ g 2g = static head + dynamic head + elevation head

H=

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:

g ΔH L T

2

1

×

N

2

T2 1

L ×

× ×

1 D

g ΔH

=

2

N 2 D2

1 L2

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:

⎛ gH ⎞ ⎜ or 2 2 ⎟ ⎝ N D ⎠

g ΔH N 2 D2

=

⎛ Q ρ N D2 ⎞ ⎟ , f⎜ 3 μ ⎠ ⎝ ND

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

g ΔH 2

N D

2

=

ρ g ΔH ρ N 2 D2

=

Δ P0

ρ N 2 D2

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

and

fluid power = mass flow rate × mechanical energy change per unit mass

Thus

=

m& ×

=

ρ Qg ΔH

η pump

=

gΔH

ρ Qg ΔH W&

⎛ Q ⎞ ⎛ g ΔH ⎞ ⎜ ⎟⎜ ⎟ ⎝ N D3 ⎠ ⎝ N 2 D2 ⎠ = ⎛ W& ⎞ ⎜ ⎟ ⎝ ρ N 3 D5 ⎠ =

Flow Coefficient × Head Coefficient Power Coefficient

Similarly, for turbines:

η turb

= =

shaft power W& = fluid power ρ Qg ΔH Power Coefficient Flow Coefficient × Head Coefficient

2.2.2

Effect of Reynolds Number We have shown that in general for incompressible flow:

g ∆H W& , , η , etc. = N 2 D2 ρ N 3 D5

æ Q ρ N D2 ö fnsç , ÷ µ ø è N D3 æ Q ö fnsç , Re÷ 3 è ND ø

=

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

g ∆H W& , , η , etc. = N 2 D 2 ρ N 3 D5

ö æ Q fnsç only÷ 3 ø è ND

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

1 − η P æ Re M ö =ç ÷ 1 − η M è Re P ø

n

(1)

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.

Reynolds Number Based on Blade Chord

Taken from: AGARD-LS-167

2.2.3

Performance Curves for Incompressible-Flow Turbomachines Relationships such as g ∆H N 2 D2

=

æ Q ö fç ÷ (neglecting Re) 3 è ND ø

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):

Coefficients

Likely "Design Point"

W& ρN 3 D 5

η g∆H N 2 D2

Q N D3 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.

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:

⎛ a 01 ⎞ ⎛ P01 ⎞ ⎜ ⎟ ⎜ ⎟ N , D, m& , ⎜ or ⎟ , ⎜ or ⎟ , μ , R, γ ⎜ ⎟ ⎜ ⎟ ⎝ T01 ⎠ ⎝ ρ 01 ⎠ where m& = mass flow rate (rather than Q as measure of flow rate)

a 01 = γ RT01

(stagnation speed of sound)

ˆ could use T01 rather than a01 P01 = ρ 01 RT01

(perfect gas)

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)

ND a 01

speed parameter (effectively the tip Mach number)

(2)

m& ρ 01 D 2 a 01

flow parameter (effectively the axial Mach number)

(3)

μD 1 ρ N D2 = or we could use 01 again μ m& Re

(4) γ =

Cp Cv

specific heat ratio (which is already non-dimensional)

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

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:

P02 P01

P02 = machine outlet total pressure

Then

⎛ ND ⎞ m& , , Re, γ ⎟ (1) fns⎜ 2 ⎝ a 01 ρ 01 a 01 D ⎠

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

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 01

=

ND

γ RT01

N

D T01 γ R

=

(2) Flow coefficient:

m& ρ 01 a 01 D 2

=

m& P01 γ RT01 D 2 RT01

=

m& T01 P01

R 1 γ D2

Then (1) can be written

P02 W& , η , etc. = , P01 ρ 01 N 3 D 5

⎛ N fns⎜⎜ ⎝ T01

D

γR

,

m& T01 P01

⎞ R 1 , Re, γ ⎟⎟ 2 γ D ⎠

(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:

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

⎛ N D m& T01 ⎞ fns⎜⎜ , , Re⎟⎟ 2 ⎝ T01 P01 D ⎠

(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:

P02 W& , , η , etc. = P01 ρ 01 N 3 D 5

⎛ N m& T01 ⎞ fns⎜⎜ , , Re⎟⎟ P01 ⎝ T01 ⎠

(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.

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: P02 P01

=

æ ND m & T01 ö ÷ , f 1 çç 2 ÷ è T01 P01 D ø

etc.

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")

P02 P01

LINE OF CONSTANT

ND SURGE LINE (UPPER LIMIT OF STABLE OPERATION)

T01

CHOKING

ND INCREASING

T01

& T01 m P01D 2

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.

(b) Turbine Characteristic: P02 P01

STATORS CHOKED

LINES OF CONSTANT

ND T01

& T01 m P01D 2

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:

& T01 m P01D 2

CONSTANT

ND T01

P02 P01

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

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 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)

Qmachine = Qload (or, for compressible flow, m& machine = m& load )

(2)

ΔH machine = ΔH load (or ΔP0,machine = ΔP0,load )

& , characteristics Thus, the operating point is where the machine and load ΔH vs Q , or ΔP0 vs m 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 the system. 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. ΔH

LOAD LINE

C

PUMP CHARACTERISTICS AT CONSTANT SPEED

B N3

A N2 N1

Q

(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). & fuel m COMBUSTOR

W& C COMPRESSOR

&C m

W& out TURBINE

&T m

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

m& T

= m& C + m& fuel

NC W&

=

T

NT = W& C + W& 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& T = η T ρ gQ ΔH 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.

CONSTANT SPEED LINES SAME SPEED, DIFFERENT BLADE SETTINGS

ΔH β1

β2

β3

LOAD LINE NEGLECTING FRICTION

LOAD LINE INCLUDING FRICTION

Q

EXAMPLE (Section 2.3): A pump is connected to the piping system shown. What flow rate of water will be pumped for the two valve settings?

VALVE K = 1, K=10 K = 0.9

K = 0.9 K = 0.9

K = 1 (EXIT LO SS) K = 0.9 6 m.

WATER

K = 0.9 PUMP

K = 0.5 (ENTRY LOSS)

Pipe diameter:

dpipe := 50

Pipe length:

L := 125

Viscosity (water):

ν := 10

−6

mm

(smooth)

m m2/s

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

Pump speed:

Flow coefficient:

4

N := 1750 RPM Q

3 3

N⋅ D Head coefficient:

Pump Characteristics

D := 30 cm

g⋅ ∆H 2 2

N D

Head Coefficient

Pump diameter:

2

(with N in revs/s in the coeffcients) 1

0

0

0.002

0.004 0.006 Flow Coefficient

0.008

0.01

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

FAMILY B FAMILY A

η =

⎛ Q ⎞ only⎟ f⎜ 3 ⎠ ⎝ ND

Q ND3

Thus, the maximum 0 will occur for this family (say family A) at some particular value of Q/ND3. For another family of machines, the maximum 0 might occur at a different value of Q/ND3. We could therefore classify 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: 1

⎛ Q ⎞2 ⎜ ⎟ 3 ⎝ ND ⎠ ⎛ g ∆H ⎞ ⎜ 2 2⎟ ⎝N D ⎠

3 4

1

=

NQ 2 3

( g ∆H ) 4

Following convention, we then define 1

Ω =

ω Q2

( g ∆H )

3 4

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

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 following table can be used to convert between the various definitions used:

AREA OF APPLICATION FANS, BLOWERS AND COMPRESSORS (BRITISH UNITS)

SPECIFIC SPEED

N S1 =

RPM cfs

EQUIVALENT S

Ω=

N S1 129

Ω=

N S2 2730

Ω=

N S3 42

3

ft 4 PUMPS (AMERICAN MANUFACTURERS)

HYDRAULIC TURBINES (BRITISH UNITS)

N S2 =

RPM USgpm ft

N S3 =

RPM HP ft

HYDRAULIC TURBINES (METRIC UNITS)

N S4 =

3 4

5 4

RPM metric HP

N S5 =

Ω=

5

m4 FANS, BLOWERS AND COMPRESSORS (METRIC UNITS)

(IF WORKING FLUID IS WATER)

RPM m 3 s 3

N S4 187

(IF WORKING FLUID IS WATER)

Ω=

N S5 53

m4

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.

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& t = η t ρ Qg ∆H

we can rewrite the specific speed as

Ω=

ω Q

( g ∆H )

3 4

=

ω W& 5

η ρ ( g ∆H ) 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 NS =

N W& 5

∆H 4 The following figure (from Shepherd, 1956) shows the variation of the power specific speed for hydraulic turbines of different geometries.

The plots shown above were based on data that is as much as 50 years old. One might expect that over 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.

1

0.9

Axial-Flow Machines

Efficiency, η

0.8

0.7

0.6

Centrifugal Machines

Positive-Displacement Machines

0.5

0.4 1 10 10

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

20

40

60

2 80 10 100

200

Specific Speed, N S

400

600

1000 103

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 For turbines

N, Q and )H (or )P0) are typically specified.

& and )H (or )P0) are typically specified. N, 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 0.5DV2 (and with gases, 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 0 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" ): 1

∆=

D( g ∆H ) 4 Q

The specific size for a given machine is then the value of ) at which it achieves its best efficiency. The value of ) depends on the machine type (i.e. S) and to some degree on its detailed design. However, in the early 1950s 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

é ω Q Ω = êê 3 êë ( g ∆H ) 4

ù ú ú úû duty

(2) From S, read ) from the Cordier diagram and size the machine such that 1 é 4 D g ∆ H ( ) ê ê Q êë

ù ú =∆ ú úû duty

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 For turbines

N, Q and )H (or )P0) are typically specified.

& and )H (or )P0) are typically specified. N, 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 as 0.5DV2 (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. 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" ): 1

∆=

D( g ∆H ) 4 Q

The specific size for a given machine is then the value of ) at which it achieves its best efficiency. The value of ) depends on the machine type (i.e. S) and to some degree on its detailed design. However, in the early 1950s 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 Q Ω=⎢ 3 ⎢⎣ ( g ∆H ) 4

⎤ ⎥ ⎥ ⎥⎦ duty

(2) From S, read ) from the Cordier diagram and size the machine such that 1 ⎡ ⎢ D( g ∆H ) 4 ⎢ Q ⎢⎣

⎤ ⎥ =∆ ⎥ ⎥⎦ duty

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, 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 sv = H abs − hv

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

⎡P V2⎤ H abs = ⎢ abs + ⎥ 2g ⎦ suction side ⎣ ρg 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,

hv =

Pvap

ρg

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

P01 1 ρ V 12 2

P

P1

P01 1 ρ V 22 2

ρ gH SV

P2

P1

Pv

Pv f (T )

o

1 ρ V 22 2

P2

ρ gH SV

critical

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

S=

ω Q 3

( gH sv ) 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:

ΔH η DATA FOR CONSTANT

Q ND 3

INCEPTION

S

Si

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 lowpressure 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: For turbines:

Si . 2.5 - 3.5 Si . 3.5 - 5.0

N.B.: near the design point

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 sv

crit

ΔH

where H sv crit is the critical value of H sv : that is, the value at cavitation inception. However, the value of σ 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:

2 1 D2 D1

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=

ω Q 3

( gH sv ) 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

σ1 =

H sv

crit ,1

Δ H1

> σ2 =

H sv

crit ,2

ΔH 2

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.

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 this turbine 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 at the draft tube outlet.

6m

3m

h

TAIL POND

DRAFT TUBE OUTLET

3.0 FUNDAMENTALS OF TURBOMACHINERY FLUID MECHANICS AND THERMODYNAMICS 3.1 STEADY-FLOW ENERGY EQUATION Consider a control volume containing a turbomachine: 1

2

& m

& m

W& shaft

Q& 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

∫E

1

∫

dm& + d Q& + W& shaft =

∫E

2

dm&

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

& 1 + Q& + W& shaft = mE & 2 mE where

(1)

m& E Q&

= mass flow rate of fluid = energy per unit mass for fluid = rate of heat transfer to the machine

W& shaft

= shaft power into the machine

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

E

⎛ P C2 ⎞ = u + ⎜ + + gz⎟ 2 ⎝ρ ⎠ thermal + mechanical =h+

where

u P/ρ C C2/2 gz h

= = = = = =

C2 + gz 2

internal thermal energy per unit mass (= CvT) flow work (“pressure energy”) per unit mass absolute velocity of fluid kinetic energy per unit mass potential energy per unit mass P/ρ + u = enthalpy per unit mass

(2)

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

⎡⎛ C2 ⎞ ⎛ C2 ⎞⎤ = m& ⎢⎜ h2 + 2 ⎟ − ⎜ h1 + 1 ⎟ ⎥ 2 ⎠ ⎝ 2 ⎠ ⎥⎦ ⎢⎣⎝

W& shaft

= m& (h02 − h01 )

where

(3a)

C2 = stagnation enthalpy 2 = C P T0 for perfect gases

= h+

h0

For general non-uniform flows, we would write

∫

∫

W& shaft = h0 dm& − h0 dm& 2

1

(3b)

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:

W& shaft

⎛ m& ⎞ ⎞ ⎛ ⎞⎤ P C2 P C2 ⎜ ⎟ ⎡⎛ = ⎜ or ⎟ ⎢⎜ u2 + 2 + 2 + gz 2 ⎟ − ⎜ u1 + 1 + 1 + gz1 ⎟ ⎥ ρ ρ 2 2 ⎠ ⎝ ⎠ ⎦⎥ ⎜ ⎟ ⎢⎝ ⎝ ρ Q⎠ ⎣

(4)

It is also common to write

u2 − u1 = H L = " total head loss" due to friction inside the machine g The total head is a measure of the total mechanical energy content of the fluid

H

= total head

=

P C2 + +z ρ g 2g

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

W& shaft

=

ρ Q g ( H 2 − H1 + H L )

=

ρ Q g ΔH + ρ Q g H L

(5)

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”.

We defined the efficiency of a pump or blower as

η pump = then

fluid power shaft power

ρQgΔH ρQgΔH + ρQgH L 1 = H 1+ L ΔH

η pump =

(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& shaft

= ρ Q g ΔH − ρ Q g H L

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.

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):

Torque applied to fluid in CV = outflow of angular momentum - inflow of angular momentum T0

=

ò

out

&− r × C dm

ò

in

& r × C dm

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

T

=

ò rC

out

where

r = Cw =

w

ò

& − rC w dm & dm in

radial distance from the axis tangential component of absolute velocity

Or using mean values T

& (rC w ) out − m & (rC w ) in = m

(7)

3.3

EULER PUMP AND TURBINE EQUATION We will use the following nomenclature in this and the subsequent sections: β (−)

ROTOR U α (+)

β (−) α (+)

C U W

STATORS

β (−)

C W U

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

Subscripts: a r w

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

Angles: α αN β βN

= absolute velocity = stator blade metal angles = relative velocity = 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 the turbomachinery 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.

Consider again the general turbomachinery rotor

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

=

T

∫ rC

w

∫

dm& − rC w dm&

2

(7)

1

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& shaft = T ω and multiplying through by ω in (7)

∫ = ∫ UC

∫ rωC

W& shaft = rω C w dm& − 2

2

& w dm

−

1

∫

& w dm

(8)

UC w dm&

1

where U = rω is the blade speed. But the SFEE also relates the shaft power, W& shaft , to the energy changes in the fluid. Equating the shaft powers from Eqns. (3) and (8)

∫ h dm& − ∫ h dm& = ∫ UC 0

2

0

1

2

&− w dm

∫ UC

& w dm

(9)

1

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

h02 − h01

= U 2 C w2 − U 1C w1

(10)

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

g ( H 2 − H1 + H L ) = U 2 C w 2 − U 2 C w 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 ΔH E = U 2 C w2 − U 1 Cw1

(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

ΔH E =

ΔH

η pump

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.

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

C =W +U 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& shaft m&

= g ΔH E = Δh0 = U 2 C w2 − U 1C w1

(12)

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

C22 = Ca22 + C w2 2 + Cr22

(a)

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

W22 = Wa22 + Ww22 + Wr22 = Ca22 + (U 2 − C w2 ) + Cr22 2

Solve (a) and (b) for Ca22 + Cr22 and equate

C22 − C w2 2 = W22 − U 22 + 2U 2 C w2 − C w2 2

(b)

Then

U 2 C w2 =

(

1 2 C2 + U 22 − W22 2

)

Similarly for the velocity triangles at the inlet, station1,

U 1 C w1 =

(

1 2 C1 + U 12 − W12 2

)

Substituting into (12)

W& shaft m&

= g ΔH E = Δh0 =

((

) (

) (

1 C22 − C12 + U 22 − U 12 + W12 − W22 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).

(

)

1 2 C2 − C12 is clearly the kinetic energy change of the fluid across the rotor. In a pump, 2 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. Term (1),

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:

⎛ P C2 ⎞ m& ⎜ 1 + 1 + u1 ⎟ + W& shaft 2 ⎝ ρ ⎠

⎛P ⎞ C2 = m& ⎜ 2 + 2 + u2 ⎟ 2 ⎝ ρ ⎠

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

P2 − P1

=

(

)

(

)

1 1 ρ U 22 − U 12 + ρ W12 − W22 − ρ (u2 − u1 ) 2 2

(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.

(

)

1 2 U 2 − U 12 is then energy transfer to the fluid due to the centrifugal compression (or 2 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. Term (2),

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

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. (+) F 2

(-)

ω

1

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

P2 − P1

=

(

1 ρ U 22 − U 12 2

)

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).

(

)

1 2 W1 − W22 represents the change in pressure energy due to the change in fluid velocity 2 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) Term (3),

(

1 P2 − P1 = ρ W12 − W22 2

)

(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 these forces (since the rotor is moving) work is being done on the fluid.

W2

W1 U

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 due to 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.

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:

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:

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 1 U 22 − U 12 + W12 − W22 2 (16) (h02 − h01 )

[(

)]

) (

which can also be written

Λ =

h2 − h1 h02 − h01

(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 =

ΔProtor ΔPstage

(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 wheel turbine, P1 = P2 = Patm and Λ = 0. A machine with Λ = 0 is known as an impulse machine. Impulse wheels are sometimes used for axial turbines, particularly steam turbines.

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 m& = ρCa Aannulus . We therefore 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 this will 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 and outlet metal angles.

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

& ΔC w W& = m& (U 2 C w2 − U 1C w1 ) = mU 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 of ΔCw (for a given U) is clearly related to Ca1 = Ca2 α1. Thus, the required W& plays a direct role in determining the velocity triangles, and ultimately the metal angles. C2

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

α 1 = α 1′ , β 2 = β 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".

Cw2 (+)

α1 (+) U

W2

C1

Cw1 (+)

ΔCw

W1

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 as follows:

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:

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 detailed design 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,max =

where ΔP V

ΔP = 0.44 1 2 ρV 2

(a)

= static pressure rise between inlet and outlet of the blade row = 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

1 1 P2 − P1 = ρW12 − ρW22 . 2 2 Substituting into (a) and simplifying,

⎛ W2 ⎞ = 0.75 . ⎜ ⎟ ⎝ W1 ⎠ 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.

3.5.4

Work Coefficient

From the Euler Equation ∆h0 = U 2 C w2 − U 1C w1 = ∆(UCw )

and for an axial machine with simple velocity triangles (so that U1. U2 = U) ∆h0 = U∆C w .

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

and

∆C w

∝ U

∆h0

∝ U2 .

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

ψ =

∆h0 U

2

=

∆(UCw ) U

2

=

g∆H E U2

For “high” R, we are taking full advantage of the blade speed and we have “high stage loading”: we will specify what constitutes “high” R for different types of machines in Section 3.5.6. For a centrifugal machine, tip speed, U2, would be used in R. For an axial machine with simple velocity triangles (so that U1. U2 = U)

ψ =

U∆C w U

2

=

∆Cw U

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

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 Euler equation, Δh0 = UΔCw , and to achieve the same Δh0 , and thus the same pressure rise, they must therefore have 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 is achieved by reducing the cross-sectional area of the machine. This change obviously has a significant effect on the rotor blade geometry. It also has aerodynamic consequences: (i) For rotor B, both the absolute and relative velocities have been increased. Since losses generally vary as 0.5ρV2 (where V = W for the rotor), rotor B will, all other things being equal, have poorer efficiency than rotor A. (ii) All other things are not equal. Note that the increase in Ca in rotor B has had the effect of increasing the de Haller number (W2/W1). Thus, the diffusion has been reduced in rotor B, which is aerodynamically favourable. We can thus identify an additional important parameter which must be chosen by the designer, the flow coefficient, φ:

φ=

Ca U

For a centrifugal compressor, we would use Cr2/U2, where Cr2 is the radial component of velocity at the rotor outlet. Note that for the compressors shown, the change in flow coefficient did not in fact change the degree of reaction. As you will show in Problem Set 3, the symmetry of the velocity triangles for both machines implies 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. Design Parameter

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 1.5 - “Highly Loaded”

Λ

0.3 6 0.7

(Not much used)

061

de Haller

>0.65 (well-designed machines with clean inlet flow) >0.80 (simple design, poor inlet flow uniformity)

See Section 6.4.3

N/A

For compressible-flow axial turbines, Smith ( S.F. Smith, "A Simple Correlation of Turbine Efficiency," J. Royal Aero. Soc., Vol. 49, July 1965, pp. 467-470.) developed a very useful figure (the “Smith chart”) 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 (for both model and full-scale machines) conducted at Rolls-Royce from 1945 to 1965. Over that period, the flow over the tip of the rotor blades ("tip leakage") was considerably reduced. The tip-leakage flow is an important source of losses and as a result there was significant improvement in efficiency. To isolate the influence of the stage loading and shape of the velocity triangles, the efficiencies were corrected back to their zero-clearance equivalents. Thus, efficiencies for actual machines can be expected to be lower than those shown by a couple of percentage points. Note that the degree of reaction is not mentioned on the Smith chart. The turbines used to generate the chart had a range of degrees of reaction. However, the performance of turbines is not strongly dependent 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 preliminary stages of design. The usefulness of the Smith chart makes it surprising that comparable charts are not more widely used by axial and centrifugal compressor designers. Part of the reason lies in the important role played by diffusion (expressed through both the degree of reaction and the de Haller number) in compressor performance. Thus a single “Smith chart” for compressors is not feasible. However, it is possible to generate a small number of charts, each for a different value of degree of reaction say, and then use these in design. In the late 1980's Casey (M.V. Casey, “A Mean Line Prediction Method for Estimating the Performance Characteristics 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 “Smith charts” for axial compressors for three values of degree of reaction: 50, 70 and 90%. Note the rapid deterioration in efficiency when the de Haller number is less than about 0.7.

“Smith” Charts for Axial Compressors: (a) Λ = 0.5, (b) Λ = 0.7, (c) Λ = 0.9. The use of the guidelines presented in this section will be illustrated in the next chapter.

3.6

EFFICIENCY OF TURBOMACHINES 3.6.1 Incompressible-Flow Machines

The definitions of efficiency used for incompressible-flow machines have been discussed briefly in earlier 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 the definitions of efficiency used for incompressible- and compressible-flow machines. These will therefore be clarified now. (a) Pumps, Fans and Blowers

From the steady flow energy equation,

& ΔH E = m& Δh0 W& shaft = mg 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 the machine

& ΔH W& fluid = mg where

ΔH

=

total head rise that is actually observed across the machine.

The Euler head and the actual total head are related by ΔH = ΔH E − H L

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

ΔP0,actual = ρ gΔH ΔP0,ideal = ρ gΔH E We then define the efficiency for a pump, fan or blower as

η pump =

ΔP0,actual Fluid power ρQgΔH ΔH = = = Shaft power ΔH E ΔP0,ideal W& shaft

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

The specification calls for the machine to raise the fluid head by ΔH, or the total pressure by ΔP0,actual = P02 − P01 = ρgΔH . With the same shaft power input per unit mass flow (Δh0), the ideal machine would raise the pressure by ΔP0,ideal = PN02 - P01. Thus, the efficiency for pumps, fans and blowers is defined by comparing the head or total pressure rises for the actual and an ideal machine that have the same shaft power input. As described below, the definition of efficiency for compressors is slightly different.

P02 h0

P02

Δh0

P01

ACTUAL IDEAL

s

(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 and & ΔH E . That is, is therefore not available to be extracted as shaft power output, W& shaft = mg ΔH = ΔH E + H L

The turbine efficiency is then defined as

η turbine =

ΔP0,ideal Shaft power ρQgΔH E ΔH E = = = ρQgΔH ΔH ΔP0,actual Fluid power

The physical interpretation can again be seen in terms of the h0 versus s diagram. The actual pressure drop is ΔP0,actual = P01 - P02 & ΔH E = m& Δh0 . In an and the shaft power extracted is W& shaft = mg ideal machine, a smaller pressure drop, ΔP0,ideal = P01 - PN02, would be needed to produce the same shaft power output. Thus, the turbine efficiency is defined in terms of two machines that have same shaft power output. The comparison is between the head or total pressure drops required to obtain that shaft power output in the ideal and actual machines. The similarity with the definition used for pumps, fans and blowers is evident.

h0 P01

ACTUAL

Δh0

IDEAL

P02 P02

s

3.6.2 Compressible-Flow Machines The efficiency of compressible flow machines is defined slightly differently. The comparison is again between ideal and actual machines. However, instead of the shaft power input or output, the common basis is the pressure rise or drop across the machines. (a) Compressors The h0-s diagram is again used to compare the processes used to define the efficiency. For compressible-flow machines, the pressure rise or drop across the machine is generally expressed in terms of the total pressure ratio. The h0-s diagram shows the ideal and actual compression processes needed to obtain the same pressure ratio, P02/P01. For the ideal machine, the shaft power required is

h0

P02

Δh0,actual P01

ACTUAL

Δh0,ideal IDEAL

W&ideal = m& Δh0, ideal while for the actual machine

s

W& actual = m& Δh0, actual . The compressor efficiency is then defined as the ratio of the shaft powers required to produce the same pressure ratio in the ideal and actual machines:

ηc =

m& Δh0, ideal Δh0, ideal W&ideal = = & Wactual m& Δh0, actual Δh0, actual

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

ηc =

C p ΔT0, ideal C p ΔT0, actual

=

T02′ − T01 T02 − T01

For any isentropic process involving a perfect gas,

P

ργ

T0

P02

T02 T02

= const .

P01

ACTUAL IDEAL

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

T02′ ⎛ P02 ⎞ =⎜ ⎟ T01 ⎝ P01 ⎠

γ −1 γ

T01

s

Then

Δh0, actual =

C p (T02′ − T01 )

ηc

γ −1 ⎛ ⎞ C p T01 ⎜ ⎛ P02 ⎞ γ ⎟ − 1⎟ = ⎟ ⎜⎜ η c ⎜ ⎝ P01 ⎠ ⎟ ⎝ ⎠

& Δh0, actual , to be and this expression allows the shaft power required to drive the actual machine, W& actual = m related to the specified pressure ratio. (b) Turbines The efficiency of compressible-flow turbines is similarly defined by comparing the shaft power produced by the expansion through the same pressure ratio for an ideal and the actual machine. Following the same procedure as for the compressor, we obtain

ηt =

W& actual Δh0, actual T02 − T01 = = T02′ − T01 Δh0, ideal W&ideal

T0

and

P01

T01 ACTUAL

Δh0, actual

γ −1 ⎛ ⎞ ⎜ ⎛ P02 ⎞ γ ⎟ = C p T01η t ⎜ ⎜ − 1⎟ ⎟ ⎜ ⎝ P01 ⎠ ⎟ ⎝ ⎠

IDEAL

P02

T02 T02

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

s

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

PRs = stage total-pressure ratio ηs = stage isentropic efficiency

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

T0 N +1

γ −1 ⎛ ⎞ γ −1 PR ( ) ⎜ ⎟ s = T01 ⎜ 1 + ⎟⎟ ηs ⎜ ⎝ ⎠

The overall pressure ratio for the N stages is

N

PRc = ( PRs )

N

and the isentropic temperature rise for the whole compressor is then γ −1 ⎛ ⎞ ⎜ T0′N +1 − T01 = T01 PRc γ − 1⎟ ⎜ ⎟ ⎝ ⎠

⎛ = T01 ⎜⎜ PRsN ⎝

(

)

γ −1 γ

⎞ − 1⎟⎟ ⎠

and thus the overall isentropic efficiency is

ηc =

T0′N +1 − T01 T0 N +1 − T01

N ( γ −1)

=

PRs

γ

−1

γ −1 γ

N

⎞ ⎛ ⎜ PRs − 1⎟ ⎟ ⎜1 + ηs ⎟ ⎜ ⎠ ⎝

−1

For example, if PRs = 1.2 and ηs = 0.9, the resulting variation of the overall pressure ratio and overall isentropic efficiency with the number of stages is shown in the figure. As EFFECT OF PRESSURE RATIO ON OVERALL ISENTROPIC EFFICIENCY seen, the overall efficiency decreases as the pressure ratio increases. When cycles for gas turbine engines are being investigated, it is normal to examine the effect of varying pressure ratio. It is evident that assuming a constant value of the overall compressor isentropic efficiency is not valid for such investigations. To account for the effect of the pressure ratio on the isentropic efficiency, the concept of the small-stage or polytropic efficiency has been introduced.

Overall Compressor Isentropic Efficiency

0.9 1

2

3

4

5

6

0.88

Number of Stages

7

8

9

10

11

0.84

Stage PR = 1.2 Stage ηisen = 0.9

0.82

0.8

1

2

3

4

5

6

7

Overall Compressor Pressure Ratio

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

dh0 = Then for an isentropic process (ds = 0)

12

0.86

dP0

ρ0

+ T0 ds

8

9

10

dh0′ =

dP0

ρ0

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

dh0′ = η p dh0

∴

dP0

ρ0

= η p dh0

Then assuming a perfect gas, h0 = CpT0 and P0 = ρ0RT0. Also Cp - Cv = R, or C p =

Rγ , and the fluid γ −1

properties are assumed constant through the process. Then

(

d C p T0 dP0 = ηp C p T0 ρ 0 C p T0 dP0

γ ρ 0 RT0 γ −1 or

= ηp

)

dT0 T0

dT0 γ − 1 dP0 = T0 η p γ P0

Integrating this between the start and end of a finite process

⎛ T ⎞ γ − 1 ⎛ P02 ⎞ ln⎜ 02 ⎟ = ln⎜ ⎟ ⎝ T01 ⎠ η p γ ⎝ P01 ⎠ γ −1

or

T02 ⎛ P02 ⎞ η pγ =⎜ ⎟ T01 ⎝ P01 ⎠

For a compression process, the isentropic efficiency is defined as

ηc =

Δh0′ Δh0

γ −1 γ −1 ⎛ ⎞ ⎞ ⎛ η γ ⎜ pcγ ⎟ ⎜ Δ h = C T PR − 1 where and Δh0 = C p T01 PR − 1⎟ where ηpc is the polytropic efficiency for 0′ p 01 ⎜ ⎟ ⎟ ⎜ ⎠ ⎝ ⎝ ⎠ the compressor. Then

ηc =

PR

γ −1 γ γ −1 η pcγ

PR

−1 −1

For a turbine,

ηt =

Δh0 Δh0′

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

ηt =

⎛P ⎞ 1 − ⎜ 04 ⎟ ⎝ P03 ⎠

η pt (γ − 1)

⎛P ⎞ 1 − ⎜ 04 ⎟ ⎝ P03 ⎠

γ

γ −1 γ

The following figure shows the resulting variation of isentropic efficiency with pressure ratio for an assumed polytropic efficiency of 0.9 and γ = 1.4, for both a compressor and a turbine. Also shown are the earlier results for the multistage compressor with stage pressure ratio of 1.2. VARIATION OF ISENTROPIC EFFICIENCY WITH PRESSURE RATIO Polytropic Efficiency, ηp = 0.9, γ = 1.4

0.94

Isentropic Efficiency

Turbine

0.92

0.9 1

2

3

4

5

6

7

8

0.88

9

10

11

12

Compressor

0.86 1

2

3

4

5

6

7

8

9

10

Pressure Ratio

The concept of polytropic efficiency should be used with caution. It is only valid if the machine can be considered to employ comparable technology and produce comparable performance as the pressure ratio is varied. For this reason, it should be applied only to explore the influence of pressure ratio on performance for multistage 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 is changed. As will be shown later, stage performance is closely related to its tip speed. For example, 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 of sound, 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: P1

y A

B C1

x

α1

Ca1

Cw1

Cm

αm

Y αm

Ca

Cwm

X

F

L

D

s α2

D C2

C

Ca2

P2

Cw2

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 tangential components X and Y. By definition, the lift generated by a turbomachinery blade L is the component of the blade force normal to the vector mean flow direction through the blade row. The drag D is the component of the blade force parallel to the vector mean flow direction. Then apply the linear momentum equation to the control volume. In the x direction:

ΣFx = m& (V x 2 − V x1 )

X + P1 ( s × 1) − P2 ( s × 1) = m& (Ca 2 − Ca1 ) Note that the pressure forces along the left and right faces of the control volume exactly balance each other in both the x and y directions. Then since we have assumed Ca1 = Ca2, the x-wise momentum equation reduces to

X = ( P2 − P1 ) s For the y direction:

(

ΣFy = m& V y 2 − V y1

(1)

)

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

direction is that due to blade, Y

Y = ρCa ( s × 1)(( − C w1 ) − ( − C w2 )) = ρCa s(C w1 − C w2 )

(2a)

Since

C w1 , Ca

tan α 1 =

tan α 2 =

C w2 Ca

we can also write

Y = ρCa2 s( tan α 1 − tan α 2 )

(2b)

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

ΔP0 = P01 − P02 = ( P1 − P2 ) +

(

1 ρ C12 − C22 2

)

From the velocity triangles,

(

) (

C12 − C22 = C w21 + Ca21 − C w2 2 + Ca22 = (C w1 + C w2 )(C w1 − C w2 )

)

since Ca1 = Ca2. Then

ΔP0 = ( P1 − P2 ) +

1 ρ (Cw1 + Cw 2 )(Cw1 − Cw 2 ) 2

substituting for Cw1 − Cw 2 from (2a)

ΔP0 = −

X 1 ⎛ Y ⎞ + ρ ⎜ ( C w1 + C w 2 ) ⎟ s 2 ⎝ ρ Ca s ⎠

and using the vector mean flow direction through the passage, tan α m =

ΔP0 =

1 (− X + Y tan α m ) s

1 ( tan α 1 + tan α 2 ) , we can write 2

(3)

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

D = Y sin α m − X cosα m

= cosα m ( − X + Y tan α m )

(4)

and substituting from (3)

D = ΔP0 s cos α m

(5)

Then from the definition of the drag coefficient

CD =

D

1 ρCm2 c × 1 2 ΔP s cos α m = 0 1 ρCm2 c 2

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

⎛1 ⎞ 1 ΔP0 = C D σ ⎜ ρCm2 ⎟ ⎝2 ⎠ cos α m

(6a)

or alternatively, using Cm = Ca/cosαm,

1 ⎞ ⎛1 ΔP0 = CDσ ⎜ ρCa2 ⎟ ⎠ cos3 α m ⎝2

(6b)

As will be seen later, some axial fan and compressor prediction procedures use the airfoil drag coefficient to express the loss performance for the blade row. Equation (6a) or (6b) can then be used to express this as a total pressure loss. Returning to the lift force, from the force triangles the lift L can be expressed as

L = X sin α m + Y cos α m

(7)

Solving (4) for X

X = Y tan α m −

D cos α m

and substituting into (7)

⎛ D ⎞ L = ⎜ Y tan α m − ⎟ sin α m + Y cos α m cos α m ⎠ ⎝ =

Y − D tan α m cos α m

Then substituting for Y from (2b)

L=

ρCa2 s ( tan α 1 − tan α 2 ) − D tan α m cos α m

By definition

CL =

L L = 1 1 ⎞ ρCm2 c 1 ρC 2 c⎛⎜ ⎟ a 2 2 2 ⎝ cos α m ⎠

(8)

then

ρCa2 s ( tan α 1 − tan α 2 ) cos α m D tan α m CL = − 1 ⎛ ⎞ 2 1 1 ρ Cm c ⎟ ρCa2 c⎜ 2 2 2 ⎝ cos α m ⎠ or

⎛ s⎞ C L = 2⎜ ⎟ cos α m ( tan α 1 − tan α 2 ) − C D tan α m ⎝ c⎠

(9a)

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

⎛ s⎞ C L = 2⎜ ⎟ cos α m ( tan α 1 − tan α 2 ) ⎝ c⎠

(9b)

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

∫

Γ = VS dS

(10)

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

Γ=

∫

B A

VS dS +

∫

C B

VS dS +

∫

D

C

VS dS +

∫

A D

VS dS

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

∫

C B

VS dS = −

∫

A D

VS dS

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

Γ = C y1 s − C y 2 s We are assuming constant axial velocity, and since tan α = C y C x (and Cx = Ca), we can write

Γ = Ca ( tan α 1 − tan α 2 ) s

(11)

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

L = ρ Cm Γ which is the expression given by the Kutta-Joukowski Theorem for an isolated airfoil. Note that in the case of the blade 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

rt rm 1

2

3

rh

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

(

)

rm2 = rh2 + rt2 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 ( m& − ΔP0 or Q − ΔH ) and using the guidelines for φ, ψ etc. from Chapter 3. To illustrate the procedure, we will use a semi-quantitative example. Assuming an incompressible flow machine, we will define the velocity triangles for 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: C φ = a = 0.5 Flow coefficient: U

gΔH E

Work coefficient:

ψ =

Degree of reaction:

Λ = 0.5

U2

= 0.4

Note that if we were using the Lewis charts from Section 3.5.6, we would probably choose a slightly higher value 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 a value for the stage efficiency η. This can be done from experience, or from the specific speed plots in Chapter 2, or from the approximate correlations shown in Section 3.5.6. Later, we will see how to calculate the efficiency of the stage we have designed. If this efficiency is different from the one we have guessed here, we will have designed the stage with an incorrect value of the ΔHE and it will not match the required performance ΔH. If this turns out to be the case, we will have to return to the beginning and revise the design. Thus, the design of a turbomachine inherently tends to be iterative: to design the machine we need its efficiency, but we do 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 our chosen value of the work coefficient ψ:

U=

gΔH E

ψ

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

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

A=

Q m& = ρ Ca Ca

We will assume “simple” velocity diagrams, as defined in Section 3.5.1. That is, we assume that the annulus is shaped such that Ca and U remain constant through the stage: U1 = U2 = U, Ca1 = Ca2 = Ca3 = Ca. Then from the Euler equation gΔH E = U 2 C w2 − U 1C w1 = UΔC w

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

ψ =

ΔC w U

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 we have chosen 50% reaction, equal amounts diffusion are occurring in the rotor and the stators. Thus the de Haller Ca1 = C a 2 numbers for the rotor and stators must be the same:

W2 C3 = W1 C2

α 2 (+ )

α 1 (+ )

On Problem Set #3, you will show that for simple velocity diagrams, this is achieved by making the velocity triangles for the inlet and outlet of the rotor symmetrical, and by designing the stators so that C3 = C1. That is, we design the stators so that the flow at the U stage 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 can determine the de Haller numbers:

W2 C3 = = 0.678 W1 C2 This is approaching the limit of about 0.65 that was

Cw 1 (+ )

C1

Cw 2 (+ ) C2 W1 W2

β 1 (− )

β 2 (− )

ΔCw

recommended in Section 3.5.6 and we will therefore have to monitor our design for the possibility of stall. As noted 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, the inlet flow must have a swirl angle α1. Thus, there must either be a stage ahead of the present one, or a set of inlet guide vanes, that leave the required amount of swirl in the flow. The flow from this stage will also leave with swirl α3 = α1, so that C3 = C1 . Suppose instead that the inlet swirl was specified. For example, if this is the first stage in the machine then we will normally have no swirl in the flow, α1 = 0. Using the same values of φ and ψ, the velocity triangles will then look as shown. We can then show that the resulting degree of reaction is Λ = 0.8. This means that the diffusion is much higher in the rotor than in the stators and this might at first be a matter for concern. However, consider the values of the de Haller numbers (we will assume that the flow leaves the stage with no swirl, C3 = C1): Rotor:

W2 = 0.699 W1

Stators:

C3 = 0.781 C2

C a1 = Ca 2 C1

α 2 (+ ) Cw 2 (+ )

ΔCw

C2

U W1

α1 = 0 W2

Cw 1 = 0

β 2 (− ) β 1 (− )

As expected, the value is lower for the rotor than for the stators. However, the diffusion is actually less than for the 50% reaction machine. As a result, this stage may be just as feasible as the earlier stage, despite the high value of degree of reaction. Having determined the velocity triangles, the next step is to define the blade geometries that will produce the required velocities.

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 the blade row parallel to the metal angle at the trailing edge of the blades: this is known as the Euler Approximation. 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 also make the leading edge metal angle parallel to the inlet flow angle. We can then define the shapes of the blades for the 50% reaction stage as follows:

As indicated on the drawing: (i) β 1′ = β 1 to bring the flow smoothly onto the leading edge of the rotor blades. (ii) In the relative frame of reference, the flow must leave the rotor blade passage at β 2 to produce the required turning. Based on the Euler approximation, the flow will leave the trailing edge at the metal angle and we therefore use β 2′ = β 2 . (iii) The stators see the flow in the absolute frame. To bring the flow smoothly onto the leading edge of the stator blades we therefore make α 2′ = α 2 . (iv) Again, the flow is assumed to leave the stators at the metal angle, and we use α 3′ = α 3 = α 1 . Note that with the assumptions made, the rotor and stator blade geometries are identical for the 50% reaction stage.

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 away from its design point. The idealized analysis can also be used to give reasonable predictions of how the stage will 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). The resulting 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 vanes ahead 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 of Ca would just be obtained from Ca = Q A , where Q is the new volume flow rate, and A is the 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 in swirl velocity ΔCw has increased. From the Euler equation gΔH E = UΔC w

and the head rise produced by the machine ΔH = ηΔH E will be increased. Equivalently, for a compressible flow machine, Δh0 , and the corresponding pressure ratio, P02 P01 , will be increased. Note that this is consistent 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. On the other hand, increasing the incidence will eventually lead to stalling of the blade. Thus, reducing the flow rate through a compressor stage will move it towards stall. Note that the incidence was also increased for the stators, bringing them closer to stall as well. Clearly, we can use the velocity triangles and the Euler equation to predict the quantitative stage characteristic for the idealized stage. It is convenient to express the characteristic in terms of the work and flow coefficients. The flow turning is ΔC w = C w2 − C w1

and from the velocity triangles (noting that Ww2 is negative for the conventional compressor velocity triangles) C w1 = Ca tan α 1

Then

C w2 = U + Ww2 = U + Ca tan β 2

ΔC w = U + Ca ( tan β 2 − tan α 1 )

and dividing by U

ΔC w C = 1 + a ( tan β 2 − tan α 1 ) U U

or

ψ = 1 + mφ

Thus, the ψ versus φ curve (effectively, the head rise versus flow rate characteristic) is a straight line with slope m = tan β 2 − tan α 1 For the present case, the symmetry of the velocity triangles implies that β 2 = − α 1 and the slope is then m = − 2 tan α 1 . For α1 > 0, as is the case here, this gives a negative slope and an inverse relationship between head rise and flow rate, as inferred above. Alternatively, since the characteristic passes through the design point (say, φD and ψD), we can write

m=

1.0

ψ

ψ D −1 φD

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

ψD3

0.5

ψD2 ψD1 INCREASING DESIGN-POINT LOADING

φD 0.0

0.5

φ

1.0

(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 the Euler Approximation again, it can be shown that the change in the velocity triangles will look as follows:

From the triangles, φ = Ca U > φ D since U has decreased. For the work coefficient, ψ = ΔC w U , ΔCw has clearly decreased, but so has U. However, ΔCw has decreased more rapidly than U; as can be seen, a small further decrease in U would reduce ΔCw to zero. We therefore conclude that ψ = ΔC w U < ψ D and φ and ψ are again seen to vary inversely. In summary, any deviation from the design point will cause the a given compressor 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 the absolute work transfer: gΔH E = UΔC w

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

gΔH E ≈ kU 2 and the head rise delivered by the stage, at a fixed value of flow rate, will change strongly with the rotational speed: 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 illustrated further in later sections.

As seen, the Euler Approximation results in an idealized ψ versus φ characteristic for the stage that is a straight line with a negative slope. We have already noted that some changes in operating point will result in positive values of the incidence at the leading edge of the airfoils. If this incidence becomes too large, we would expect the airfoils to stall. Also, we would expect the efficiency of the stage to be best when the rotor and stator blades are operating at the design point. We can therefore project what the actual stage characteristic is likely to be based on the idealized characteristic: MAXIMUM

ψ

ψD

η

η

STALL

USING EULER APPROXIMATION

LIKELY ACTUAL

φD

φ

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

η max

ΔH

η

CONSTANT

N

Q

CONSTANT

On each of the constant speed lines, there will be a point that corresponds to the design point values of φ and ψ on the non-dimensional characteristic. At each of those points, the velocity triangles will be similar, as indicated in the drawing. In each case, the relative velocity vector at the rotor inlet is lined up with the metal angle and the flow comes smoothly onto the leading edge. As shown, we would therefore expect that the machine will operate at its maximum efficiency at each of those points, apart perhaps for some small effect of differing Reynolds numbers. Also, as we will see later, frictional losses vary as V2 and thus the higher flow velocities with increasing rotational speed will result in higher frictional losses. This effect will be 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 performance characteristics 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 vary across the span. For this example, we will take the case with no inlet swirl from Section 4.3.1. At the mean radius, 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 mean radius known and the mean blade speed Um fixed by the choice of work coefficient ψ, we have the rotational speed, ω = Um/rm. To define the resulting spanwise geometry, we assume that the inlet axial velocity Ca is constant across the span and that we want the same total head rise, gΔH E = UΔCw , at every spanwise section. This 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 the performance parameters.

C1 rt W1

C2

U

W1

W2

C1

rm

C2 W1 U

W2

C1 rh W1 U C2

W2

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 de Haller numbers there also indicate too much diffusion. The blade will need to be redesigned. If the stage is still 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 design parameters is a function of the hub-to-tip ratio, HTR = rh/rt. Increasing the HTR will make the blade more uniformly loaded across the span, but since the cross-sectional area is fixed (by the choice of φ), this has consequences for the tip diameter of the machine and the rotational speed. This is demonstrated in the following sketch, which shows three different blades with the same annulus cross-sectional area but different values of HTR. In multi-stage compressors, the HTR will normally increase along the machine since the cross-sectional area is decreased to keep the axial velocity high. This is illustrated by the cross-section of the compressor from the GE LM2500+ gas turbine engine (17 stages, PR = 23.3).

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

0.3

RPM

Higher

0.5

0.8 Lower

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 has shown that it is possible to design a stage of good efficiency if the guidelines for those design parameters are followed. 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 other words, 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 is required to turn. From the control volume analysis in Section 4.2, the resulting lift coefficient was given by

⎛ s⎞ C L = 2⎜ ⎟ cosα m ( tan α 1 − tan α 2 ) ⎝ c⎠ ⎛ 1⎞ = 2⎜ ⎟ cosα m ( tan α 1 − tan α 2 ) ⎝σ ⎠ and it is seen to vary directly with spacing, or inversely with the solidity. Just as for an isolated airfoil, there is an upper limit to the lift that a turbomachinery blade can develop before it stalls. For a given set of inlet and outlet flow angles, it is possible to stay below the loading limit by making the solidity of the blade row large enough. Thus, the solidity of the blade row is selected on the basis of a blade loading limit. This is in contrast to 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 lift coefficient (Horlock, 1958). In the early 1950s, Howell suggested that a well-designed compressor airfoil will stall at 3

⎛C ⎞ C L ⎜ 1 ⎟ ≈ 3.3 ⎝ C2 ⎠ and designers of low-solidity fans have sometimes used the criterion

⎛ c⎞ C L ⎜ ⎟ ≤ 11 . ⎝ s⎠ However, expressing the loading limit simply in terms of CL has been found to be unreliable. Recent practice has therefore taken a somewhat different approach. Howell (British Practice) In the 1950s, Howell conducted an extensive series of cascade measurements on the compressor airfoils that were commonly used in British compressor design. The performance was measured for a wide range of the design parameters, including the flow turning angle and solidity. Howell varied the amount of flow turning up to the onset of stall. The corresponding total-pressure losses were also measured. Howell suggested that a suitable design turning angle for a blade row was that which corresponded to about 80% of the 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 blade row operating at 80% of the stalling turning angle. This correlation is shown in the next figure (taken from Saravanamuttoo 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 cascade measurements to determine the performance of compressor airfoils for a wide range of geometric and aerodynamic parameters. As described later, these results became the basis for a compressor design system which 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

C2 SUCTION SURFACE

Cmax

C1

C1

C2

0 0

x/c

1.0

The performance of the blade is limited by the deceleration (that is, the diffusion or adverse pressure gradient) on the suction surface of the airfoil. If the diffusion is too great, the boundary layer separates, the blade stalls, and the losses increase significantly. Lieblein proposed a parameter to measure the severity of the diffusion: − C2 C D = max (1) C1 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 be known early in design. However, the larger the lift (or circulation) being generated by the airfoil the larger Cmax must be. From Section 4.2.2, the circulation is given by

Γ = s(C w1 − Cw2 ) = sΔC w and thus we can write

C max = C1 + f ( Γ )

= C1 + f ( sΔC w )

Substituting into (1),

D =1−

C2 1 + f ( sΔC w ) C1 C1

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

D =1−

ΔC w C2 + C1 2σ C1

(2)

This parameter is known as the diffusion factor. Note that (2) depends only on the upstream and downstream velocities, which are known once the velocity triangles are established. The figure (taken from NASA SP-36, 1966) shows the variation of the total pressure loss coefficient, ω1, with D. As seen, the losses rise sharply for D > 0.65, implying the onset of stall. At the design point, the diffusion factor should therefore be less than this. A suitable value might be D = 0.3 - 0.4. With D chosen, the only unknown in (2) is the solidity and it can therefore 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 fully satisfied 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 angle necessarily result in the lowest losses. Finally, we need methods for estimating the losses generally, in order predict 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 fairly widely. The British system, connected mainly with the name of Howell, will be discussed since it is relatively easy to apply in hand calculations. However, it omits the influence of a number of blade geometric parameters, does not directly apply to all the families of blade geometries that are in common use, and has somewhat 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 the 1950s and 60s. This system is summarized in a famous document, NASA SP-36, “Aerodynamic Design of Axial-Flow Compressors” published in 1965. SP-36 continues to form the basis for much practical axialcompressor design, both in North America and outside. The correlations presented in SP-36 have also been re-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, General Electric and Rolls-Royce) have to some extent developed their own compressor design systems that reflect their in-house design philosophies and proprietary blade profile designs. However, these systems are often structured in similar ways and strongly influenced by the design systems that are available in the open literature.

4.5.2 Blade Design and Analysis Using Howell’s Correlations The figure shows the nomenclature used by Howell:

Nomenclature: s c ζ θ a t

= = = = = =

i = δ = ε =

blade spacing blade chord (solidity σ = c/s) stagger angle α1' - α2' = camber angle distance of maximum camber aft of blade leading edge maximum thickness of blade incidence = α1 - α1' deviation = α2 - α2' = difference between outlet flow angle and metal angle flow turning = α1 - α2

The nomenclature applies for a stationary blade row. For a rotor, replace α by β and use the relative components of velocity. Typical results obtained by Howell for a particular cascade geometry are shown in the following figure. The figure (taken from Horlock, 1958) shows the variation of flow turning, ε and the total pressure loss as a function of the incidence, i.

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

ε , δ , losses = f (blade geometry , cascade geometry , = f(

a c, θ ,

s c,

flow conditions) i, α 2

)

He also found that the results collapse well onto universal curves if they are normalized in terms of the results at the "nominal" (or "design"or "reference") flow condition for each cascade. The nominal condition is defined, somewhat arbitrarily, as the condition at which the flow turning, ε, is 0.8 of the value at stall. Stall is the 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 to turn the flow. For convenience, Howell defined the stalling incidence as the positive incidence at which the losses have increased to twice their minimum value. This definition is fairly easy to apply to experimental data. As the figure above indicates, it also seems to correspond fairly well to the point of maximum flow turning. 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:

⎛ 1⎞ δ * = mθ ⎜ ⎟ ⎝σ ⎠

n

(1)

where 2

α 2* ⎛ a⎞ m = 0.23⎜ 2 ⎟ + ⎝ c⎠ 500

(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 a compressor stage, Howell suggested using n = 1.0 and a constant value of m = 0.19. Unlike typical compressor rotor and stator blades, IGVs form an accelerating flow passage. They therefore behave more like a 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

s⎞ ⎛ ε * = f ⎜ α 2* , ⎟ ⎝ c⎠ The correlation is usually presented graphically ( Fig. 5.14 from Saravanamuttoo et al.) and was used in Section 4.4 to select the solidity. The blade will often be used at other than the nominal (“design”) flow conditions. Howell was able to correlate fairly successfully the “off-design” behaviour of the cascades by plotting the results against the nondimensional relative incidence, irel = (i - i*)/ε*. Figure 3.17 (taken from Dixon) shows the normalized flow turning, ε/ε* as a function of irel. The figure also shows the variation of the losses (expressed as a “drag coefficient”) 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 chord and upstream velocity). The resistance of the suction-surface boundary layer to separation is a function of the thickness of the boundary layer and whether it is laminar or turbulent. Thus, the flow turning behaviour of the blade row is a function of the Reynolds number, particularly at low values. Howell examined the dependence of the flow turning on the Reynolds number. Figure 3.3 (taken from Horlock) shows the effect of Reynolds number on the nominal turning.

The correlations presented to this point can be used to predict the flow turning capability of a compressor 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 be illustrated 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 2* = α 2 ′ + δ * Compare this value with the assumed α2*, revise as necessary and repeat until α2* and δ* are consistent. (ii) Read the value of ε* from Fig. 5.14. Then

α 1* = α 2* + ε * i * = α 1* − α 1′ 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 correction would 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) which will give the desired outlet flow angle, α2. This application of the correlations is a little more complicated since 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 since nominal conditions give near-minimum losses and provide some stall margin. Then

α 2* = α 2 α 1* = α 1

and ε * = α 1 − α 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 was used in Section 4.4). Since the blade row is operating at the nominal conditions, the deviation will also be that given 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

ε =i +θ −δ

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

Thus, the value of the camber will depend on the choice made for i*. Howell’s correlations indicate that there is no unique choice for the design incidence, although he recommends that a value be chosen of a few degrees at most. Reductions in camber can be compensated for by increases in incidence, and vice versa. Note that these changes will also result in a change in the stagger of the blade row. In summary, according to the Howell’s correlations a variety of blade geometries can produce identical aerodynamic performance. This gives the designer some freedom to tailor the blade geometry to meet other possible requirements: eg. to simplify the spanwise variation in the blade geometry, to alter a natural frequency, or to alter the stress level in some 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 predicted performance if the same blade profiles are used in the machine. The C4 profile, one of the most widely used of 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 for the Howell’s correlations have been fitted by polynomials. These curve and surface fits are also given in an appendix.

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 the NACA 65-series airfoil profile shape (Emery et al., "Systematic Two-Dimensional Cascade Tests of NACA 65-Series Compressor Blades at Low Speeds," NACA Report 1368, 1958). The results are correlated and design 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 design transonic compressors. As noted, Howell’s correlations do not give clear guidance for the choice of design incidence. While the nominal incidence, i*, is a reasonable choice for the design point, it is also clear from Fig. 3.17 that using i* does not in general minimize the profile losses. Howell’s correlations also do not take into account some geometric parameters which are known to affect the blade performance, such the ratio of maximum-thicknessto-chord, tmax/c. Finally, the Howell’s correlations are most suitable for analyzing the performance of a blade row of specified geometry ("analysis mode") rather than determing a geometry which gives a desired performance ("design mode"). By comparison, the NASA correlations are intended particularly for use in design mode, although they can also be used for analysis. They guide the designer to a choice of design incidence which nominally minimizes the profile losses. The correlations also account for more aspects of the blade geometry. The drawback to using the NASA correlations is that reference must be made to more graphs than for the Howell’s correlations. For consistency with the SP-36 graphs, the procedures will be described in terms of the nomenclature used by NASA. As with the Howell correlations, the incidence and deviation are defined in terms of some reference flow condition, although the definition of this condition is slightly different. Fig. 131 (from SP-36) shows the definition of the reference incidence, iref. It is the incidence half way between two off-design values of incidence at which the losses are equal. SP-36 usually refers to this as the “minimum-loss incidence” although the losses will only be a minimum if the loss “bucket” is symmetrical. As evident from Fig. 3.17, this is not normally the case. Nevertheless, the reference condition will be near minimum loss and thus would be a reasonable choice for the design point. The deviation produced at the reference incidence is designated as δref. For specified inlet and outlet flow angles, β1 and β2, the required flow turning, Δβ = β1 - β2, is related to the camber, incidence and deviation by

Δβ = θ + i − δ If we use the reference values of incidence and deviation then

Δβ = θ + i ref − δ ref

(1)

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

i ref = i 0 + nθ

δ ref = δ 0 + mθ where i0 and δ0 are the values for the same blade when it has zero camber. Substituting into (1), the required camber is given by

θ=

Δβ + δ 0 − i 0 1+ 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 0 = ( K i ) sh ( Ki ) t (i 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 are being 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 the following 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 same correlations 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 0 = ( Kδ ) sh ( Kδ ) t (δ 0 ) 10

(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 are being 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 and DCA 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 rule similar to that used in the Howell’s correlations:

m=

mσ =1.0

σb

(5)

where mσ=1.0

b

=

value of m for a solidity σ = 1.0

=

f6(β1)

Fig. 163

=

f7(β1)

Fig. 164.

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

⎛ dδ ⎞ δ = δ ref + (i − i ref )⎜ ⎟ ⎝ di ⎠ ref

(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 which will result in the required performance: that is, they are suitable for use in design mode. Of course, some decisions must already have been made concerning the type of blading (C-series, 65-series, DCA etc.), the camber 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 the following approach would be used: (i) For the specified geometry and design inlet-flow direction, β1, the reference conditions are first determined. (ii) For an off-design inlet value of β1, Eqn. (6) would then be used to predict the deviation. This defines 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 an Appendix.

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. In addition to the frictional effects in the boundary layers on the surfaces of the rotor blades, there are a number of other flow features that can generate losses. The losses due to each of these features are normally estimated individually 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 the sudden expansion in area at the trailing edge, and by the mixing out of the wake downstream of the blade. (ii) Secondary losses: The slower-moving flow in endwall boundary layers is "over turned" by the blade-toblade pressure field, as shown in the drawing. The fluid swept towards the low pressure (“suction”) side of the passage is blocked by the blade surface and rolls up into a "passage vortex" that generates additional losses through high shear stresses at the endwalls and as it mixes with the downstream flow. The boundary-layer separation 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 of the blade passage. The endwall losses inside the passage are normally assigned to the secondary losses. (iv) Tip-leakage losses: There must be some clearance between the rotor blade tips and the compressor casing. The flow that is driven through the tip gap rolls up into a "tip-leakage vortex" 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 through downstream mixing with the surrounding fluid. In transonic and supersonic compressors, there will be additional losses due to the presence of shock waves.

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 being generated by the airfoils, since this determined the pressure difference that drives the flow across the passage to form the secondary flow. Thus C DS = 0.018 C L2

where from Section 4.2.1 the blade lift coefficient is given by

æ sö C L = 2ç ÷ ( tan α 1 − tan α 2 ) cos α m è cø

and

æ tan α 1 + tan α 2 ö α m = arctanç ÷ è ø 2 For the rotor flow, we would use the relative flow angles, $1 and $2, as usual. (iii) Annulus Losses: C DA = 0.02

s sc 0.02 = 0.02 = h c h σ AR

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 J compared with the blade span h. Howell suggested that a 1% increase in the rotor clearance gap would reduce the stage efficiency by 3%: ∆η clearance = 3

τ h

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

1 ö öæ æ1 ∆P0, loss = C D σ ç ρ Ca2 ÷ ç ÷ ø è cos 3 α m ø è2

(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 for total pressure loss coefficient, rather than for drag coefficient.

4.6.3 Loss Estimation Using NASA SP-36 Correlations

(i) Profile Losses C

The profile loss system presented in NASA SP-36 is associated with the name of Lieblein, as were the blade loading limits presented in Section 4.4. In that section, it was seen that the profile losses correlated quite well with the diffusion factor defined by Lieblein, which was defined as C − C2 D = max C1

C2 SUCTION SURFACE

Cmax

C1

C1

C2

0 0

(1)

x/c

1.0

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

C max C2

(2)

Note that Deq resembles the deHaller number. Whereas the deHaller number defines the net diffusion between the inlet and outlet of the blade row, Deq defines the local diffusion on the suction side of the airfoil. Lieblein then 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 the airfoil is known. For use in the early stages of design, an approximate value of Deq, estimated from the circulation, is therefore used. The following correlation appears to be widely accepted: Deq =

(cosα 1 ) 2 tan α − tan α ö÷ cosα 2 æç . + 0.61 112 ( 1 2 )÷ cosα 1 çè σ ø

(3) y

The profile losses are reflected in a momentum deficit in the wake, as measured by the momentum thickness 2 downstream of the airfoil:

C2,ref s C 2 (y )

θ =ò

s 0

C2 C2 ,ref

æ C çç 1 − 2 C2 ,ref è

ö ÷÷ dy ø 0

where C2,ref is the velocity outside the wake. The corresponding total-pressure loss coefficient is then given by 2

æ θ ö σ æ cosα 1 ö ω = 2ç ÷ ç ÷ (4) è c ø cosα 2 è cosα 2 ø

where

ω=

P01 − P02 1 ρC12 2

(5)

The loss correlation is then expressed in terms of the variation of the momentum thickness ratio, 2/c, with equivalent diffusion ratio, Deq:

θ = f ( Deq ) c

(6)

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

AXIAL COMPRESSOR PROFILE LOSSES AT DESIGN INCIDENCE Comparison of Correlations with Lieblein Data

Wake Momentum Thickness Ratio, θ//c

0.12

Aungier Wilson & Korakianitis Koch & Smith Casey/Starke Konig et al

0.1

0.08

0.06 Lieblein Data

0.04

0.02

0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Equivalent Diffusion Ratio, D eq

Recently, Konig et al. (W.M. Konig, D.K. Hennecke & L. Fottner, “Improved Blade Profile Loss and Deviation 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 Lieblein correlation approach worked equally well for more recent compressor airfoil shapes. Their data are shown in the next figure, along with the same curve fits.

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

Wake Momentum Thickness Ratio, θ/c

0.12

Aungier Wilson & Korakianitis Koch & Smith Casey/Starke Konig et al

0.1

0.08

0.06

0.04 Konig et al. Data

0.02 +

+

0

1

1.2

1.4

1.6

+

+

1.8

+

2

2.2

2.4

2.6

Equivalent Diffusion Ratio, D eq

Although there is some evidence that more recent blade designs can tolerate somewhat higher values of Deq before stalling, the curve fit suggested by Aungier (R.H. Aungier, Axial-Flow Compressors, ASME Press, 2003) seems as reasonable as any, for both data sets: 2 8 θ = 0.004 éê10 . + 31 . ( Deq − 1) + 0.4( Deq − 1) ùú ë û c

(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, 2/c, using (7). (3) The total-pressure loss coefficient T is then calculated from (4).

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

(cosα 1 ) 2 tan α − tan α + a i − i * cosα 2 æç . + 0.61 112 ( 1 2) cosα 1 çè σ

(

)

1.43

ö ÷ ÷ ø

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

(ii) Endwall Losses NASA SP-36 does not provide clear guidance for estimating either the secondary losses or tip clearance losses for the purposes of meanline analysis. Instead, most recent text books (eg. Japikse & Baines) and papers seem to recommend a method developed by Koch & Smith at General Electric (Koch, C.C. and Smith, L.H., “Loss Sources and Magnitudes in Axial-Flow Compressors,” ASME J. Eng. for Power, Vol. 98, 1976, pp. 411-424). The method provides combined estimates for the effects on stage efficiency of both secondary flows and tip leakage. This is physically 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 a fairly detailed knowledge of the stage geometry. It is also necessary to specify how close the stage is to stall at the operating point for which the loss estimates are being made. Nevertheless, because of the importance of endwall 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

⎛ 2δ * ⎞ 1− ⎜ ⎟ η ⎜ h ⎟ = η P ⎜ 1 − 2ν ⎟ ⎜ ⎟ ⎝ h ⎠ where

ηP δ* ν

= = =

(1)

stage efficiency as calculated from the profile losses only average displacement thickness of the two endwall boundary layers 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 the lower fluid velocity present in the endwall boundary layers. Koch & Smith provide correlations, derived from very wide-ranging tests conducted on a large, lowspeed compressor test rig, for estimating the values of δ* and ν. The drawing defines some of the geometric parameters that appear in the correlations. s λ g

= = = =

spacing stagger angle staggered spacing

λ

s cos λ

In addition, the following are used ε h ξ

= = =

tip clearance blade span axial gap between rotor and stators

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

s g

c

λ

from the casing wall and have a clearance at the hub wall. If the stators are variable pitch, they will also need clearance. 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 of the clearance and the pressure rise ratio:

⎛ ΔC P 2δ * ε⎞ = f⎜ , ⎟ g ⎝ ΔC P ,max g ⎠ where

ΔC P =

ΔP q

with ΔP the static pressure rise across the stage and q the average of the inlet dynamic pressures for the rotor and stator rows. ΔC P,max is the maximum value of the static pressure rise coefficient for the same stage, corresponding 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 has been chosen. For example, if the deHaller numbers are low and the solidities have been selected to give relatively high values of the diffusion factors, the pressure rise ratio would be expected to be towards the higher end of the scale.

0.55 0.5 0.45 0.4

2δ*/g

0.35 0.3

ε/g = 0.10 0.075

0.25

0.050

0.2

0.025

0.15

ε/g = 0.0

0.1 0.05 0

0.7

0.75

0.8

0.85

ΔC P/ΔCP,max

0.9

0.95

1

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

1.1

1.05

2δ*/(2δ*)ref

1

0.95

0.9

0.85

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Axial Gap/Blade Spacing, ξ/s

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

0.9 0.8 0.7

2ν/2δ*

0.6 0.5 0.4 0.3 0.2 0.1 0

0.75

0.8

0.85

ΔCP/ΔCP,max

0.9

0.95

1

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 hc s = g hcsg ε AR σ = h cos λ

(2)

Values of the tip clearance ε are often specified as a fraction of the blade span h. Therefore, reasonable values of ε/h would be known early in design. Eqn. (2) also implies that the Koch & Smith correlation can be used to conduct parametric studies to investigate the influence on the endwall losses of common design parameters such as the solidity σ and the blade aspect ratio AR = h/c. For use in Eqn. (1), note that

2δ * 2δ * g s c = h g s ch =

2δ * cos λ g σ 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 included the 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 the blade section. In addition, the Howell results apply only for low subsonic values of the inlet Mach number. The loss behaviour 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 some representative results are presented to illustrate the complexities. The figure (taken from SP-36) shows the variation of profile losses with both incidence and inlet Mach 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 camber lines 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 transonic compressors. It is seen that their sensitivity to Mach number is delayed to a higher inlet Mach number than some of the other shapes.

The strong influence of the detailed airfoil geometry on the behaviour at both off-design incidence and with increasing inlet Mach number obviously makes it more difficult to devise simple correlations for the losses, 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 Transonic Compressor 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 ΣP0,loss determined from the correlations described in Section 4.6.2 or 4.6.3, the total-pressure rise across the stage is then: ∆P0 = ∆P0, ideal − Σ∆P0 , loss

where )P0,ideal is the pressure rise that would have been obtained in an ideal machine having the same work input. (i) For compressible flow of a perfect gas, the ideal pressure rise is γ

∆P0, ideal = P03′ − P01

where

P03′ æ T03 ö γ − 1 =ç ÷ P01 è T01 ø

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

T0

P′03

P03 T03 T03′

P01

T01

s

Then the actual P03 is obtained from P03 = P03′ − ΣP0, loss

where ΣP0,loss is as predicted for the rotor and stators together. The corresponding efficiency can be calculated by determining the power required to compress through the same pressure ratio, P03/P01, in an isentropic machine: T03′ æ P03 ö =ç ÷ T01 è P01 ø

γ −1 γ

Then

η=

T03′ − T01 T03 − T01

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

η≅

∆P0 ∆P0, ideal

If any of the loss components is expressed in terms of an efficiency decrement, as is sometimes the case with tip-leakage loss, its contribution to the total pressure losses can be estimated from ∆P0, TL ≅ ∆η clearance ∆P0, ideal

and this would be included in the earlier summation of losses. (ii) For incompressible flow, we can use ∆P0 , ideal = ρ g ∆H E = ρ (U 2 C w 2 − U 1 C w1 ) ≅ ρ U ∆C w

and

η= where, as before

∆P0 ∆P0, ideal

∆P0 = ∆P0,ideal − Σ∆P0,loss

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 velocity triangles show that the incidence at the leading edge of the rotor blades is increased. If the incidence becomes too large, the blades may stall. The disorganized flow in the stalled region partially blocks the blade passage. As a result some of the fluid that was previously passing through the stalled passage is diverted to the adjacent passages, as indicated in the middle figure. This has the effect of increasing the incidence at the airfoil lying next to the stalled region while reducing the incidence at the other adjacent airfoil. The increase in incidence on the one adjacent airfoil may cause it to stall in turn. On the other hand, the airfoil which has its incidence reduced will move further away from stall, or may unstall if it was previously stalled. Thus, there is a tendency for the stall cell to migrate from blade passage to blade passage in the opposite direction to the rotation. This phenomenon is known as rotating stall. The stall cells move in the opposite direction to the rotation at a relative speed which is about half the rotational speed.

MIGRATION OF STALL CELL

NORMAL ATTACHED FLOW

SEPARATION OF BLADE BOUNDARY LAYER - BLADE STALL

ROTATING STALL

The rotating stall can take a number of patterns. It may involve only one blade passage, or a large number 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 considerably across the span and it will be the hub region that will have the highest loading and therefore be the most likely to stall. In that case, the may stall cells may only involve part of the span of the blade. If the blades have high hub-to-tip ratio, the stall is more likely to extend across the full span.

LOW HTR ROTATING STALL

HIGH HTR FULL SPAN STALL, ALL PASSAGES

Particularly if the rotating stall occurs at low speed and only involves part of the span, it may not be a danger 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 of noise. For reasons discussed in Section 4.8, it is fairly common for the early stages of multi-stage axial compressors to experience some rotating stall at low rotational speeds. If it is present only during start-up and shut-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 & characteristic becomes positive. As will be shown in Chapter 7, if this occurs the system of which versus m the compressor is a part can become dynamically unstable. If this instability is triggered, the result is known as surge. The peak point of the compressor characteristic is therefore often identified as the surge point.

ΔP0 SURGE

ROTATING STALL

N const .

& m

In Chapter 7, an approximate, unsteady-flow analysis is developed for a compression system. This analysis identifies the various factors that will make the system more or less prone to surge. However, a simple 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 is being “drained” through the turbine. At any instant in time, there is a fairly large mass of gas present in the volume of the combustor. Now suppose that the last stage of the compressor suddenly stalls. This might be due to some disturbance that causes a drop in the mass flow rate through the machine. Since the last stage normally has a high hub-to-tip ratio, the stall may involve the full span of the blades and all of the passages, as described in the last section.

If the stall is very extensive, there will be an abrupt drop in the pressure of the gas delivered by the compressor. The flow area through the turbine is relatively small and this limits the outflow through the turbine. Consequently, the pressure in the combustor will drop somewhat gradually. It the rate of pressure drop is too slow, the situation can occur that the pressure in the combustor is higher than the pressure at the compressor outlet. Since fluid tends to flow from a region of high pressure to one of low pressure, it is therefore 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 point the flow in the compressor may re-establish itself. However, if the conditions that led to the initial stall are still present, the whole process can repeat. The cyclic flow reversal in the compressor can result in very large fluctuating forces on the blades which can destroy the machine. In the gas turbine engine, the abrupt drop in compressed air supplied to the combustor can also lead to over-temperatures and resultant serious damage to the 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 with identical aerodynamic characteristics and thus identical stage design points. Therefore, at design point for the machine as a whole, each of the stages will be running at their individual design points, which occur for the same value of the flow coefficient φ= Ca/U for all of the stages. Assume also that the mean radius, and thus the blade speed U, is the same for all four stages. Since the density of the gas increases across each successive stage, to maintain the constant axial velocity Ca needed to keep φ constant it is necessary to reduce the annulus area along the machine. This variation in the cross-sectional area would be determined at the design-point flow conditions. Ca U U

Ca

ψ

FOUR IDENTICAL STAGES

DESIGN N 1

m& = ρ Ca A

2

ψD

3 OR

4

Ca =

m&

ρ

ρ VARIATION DUE TO COMPRESSION

ρA

Ca

TARGET Ca VARIATION

φ

φD

A AREA VARIATION TO ACHIEVE TARGET Ca

DESIGN POINT

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

STAGE I

1

2

STAGE II

3

Since the annulus area has been adjusted such that at the design point the two stages have the same flow coefficient: φ I D = φ II D = φ D For stage I

φI = D

CaI D UD

and the corresponding pressure rise is

ΔPI D = P3 − P1 and the outlet density ρ 3D =

P1 + ΔPI D RT3D

Then for stage II

Ca 3 D =

m& D ρ 3D A3

φ II = D

Ca 3 D UD

where, as mentioned, A3 was adjusted to give φ I D = φ II D . Now consider the effect of halving N (ie. halving U) while also halving m& (ie. halving Ca) to keep Stage I operating at its design φ:

1 C a 1D 2 φI = = φ ID 1 UD 2 Thus, Stage I will also be operating at its design ψ. However, the absolute Δh0 (and thus ΔP0} varies as U2 and the pressure rise is therefore reduced to 1 P1 + ΔPD 1 4 ΔP ≅ ΔPD and ρ 3 = RT3 4 Neglecting the changes in T3, which will be relatively much smaller than the changes in P3, then

1 P + ΔPD ρ3 4 ≅ . < 10 ρ3 P + ΔPD D

The corresponding change in Ca3 is then

Ca 3 Ca 3 D

m& ρ 3 m& ρ A ⎛ 1⎞ = 3 3 = D = ( > 10 . )⎜ ⎟ = k m& D ⎝ 2⎠ ρ 3 m& D ρ 3D A3

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

φ II =

Ca 3 kCa 3D k = = φ II D 1 1 U UD 2 2

and since k > 1/2, φ II > φ II D . Thus, the non-dimensional operating point for Stage II shifts to a lower value of 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 only increases in the subsequent stages. For the 4-stage compressor with four identical stages we would therefore expect to see the following pattern of operating points:

Ca U U

ψ

Ca LOW N, STAGE 1 AT DESIGN 1 2

ψD

3

m& = ρ C a A < m& DES

4

Ca =

φD

m& ρA

ρ

ρ VARIATION (UNDERCOMPRESSION)

φ Ca

DESIGN Ca VARIATION NEW Ca VARIATION

A AREA VARIATION (FIXED)

REDUCED RPM, REDUCED

If we now reduce the mass flow rate at the low speed operating point, keeping N constant, the flow coefficient for Stage I will be lowered. Stage I will then be producing slightly higher pressure rise. However, the effect of the low U2 is much greater than the small increase in ψ and Stage I will still be producing much lower pressure rise than at the design N. Consequently, Stage I is still undercompressing the fluid and the downstream stages will again be at successively higher values of φ. We therefore conclude that if we throttle the flow further, Stage I will be the first to reach its stalling value of φ.

m&

ψ LOW N, THROTTLED 1 2 3

ψD

4

φD

φ

ψ

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

HIGH N, STAGE 1 AT DESIGN 4 3 2

ψD

1

φD

φ

Combining these arguments, we can plot the expected map for the compressor as a whole. 1

P02 P01

2

3 4

STAGE STALL LINE

HIGH-SPEED OPERATING POINT

ηmax

HYPOTHETICAL STEADY-STATE OPERATING LINE

DESIGN POINT

COMPRESSOR SURGE LINE

η CONSTANT SPEED LINE

N

4

T01 3

2

1

m& T01 P01

Note that: (i) Over most of the map, we assume that the stalling of any stage results in compressor surge. As a result, when the onset of the stall switches from the front to the back of the machine (near the design N), 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 more likely to be part-span, rotating stall (as discussed in Section 4.7). This, combined with the fact that the absolute forces on the blades will be low at low N, means that some degree of rotating stall is acceptable at low N. As a result, at the low end of the map the surge line has a “kink”, indicating that some early-stage stall is allowed. (ii) At the design point of the compressor, all of the individual stages are operating at their design points and therefore have their maximum efficiencies. From the earlier discussion, it is evident that at any other operating point at most one of the stages will be operating at best efficiency. Therefore, the efficiency of the overall compressor will be less than its value at design. For this reason, the lines of constant efficiency are shown as closed contours surrounding the design point.

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

NASA 8-Stage Research Compressor

Pratt & Whitney TF30 LP Compressor

4.9

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

For solidities, F, less than about 0.4 each blade can be treated as an isolated airfoil. Note that F = 0.4 was the lowest value of solidity that appeared on the NASA SP-36 correlations (Section 4.5.3). Usually, the blade is divided into a series of spanwise segments or blade elements. Three-dimensional flow effects in the form of spanwise flows are usually neglected, although the downwash induced by the trailing vortex system is sometimes taken into account. This approach, known as the "blade-element method", is commonly used to design propellers and low-performance axial fans. Consider the flow relative to a blade element. The element behaves like an isolated airfoil in a stream in the direction of the vector mean of the inlet and outlet flows:

ZLL Wm

= =

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

Wm =

P )L

= = =

)D

=

)X

=

)Y

=

CL, CD =

Ca Q æ tan β 1 + tanβ 2 ö ; β m = arctanç ÷ ; Ca = è ø A cos β m 2

angle of attack of blade element = angle between ZLL and Wm lift force on blade element (perpendicular to Wm) 1/2DWm2c)rCL (1) where c = chord length of blade element )r = radial width of blade element CL = lift coefficient of blade element (as obtained from airfoil characteristics and P) drag force on blade element (parallel to Wm) (normally )D )L) axial component of force on blade element )X . )L sin$m (see note at end of section) tangential component of force on blade element fns [P, section shape, Re] - as obtained from airfoil data

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

(2)

Fx = N B ∆X = A∆P = (2π r∆r ) ∆P where NB )P

= =

(3a)

no. of blades static pressure difference across the blade row

Substituting for )X in terms of the lift coefficient

ö æ1 N B ç ρWm2 c∆rC L ÷ sin β m ≅ 2πr∆r∆P ø è2 ö æ1 N B ç ρWm2 cC L ÷ sin β m ≅ 2πr∆P ø è2

(3b)

For the input power, from the energy equation

∆Q∆P0 ∆W&in = ∆Tω = ≅ ρ∆Q∆h0 ≅ ρ∆Q(U∆C w ) ηR

(4a)

assuming )P0 is small so that )P0/D0R . )h0, and where

)T )Q 0R )P0

= = = =

torque applied to flow through annulus width )r volume flow rate through annulus width )r rotor efficiency (0R =1 if CD = 0) 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 = NBr)Y)

∆Q∆P0 é1 ù ∆W&in = N B ê ρWm2 c∆r ( CL cos β m + CD sin β m )r úω = = ρ ∆Q(U∆Cw ) (4b) ηR ë2 û Rotor and stator blade rows can then be designed using Eqns. (1) - (4). Iteration will generally be necessary since W2 is a function of )L, which is a function Wm, which in turn is a function of W2. The analysis 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 the trailing tip vortices from the blades. The downwash would slightly alter the effective flow incidence seen by the blade and thus the lift it develops. To make the velocity triangle diagram clearer, the blade was sketched with somewhat lower stagger angle than would normally be found in practice. The diagram shows the force triangles for a more realistic value of the stagger angle:

y

∆D

x

∆L ∆X ∆F

Wm

∆Y

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

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, except that camber is usually much larger. The stage consists of a set of stators ("nozzles") followed by a rotor. The nozzles control the swirl in the flow entering the rotor and the rotor then extracts work from the fluid by removing swirl. This arrangement of components results in stage aerodynamic characteristics that are very different 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 mean radius, 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 velocity triangles:

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

φ=

Ca < φD U

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

ψ =

Δh0 ΔC w = U U2

φD

ψ =

Cw U

>ψ D

and the same trend is found as when the mass flow rate was changed. Now consider the absolute output. From the Euler equation Δh0 = UΔC w and from the velocity triangles, ΔCw increased as U decreased. It is not entirely clear whether the product UΔCw 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 axial compressors and turbines differ in two main ways: (i) ψ versus φ, and therefore ΔP0 versus m& , is negative for compressors, positive for turbines. (ii) The energy transfer Δh0 is a strong function of U for compressors, but only a weak function for turbines.

The following figures show the actual characteristics of the gas-generator turbine of the Orenda OT-2 gas turbine engine. Note that it is conventional to use the pressure ratio as the independent variable for plotting turbine aerodynamic characteristics. The characteristics confirm that the mass flow-pressure ratio characteristic is only a weak function of the rotational speed. However, this does not mean that the rotational speed is not important in order to have a high output of useful work. As seen from the velocity triangles, if the rotational speed is reduced below the design value, the energy released by the fluid, Δh0 = UΔCw, may not be changed very much, but this is also accompanied 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 recover most 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 axial compressors. Again, primarily British results will be presented, but these are also widely used in North America.

5.3.1

Flow Outlet Angle

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

æ sö δ = mθ ç ÷ è cø

n

where 2 = 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 of 1/2 used for compressor rotor and stator blades). However, the Carter & Hughes correlation tends to overestimate the deviation for most modern turbine blades. A more satisfactory (but less convenient) correlation is that due to Ainley & Mathieson (A-M). Their correlation uses the so-called gauge angle 2g as a reference angle to which the actual outlet angle is related:

æ oö θ g = cos −1 ç ÷ è sø

where o = throat opening and s = blade spacing. For an infinitesimally thin blade which is straight from the throat to the trailing edge, the gauge angle would define the direction normal to the throat line. For low speed flow, A-M correlated the outlet flow angle "2 with the gauge angle. Fig. 7.13 (from Saravanamuttoo et al.) shows the variation for a “straight-backed” blade: that is, a blade for which the suction side is straight from the throat point to the trailing edge. The curve in the figure can be approximated by

o

θg s

æ oö α 2 = 11625 . cos −1 ç ÷ − 12 è sø

However, most turbine blades are not straight backed. Instead they have a certain amount of “unguided turning” as defined by the angle 2u. In A-M’s day, if the suction surface was not straight from the throat to the trailing edge, it was usually defined by a circular arc. AM therefore corrected the outlet angle as follows:

æ oö æ sö α 2 = 11625 . cos −1 ç ÷ − 12 + 4ç ÷ è sø è eø where e is the suction side radius of curvature. Unfortunately, modern turbine blades usually do not use circular arcs to define their surface shapes. As a result, e is not constant and generally not known. To use the AM correlation it is therefore necessary to obtain an “equivalent” value of e. An approximate value can be calculated from the unguided turning angle as follows:

s = e

πθ u æ oö 180 1 − ç ÷ è sø

for 2u measured in degrees.

2

θg

(ii) Sonic Condition For M2 = 1.0 and a straight-backed blade, A-M indicated that the outlflow angle would be equal to the gauge angle: æ oö α 2 = cos −1 ç ÷ è sø

For a curved-back blade, this was again corrected for the suction side radius of curvature. The results were presented graphically but can be approximated by the following curve fit: s 1.787 + 4 .128 ö æ e æ oö çæ sö ÷ sin −1 æ o ö α 2 = cos ç ÷ − ç ç ÷ ç ÷ ÷ è sø è eø è sø è ø −1

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

(

α 2 = α 2 M 2 = 0.5 − (2 M 2 − 1) α 2 M 2 = 0.5 − α 2 M 2 =1.0

)

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 depth in the z direction.

Y y

x X

P1

C1

α1 Cw1

s

Ca1 P2 Ca2 α1 cx

Cw2 C2

Apply the linear momentum equation in the y direction:

(

ΣFy = m& V y 2 − V y1

)

(1)

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

Y = m& (C w 2 + C w1 )

(2)

From the velocity triangles, C w1 = Ca1 tanα 1

C w 2 = Ca 2 tanα 2

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

⎛ ⎞ C Y = ρ 2 sCa22 ⎜ tanα 2 + a1 tan α 2 ⎟ Ca 2 ⎝ ⎠ or, since Ca 2 = C2 cosα 2 ,

⎛ ⎞ C 1 Y = ρ 2 C22 (2 s)cos 2 α 2 ⎜ tanα 2 + a1 tanα 2 ⎟ Ca 2 2 ⎝ ⎠

(3)

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

P

"IDEAL" DISTRIBUTION

P0

P1

1 P0 − P2 = ρC22 2

PS

ACTUAL DISTRIBUTION

P2 SS

x

0

Y=∫

cx

0

cx

( PPS − PSS )dx

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

1 Yideal = ( P0 − P2 )c x ×1 = ρ 2 C22 c x 2

(4)

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

Z= Substituting from (3) and (4) then

Y Yideal

⎛ s⎞ ⎛ ⎞ C Z = 2⎜ ⎟ cos 2 α 2 ⎜ tanα 2 + a1 tanα 1 ⎟ Ca 2 ⎝ cx ⎠ ⎝ ⎠

(5)

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

⎛ s⎞ ⎛ ⎞ C Z = 2⎜ ⎟ cos 2 α 2 ⎜ tanα 2 − a1 tanα 1 ⎟ Ca 2 ⎝ cx ⎠ ⎝ ⎠ As usual, for rotor blades β replaces α. The normal definition of the solidity is σ = c/s. The way the Zweifel coefficient 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 the drawing, where ζ is the stagger angle: C X

ζ

S

C

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 the value 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 solidity for axial compressors. Since Zweifel’s time, profile design has improved and today turbines are often designed with considerably higher values of Z ( Z = 1.00-1.05 is common).

5.3.2.2 Ainley & Mathieson Correlation Ainley & Mathieson developed a widely used loss system (see next section), based on British turbine cascade data from the 1940s and 1950s. They likewise identified the geometries that gave minimum profile losses for different combinations of inlet and outlet flow angles. These optimum geometries, expressed as optimum 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 the Zweifel criterion).

Unfortunately, Zweifel and Ainley & Mathieson expressed “solidity” differently: cx/s versus c/s. This makes it difficult to compare the geometries that would be obtained using each approach, for the same set of velocity triangles. The two ratios are related through the stagger angle, ζ, of the blade row (see the figure on the previous page), since cosζ = cx c . However, the value of the stagger angle is not fixed by the inlet and outlet flow (or metal) angles. This is illustrated in the figure at the right, which shows two actual, very highly-loaded (Z = 1.37) low pressure turbine blade rows that were designed for identical inlet and outlet flow angles (α1 = 35o, α2 = 60o). The two blades clearly have very different stagger angles. This is the result of different decisions regarding the detailed pressure distributions around the blades. The blade with the high stagger angle was designed to be “forward-loaded”: that is, to develop most of its lift on the forward part of the airfoil. The one with the lower stagger angle is much more “aft-loaded”. The two airfoils have identical 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 have the “optimum” geometry according to Fig. 7.14. Despite these difficulties, it is possible to make an approximate Z = 1.37 comparison between the results obtained by the two different approaches to 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 different combinations of α1 and α2:

Using values of stagger angle obtained from K-O Fig. 5, the following figure shows the values of the Zweifel coefficient for selected combinations of inlet and outlet flow angles. It is evident that the optimum geometries based on the Ainley & Mathieson correlations lead to higher values of Z than Zweifel originally recommended. 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 results suggest that relatively higher values of Zweifel coefficient can be tolerated in the rotor blades for stages with low values of degree of reaction, especially if the total flow turning is low.

Z = 1.028 Z = 1.034

Z = 0.911

Z = 1.56 Z = 1.26

Z = 0.990 Z = 0.909

5.3.3

Losses

In both North America and Europe, most loss estimates for axial-flow turbines are based on a loss system developed by Ainley & Mathieson (AM) in the UK in the early 1950s (ARC R&M 2974, 1957; see also Saravanamuttoo et al.). The AM system has been updated a couple of times to reflect improvements in blade 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 Turbine Efficiency”, 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 curves or surfaces and these fits are given in Appendix E. For turbines, the total-pressure loss coefficient Y is defined as

Y=

ΔP0,loss P02 − P2

(1)

Note that in this case, the loss is non-dimensionalized by the outlet dynamic pressure, whereas the inlet value is used in the loss coefficients for axial compressors. As for compressors, losses are again divided into components and these are then added linearly to obtain the total losses:

YTotal = YP f ( Re) + YS + YTET + YTC

(2)

where the subscripts designate the components as follows: P = profile, S = secondary, TET = trailing-edge thickness, TC = tip clearance. f(Re) represents a correction for the effects of Reynolds number on the profile losses. The effect of Reynolds number on the other loss components is not well documented but it is believed to be small. The following figure shows the blade nomenclature used in the KO system. Note that they do not follow the sign convention we defined earlier. Using that convention, the inlet and outlet flow and metal angles will often have opposite signs because of the high turning that is normally present in turbine blade rows. It becomes a nuisance to keep track of the signs and therefore it is common practice by turbine designers 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 same outlet flow angle: a nozzle blade (maximum blade-passage acceleration) and an impulse blade (zero acceleration). In the original AM system, the expression took the form:

YP , AM

2 ⎡ ⎛ β1 ⎞ = ⎢YP ,nozzle + ⎜ ⎟ YP ,impulse − YP ,nozzle ⎝α2 ⎠ ⎢⎣

(

)

β1

⎤⎛ t c ⎞ α2 ⎥⎜ max ⎟ ⎥⎦⎝ 0.2 ⎠

(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) and Fig. 2 (for impulse blades). Note also that for a given value of the outlet angle α2 there is a value of solidity σ = c/s that minimizes the profile losses. This was the origin of the "optimum σ" that is plotted on Fig. 7.14 in Section 5.3.2.2.

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

YP , KO =

2 (0.914YP , AM ) 3

(4)

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

As seen, Eqn. (3) includes a correction for the maximum thickness of the airfoil: the data in Figs. 1 and 2 apply for a maximum thickness of 20% of the chord length. Decisions about the maximum airfoil thicknesses would not normally be made at the stage of a meanline analysis for the blade row. However, KO examined the range of maximum thicknesses observed for a number of recent actual designs and provided the correlation shown in Fig. 4. Knowing the flow turning from the velocity triangles, this figure can then be used to 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 turbines in gas turbine engines normally operate under compressible flow conditions. The Mach number levels encountered 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 and drive the high pressure compressor. To minimize the number of stages, HPTs are typically designed to operate at transonic outlet flow conditions. Low Pressure Turbine (LPT). The LPT drives the low pressure compressor, and the fan stage in a turbofan engine. The fan has a large tip diameter and to keep the tip Mach numbers acceptable, the fan shaft must rotate at a much lower speed than the high-pressure spool. The tip diameter of the LPT is much smaller than that of the fan and as a result it runs at a relatively low blade speed. This in turn results in lower flow velocities generally. It is therefore normal for the flow around LPT airfoils to be subsonic everywhere. As a result of these differences, the strongest effects of compressiblity are normally seen in HPTs. The following Schlieren photos (taken from E. Detemple-Laake, “Measurement of the Flow Field in the Blade Passage and Side Wall Region of a Plane Turbine Cascade,” AGARD-CP_469, 1989) show the flow through an 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 blades result in high local velocities in this region. For inlet relative Mach numbers as low as 0.6, patches of supersonic 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 accelerating flow channel. As the outlet Mach number increases, there is a tendency for the blade surface boundary layers to be thinned and their contribution to the losses actually decreases slightly. As the outlet Mach number approaches 1.0, patches of supersonic flow, terminating in shocks, may begin to appear on the aft suction surface. Finally, as the outlet Mach number becomes supersonic, expansion waves and shocks appear in the trailing edge region. In addition to directly contributing additional total pressure losses, it is common for one or more of the shocks to impinge on the surface of the adjacent 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 operating at an outlet Mach number of 1.30.

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

The complexity of the compressibility effects makes it difficult to predict their influence on the losses. Kacker & Okapuu provide procedures for estimating the contributions to the profile losses; see the paper for details. Finally, KO give the following Reynolds number corrections for profile losses: ⎛ Re c ⎞ f ( Re) = ⎜ ⎟ ⎝ 2 × 10 5 ⎠

− 0. 4

for

= 10 .

Re c ≤ 2 × 10 5

for 2 × 10 5 < Re c < 10 6

⎛ Re ⎞ = ⎜ 6c ⎟ ⎝ 10 ⎠

− 0.2

for

Re c > 10 6

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 secondary losses in axial turbines are a function of CL2: 2

⎛ cos α 2 ⎞ ⎛ C L ⎞ cos 2 α 2 YS = 0.04 f ( AR)⎜ ⎟⎜ ⎟ ⎝ cos β 1 ⎠ ⎝ s c ⎠ cos 3 α m

(5)

where CL = 2( tan α 1 + tan α 2 ) cos α m sc

1 α m = tan −1 ⎛⎜ ( tan α 2 − tan α 1 )⎞⎟ ⎝2 ⎠ and as before, all angles are taken as positive. The loss coefficients give the total-pressure losses as averaged over the total mass flow rate through the blade passage. As the aspect ratio of the blade becomes larger, a smaller fraction of the span is occupied by the secondary flow and the loss associated with it becomes averaged over an increasingly larger mass flow rate. Consequently, the mass-averaged loss coefficient varies inversely with the aspect ratio. This effect is embodied in the aspect ratio correction, f(AR) in Eqn. (5). Kacker & Okapuu found that the AM loss system tended to over-estimate the effect of aspect ratio on blades of very low aspect ratio (which are often used in modern HPTs). In the KO loss system, the correction for blade aspect ratio therefore takes the following form:

f ( AR) =

1 − 0.25 2 − h c

1 = hc

hc

for h c ≤ 2 (6)

for h c > 2

Aspect Ratio Correction, f(AR)

2.5

2 Ainley & Mathieson

1.5

1

Kacker & Okapuu (Eqn. (6))

0.5

0

0

0.5

1

1.5

2

2.5

3

Aspect Ratio, h/c

Kacker & Okapuu also provide a compressibility correction for the secondary losses (see the paper). Trailing-Edge Losses: Due to the finite thickness of the trailing edge, the streamtube experiences a sudden increase in area as it leaves the blade passage. The resulting sudden-expansion loss is correlated in terms of an alternative form of loss coefficient, known as an energy loss coefficient, Δφ2, as a function of the ratio of the trailing-edge thickess to the throat opening. KO correlated the values for nozzle blades and impulse blades separately, as shown in Fig. 14.

In the same way as for the profile losses, the trailing-edge loss for an arbitrary blade is expressed as the weighted average of the values for nozzle and impulse blades: 2

Δφ

2 TET

= Δφ

2 TET ( β1 = 0 )

(

⎛β ⎞ 2 2 + ⎜ 1 ⎟ Δφ TET ( β1 =α 2 ) − Δφ TET ( β1 = 0) ⎝α2 ⎠

)

(7)

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

YTET =

⎡ γ −1 2⎛ ⎞⎤ 1 M2 ⎜ − 1 ⎟⎥ ⎢1 − 2 2 ⎝ 1 − Δφ TET ⎠ ⎥⎦ ⎢⎣

γ −1 2⎞ ⎛ M2 ⎟ 1 − ⎜1 + ⎝ ⎠ 2

−

−

γ γ −1

−1

γ γ −1

and for incompressible flow this reduces to YTET =

1 2 Δφ TET

−1

Tip-Clearance Loss: For unshrouded blades, KO express the effects of tip-clearance losses as a correction to the efficiency:

Δη

η0

= 0.93

RTip k h cos α 2 R Mean

(8)

where η0 is the efficiency for zero tip clearance and k is the tip clearance. Note that Eqn. (8) indicates that a 1% increase in tip clearance, relative to blade span, will result in a 1% reduction in efficiency. This is considerably lower sensitivity than the 3% reduction that is predicted by Howell’s correlation for axial compressors. As seen, KO also found the loss to be a function of the hub-to-tip ratio of the blade, since 1 R Mean = RTip 1 + HTR 2 , where HTR = RHub/RTip. 2

(

)

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

YTC

c ⎛ k ′⎞ = 0.37 ⎜ ⎟ h⎝ c ⎠

0.78

2

⎛ C L ⎞ cos 2 α 2 ⎜ ⎟ ⎝ s c ⎠ cos 3 α m

(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 loss components for two different turbine stages, one subsonic and one transonic, will be quoted (taken from Moustapha et. al., Axial and Radial Turbines, Concepts NREC, 2003, pp. 89-90). The table summarizes the 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

Exit Mach Number

Stators

Rotor

Stators

Rotor

0.67

0.82

1.1

1.14

60

78

76

124

0.71

1.25

0.70

1.44

Total Flow Turning (o) Blade Aspect Ratio, h/c Tip Clearance, k/h (%)

1.5

Zweifel Coefficient

0.74

1.5

0.88

0.84

0.76

The figures show the resulting values of the loss coefficients:

Transonic Turbine

Subsonic Turbine 0.4

0.3

0.4 Stators

0.35

Rotor

Loss Coefficient, Y

Loss Coefficient, Y

0.35

0.25 0.2 0.15 0.1 0.05

0.3

Stators Rotor

0.25 0.2 0.15 0.1 0.05

0 Profile

Trailing Edge

Secondary

Tip Clearance

Loss Component

Total

0 Profile

Trailing Edge

Secondary

Tip Clearance

Loss Component

Total

PW100 Turboprop

2

95

1.8

90

1.6 η, (Rotor Only)

η, Efficiency (%)

85

1.4 Diffuser Loss

80 75

1.2 1

η (Rotor + Diffuser)

70

0.8

65

0.6

60

0.4

55

0.2

50

0

5

10

Tip Mach Number

100

0 15

Compressor Pressure Ratio

Data for rotor only: Senoo, Y., Hayami, H., Kinoshita, Y. and Yamasaki, H., "Experimental Study on Flow in a Supersonic Centrifugal Impeller," 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 Ratio Centrifugal 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 in Advanced 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 swirl in the rotor inlet flow and that the fluid is incompressible. Assume also the Euler Approximation so that the flow leaves the rotor parallel to the metal angle at the vane trailing edge. U2 W2 β2 (+)

C2

Cr2

β'2 (+)

Cw2

ω

From the Euler equation ∆h0 = g∆H E = U 2 C w 2 − U 1 C w1 = U 2 Cw2

and

tan β 2 =

Cw2 − U 2 Cr 2

or C w 2 = U 2 + Cr 2 tan β 2 Q = U2 + tan β 2 A2

Then g∆H E = U 22 + U 2

Q tan β 2 A2

or, dividing by N2D2

g∆H E Q = K1 + K 2 2 2 N D ND 3 2

where

U2 æπ ö K1 = 2 2 2 = ç ÷ = const . è 60 ø N D

(1)

and K2 =

U2D tan β 2 NA2

Equation (1) is the equation for the idealized head rise versus flow rate characteristic. Within the Euler Approximation ($N2 = $2), the slope of the characteristic, K2, is constant and has the same sign as $2, as shown in the sketch:

Note that whereas the slope of the )H vs Q characteristic for axial machines was always negative (assuming R < 1.0, which experience has shown is necessary), radial machines can have charateristics with either 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 centrifugal compression, even with no flow. From Eqn. (1), for Q = 0, )HE = U22/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: from specific 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 turbomachine that is suitable for a particular application. The following figures (from C. Rodgers, “Specific Speed and Efficiency 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 geometric and aerodynamic parameters.

The definition of specific speed used here is based average density (or average volume flow rate): 1

NS =

æ Q + Q2 ö 2 ωç 1 ÷ è 2 ø 3

( g∆H ) 4

Compare this with the usual non-dimensional specific speed:

Ω=

1 2 1

ωQ

3

( g∆H ) 4

Note that for a typical case, S will be slightly larger than N S . If there are no constraints on the rotational speed, then one would normally choose the value of N S that gives the highest 0 ( N S ≅ 0.6 − 0.8 ). With N S chosen and S estimated, the Cordier diagram can be 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 coefficient for industrial compressors of several configurations: with shrouded and unshrouded impellers; and with vaneless and vaned diffusers. The correlations are based on results for compressors with pressure ratios up to about 3.5, but can probably be extrapolated to somewhat higher values. Aungier defines the non-dimensional parameters as follows: Flow Coefficient:

φ=

& m Work Coefficient: ρ 1πr22 U 2

µP =

∆h0ref U 22

where )h0ref is the total enthalpy rise for the reversible process with the same pressure ratio. 0P is the stage polytropic efficiency. Aungier’s correlations are presented in the following two figures:

The figures are applied as follows: (a) Select N to give the best stage polytropic efficiency, 0P, and read the corresponding work coefficient, :P. From

µP =

∆h0ref U 22

calculate U2. (b) Then from the chosen N

φ=

& m ρ 1πr22U 2

calculate r2. With the tip radius and tip blade speed defined, the rotational speed is known. If the rotational speed is constrained (eg. driving motors are only available for certain speeds) then Fig. 6-1 or Fig. 6-2 can be used to select a compromise size and rotational speed that minimizes the impact on the stage efficiency.

6.3.2 Rotor Inlet Geometry From the Euler equation Δh0 = gΔH E = U 2 C w 2 − U 1 C w1

If there are no IGVs, Cw1 = 0 and the work transfer depends entirely on the rotor tip or outlet conditions. For good efficiency, the impeller inlet must nevertheless be well designed (eg. the inducer inlet metal angle must be matched to the inlet relative flow vector) and correctly sized. Consider three rotors designed for the same m& , U2 and with the same outlet geometry so that all three give the same Δh0. The critical region for frictional losses (which vary as V2), cavitation and compressibility effects is at the vane tip at the inlet, since that is where the relative velocity is the highest and static pressure the lowest. The drawing shows the resulting inlet tip velocity triangles for three different inlet sizes:

SMALL EYE LOW U1t HIGH C1t

LARGE EYE HIGH U1t LOW C1t

r2

r1t

r1h

C1t

W1t

C1t

U1t

W1t

C1t U1t

W1t

U1t

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 the inlet relative velocity and Mach number:

W1t

OPTIMUM

M1t

r1t

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

m& = ρ 2 Cr 2 A2 = ρ 2 Cr 2 (2πr2 b2 ) and thus for a fixed m& , the choice of b2 determines the radial component of velocity at the rotor outlet:

Cw 2 U2 C2

W2

Cr 2

SMALL b2

β2 LARGE b2 C2

W2

b2 r2

ω

Summarizing the effect of different choices of b2:

LARGE b2

SMALL b2

C2

Lower (Good)

Higher (Bad - Larger diffusion required 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 rotor passage and high diffusion in the downstream diffuser (which serves the same function as the stators in an axial compressor stage). Note that W2/W1 is again the de Haller number. Various papers and textbooks provide guidelines for

choosing the de Haller number for centrifugal fan, compressor, and pump rotor passages: (1) Aungier (2000)

Recommended: Never exceed:

W2/W1 > 0.75 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 > 0.8 High 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. The corresponding metal angle $2Ndepends on the deviation, which is called “slip” in centrifugal machines. The slip in turn depends on the rotor “solidity”: that is, the number of vanes, Z. Thus, the choices for $2N and Z are inter-related. Consider a backswept rotor:

Because of slip, the rotor imparts less swirl to the flow than for the “ideal” case, for which $2 =$2N (that is, the Euler approximation is taken to hold in the ideal case). Since Cw2 < Cw2N, the )h0 is reduced by this effect. We then define the slip factor F as

σ=

Cw 2 C ′ w2

where F # 1.0. A number of correlations have been proposed for F. The one due to Stodola has been widely used:

π ′ cosæç β 2 ö÷ ø è Z σ = 1− ′ 1 − φ 2 tanæç β 2 ö÷ ø è

where N2 = Cr2/U2 and $2N is the backsweep or forwardsweep angle (taken as positive in both cases). Stanitz suggested a slightly simpler form:

0.63

σ = 1−

π Z

1 − φ 2 tan β 2

′

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 the Stodola, Stanitz and similar correlations are only valid for impellers with long blades. Wiesner recommended the 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 1

ε lim it = e

æ 8.16 cos β ′ ö 2 ÷ ç ç ÷ Z è ø

then for , # ,limit (ie. longer vanes)

σ = 1−

sin β 2

′

Z 0.7

and for , > ,limit (ie. shorter vanes)

æ ç σ = ç 1− ç è

3 ö cos β 2 ′ ÷ æ æ ε − ε lim it ö ö ç 1− ç ÷ ÷ Z 0.7 ÷÷ çè è 1 − ε lim it ø ÷ø ø

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

6.3.5 Choice of Number of Vanes - Vane Loading Wilson & Korakianitis (Design of High-Efficiency Turbomachinery and Gas Turbines, 2nd ed., PrenticeHall, 1998) provide a broad guideline for selecting the number of blades, as function of the vane angle at the tip, as shown in the figure at right. More recently, Rodgers (2000) presented a correlation for the number of vanes which, according to his loss estimates, gives the best rotor efficiency:

Z=

25cos β 2 Ω

′

where Ω is the usual non-dimensional specific speed. Comparison with the Wilson & Korakianitis figure suggests the Rodgers’ correlation is very conservative, leading to very large numbers of vanes. Aungier (2000) outlines a method of selecting the number of vanes based directly on the vane loading. He suggests the following limit:

CENTRIFUGAL COMPRESSOR - NUMBER OF VANES

2 ΔW ≤ 0.9 W2 + W1

45

40

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

2π D2 U 2 ψ Z LB

30

where ψ = work coefficient = Δh0/U22 and LB is the length of the vane along the mean camber line. A reasonable initial estimate of LB can be obtained from

b ⎞ 1 ⎛ D − D1 ⎞⎟ ⎛ L B = ⎜ Δz I − 2 ⎟ + ⎜ 2 ⎝ 2 ⎠ 2 ⎜⎝ cos β ′ ⎟⎠

Number of Vanes, Z

ΔW =

35

25

20

15

10

2

5

where ΔzI is the axial length of the rotor. 0

0

10

20

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

30 40 Backsweep Angle (Deg.)

50

60

70

6.3.6 Losses

The actual stage characteristics are different from ideal due to slip and losses. Slip reduces output but does not affect efficiency since the required input power is reduced along with the output. Sources of losses: (1) Disc friction:

(2) Leakage:

(3) Inlet:

(4) Impeller: (5) Diffuser/Volute:

- friction on outer surface of impeller - since this torque is not exerted on the through-flowing fluid, it does not appear in the Euler work, DgQ)HE - 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 the inlet, to be recompressed over and over again - thus, more fluid is compressed than is delivered by the machine, increasing the power required and showing up as an apparent loss - at other than design Q, flow angle and metal angle will be mismatched at the leading edge, resulting in separation and additional losses - a simple, inexpensive machine with no inducer will have significant inlet losses at all operating conditions - frictional and separation losses inside the impeller channels - roughly " Q2 - 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, for backward-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 stable operating 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& = ρgQ∆H E ( th ) + ρ gQl ∆H E ( th ) + Disc & Bearing Friction Power

where

)HE(th) = Ql =

theoretical Euler head (Euler head with slip but no losses) 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 E ( th ) − ∆H L

where

)HL

=

sum of losses (3) + (4) + (5)

and the corresponding efficiency is

η overall =

ρgQ∆H W& shaft

& for DQ and )h0 for g)H. As usual, for compressible flow substitute m

CHAPTER 7 Static and Dynamic Stability of Compression Systems

7.1 INTRODUCTION

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

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

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

It was also noted that surge is a dynamic instability which depends on not just the characteristics of the compressor but also on the aerodynamic characteristics of the other components to which it is connected. It is possible to develop a simple lumped-parameter analysis for a compression system. Such an analysis can provide useful insights into which characteristics of the system encourage or delay the onset of surge. For further information see Stenning (1980), Greitzer (1980, 1981) and Cumpsty (1989).

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

7.2 STATIC STABILITY 7.3 DYNAMIC STABILITY - SURGE A system is statically stable if, when it is disturbed by a small amount from its equilibrium operating point, a reaction arises which tends to restore it to the equilibrium condition. Static stability is normally a necessary, but not sufficient condition for dynamic stability

A simple analysis can be developed to predict approximately the dynamic stability characteristics of a compression system. A compressible flow system will be examined. Only minor modifications are needed to make it apply to an incompressible flow system.

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

Fig. 7.2 shows schematically a simple system consisting of 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 first approximation, the throttle could also represent the turbine in a gas turbine engine.

Fig. 7.2 Four-component compression system.

Fig. 7.1 Compressor operating points.

1

The flow through the components is treated as onedimensional. Thus, the flow at any point is characterized by & etc. (if necessary, these would a single value of P, T, C, m be interpreted as the local average values). The analysis will consider perturbations about an equilibrium operating point and the perturbations will be assumed to be small. The instantaneous value of any flow quantity is represented by the sum of the mean value plus the instantaneous (small) perturbation:

m2 = m2 + m2′

d ( P2 − P01 ) dm1

=

dC =c dm1

and since we are assuming that the perturbations from the operating point are small, we can assume that c is constant in our analysis. That is, we linearize the compressor characteristic at the operating point of interest.

P3 = P3 + P3′ etc.

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

where m2 is the mass flow rate at plane 2 (the dot is omitted for convenience). The goal of the analysis is to determine the behaviour of the perturbations over time after some initial disturbance has occurred. If m2N, P3N etc. eventually decrease to zero, the system is dynamically stable at the operating point in question.

dP2 dP01 dP2 − = =c dm1 dm1 dm1

(3)

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

The approach used is known as the lumped-parameter method: equations for the behaviour of each component are developed separately and they are then linked by the flow conditions at the interfaces between the components. Consider each component in turn:

P2

m1

∫ dP = ∫ cdm 2

P2

1

m1

(

P2 − P2 = c m1 − m1

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

P2 − P01 = C(m1 )

(2)

)

From the definition of the perturbations, this can be written

P2′ = cm1′

(1)

(4)

This is then the perturbation equation for the compressor.

where C is the function which defines the compressor characteristic (see Fig. 7.3). The gradient at any operating point along the characteristic is

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

P2′ = cm2′

(5)

(2) Duct We assume that the losses in the system occur primarily in the throttle so that we can neglect the frictional losses in the duct. We also neglect the volume of the duct relative to the volume of the plenum. Therefore, the duct introduces only inertia: a pressure difference is present between stations 2 and 3 only when the fluid in the duct is being accelerated or decelerated. The equation governing the behaviour of the duct can be obtained either by performing a force balance on the free body consisting of the cylinder of fluid in the duct or by

Fig. 7.3 Compressor characteristics.

2

applying the unsteady momentum equation to a control volume occupying the duct. For both analyses, we will neglect density changes along the duct.

P2′ − P3′ =

L dm2′ A dt

(7)

(i) Force balance: (3) Plenum

d (mu) dt d P2 A − P3 A = ( ρ A Lu) dt d = L ( ρ Au) dt Σ Fx

=

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

m2 − m3 = V

L dm2 A dt

(8)

where m3 = mass flow rate through the valve. Changes in the mass in the plenum will be reflected in the density of the stored gas. In a pump system, a reservoir with a free surface or a surge tank would similarly act as a mass storage component.

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

P2 − P3 =

dρ3 dt

(6) If the compression or expansion process is isentropic, then

(ii) Control volume analysis:

P

ργ

For the control volume in the duct

ΣFx =

∫ u ρ dV + (mu)

d dt

V

out

− (mu)in

Differentiating with respect to time and assuming a perfect gas

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

ΣFx =

d dt

∫

L

u ρ Adx =

0

d⎛ ⎜ m2 dt ⎜⎝

dρ3 ρ dP3 = 3 dt P3 γ dt 1 dP3 = γ RT3 dt

⎞ dm dx⎟ = L 2 ⎟ dt 0 ⎠

∫

= const.

L

=

1 dP3 a32 dt

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

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

m2 − m3 = We then substitute into (6) in terms of the perturbations

(

) (

)

P2 + P2′ − P3 + P3′ =

(

L d m2 + m2′ A dt

)

V dP3 a32 dt

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

m2′ − m3′ = and since there are no losses in the duct, the mean inlet and outlet pressures must be the same. Thus, the perturbation equation for the duct becomes

V dP3′ a32 dt

(9)

(4) Throttle The throttle is handled in exactly the same way as the compressor: the load line is linearized at the equilibrium operating point. If the valve is choked, the mass flow rate

3

through it is a function of only the upstream pressure, P3. If it is not choked, the pressure downstream is assumed to be constant. Then the perturbation equation for the throttle becomes:

P3′ = f m3′

dP3′ dP2′ L d 2 m2′ = − dt dt A dt 2

(10)

and from (11)

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

dm′ dP2′ =c 2 dt dt

Characteristic Equation for the System

Thus

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

P2′ = cm2′

dP3′ dm′ L d 2 m2′ =c 2 − dt dt A dt 2

(11)

L dm2′ P2′ − P3′ = A dt V dP3′ m2′ − m3′ = 2 a3 dt

or

(12)

f V dP3′ c f V dm2′ f V d 2 m2′ = − a32 dt a32 dt a32 dt 2

(13)

P3′ = f m3′

(14)

Substituting (18) into (17) and rearranging

These are four equations in the four unknowns m2N, m3N, P2N and P3N. Solving for any one of the unknowns from (11) (14) leads to a second-order ordinary differential equation for the variation in time for that unknown.

f V L d 2 m2′ ⎛ L c f V ⎞ dm2′ + ( f − c) m2′ = 0 (19) + ⎜⎜ − 2 ⎟⎟ a32 A dt 2 a3 ⎠ dt ⎝A

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

cm2′ − f m3′ =

L dm2′ A dt

This is seen to be a second-order ordinary differential equation in m2N. It can be shown that the corresponding equation for any of the other three perturbations would have the save coefficients as (19).

(15)

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

f m2′ − f m3′ = f

V dP3′ a32 dt

Within the assumptions of the analysis, the coefficients are constant and, given initial conditions for m2N and dm2N/dt, (19) can readily be solved to determine the response of the system. As noted earlier, if m2N tends to 0 with increasing time, the system is dynamically stable.

(16)

A useful analogy can be drawn between the present system and a mass-spring-damper system for which the governing equation is (for free vibrations)

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

V dP ′ L dm2′ f m2′ − cm2′ = f 2 3 − A dt a3 dt

(18)

m

d2x dt

(17)

2

+s

dx + kx = 0 dt

(20)

where s = damping coefficient, k = spring constant. Two conditions must be met for the system governed by (20) to be stable:

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

(i) k > 0 - that is, the spring constant must be positive The equivalent condition in (19) is that f > c, which is

4

It is not immediately clear whether it requires a large positive value of c (large in comparison to f, for example) to destabilize the system, but note that as L tends to zero c also tends to zero. Therefore, a compressor which is connected to a plenum by a very short length of duct will become unstable essentially at the peak of the compressor characteristic. That is why the latter is often used as a criterion for predicting surge. In general, we would expect to encounter the condition for dynamic instability near the peak of the characteristic and probably long before we reach the condition for static instability (operating point C on the & diagram) . original )P0 versus m

precisely the requirement for static stability which we arrived 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: the system has a critical value of the damping coefficient, sc, given by

sc = 2 k m

If s < sc, the system is under-damped and the solution takes the form

x = Ae

−

s sc

k t m

⎛ 2 ⎛ s⎞ ⎜ sin⎜ 1 − ⎜ ⎟ ⎝ sc ⎠ ⎜ ⎝

The relationship between stall and surge now is a little clearer. For a typical compressor characteristic, as the flow rate through the machine is reduced the output peaks and eventually begins to reduce. This is generally the result of increasingly extensive stall: perhaps an increasing number of rotating stall cells and/or cells of increasing spanwise extent as the flow rate is reduced. Stall thus prepares the conditions for surge. Note that the appearance of stall is a phenomenon of the compressor itself, not the system. On the other hand, surge is an unstable condition in compression system in which flow quantities, including the compressor mass flow and delivery pressure, undergo oscillatory fluctuations which grow over time. In systems such as gas turbine engines, these fluctuations can reach destructive magnitudes in a very small number of cycles.

⎞ ⎟ k t + φ⎟ m ⎟ ⎠

Thus, the system oscillates sinusoidally in time, with the magnitude of the peak displacement being controlled by the exponential term. Since m, k, and sc are all positive, if s > 0 the exponential term decreases in time, the magnitude of the fluctuations decays, and the system is seen to be dynamically stable. If s > sc, the system is over-damped and the solution is no longer oscillatory but it again includes exponential terms which are a function of s. Again, if s is negative the exponential terms grow in time and the system moves away from the equilibrium point in an unstable way.

References Cumpsty, N.A., 1989, Compressor Aerodynamics, Longman, Harlow. Greitzer, E.M., 1980, “Review - Axial Compressor Stall Phenomena,” ASME J. Fluids Engineering, Vol. 102, June 1980, pp. 134-151.

Applying these ideas to the compressor system, it is seen that there are two contributions to the system damping:

Greitzer, E.M., 1981, “The Stability of Pumping Systems,” ASME J. Fluids Engineering, Vol. 103, June 1981, pp. 193242.

(a) positive (stabilizing) damping is supplied by the inertia of the fluid in the duct (the L/A term), and (b) potentially negative damping is supplied by the term involving the slopes of the compressor and throttle characteristics.

Stenning, A.H., 1980, “Rotating Stall and Surge,” ASME J. Fluids Engineering, Vol. 102, March 1980, pp. 14-20.

Since f is always positive, the sign of the damping term is controlled by the sign of the slope of the compressor characteristic, c. If c < 0 (as it normally is at higher flow rates) strong positive damping will be present and the system will be stable. The condition for instability is then

L cfV − 2 >0 A a3

or

c>

La32 AV f

Thus, the system will become unstable for some positive value of the slope of the compressor characteristic, the precise magnitude being a function of a number of system parameters.

5

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/F): With: A = 33.5293 D = 0.00209610

B = -0.530812 E = -0.677212

C = -15.2599 F = 0.187148

,*("2,s/c) = A + B"2 + C ln(s/c) + D"22 + E(ln(s/c))2 + F"2ln(s/c) Applies for: 0 < "2 < 70o, 0.5 < s/c < 1.5 (or 0.666 < F < 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: 1

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 + 0.05002irel2 + 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 < F < 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 + 0.06477irel3 + 0.3949irel4 + 0.3426irel5

2

APPENDIX B: C4 Compressor Blade Profiles

Like NACA 4-digit airfoils, the C-series compressor blades are defined by a symmetrical thickness distribution which is superimposed on a specified mean, or camber, line. As indicated in the Howell correlations, both circular arc and “parabolic” arc camber lines have been used with C-series blades. For the blade with a parabolic arc camber line, the point of maximum camber lies at other than mid-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:

(1)

and

The term parabolic arc camber line is somewhat misleading. The mean line is not defined by a single parabola, or even by two joined parabolas. For example, to define a polynomial which passes 1

through (0,0) with slope tan21 and through (a,b) with zero slope requires at least a cubic. The following discussion will consider mainly the circular arc camber line. Setting a/c = 0.5 in Eqn (1),

(2)

The equations of the camber line and its inclination, Nc, are then

(3)

and (4)

The co-ordinates of the upper and lower sides of the blade are then

(5)

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 blade designated 10C4/30C50 refers to a blade with a C4 profile and: 10% maximum thickness, circular arc camber, camber angle 30o and maximum camber at 50% chord (the last piece of information is redundant 2

since circular arc camber has already been specified). The resultant geometry is shown:

3

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 as a function of inlet flow angle, $1, and solidity, F: With:

A00 = -0.13571 A10 = 0.015986

A01 = 0.075795 A11 = 0.074959

A02 = 9.1315x10-4 A20 = -2.4954x10-4

i0(10)($1,F) = A00 + A01F + A02F2 + A10$1 + A11$1F + A20$12 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 A03 = 0.033568 A11 = 7.402x10-3 A20 = -3.3001x10-5 A30 = 8.0286x10-7

A01 = 0.05897 A04 = -7.1706x10-3 A12 = -2.5749x10-3 A21 = -3.084x10-5 A31 = -1.2016x10-7

A02 = -0.054019 A10 = -6.0476x10-3 A13 = 2.6067x10-4 A22 = 1.3955x10-5 A40 = -9.1961x10-9

n1($1,F) = A00 + A01F + A02F2 + A03F3 + A04F4 + A10$1 + A11$1F + A12$1F2 + A13$1F3 n2($1,F) = A20$12 + A21$12F + A22$12F2 + A30$13 + A31$13F + A40$14 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.

1

(d) Zero Camber Deviation Angle, *0 (SP-36 Fig. 161) With:

A00 = 0.053535 A03 = -0.75902 A10 = -3.838x10-3 A13 = 3.4149x10-3 A21 = 2.0917x10-4 A30 = -1.3124x10-5 A40 = 2.3356x10-7

A01 = -0.29275 A04 = 0.3706 A11 = 0.02838 A14 = 5.8448x10-4 A22 = 3.0519x10-4 A31 = -1.0755x10-5 A41 = 1.1718x10-7

A02 = 0.71879 A05 = -0.067233 A12 = -0.02068 A20 = 3.5333x10-4 A23 = -1.2273x10-4 A32 = 1.7229x10-6 A50 = -1.4651x10-9

*o1($1,F) = A00 + A01F + A02F2 + A03F3 + A04F4 +A05F5 + A10$1 + A11$1F + A12$1F2 + A13$1F3 + A14$1F4 *o2($1,F) = A20$12 + A21$12F + A22$12F2 + A23$12F3 + A30$13 + A31$13F + A32$13F2 + A40$14 + A41$14F + 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$13 b($1) = 0.965 - 2.5464x10-3$1 + 4.2695x10-5$12 - 1.3182x10-6$13

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) 2

Valid for: 0.4 < F < 1.8, 0.0 < $1 < 70.0.

3

APPENDIX D: NACA 65-Series Compressor Blade Profiles The 65-series blade geometry is not represented by closed-form analytical expressions. Instead, it is 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)

1

The thickness distribution is given for a NACA 65-010 blade which has been modified to give a finite trailing-edge thickness of 0.3% of the chord length. The baseline thickness distribution has zero thickness at the trailing edge and therefore cannot be manufactured. The nominal maximum thickness is 10% of chord. For blades with other values of maximum thickness, the tabulated distribution is simply scaled accordingly. The table indicates that maximum camber is at 50% of chord. However, the camber line is not a simple 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 sensitivity to incidence. Because of the camber line shape, there is no simple relationship between the camber angle, as defined earlier, and the magnitude of the maximum camber. Instead, the camber line shape is related to the nominal maximum lift coefficient which the blade shape would achieve as an isolated airfoil. The camber line shape quoted applies for a nominal lift coefficient CL = 1.0. To generate compressor blades with a desired camber angle, the following can be used to relate an equivalent circular arc camber angle to the nominal CL:

(1)

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 thickness distribution (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 thickness and the equivalent camber:

2

APPENDIX E: Curve and Surface Fits for Kacker & Okapuu Loss System for Axial Turbines Kacker & Okapuu ("A Mean Line Prediction Method for Axial Flow Turbine Efficiency," ASME Trans., J. Eng. for Power, Vol. 104, January 1982, pp. 111-119) presented an updated version of the Ainley & Mathieson loss system for axial turbines. The Kacker & Okapuu (KO) system presents a basis for estimating the complete losses, and thus the efficiency, of an axial turbine at its design point. For a complete outline of the loss system see the paper. Some aspects of the loss system are presented only in graphical form in the paper. Therefore a number of figures have been digitized and curves or surfaces fitted to the data. This appendix documents the curve fits and, in some cases, demonstrates the quality of the fits graphically. The figure numbers refer 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 is obtained as a weighted average of the values for a nozzle blade ($1 = 0) and an impulse blade. These values are obtained from the plots shown in Figures 1 and 2. The data in these figures have been fitted to polynomial surfaces of the form:

The values of the coefficients follow: (i)

Nozzle Blade,

(Fig. 1)

a0,0 = 0.358716 a0,1 = -1.43508 a0,2 = 1.57161 a0,3 = -0.496917 a1,0 = -0.0112815 a1,1 = 0.0548594 a1,2 = -0.0555387 a1,3 = 0.014165 a2,0 = 0.000175083 a2,1 = -0.000824937 a2,2 = 0.000652287 a2,3 = -7.30141E-05 a3,0 = -8.61323E-07 a3,1 = 3.95998E-06 a3,2 = -1.89698E-06 a3,3 = -4.9954E-07

1

(ii)

Impulse Blade,

(Fig. 2)

a0,0 = 0.0995503 a0,1 = 0.182837 a0,2 = 0.01603 a1,0 = 0.00621508 a1,1 = -0.0283658 a1,2 = 0.011249 a2,0 = -7.10628E-05 a2,1 = 0.000327648 a2,2 = -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 true chord. KO present an approximate correlation for the stagger angle as a function of the inlet and outlet angles. The true chord can then be calculated from the axial chord. The graphical data are again fitted to a surface, using a polynomial of the form:

with coefficients, a0,0 = -2.90463 a1,0 = 0.412797 a2,0 = 0.593956E-02

a0,1 = 0.307036 a1,1 = -0.355369E-01 a2,1 = 0.389157E-03

The surface fit and the digitized values are compared over.

2

a0,2 = 0.370176E-02 a1,2 = -0.194938E-03 a2,2 = 1.74147E-06

(c) Inlet Mach Number Ratio (Fig. 6) A correction is made for shock losses at the leading edge of the blade. Since the Mach number tends to be higher at the hub than at midspan, KO present a correlation for the hub Mach number as a function of the midspan value and the hub-to-tip ratio. The shock loss is then calculated from the estimated hub Mach number. The following polynomials were fitted to the curves of Figure 6: (i)

Rotors

(ii)

Nozzles

3

(d) Trailing-Edge Energy Coefficient (Fig. 14) The trailing-edge losses are expressed in terms of the energy coefficient. This was correlated with the ratio of the trailing-edge thickness to the throat opening. Curves were presented for nozzle and impulse blades. The values from these curves are then averaged in a weighted way to give the coefficient for a blade of arbitrary inlet and outlet flow angles. (i)

Impulse (Rotor) Blade

(ii)

Nozzle

4

5

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 often conflicting considerations: aerodynamics, heat transfer, materials, stresses, and vibrations (not to mention cost). While our focus is on the aerodynamics, it is obviously wasteful to develop even a preliminary aerodynamic 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 steady stress determines the margin which is available for these unsteady stresses. In turbines, creep distortion is an 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 be mechanically feasible. Fortunately, the steady centrifugal stresses in the rotor blades can be estimated fairly 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 stresses are particularly high in low hub-to-tip ratio fan blades and in turbine blades; they are much lower in normal compressor blades. A survey of typical, industrial axial-flow fans from several manufacturers shows that peak tip speeds are consistently below 120 m/s. It is believed that this limit is related to the stresses which can be sustained by the rather simple blade attachments, rather than stresses in the actual blades. Higher tips speeds can be used but these require a switch to a considerably more expensive method of attachment. 2.0

Steady Centrifugal Stresses

1

Consider the forces on the small blade element shown.

Then (1)

and this can then be integrated from radius R to the tip, RT, (with a specified blade area variation) to obtain the centrifugal stress at R. Constant Section Blade: With dA = 0, integrating (1):

and the maximum stress occurs at the root: (2)

Tapered Blades: The cross-sectional area of turbomachinery blades often varies from hub to tip. If the area decreases, the root stress will be reduced from the value given by (2). Taking into account the taper, the hub stress can be written

(3)

where K depends on the nature of the taper in cross-sectional area: (a)

Blade with constant cross-section.

(b)

Blade with linear taper.

2

where

The cross-sectional area of the blade is roughly proportional to the product of the chord length (c) and the maximum 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, to maintain constant maximum thickness-to-chord ratio, the cross-sectional area will in fact vary parabolically. 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, (4)

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 the proposed design is feasible mechanically. The following table gives values of KAN2 which are 3

reasonably representative of the current stress limits for axial turbomachines:

KAN2

KAN2

(A in inches2, N in RPM)

(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

MACHINE TYPE

4

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