23217.Axial-Flow Compressors a Strategy for Aerodynamic Design and Analysis

AXIAL-FLOW COMPRESSORS A STRATEGY FOR AERODYNAMIC DESIGN AND ANALYSIS

Ronald H. Aungier

NEW YORK

ASME PRESS

2003

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-a

© 2003 by The American Society of Mechanical Engineers Three Park Avenue, New York, NY 10016 ISBN: 0-7918-0192-6 Co-published in the UK by Professional Engineering Publishing Limited, Northgate Avenue, Bury St Edmunds, Suffolk, IP32 6BW, UK ISBN: 1-86058-422-5 All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. Statement from By-Laws: The Society shall not be responsible for statements or opinions advanced in papers . . . or printed in its publications (B7.1.3) INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. For authorization to photocopy material for internal or personal use under circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, Tel: 978-750-8400, www.copyright.com. Library of Congress Cataloging-in-Publication Data Aungier, Ronald H. Axial-Flow compressors : a strategy for aerodynamic design and analysis / Ronald Aungier. p. cm. ISBN 0-7918-0192-6 1. Compressors—Aerodynamics. 2. Compressors—Design and construction. I. Title. TJ267.5.C5 A94 2003 621.5’1—dc21

20020385720

Cover photo: Courtesy of Elliott Turbomachinery Co., Inc., Ebara Group


To Anne



TABLE OF CONTENTS

Preface

1

Introduction 1.1 1.2 1.3 1.4 1.5 1.6

1

Axial-Flow Compressor Basics .........................................................3 Basic Velocity Diagrams for a Stage................................................5 Similitude and Performance Characteristics...................................7 Stage Matching and Stability........................................................11 Dimensionless Parameters.............................................................13 Units and Conventions ..................................................................14

2 Thermodynamics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

xi

17

First and Second Laws of Thermodynamics................................18 Efficiency.......................................................................................20 Fluid Equation-of-State Fundamentals.......................................22 The Caloric Equation of State .....................................................24 Entropy and the Speed of Sound................................................25 The Thermal Equation of State for Real Gases ..........................26 Thermodynamic Properties of Real Gases ..................................30 Thermally and Calorically Perfect Gases .....................................31 The Pseudo-Perfect Gas Model ...................................................32 Component Performance Parameters ........................................33 Gas Viscosity .................................................................................37 A Computerized Equation-of-State Package .............................37


vi • Table of Contents

3

Fluid Mechanics 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Flow in a Rotating Coordinate System .........................................43 Adiabatic Inviscid Compressible Flow...........................................46 Adiabatic Inviscid Compressible Flow Applications .....................48 Boundary Layer Analysis................................................................50 Two-Dimensional Boundary Layer Analysis..................................51 Axisymmetric Three-Dimensional Boundary Layer Analysis........54 Vector Operators in Natural Coordinates.....................................57

4 Axial-Flow Compressor Blade Profiles 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

77

The Blade-to-Blade Flow Problem ................................................79 Coordinate System and Velocity Components .............................81 Potential Flow in the Blade-to-Blade Plane .................................83 Linearized Potential Flow Analysis ...............................................92 The Time-Marching Method .........................................................96 Blade Surface Boundary Layer Analysis......................................107 Summary.......................................................................................113

6 Empirical Performance Models Based On Two-Dimensional Cascade Tests 6.1 6.2

59

Cascade Nomenclature ................................................................60 NACA 65-Series Profile.................................................................62 Circular-Arc Camberline...............................................................65 Parabolic-Arc Camberline ............................................................66 British C.4 Profile..........................................................................68 Double-Circular-Arc Profile..........................................................69 NACA A4K6 63-Series Guide Vane Profile ...................................70 Controlled–Diffusion Airfoils.......................................................71 Blade Throat Opening .................................................................73 Staggered Blade Geometry .........................................................75

5 Two-Dimensional Blade-to-Blade Flow Through Cascades of Blades 5.1 5.2 5.3 5.4 5.5 5.6 5.7

41

117

Cascade Geometry and Performance Parameters....................119 Design Angle of Attack or Incidence Angle .............................121


Table of Contents • vii

6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Design Deviation Angle.............................................................125 Design Loss Coefficient and Diffusion Factors .........................128 Positive and Negative Stall Incidence Angles...........................134 Mach Number Effects.................................................................136 Shock Wave Loss for Supersonic Cascades................................138 Off-Design Cascade Performance Correlations ........................141 Blade Tip Clearance Loss............................................................146 Shroud Seal Leakage Loss..........................................................147 Implementation, Extensions and Alternate Methods .............149

7 Meridional Through-Flow Analysis 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Meridional Coordinate System ...................................................154 Inviscid, Adiabatic Flow on a Quasi-Normal...............................157 Linking Quasi-Normals ................................................................162 Repositioning the Stream Surfaces.............................................164 Full Normal Equilibrium Solution ...............................................165 Simplified Forms of the Through-Flow Analysis ........................167 Annulus Sizing .............................................................................169 Numerical Approximations .........................................................171

8 End-Wall Boundary Layer Analysis 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

153

175

Historical Development of End-Wall Boundary Layer Theory .....177 The End-Wall Boundary Layer Equations ...................................181 The Boundary Layer Velocity Profile Assumptions ....................183 Empirical Models for Entrainment and Wall Shear Stress .........184 The Blade Force Defect Thicknesses ...........................................187 Seal Leakage Effects for Shrouded Blades .................................191 Boundary Layer Jump Conditions ...............................................193 Solution Procedure ......................................................................194 Typical Results ..............................................................................195

9 Aerodynamic Performance Analysis

199

9.1 Geometry Considerations............................................................200


viii • Table of Contents

9.2 9.3 9.4 9.5 9.6

Cascade Performance Considerations.........................................203 Stall and Compressor Surge Considerations...............................204 Approximate Normal Equilibrium Results..................................207 Full Normal Equilibrium Results..................................................211 Concluding Remarks ....................................................................213

10 Compressor Stage Aerodynamic Design 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

Dimensionless Performance Parameters.................................217 Application to Stage Design....................................................219 Blade Design.............................................................................221 Selecting the Stage Performance Parameters ........................222 Selecting the Swirl Vortex Type...............................................229 Free Vortex Flow ......................................................................230 Constant Reaction Vortex Flow...............................................235 Constant Swirl and Exponential Vortex Flow.........................242 Assigned Flow Angle Vortex Flows .........................................245 Application to a Practical Stage Design .................................245 A Repeating Stage Axial-Flow Compressor............................251 A Computerized Stage Design System ...................................257

11 Multistage Axial-Flow Compressor Aerodynamic Design 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

215

259

The Basic Compressor Design Approach ..................................261 Aerodynamic Performance Specifications................................262 Blade Design ..............................................................................264 Refining the Compressor Design ..............................................266 An Axial-Flow Compressor Design Example.............................268 The Distribution of Stage Performance Parameters................272 The Swirl Vortex Type ................................................................280 Risks and Benefits ......................................................................284

12 Quasi-Three-Dimensional Blade Passage Flow Field Analysis

287

12.1 Quasi-Three-Dimensional Flow .................................................289 12.2 Hub-to-Shroud Flow Governing Equations ..............................291


Table of Contents • ix

12.3 12.4 12.5 12.6 12.7

Numerical Integration of the Governing Equations................294 Repositioning Stream Surfaces .................................................297 The Hub-to-Shroud Flow Analysis.............................................298 Coupling the Two Basic Flow Analyses .....................................299 Boundary Layer Analysis............................................................302

13 Other Components and Variations 13.1 13.2 13.3 13.4 13.5

309

Adjustable Blade Rows ..............................................................311 The Exhaust Diffuser..................................................................316 The Scroll or Collector ...............................................................322 Reynolds Number and Surface Roughness Effects...................328 The Axial-Centrifugal Compressor............................................328

Answers to the Exercises

333

References

349

About the Author

355

Index

357



PREFACE

Books on compressor aerodynamics approach the subject in various ways depending on the intended audience. Introductions to the fundamentals are available to students and newcomers to the field. The experienced aerodynamicist can find more comprehensive overviews of the core technologies that review alternate approaches and summarize important contributions that have significantly advanced the state of the art. Users of compressors can find books that address the alternatives and critical issues of particular interest to the application, selection, procurement and operation of compressors. But in recent years, the aerodynamicist seeking detailed information and practical guidance on the effective application of basic technology to compressor aerodynamic design and analysis has been less fortunate. When I started working in this field, books of that type were available, although they were somewhat dated. The Centrifugal Compressor Stage (Ferguson, 1963), Axial Flow Compressors (Horlock, 1958) and Aero-Thermodynamics and Flow in Turbomachines (Vavra, 1960) were my particular favorites, and they provided valuable guidance during my early efforts to formulate compressor aerodynamic design and analysis systems. Unfortunately, their modern equivalents have simply not appeared. With his usual insight, Cumpsty (1989) recognizes and discusses this change in emphasis in recent books in his preface to Compressor Aerodynamics. He attributes it to the expanded roles of the computer and of proprietary industrial research and development, which have combined to make it rather impractical for authors to provide a general and detailed description of modern compressor aerodynamic design and analysis methods. Several years ago, similar reasoning led me to consider a different type of book to meet this need. If a general description of aerodynamic design and analysis methods is no longer practical, a detailed description of a specific aerodynamic design and analysis system might be a viable alternative. Although unsure of how it might be received, I decided to write a book on centrifugal compressor aerodynamics and managed to find a publisher willing to try something different (Aungier, 2000). The feedback received since my book was published confirms that there is interest in this type of book as well as a significant audience. Aerodynamic design and analysis systems evolve through a trial-and-error process while we painfully learn what capabilities should be included, how to use them effectively and how they can interact efficiently to support the overall process. The conversion of basic technology into a working design and analysis


xii • AXIAL-FLOW COMPRESSORS

system is a very complex process involving many decisions and false starts. Along the way, all developers expend a lot of time and effort formulating virtually identical models for phenomena fundamental to the process. These critical details are quite important to aerodynamicists who must develop, maintain or improve an aerodynamic design and analysis system. The present book approaches axialflow compressor aerodynamics in a manner similar to that of my previous book on centrifugal compressors. The approach used is a description of a comprehensive aerodynamic design and analysis system in sufficient detail so that readers can readily implement the complete system or any of its components. This proved to be more difficult for axial-flow compressors than was the case for centrifugal compressors. Centrifugal compressor stage designs tend to be fairly unique and specific to the design objectives. Most current design and analysis systems share many common features and are used in a wide range of applications. The basis for axial-flow compressor design is far more varied and application-dependent, often based on proprietary information that is not available in the open literature. Indeed, it is now quite common to find designs based on proprietary and customized airfoil families, such as the popular controlled diffusion airfoil. The geometry and performance characteristics of these proprietary airfoils are well known only to the organizations that developed them. There are also inherent differences in the technology used on industrial axialflow and centrifugal compressors. The development of a completely original industrial axial-flow compressor design is relatively rare. These compressors are almost always unique, one-of-a-kind designs that must rely on variations of standard components to minimize risks while maintaining acceptable development and manufacturing costs. Hence, the variety of application experience is far more limited for the axial-flow compressor design and analysis system than was the case for the centrifugal compressor system. I decided that the desired objectives could still be achieved by adopting the classical design approach based on the systematic application of standard airfoil families to develop the blade geometry used in the compressors. The basic principles of the design process described here remain applicable when proprietary airfoil families are in use. But it is likely to be necessary to adapt them to reflect the specific geometry and performance characteristics of those airfoils. Aerodynamic performance prediction accuracy is established by comparing predictions with experimental data for several axial-flow compressors. That established performance prediction accuracy is then used to demonstrate the effectiveness of the overall design and analysis system. A substantial number of design examples are included to illustrate the use of this design and analysis system, as well as to provide some evaluation of alternate design approaches suggested in the literature, or which I have found to be effective. Considerable care is taken to provide complete and detailed descriptions of this comprehensive aerodynamic design and analysis system for axial-flow compressors. The basic principles of thermodynamics and fluid mechanics required are presented in a form particularly well-suited to the axial-flow compressor application. Well-defined empirical models are used to augment these basic principles to address the essential problem areas of performance analysis, stage design, compressor design and internal flow analysis. Descriptions of numerical methods used are included as well as other critical considerations important to


Preface • xiii

readers who may wish to apply these methods. In a few cases where components are common to both my centrifugal and axial-flow compressor design and analysis systems, I refer to Aungier (2000) rather than repeat some rather lengthy and detailed descriptions that will be of interest only to readers who choose to implement those specific methods. Some important topics have received inadequate treatment or have been completely omitted. Surge and stall are discussed only in the context of estimating the expected limits of stable operation, while noise and blade vibration are not discussed at all. I prefer to limit myself to topics on which I can offer at least some original ideas. I always feel uncomfortable when presenting ideas obtained almost entirely from others, even when I have considerable confidence in the sources. I much prefer that readers obtain such information from qualified authors, even if it does display my own limitations.


Chapter 1

INTRODUCTION

Compressors are commonly classified as either positive displacement or dynamic compressors. The positive displacement compressor achieves its pressure rise by trapping fluid in a confined space and transporting it to the region of higher pressure. The dynamic compressor develops its increase in pressure by a dynamic transfer of energy to a continuously flowing fluid stream. There are two basic types of dynamic compressors: axial-flow compressors and centrifugal (radialflow) compressors. The flow streamlines through rotating rows in an axial-flow compressor have a radius that is almost constant, whereas they undergo a substantial increase in radius in a centrifugal compressor. For this reason, the centrifugal compressor can achieve a much greater pressure ratio per stage than the axial-flow compressor. But the axial-flow compressor can achieve a significantly greater mass flow rate per unit frontal area. Figure 1-1 compares normalized discharge pressure, P, versus flow rate, Q, for these two compressor types to illustrate the differences in their performance characteristics. The axial-flow compressor approximates a variable pressure ratio—constant flow machine, whereas the centrifugal compressor is closer to a constant pressure ratio—variable flow machine. The performance data displayed in Fig. 1-1 are for a singlestage centrifugal compressor and a five-stage axial-flow compressor, both of which have about the same design pressure ratio. This demonstrates the superior pressure ratio-per-stage capability of the centrifugal compressor. Traditionally, the centrifugal compressor has been the more rugged and lower-cost type, while the axial-flow compressor has offered better efficiency. Those differences have become much less significant in recent years due to advances in technology, particularly with regard to efficiency. Presently, the compressor type selected is more likely to be based on the performance characteristics, size and cost that is best suited to the application.

NOMENCLATURE a C cp cv H

= = = = =

sound speed absolute velocity specific heat at constant pressure specific heat at constant volume total enthalpy and compressor head


2 • AXIAL-FLOW COMPRESSORS

h k M ˙ m N P Q R Re r T U W β γ δ η θ µ ρ φ ψ ω

= = = = = = = = = = = = = = = = = = = = = = =

static enthalpy ratio of specific heats = cp / cv Mach number = C / a mass flow rate rotation speed (rpm) pressure ˙ / ρt volume flow rate = m Gas constant and stage reaction Reynolds number radius temperature local blade speed, ωr relative velocity flow angle stagger angle Pt0 / Pref efficiency polar (tangential) coordinate and Tt0 / Tref viscosity density stage flow coefficient stage work coefficient rotation speed (radians/second)

FIGURE 1-1 Performance of Dynamic Compressors


Introduction • 3

Subscripts d ref rev t z θ 0 1 2 3

= = = = = = = = = =

parameter at compressor discharge reference thermodynamic conditions reversible thermodynamic process total thermodynamic condition axial component swirl (tangential) component parameter at guide vane inlet or compressor inlet parameter at guide vane exit and rotor inlet parameter at rotor exit and stator inlet parameter at stator exit

Superscripts ′ = relative condition

1.1 AXIAL-FLOW COMPRESSOR BASICS Figure 1-2 illustrates the basic configuration of an axial-flow compressor. The first blade row shown is an inlet guide vane (IGV) to develop the swirl (tangential) velocity for which the first rotating row (R1) was designed. If the first rotating row is designed for no inlet swirl, the inlet guide vane will normally be omitted. This is followed by a series of stages (two in this illustration), where a stage refers to a rotating row, or rotor, in combination with its downstream stationary row, or stator (e.g., R1 and S1). The rotor row imparts energy to the fluid by increasing the swirl velocity. The stator row removes the swirl developed by the rotor to convert kinetic energy to static pressure and to establish the proper swirl velocity for the flow to enter the next rotor. Typically, an exit guide vane (EGV) follows the last stage to remove any residual swirl velocity to convert that kinetic energy to static pressure. Like the inlet guide vane, this row may be omitted if the last stator is designed to remove all of the swirl velocity. Although not shown on the figure, a diffuser-combustor (gas turbine) or diffuser-collector (industrial compressor) will follow the exit guide vane to recover as much kinetic energy as possible, as well as to direct the flow to its intended destination. Similarly, an inlet passage will precede the inlet guide vane. This can range from a smooth axial bell-mouth inlet to a complex side inlet, depending on the compressor’s application. Figure 1-3 illustrates the blade profiles for a stage viewed on a polar stream surface between adjacent blades. The rotor row is rotating with a velocity, U = ωr, where ω is the rotation speed and r is the radius. Viewed in a frame of reference rotating with the rotor, the upstream velocity, W, is referred to as the relative velocity. The rotor deflects the flow such that the velocity in the stationary frame of reference of the stator (the absolute velocity), C, is properly aligned to enter the stator row. This process repeats in subsequent stages, with each stage adding energy to the fluid to achieve the overall pressure ratio required. Axial-flow compressor design strategies are quite varied. Gas turbine compressors are normally intended for use in many identical units. Extensive design



FIGURE 1-2 Axial-Flow Compressor Configuration

FIGURE 1-3 Polar Surface View of a Stage


Introduction • 5

and development programs are common and may include unique blade designs for all blade rows. By contrast, industrial compressors are usually designed specifically for a unique application, rarely involving any duplicate units. Here, a repeating stage approach is more common, where one or more basic stage designs are used for all compressors. Aerodynamic and mechanical flexibility are obtained by minor adjustments that do not compromise the basic stage’s performance. Blades may be scaled to longer and thicker blades for mechanical integrity with a corresponding adjustment of the number of blades per row to preserve aerodynamic similarity. Blades usually must be restaggered, i.e., rotated on their base to change the stagger angle, γ, to achieve different performance levels. Otherwise, the intended duty would normally require a non-integer number of stages. Often the inlet guide vane and some of the stator blades may be adjustable so they can be restaggered by a control system while the machine is in operation to broaden the compressor’s application range. This approach allows these “one-of-a-kind” compressors to be designed within practical cost. It also allows each compressor’s design to be based on a well-established performance history. This is important, since these compressors cannot be confirmed by performance testing until after they are manufactured. Figure 1-2 illustrates normal cantilevered blades that are attached at the root, with a clearance between the blade tip and the adjacent end-wall. Figure 1-4 shows a different style often used for stator blades. Here, a shroud band is attached to the blade tips to connect them together. This is often done for reasons of mechanical integrity. To reduce fluid leakage from the blade discharge back to the blade inlet, seal fins are normally attached to the shroud band. These provide a reduced clearance to retard leakage, yet are thin enough to minimize damage in the event that a rotor shaft excursion or “rub” causes the seals and shaft to come into contact. To minimize damage to the shaft, the stator blades and stator shrouds, the seal fins will be sacrificed in the event of a rub.

1.2 BASIC VELOCITY DIAGRAMS FOR A STAGE The construction of velocity diagrams is a very useful concept for axial-flow compressor design. Here they will be used to illustrate the velocity vectors entering and leaving blade rows in a stage. It will be necessary to use both absolute and relative velocities, where relative velocities are viewed in a frame of reference rotating with the compressor’s rotation speed, ω. Designating the relative and absolute tangential velocities as Wθ and Cθ, respectively, the two are related by Wθ = Cθ − ω r

(1-1)

where r is the local radius. The axial components of velocity are identical in both frames of reference, i.e., Wz = Cz

(1-2)

Therefore, the absolute and relative velocities are



FIGURE 1-4 Shrouded Stator Blades

C = Cz2 + Cθ2

(1-3)

W = Cz2 + Wθ2

(1-4)

The absolute and relative flow angles are designated as β and β′, respectively. They are defined by tan β = Cθ / Cz

(1-5)

tan β ′ = Wθ / Cz

(1-6)

Figure 1-5 illustrates the velocity diagrams for an inlet guide vane. The flow enters the guide vane with no swirl, i.e., Cθ 0 = 0, C0 = Cz0. The guide vane deflects the flow by an angle, β1. If Cz1 is known, this defines the swirl velocity component, Cθ1. Then Eq. (1-1) is applied in vector form to subtract ωr from Cθ1 to define the swirl velocity component in the relative frame, Wθ1 and the relative flow angle, β′1. Hence, the complete velocity diagram for the entrance to the downstream (rotating) rotor blade row is known. Figure 1-6 shows the velocity diagram construction for the rotor blade row. The inlet velocity diagram is the same as that determined for the guide vane exit. The rotor blade deflects the flow in the relative frame of reference from β′1 to β′2 to produce the discharge swirl velocity, Wθ2. If Cz2 is known, Wθ 2 can be computed. Then vector addition of ωr to


Introduction • 7

FIGURE 1-5 Guide Vane Velocity Diagrams

Wθ2 yields the absolute swirl component Cθ 2. Hence, all velocity components and the flow angle in the absolute frame of reference can be computed to define the inlet conditions for the downstream (stationary) stator blade row. Construction of the stator and exit guide vane blade row velocity diagrams is accomplished in a similar fashion and will be left as an exercise for the reader. The important thing to note is that construction of this simple velocity diagram is a fundamental technique commonly used by turbomachinery aerodynamicists to convert between absolute and relative flow conditions. Here, Cz has been treated as known. In practice, values of Cz may be specified design conditions from which the flow passage areas will be computed to conserve mass. This will be referred to as the design mode. Alternatively, Cz may be computed from basic mass and momentum conservation for specified passage areas and the mass flow rate. This will be referred to as the analysis mode.

1.3 SIMILITUDE AND PERFORMANCE CHARACTERISTICS Similitude or similarity is one of the most useful concepts in turbomachinery aerodynamics. Two turbomachines are completely similar if the ratios of all



FIGURE 1-6 Rotor Blade Velocity Diagrams

corresponding length dimensions, velocity components and forces are equal (Sheppard, 1956). If two turbomachines are completely similar, it is possible to present their performance on a common performance map by selecting appropriate equivalent performance parameters. Equivalent performance requires that the two compressors have similar velocity diagrams throughout. To maintain similar ˙ , the ratio of gas densityvelocity diagrams while conserving the mass flow rate, m to-inlet density must also be similar throughout. This means that inlet volume flow, Q0, is the relevant equivalent flow rate parameter, where ˙ / ρt Q=m

(1-7)

The local axial velocity is actually given by ˙ / ( A0ρ0 ) Cz0 = m

(1-8)

where A is the passage area and ρ0 is the inlet gas density, which is unknown. But the exercises in Chapter 2 will show that the ratio ρ0 / ρt0 is a function of Cz0,


Introduction • 9

where ρt0 is the known inlet total gas density. Hence, unique velocity diagrams are associated with a unique Q0 / A0, but can correspond to many values of m˙ /A0 simply by altering ρt0. For this reason, all dynamic compressors are commonly referred to as volume flow machines. The exercises in Chapter 2 will show that the requirement for density ratio similarity requires that the Mach numbers be similar throughout, where the Mach number is the ratio of fluid velocity to the local sound speed, a. It will be shown that the ratio of a0 / at0 is, itself, a function of the Mach number, so the unknown a0 can be replaced by the known inlet total sound speed, at0. Figure 1-7 shows an equivalent performance map based on these requirements. The flow parameter used is volume flow normalized by the inlet total sound speed, at0, and the inlet area, A0. This ensures that the inlet axial Mach numbers will be similar. Three performance characteristics, or speed lines, are shown for three different rotation speeds: N, multiplied by a characteristic diameter, D, and normalized by the inlet total sound speed. This will ensure similarity of the tangential Mach numbers. If two axial-flow compressors are geometrically similar, and use the same working fluid, this performance map will apply to both machines. This, in turn, ensures that the pressure ratios will be the same for both, so pressure ratio is a reasonable choice for the other performance parameter. The situation becomes more complicated if the two compressors use different working fluids. In that case, complete similarity usually cannot be achieved, since different working fluids may produce different gas density (or specific volume) ratios for the same blade row velocity diagram. This “volume-ratio” effect will compromise similarity after the first rotor row, since

FIGURE 1-7 An Equivalent Performance Map



the differences in density will necessitate differences in axial velocity to conserve mass. Indeed, even with identical working fluids, non-ideal gas behavior may compromise similarity. True equivalent performance is assured only for working fluids that obey the perfect gas equation of state discussed in Chapter 2, which can be expressed as: P = ρRT

(1-9)

where P is pressure, T is temperature and R is the gas constant. An exercise in Chapter 2 will also show that achieving equivalent density ratios requires that the two working fluids have the same ratio of specific heats, k = cp / cv. Since axialflow compressors are widely used in aircraft engine gas turbines, discussions on this topic often overlook these more subtle effects. That application of compressors deals exclusively with air as the working fluid, which is very nearly a perfect gas, offering little chance of any volume-ratio effect. Consequently, equivalent performance maps must be used with caution when a perfect gas model with a constant value of k cannot approximate the working fluid. Similarity is also compromised when the two compressors operate at substantially different Reynolds number, Re. Reynolds number is a measure of the inertia forces to viscous forces, Re = ρCL/µ, where L is a characteristic length and µ is the gas viscosity. The Reynolds number directly affects wall friction, which can alter the compressor’s performance. In most cases, effects of the Reynolds number are small enough to be neglected. When that is not the case, suitable Reynolds number corrections (e.g., Wassell, 1968) may be applied to adjust the performance. A common use of equivalent performance maps is to define performance of a specific compressor that is operated at various inlet total thermodynamic conditions. For that case, D and A0 are constant and can be omitted on the map. Another common use is to relate performance of different compressor frame sizes, derived by directly scaling the geometry. Then, D and A0 are included so the map defines the change in speed, N, needed to preserve Mach number equivalence and the flow rate supplied for each pressure ratio on a speed line. The equivalent flow rate used in Fig. 1-7 is the true similarity parameter. When the application is to a perfect gas with constant R and k, Eq. (1-9) can be used to derive alternate equivalent flow parameters, i.e., ˙ Tt0 / Pt0 ∝ m ˙ θ /δ Q / at0 ∝ m

(1-10)

where the sound speed has been replaced by a perfect gas relation from chapter 2, a = kRT

(1-11)

and θ and δ relate inlet total conditions to reference conditions (Tref, Pref), such as standard atmosphere conditions, i.e.,

θ = Tt0 / Tref

(1-12)

δ = Pt0 / Pref

(1-13)


Introduction • 11

Similarly, the equivalent speed parameter can be replaced, using N / at0 ∝ N / Tt0 ∝ N / θ

(1-14)

The alternate flow rate and speed parameters in Eqs. (1-10) and (1-14) are commonly used, but have less fundamental significance than Q0 / at0 and N / at0. Similarly, compressor head, ∆Hrev, can be used in place of pressure ratio, where head is defined as the total enthalpy increase required to produce the actual pressure rise by an ideal, reversible process, i.e., ∆Hrev =

∫

rev

dp ρ

(1-15)

It can be shown that the appropriate equivalent head parameter is ∆Hrev / (at0)2. The use of an equivalent head is common practice for centrifugal compressors, but is much less common for axial-flow compressors. Figure 1-7 supplies only part of the performance information required. In addition to the pressure ratio and flow produced, it is necessary to know the work required to drive the compressor. Hence, a second equivalent performance map is required to completely define the compressor’s performance. The most common parameter for this purpose is efficiency, η, defined as the compressor head or ideal (no loss) total enthalpy rise divided by the actual total enthalpy rise, i.e.,

η = ∆Hrev / ∆H

(1-16)

Alternate reversible processes that can be used to define ∆Hrev and η are discussed in Chapter 2. Figure 1-8 shows an equivalent efficiency map to be used in conjunction with Fig. 1-7. In some cases, it may be appropriate to use the exit static thermodynamic conditions rather than total values to define η and PR. This is appropriate when the kinetic energy available at the compressor discharge serves no useful purpose for the specific application to which the compressor will be applied.

1.4 STAGE MATCHING AND STABILITY Each blade row in a compressor will achieve its best performance for a specific inlet flow angle where losses are minimum. Basically, the designer seeks to “match” succeeding blade rows such that all operate close to their optimum inlet flow angles at a specific operating condition, commonly called the compressor’s design point or match point, defined by the design flow rate and design speed. Hence, at design speed, losses can be expected to increase and performance to deteriorate as the compressor operates farther from its design flow rate. At flow rates less than the design flow rate, losses will increase to a point that the pressure-flow rate characteristic reaches a maximum. At lower flow rates, the characteristic will have a positive slope, which is theoretically unstable. The onset of



FIGURE 1-8 An Equivalent Efficiency Map

this severe unstable operation is commonly called surge. The limit of stable operation is referred to as the surge line as illustrated in Fig. 1-7. Surge is a very complex phenomenon, which depends on the entire system, not just the compressor. So associating it with a maximum on the pressure–flow rate characteristic is an oversimplification, but a useful one. In some cases, an approach to zero-slope near surge is evident, such as for speed line N3 in Fig. 1-7. In other cases, the compressor may experience an abrupt stall, such that the characteristic appears to be quite steep at surge, similar to speed line N1 in Fig. 1-7. This is mainly because the drop in pressure with reduced flow is so abrupt that it cannot be resolved in a performance test. Indeed, estimation of the onset of surge during the design phase is based more on the expected blade loading limits at the onset of stall than on the predicted shape of the pressure-flow characteristics. Similarly, at flow rates greater than the design flow rate, the increase in loss will eventually result in no rise in pressure. This condition is commonly referred to as choke, although it may be caused by large losses due to off-design operation rather than a true aerodynamic choke condition. When the compressor is operated at off-design speeds, operation at different Mach number levels will compromise the stage matching, similar to the volumeratio effect mentioned previously. Consequently, it is unlikely that all stages will be close to their optimum operating conditions at any flow rate for off-design speeds. Rather, optimum performance will occur at the flow rate offering the best compromise on stage matching. Performance will deteriorate for flows different from this optimum, much as described for the design speed performance.


Introduction • 13

The more speed deviates from the design speed, the greater the compromise of the stage matching. In general, at speeds lower than the design speed, the front stages are required to supply a greater portion of the rise in pressure while the rear stages become less effective. The inverse is true for speeds greater than the design speed. This stage mismatching can be alleviated to some degree if some of the stationary blade rows are adjustable during operation. Closing some of the stationary rows (i.e., increasing their stagger angles) in a controlled fashion will shift the optimum matching condition to lower speeds to reduce the mismatch at low speeds. The Mach number level has a definite influence above and beyond its pronounced effect on stage matching. As a blade row’s inlet Mach number increases, its low-loss operating range will decrease. At sufficiently high values, the blade row will start to experience aerodynamic choke in the blade row to significantly reduce its maximum flow capacity. Even the minimum loss levels will increase when the inlet Mach number becomes high enough to produce shock waves that are strong enough to induce boundary layer separation or to produce significant bow shock losses. Consequently, as the equivalent speed increases, pressure-flow characteristics become steeper, with less flow range from surge to choke as illustrated in Fig. 1-7, and the maximum achievable efficiency can be limited by Mach number levels, similar to speed line N1 in Fig. 1-8.

1.5 DIMENSIONLESS PARAMETERS In addition to the dimensionless parameters associated with similitude, axialflow compressor design is often based on a number of other useful aerodynamic dimensionless parameters. To introduce these parameters, it is necessary to anticipate some results that are developed in more detail in Chapters 2 and 3. In particular, the total enthalpy rise for simple axial flow through a rotor blade row is expressed by the well-known Euler turbine equation, i.e., ∆H = U (Cθ 2 − Cθ1)

(1-17)

where U = ω r is the local blade speed and H is the total enthalpy. Similarly, it will be shown in Chapter 2 that the static enthalpy, h, is related to H by H = h + 12 C 2

(1-18)

It is often useful to introduce dimensionless stage performance parameters expressed for a “repeating” stage, i.e., a stage designed to be followed by another identical stage. This means that the velocity diagrams for the rotor inlet (station 1) and the stator exit (station 3) must be identical. Then, the stage work coefficient, ψ, can be defined as

ψ = ∆H / U 2 = (Cθ 2 − Cθ1) / U

(1-19)

where all data correspond to a constant, mean radius, or “pitch line” for the stage. The stage flow coefficient, φ, is defined by



φ = Cz1 / U

(1-20)

The stage reaction, R, is defined as the fraction of the stage static enthalpy rise that occurs in the rotor, i.e., R = ( h2 − h1) / ( h3 − h1)

(1-21)

Substituting P for h in Eq. (1-21) yields an alternate definition of reaction, in terms of static pressures, that is sometimes used. Requiring Cz to be constant through the stage, Eqs. (1-17), (1-18) and (1-21) can be combined to yield R = 1 − (Cθ 2 + Cθ1) / (2U )

(1-22)

In Chapter 10, it will be seen that parameters φ, ψ and R provide useful guidance for stage design. Stage design involves defining blade geometry that will produce the desired performance. These dimensionless performance parameters define performance in a form general to any stage design problem. They are normally used to specify the performance objectives the stage should achieve at its mean radius or pitch line. While there are no fixed rules for selecting values for them, preferred values can normally be established based on the design goals most important to the designer, supported to some degree by simple logic. For example, 50% reaction stages (R = 0.5) are quite common, prompted mainly by the intuitive judgment that it is best to share the flow diffusion load equally between the rotor and the stator. Once specified, these parameters can be used to define the stage velocity diagrams from which the blades can be designed. For example, Eqs. (1-19) through (1-22) can be combined to yield tan β1′ = −(ψ / 2 + R) / φ tan β2′ = (ψ / 2 − R)/φ tan β1 = (1 − R − ψ / 2)/φ tan β2 = (1 − R + ψ / 2)/φ

(1-23) (1-24) (1-25) (1-26)

and the velocity diagrams for the stator exit and rotor inlet are identical for a repeating stage. These parameters have defined the velocity diagrams at the pitch line only. It is necessary to supply additional design specifications and use fundamental fluid dynamics relations to generate the velocity diagrams at other radial locations.

1.6 UNITS AND CONVENTIONS This book assumes consistent units throughout, such that the reader may use any set of consistent units preferred. For historical reasons, many turbomachinery organizations do not use consistent units, often using different units for different disciplines such as aerodynamics and thermodynamics. For example, it is not


Introduction • 15

uncommon to find energy terms, fluid velocity and equation-of-state parameters expressed in inconsistent units, necessitating conversion factors in expressions such as Eq. (1-18). It will be left to the reader to recognize the need for those conversion factors. Flow angles and blade angles will be measured from the meridional plane, i.e., a plane of constant polar angle, θ, in a cylindrical coordinate system. These angles, and the associated swirl velocity components Cθ and Wθ, are considered to be positive in the direction of rotation. While the nomenclature is reasonably consistent throughout, the wide range of topics covered does not permit unique symbols for every parameter. Consequently, each chapter will include its own list of nomenclature to avoid confusion.

EXERCISES 1.1 An axial-flow compressor is to be operated with a different working fluid, which can be modeled as a perfect gas, but has values of gas constant, R, and ratio of specific heats, k, that are different from the normal working fluid. Develop new equivalent speed and flow rate ˙ , Tt0 and Pt0 that will ensure Mach number parameters, in terms of m equivalence at the compressor inlet. 1.2 Free vortex stages with β1 = β3 = 0 are often used for axial-flow compressors. Derive an expression for work coefficient as a function of reaction for this type of stage. If the stage is also to have 50% reaction, specify the range of values for ψ and β′1 that can be used. If the resulting stage is to be used as a repeating stage in a multistage compressor, what type of inlet and exit guide vanes will be needed? 1.3 All dimensions of the compressor producing the performance map shown in Fig. 1-7 are scaled by a factor of 1.2 and both compressors are operated with the same inlet conditions and working fluid. If operating points for the original compressor are denoted as NA and QA, develop expressions for equivalent operating conditions NB and QB for the scaled compressor. If the original compressor operates at a speed of 3,600 rpm, what speed must be used for the scaled compressor? How much additional flow capacity will the scaled compressor have? 1.4 What scale factor should be applied to the compressor producing the performance map shown in Fig. 1-7 to increase the compressor’s flow capacity by 20% for the same inlet conditions and working fluid? What adjustment in speed will be needed?


Chapter 2

THERMODYNAMICS

This chapter highlights some of the fundamental concepts from thermodynamics that are essential to the aerodynamic design and analysis of axial-flow compressors. These concepts provide the basis for defining and evaluating the energy transfer processes and for modeling the state properties of the working fluid. Unlike the centrifugal compressor, axial-flow compressors are most often applied to rather ideal working fluids, the most common being air. But the application of axial-flow compressors to non-ideal working fluids is becoming more common. Consequently, a basic description of non-ideal gas modeling is included. This description is sufficient for application of the techniques, but is less detailed than this author’s previous description for centrifugal compressors (Aungier, 2000). Readers interested in more detail are referred to that earlier reference or to Aungier (1995, 1998).

NOMENCLATURE A a b C c cp cv H h k M ˙ m n P q q˙ R RU s

= = = = = = = = = = = = = = = = = = =

Helmholtz energy, dA = -PdV sound speed and gas constant defined in Eq. (2-32) gas constant defined in Eq. (2-33) velocity gas constant defined in Eq. (2-34) specific heat at constant pressure and pressure recovery coefficient specific heat at constant volume total enthalpy enthalpy ratio of specific heats = cp / cv molecular weight mass flow rate exponent in Eq. (2-34), defined in Eq. (2-35) pressure specific heat transfer heat transfer rate gas constant universal gas constant specific entropy



T u V w ˙ w z η µ ρ ω ξ – ω

= = = = = = = = = = = =

temperature specific internal energy specific volume specific work input power input rate compressibility factor efficiency gas viscosity gas density acentric factor parameter defined in Eq. (2-69) loss coefficient

Subscripts ad c d i p r ref t 0 1 2

= = = = = = = = = = =

adiabatic-reversible (isentropic) process critical point parameter discharge condition inlet condition polytropic process reduced parameter (normalized by its critical point value) reference condition total condition condition ahead of an inlet guide vane condition ahead of a rotor blade row condition following a rotor blade row

Superscripts 0 = condition where the ideal gas model applies

2.1 FIRST AND SECOND LAWS OF THERMODYNAMICS The first law of thermodynamics covers the basic principle of conservation of energy. The first law can be applied to compressors with one restriction. Since a compressor is an open system, steady flow is the only case to which the first law ˙ is the power input to the compressor and q˙ is the heat transfer is applicable. If w between the compressor and its surroundings, the first law of thermodynamics can be written ˙ =m ˙ ∆[u + 1 C 2 + P / ρ] q˙ + w 2

(2-1)

˙ is the mass flow rate, u is the specific internal energy, C is velocity, P is where m pressure, ρ is density and any change in potential energy due to gravitational


Thermodynamics • 19

force is neglected. The term “specific” designates a parameter measured per unit mass of fluid. The terms in brackets specify the internal energy, kinetic energy and the flow work, P/ρ. The term flow work refers to the work necessary to move the fluid across the boundaries of the system. Only when steady flow crosses the system’s boundaries does P/ρ provide a direct measure of the flow work, which restricts Eq. (2-1) to the steady flow case. The normal practice in fluid dynamics applications is to combine the internal energy and flow work terms into a single parameter called the enthalpy, h. h = u+ P / ρ

(2-2)

The above thermodynamic conditions are simple state variables or static conditions. It is useful to introduce a special class of thermodynamic conditions referred to as total or stagnation conditions. A total thermodynamic condition is defined as the value of a parameter that will exist if the fluid is brought to rest with no transfer of heat or external work, i.e., all kinetic energy is completely recovered. Total conditions will generally be designated with a subscript, t. Total enthalpy occurs so often that an exception will be made and H will be used instead of ht. Total enthalpy is given by H = h + 12 C 2

(2-3)

Heat transfer can normally be neglected for flow through a compressor. There are obvious exceptions, such as when a heat exchanger is included in the system. If heat transfer is neglected, the flow is called adiabatic and Eqs. (2-1) through (2-3) combine to yield ˙ =m ˙ ( Hd − Hi ) w

(2-4)

where the subscripts i and d refer to the compressor’s inlet and discharge conditions, respectively. Equation (2-4) is the basic energy equation for steady, adiabatic flow through a compressor. The second law of thermodynamics introduces the concept of a reversible process. A process is referred to as reversible if the system and its surroundings can be returned to their original states after the process has occurred. If that is not the case, the process is called irreversible. Processes influenced by heat transfer or friction effects are common examples of irreversible processes. The specific entropy, s, is defined as ds =

dqrev T

(2-5)

where T is the temperature, q is the specific heat transfer and the subscript, rev, designates a reversible process. The second law of thermodynamics can be expressed as



∆s ≥ 0

(2-6)

Hence, an adiabatic, reversible process is also a constant entropy or isentropic process. Indeed, it is fairly common practice in turbomachinery to use the terms adiabatic and isentropic interchangeably, although the latter is clearly the more precise term. Entropy provides a fundamental measure of the irreversibility of a process, i.e., the inefficiency or losses associated with the process. If the first law of thermodynamics is applied to a fluid element in a closed system, ( dq)rev = Tds = du + dw = du + PdV

(2-7)

where w is the specific work and V = 1/ρ is the specific volume. Equations (2-2) and (2-7) provide a fundamental thermodynamic equation for entropy that is valid for any process, i.e., Tds = dh − VdP

(2-8)

2.2 EFFICIENCY The aerodynamic quality of a compressor or a component of a compressor is commonly measured in terms of efficiency, which is a measure of actual performance relative to an ideal performance that would be achieved by some reversible process. Figure 2-1 illustrates a typical enthalpy-entropy diagram for a compressor or a portion of a compressor that includes at least one rotor row, such that work has been done on the fluid. The inlet conditions are designated as Pti, Tti, si and Hi. The compressor does work on the fluid to produce discharge conditions Ptd, Ttd, sd and Hd. Note that static and total conditions are, by definition, related by a reversible process. Hence, there is no difference between total and static entropy and the subscript, t, can be omitted for s. One measure of efficiency is to compare the actual process to an ideal adiabatic (isentropic) process. As seen in Fig. 2-1, an isentropic process could produce the change in total pressure with an enthalpy rise ∆Had, commonly referred to as the adiabatic head. The actual process required an enthalpy rise of ∆H. Since lines of constant pressure always diverge on an h-s diagram, ∆H is always larger than ∆Had for a non-isentropic process. Hence, the adiabatic or isentropic efficiency, ηad, is defined as

ηad =

∆Had ∆H

(2-9)

where ∆Had is given by ∆Had = ∫

Ptd

Pti

dP ; (s = const) ρ

(2-10)

This development of efficiency has considered a compressor operating between inlet and discharge total conditions. Hence it is often called the total-to-total



adiabatic efficiency. It is often the case that the fluid kinetic energy at the compressor discharge serves no useful purpose to a specific application. In that case, Ptd may be replaced by the static pressure, Pd, in Eq. (2-10) to yield the total-tostatic adiabatic efficiency. The additional substitution of Pi for Pti yields the static-to-static adiabatic efficiency. Some care is required to understand which basic definition is being used when interpreting efficiency data. It is not uncommon for the term adiabatic efficiency to be used for any of the above three types without qualification. Adiabatic efficiency is the most common definition used for axial-flow compressors. But it has a definite weakness as a means of evaluating the aerodynamic quality of a design. As illustrated in Fig. 2-1, constant pressure lines diverge on an h-s diagram. This means that two compressors having the same basic aerodynamic design quality, but operating at different pressure ratios, will have different adiabatic efficiencies. Hence, adiabatic efficiency is not particularly useful to an aerodynamic designer seeking to evaluate the true aerodynamic quality of a compressor or a stage design. Another consequence of this thermodynamic effect is that the adiabatic head of a multistage compressor is not equal to the sum of the stage adiabatic heads. Polytropic efficiency is a more useful definition, which eliminates this undesirable thermodynamic effect. Polytropic efficiency is sometimes referred to as the “small-stage” or “true aerodynamic” efficiency. Instead of using a path of constant entropy as the reversible path, polytropic efficiency uses a path of constant efficiency defined by

FIGURE 2-1 An Enthalpy-Entropy Diagram



ηp =

1 dP ρ dh

(2-11)

where ηp is defined such that the path passes through the two end points of the process, e.g., (Pti, Hi) and (Ptd, Hd). This is straightforward in principle, but less so in practice. For many years, common practice was to approximate a polytrope by a path defined by P/ρ e = constant. This approximation is appropriate for ideal gases, but can introduce large errors for non-ideal gases. Models that use polytropic efficiency with non-ideal gases have been reported by Shultz (1962) and Mallen and Saville (1977). Huntington (1985) studied the problem in detail and showed that the Mallen-Saville model yields excellent accuracy. Huntington proposed a slightly better model by including an intermediate point on the path, but, for convenient use, this had the disadvantage of requiring a numerical method. Hence, this writer employs the Mallen-Saville model in all cases. This model uses an empirical path defined by T

ds = constant dT

(2-12)

Equations (2-8) and (2-12) can be combined to yield the polytropic head, ∆Hp, as ∆H p = ∆H − ( sd − si )(Ttd − Tti ) / ln(Ttd / Tti )

(2-13)

to provide the total-to-total polytropic efficiency, ηp, by

ηp =

∆H p ∆H

(2-14)

Extension of Eqs. (2-13) and (2-14) to total-to-static and static-to-static efficiency is analogous to adiabatic efficiency as previously discussed. Basically, it is now possible to employ polytropic efficiency with no more difficulty than adiabatic efficiency. It is only necessary to have an appropriate equation of state to use either model.

2.3 FLUID EQUATION-OF-STATE FUNDAMENTALS Thermodynamics contributes one of the fundamental governing equations for compressor aerodynamic design and analysis, commonly called the equation of state. To be more precise, there are actually two equations of state required. The first is the thermal equation of state, which supplies a relationship among the fundamental state variables, typically in the functional form P = P(ρ, T). The second is the caloric equation of state, which relates the energy content of the fluid to state variables, typically in the functional form h = h(T, P) or u = u(T, P). These equations of state may be derived from kinetic theory or statistical mechanics, or they may be developed empirically from experimental data.



Axial-flow compressors are usually applied to rather simple fluids, the most common being air. Consequently, books on axial-flow compressors usually provide a very limited discussion of the equation of state applicable only to very ideal working fluids. But the application of axial-flow compressors to very non-ideal fluids is becoming more common, so this chapter provides a broader discussion. A fluid is considered to be an ideal or thermally perfect gas if P, T and ρ can be related by the simple linear thermal equation of state P = ρRT

(2-15)

where R is a constant for the fluid. The gas constant, R, is related to the universal gas constant, RU, and the fluid’s molecular weight, M R = RU / M

(2-16)

where RU = 8314 Pa-m3/(kmole-°K) in metric units. All working fluids exhibit non-ideal behavior under appropriate conditions. Figure 2-2 is a schematic of a pressure-enthalpy diagram for any working fluid. It is seen that the fluid may be liquid, vapor or both at various state points. Clearly, it will not be possible for a thermally perfect gas equation of state to model all possible state points. This is true even when the equation is restricted to the vapor phase. Figure 2-2 illustrates the location of the fluid’s critical point, which is defined as the highest

FIGURE 2-2 A Pressure-Enthalpy Diagram Schematic



temperature at which liquid and vapor can coexist. Experimental measurements of critical-point properties (Tc, Pc, ρc) are commonly made and almost never conform to Eq. (2-15). Normally, the thermally perfect gas equation of state can provide a reasonable approximation if T is much greater than Tc and P is much less than Pc. At pressures well above Pc, the fluid is said to be in the supercritical regime. Again, it is known from experiment that supercritical fluids almost always show non-ideal gas behavior. Yet the role of the thermally perfect gas equation of state is far more significant than just providing a useful approximation. Indeed, at sufficiently low values of density, all fluids follow the perfect gas law. This provides a dramatic simplification to the process of modeling non-ideal gases. Under conditions where the gas is thermally perfect, it can be shown that the energy content of a fluid is independent of pressure, i.e., h0 = h0(T) and u0 = u0(T), where the superscript, 0, designates conditions where the fluid is thermally perfect. This means it is always possible to define the caloric equation of state as a function of only one state variable.

2.4 THE CALORIC EQUATION OF STATE From the previous discussion, it is seen that the caloric equation of state can be specified in a general form for use in either an ideal or non-ideal gas model. Under low-density conditions where the gas is thermally perfect, the specific heats at constant pressure and at constant volume are defined as  ∂h0  c0p (T ) =    ∂T  P

(2-17)

 ∂u0  c0v (T ) =    ∂T  V

(2-18)

For a thermally perfect gas, it can be shown that c0p (T ) − c0v (T ) = R

(2-19)

The caloric equation of state can be specified by supplying either c0p(T) or c0v(T) and using Eqs. (2-17) through (2-19), i.e., h0 (T ) = h0 (Tref ) =

T

∫

c0p (T )dT

(2-20)

c0v (T )dT

(2-21)

Tref

u0 (T ) = u0 (Tref ) =

T

∫

Tref

where h0 and u0 can be assigned any desired values at a reference state point (Tref, Pref). Specific heat correlations as a function of temperature are readily available for most fluids of interest (e.g., Ried, et al., 1977, 1987; and Yaws, 1999). For the



special case where the specific heats are constant, the gas is referred to as a calorically perfect gas. In that case, Eqs. (2-20) and (2-21) can be directly integrated to yield analytical equations, i.e., h0 (T ) = h0 (Tref ) + c0p (T − Tref )

(2-22)

u0 (T ) = u0 (Tref ) + c0v (T − Tref )

(2-23)

Most books on axial-flow compressors limit attention to thermally and calorically perfect gases. This simplifies the writer’s task by reducing some concepts to simple, analytical forms. This practice is avoided in this book because it results in some unfortunate consequences. Even experienced turbomachinery engineers have occasionally applied these simplified concepts beyond their limits of validity, resulting in some unfortunate conclusions. It has often encouraged developers of computerized aerodynamic analyses to incorporate special thermodynamic relations that are limited to this special class of fluids. A numerical analysis rarely benefits from these simplified methods, and it can be very difficult to generalize such an analysis later, should that become necessary. It is quite possible to have hundreds of such relations scattered through an analysis, presenting a formidable problem when attempting to generalize it.

2.5 ENTROPY AND THE SPEED OF SOUND An entropy equation is essential to all aspects of compressor aerodynamic design and analysis. The efficiency definitions introduced previously in this chapter all require an entropy equation. Even the simplest aerodynamic analyses require an entropy equation to convert between total and static thermodynamic conditions. This conversion is an isentropic process, accomplished by imposing a change in enthalpy (the kinetic energy) at constant entropy. An entropy equation is also required to impose a non-isentropic loss mechanism such as a total pressure loss. It is useful to develop the entropy equation consistent with the caloric equation of state defined for a thermally perfect gas. This will be needed for the non-ideal gas model described later in this chapter. For any pressure where the gas is thermally perfect, the specific entropy is given by Eq. (2-8), i.e., s0 (T , P ) = s0 (Tref , Pref ) +

T

∫

Tref

c0p (T ) T

dT − R ln( P / Pref )

(2-24)

where s0 can be assigned any desired value at any reference state point (Tref, Pref). For a calorically perfect gas, this simplifies to s0 (T , P ) = s0 (Tref , Pref ) + c0p ln(T / Tref ) − R ln( P / Pref )

(2-25)

The speed of sound is also an essential parameter governing the performance of a compressor. From fundamental thermodynamics, the speed of sound, a, for any gas is given by



 ∂P   ∂P  a 2 =   = k   ∂ρ  s  ∂ρ  T

(2-26)

where k is the ratio of specific heats. k = cp / cv

(2-27)

For a thermally perfect gas, this yields a0 = kRT

(2-28)

2.6 THE THERMAL EQUATION OF STATE FOR REAL GASES The general thermal equation of state for a real gas is P / ( ρRT ) = z(T , P )

(2-29)

where z is the compressibility factor. This equation applies to any fluid, with z = 1 for the special case of a thermally perfect gas. The compressibility factor can be obtained from generalized tabular data (e.g., Nelson and Obert, 1954; and Pitzer et al., 1955). But that is rarely done today since many excellent real gas equations of state are available. The real gas equation of state directly provides the many other thermodynamic parameters required and yields much better computational speed when used in numerical methods. The simple two-parameter equations-ofstate are a good choice for general aerodynamic design and analysis. They offer good accuracy, excellent computational speed and easy access to the required gas property data for almost any working fluid or fluid mixture. The Redlich-Kwong equation (Redlich and Kwong, 1949) and various modifications to it (Aungier, 1994, 1995; Barnes, 1973; Soave, 1972; and Wilson, 1966) are recognized as being among the most accurate of the two-parameter equations. Aungier (1994, 1995) evaluated these five equations of state in considerable detail for twelve different compounds over a wide range of temperatures and pressures. Specific emphasis was placed on covering a wide range of accentric factors, ω, since that parameter is used by the various modified Redlich-Kwong equations to improve the prediction accuracy. Accentric factor (Pitzer et al., 1955) is defined as

ω = − log10 ( Pv / Pc ) − 1 ; T / Tc = 0.7

(2-30)

where Pv is pressure on the vapor saturation line and the subscript c designates a critical point property (Fig. 2-2). Table 2-1 lists the compounds investigated, together with their accentric factors. Based on this evaluation, it was concluded that only the original Redlich-Kwong equation and Aungier’s modified RedlichKwong are suitable for general turbomachinery aerodynamic design and analysis. The modified equations of Barnes, Soave and Wilson all showed improved


Thermodynamics • 27 Table 2-1 List of Compounds

ω

Compound Ammonia Carbon Dioxide Ethylene Helium Hydrogen I-Butane Methane N-Pentane Nitrogen Propane Refrigerant R134a Steam

0.2550 0.2250 0.0868 –0.464 –0.220 0.1848 0.0080 0.2539 0.0400 0.1520 0.3254 0.3440

accuracy over the original equation for the types of compounds and thermodynamic property ranges for which they were developed. But they were found to have serious deficiencies for ω < 0, and they exhibited reduced accuracy for supercritical fluids (i.e., P > Pc, Fig. 2-2). Consequently, only the original RedlichKwong equation and Aungier’s modified form will be described here. The original Redlich-Kwong equation is P=

RT a − V − b V (V + b) Tr

(2-31)

where Tr = T / Tc is the reduced temperature, and a = 0.42747 R2Tc2 / Pc

(2-32)

b = 0.08664 RTc / Pc

(2-33)

Equations (2-32) and (2-33) are derived from the thermodynamic stability condition, which requires that the first and second partial derivatives of P with respect to V must both equal zero at the critical point. For its application, this equation requires only the critical temperature and critical pressure for the fluid (and the caloric equation of state data). It offers very good accuracy over a wide range of thermodynamic conditions. It does have one well-known deficiency near the critical point for the fluid. For any fluid, this equation yields z = 1/3 at the critical point, which is not typical of most fluids. Basically, this equation should never be used for points close to the critical point. Aungier’s modified Redlich-Kwong equation of state is P=

RT a − V − b + c V (V + b)Trn

(2-34)

where a and b are given by Eqs. (2-32) and (2-33), respectively, n is given by



n = 0.4986 + 1.1735ω + 0.4754ω 2

(2-35)

and the constant, c, is calculated directly by applying Eq. (2-34) at the critical point with all critical point properties specified. It is included to remove the deficiency in the original equation for points near the critical point. Aungier’s equation requires two more specified parameters than does the original equation, i.e., ω and either Vc or zc. Note that if c = 0 and n = 0.5, Aungier’s equation reduces to the original equation. If a = b = c = 0, the thermally perfect gas equation of state is obtained. This makes it very easy to develop a computerized equation-of-state package for use in any aerodynamic analysis, which can offer all three equations of state. Figure 2-3 illustrates the benefits of Aungier’s modified Redlich-Kwong equation over the original form. This figure shows a comparison of the prediction accuracy of the two equations for about 25% of the data points considered in the evaluation in Aungier (1994, 1995). Tabular (P, V, T) data from the literature were used for that purpose. The two equations of state were used to predict P from the tabular values of T and V, which were compared to the tabular value of P to compute the error. It is seen that Aungier’s model provides a significant improvement in prediction accuracy for a vast majority of the points considered. Aungier (1994, 1995) notes that Aungier’s model resulted in about a 50% reduction in the root-mean-square error for the complete set of data considered. This writer uses Aungier’s model for all real gas problems, but maintains the original model as a

FIGURE 2-3 State-Point Prediction Accuracy



viable alternative. No equation-of-state is infallible, so it is a good idea to have an alternative available. Either of the above equations of state can be used if the accentric factor and the critical point properties are known. This is a real advantage offered by these equations, since those data are readily available for virtually any compound likely to be encountered in turbomachinery applications. Ried and Sherwood (1966), and Reid et al. (1977, 1987) and Yaws (1999) are good sources for these gas property data. Equations (2-31) and (2-34) can also be applied to gas mixtures. If the subscript, i, designates the ith compound in a mixture of N compounds and xi is its mole fraction, the constants for the mixture are given by N

a = ∑ xi ai

(2-36)

i =1 N

b = ∑ xi bi

(2-37)

zc = ∑ xi zci

(2-38)

ω = ∑ xiω i

(2-39)

i =1 N

i =1 N

i =1 N

M = ∑ xi Mi

(2-40)

cp = ∑ xi cpi

(2-41)

i =1 N

i =1

The gas constant for the mixture is given by R = RU / M. The effective values of Tc and Pc for the mixture can be computed from a, b and R, using the definitions in Eqs. (2-32) and (2-33). Then c can be computed from Eq. (2-34). Note that Eq. (2-41) requires that cp be the specific heat per mole rather than per unit mass. For the original Redlich-Kwong equation, c = 0, and Eqs. (2-38) and (2-39) are not used. A mixture equation of state formed in this manner is applicable only to the vapor phase of the fluid. Although the above real-gas models apply to the more general two-phase flow problem, the composition of the two phases may be different and vary with state-point conditions, requiring special treatment. Axialflow compressors are not intentionally applied to two-phase flows, so the restriction to the vapor phase is not a concern in this application. Centrifugal compressors often encounter a special case of two-phase flow when intercoolers are located between stages. These intercoolers may cause some components to liquefy and drop out of the mixture. This is commonly referred to as liquidknockout, and can be modeled by an equilibrium flash calculation. This is not commonly needed for axial-flow compressors and will not be discussed in this book. Should this capability be needed, Aungier (2000) describes the equilibrium-flash calculation using the equations of state presented here.



When the real-gas equation of state is restricted to the vapor phase, it is prudent to avoid accidental excursions into the “wet region” where liquid may form. In a numerical analysis, this can easily occur due to numerical errors in iterative solutions that are far from convergence, often terminating the analysis with a fatal error. The vapor saturation pressure, Pv, is known to vary proportional to 1/T (Ried and Sherwood, 1966). The gas property specifications required for Aungier’s model include two points on the vapor saturation line shown in Fig. 2-2. One point is the critical point, while the other follows from the definition of the accentric factor in Eq. (2-30). Thus, the vapor saturation pressure can be approximated by log10 ( Pv / Pc ) = 7(1 + ω )(1 − Tc / T ) / 3

(2-42)

This equation can easily be inverted to predict the vapor saturation temperature as a function of pressure. These relations can be used to limit the relevant independent thermodynamic conditions in all calculations intended only for the vapor phase to prevent fatal errors from these accidental excursions into the wet region. In some aerodynamic analyses, this simple precaution can yield a dramatic improvement in the reliability of the analysis.

2.7 THERMODYNAMIC PROPERTIES OF REAL GASES Specification of the caloric equation of state, h0 or u0, has been limited to state points where the fluid is thermally perfect. For non-ideal fluids, h and u are functions of pressure as well as temperature. The calculation of the thermodynamic properties of a non-ideal fluid is best accomplished using departure functions (Ried et al., 1977). Departure functions are defined as the difference between the actual value of a parameter and its value under conditions where the fluid is thermally perfect, e.g., the quantity h – h0 is the enthalpy departure function. To employ this process, the pressure at which the fluid can be considered to be thermally perfect, P0, must be specified. The precise value is not important, but it will be a low (but non-zero) value of pressure where the thermally perfect gas approximation can be considered to be valid. Equation (2-15) supplies the corresponding specific volume, i.e., V0 = RT / P0. If A = the Helmholtz energy, Ried et al. (1977) shows that the relevant departure functions are V

A − A0 = − ∫ ( P − RT / V )dV − RT ln(V / V 0 )

(2-43)

 ∂( A − A0 )  s − s0 = −   ∂T  V

(2-44)

∞

h − h0 = ( A − A0 ) + T ( s − s0 ) + RT ( z − 1) 0

0

0

u − u = ( A − A ) + T (s − s )

(2-45) (2-46)

For the Redlich-Kwong equation of state, the departure functions are



h − h0 = PV − RT −

V + b a ( n + 1)Tr− n ln   b  V 

(2-47)

 V V − b + c  na − n  V + b  s − s0 = R ln  0  − bT Tr ln  V  V V   

(2-48)

where c = 0 and n = 0.5 for the original Redlich-Kwong equation of state. Basic thermodynamics provides the other parameters commonly required for aerodynamic analysis, i.e.,  ∂h  cp =    ∂T  P   ∂u   ∂z   cv =   = cp − R  z + T     ∂T  V  ∂T  P    ∂P  a 2 = k  =  ∂ρ  T

(2-49) 2



/ z − P ∂∂Pz 

  T 

kzRT P  ∂z  1−   z  ∂P  T

(2-50)

(2-51)

2.8 THERMALLY AND CALORICALLY PERFECT GASES When the fluid can be considered thermally perfect (z = 1) and calorically perfect (cp, cv, and k are constants), equation-of-state calculations are greatly simplified. If the subscript, ref, designates conditions at an arbitrary reference state point (Tref, Pref) where h and s can be assigned arbitrary values, Eqs. (2-22) and (2-25) can be written h = href + cp (T − Tref )

(2-52)

s = sref + cp ln(T / Tref ) − R ln( P / Pref )

(2-53)

Combining Eqs. (2-3) and (2-52), the total and static temperatures are related by cp (Tt − T ) =

1 C2 2

(2-54)

For an isentropic or adiabatic-reversible process Eq. (2-53) requires T / Tref = ( P / Pref )

k −1 k

= ( ρ / ρ ref )k −1

(2-55)

Equations (2-54) and (2-55) provide simple analytical expressions to relate total and static conditions. Similarly, the efficiency calculations in Eqs. (2-9) and (2-14) simplify to


32 • AXIAL-FLOW COMPRESSORS k −1

ηad

(P / P ) k − 1 = td ti Ttd / Tti − 1

(2-56)

k −1 ln( Ptd / Pti ) k ln(Ttd / Tti )

(2-57)

ηp =

Reduction of these thermodynamic relations to simple analytical form can yield substantial reductions in computation time for computerized aerodynamic analyses. If the fluid is calorically imperfect, numerical integration of Eqs. (2-20) and (2-21) will normally be necessary. If the fluid is thermally imperfect, the departure functions of Eqs. (2-47) and (2-48) must also be applied. In either case, numerical iteration is required for isentropic processes to converge to the required constant entropy condition. This is the primary source of increased computation time. Typically, an analysis will perform a massive number of isentropic calculations to relate total and static conditions and to relate conditions in rotating and stationary coordinate systems. A calorically imperfect, thermally perfect fluid requires nearly as much computation time as a general non-ideal gas. Hence, the thermally and calorically perfect gas model offers significant advantages when it can be employed.

2.9 THE PSEUDO-PERFECT GAS MODEL The pseudo-perfect gas model (Aungier, 1998, 2000) is a very useful concept that often provides all of the benefits of the thermally and calorically perfect gas model for cases where that model is not adequate. Its use requires some care in formulating an aerodynamic analysis. It is necessary to avoid use of any state relations that assume the relationships between cp, cv, R and k expressed in Eqs. (2-19) and (227). This really does not complicate the analysis, but the advantages of the pseudoperfect gas model cannot be exploited unless this is done. The concept is to use fictitious values of cp, cv, R and k in an otherwise standard thermally and calorically perfect gas model. If the fictitious constants are represented with an overbar, they can be calculated from any two state points, designated by subscripts 1 and 2. R = R z1z2

(2-58)

cp = ( h2 − h1) / (T2 − T1)

(2-59)

cv = (u2 − u1) / (T2 − T1)

(2-60)

k = ln( P2 / P1) / ln( ρ2 / ρ1)

(2-61)

Equation (2-61) yields the isentropic exponent, which can be quite different from the ratio of specific heats for a real gas. The pseudo-perfect gas model is generally adequate as long as the values of z1 and z2 are not too different. This is usually true for compressor components operating on a non-ideal fluid. It can also be applied to applications such as multistage compressor performance analysis,



simply by recomputing the fictitious constants between stages or between blade rows. The modest stage pressure ratios in axial-flow compressors almost always ensure that the model is sufficient. Indeed, this writer has used it extensively for industrial centrifugal compressor performance analysis, where stage pressure ratios are much higher and non-ideal working fluids are common. No case has been encountered to date where the pseudo-perfect gas model yields results significantly different from a full non-ideal gas model. This writer’s normal practice is to select the inlet total conditions and the conditions corresponding to sonic flow as the two state points to define the fictitious constants—that includes most conditions likely to be encountered in a compressor blade row. The full real gas model is used to compute the constants. Then the pseudo-perfect gas model is used in the actual analysis. This writer uses a computerized equation-of-state module that contains the pseudo-perfect gas model as well as the other equations of state described in this chapter. This module is supported by a gas property database from which the equation-of-state for the fluid or fluid mixture is easily formulated. All aerodynamic design and analysis computer programs use this module to perform all equation-of-state calculations. Even for thermally perfect fluids, the pseudo-perfect gas model yields substantial reductions in computation time by using a calorically perfect gas approximation. Its adequacy for those cases is a virtual certainty.

2.10 COMPONENT PERFORMANCE PARAMETERS The adiabatic and polytropic efficiencies introduced previously in this chapter can be used to evaluate the performance of a compressor, a compressor stage or even a single rotor blade row. But they have no meaning for a stationary compressor component, since total enthalpy is constant for those components. Yet, it is useful to be able to evaluate a stationary component’s performance against some ideal, reversible process. Most components in a compressor serve to diffuse the flow to convert kinetic energy into static pressure. The most important of these is the exhaust diffuser, although stator and exit guide vane rows are also stationary, diffusing components. One method of evaluating a diffusing component is known as the diffuser efficiency. Its definition is really quite similar to the adiabatic efficiency introduced previously, except that static enthalpy is used as its basis. Figure 2-4 presents a schematic of an h-s diagram for a diffuser. Flow enters the diffuser with a velocity Ci and exits the diffuser with a lower velocity Cd. Since the total enthalpy, H, is constant, Eq. (2-3) can be used to compute the static enthalpy at the inlet, hi, and at the discharge, hd. It is seen from Fig. 2-4 that this process results in an increase in static enthalpy of ∆h. Since irreversible losses such as friction occur, the entropy increases from si to sd. Now, trace the line of constant pressure for the discharge pressure back to the inlet entropy. This shows that the static enthalpy increase required for an adiabatic, reversible process to produce the same pressure increase is ∆had. Therefore, a diffuser efficiency can be defined as

ηdiff =

∆had ∆h

(2-62)



FIGURE 2-4 A Diffuser h-s Diagram

A more common parameter used to evaluate the performance of a diffuser is the static pressure recovery coefficient, cp. This expresses the ratio of the static pressure rise to the available kinetic energy at the inlet. Usually, the available kinetic energy is expressed as a “velocity pressure,” which is the difference between the total and static pressure. cp =

Pd − Pi Pti − Pi

(2-63)

Hence, cp is the fraction of the available kinetic energy that has been recovered as static pressure by the diffuser. Although compressors are basically diffusing machines, there are some compressor components that accelerate the flow. Inlet passages and inlet guide vanes are typical examples. In those cases, the component can be evaluated using the nozzle efficiency. Figure 2-5 illustrates the parameters used to define it. The flow accelerates from the inlet velocity, Ci, to a higher discharge velocity, Cd. Due to irreversible processes such as wall friction, the entropy increases from si to sd and the total pressure decreases from Pti to Ptd. If the flow is adiabatic (no heat transfer), the total enthalpy, H, is constant for this process. Equation (2-3) relates H, h and C at the inlet and the discharge. This is illustrated in Fig. 2-5, along with the ideal discharge velocity, Cad, which would be produced by an isentropic process between



FIGURE 2-5 A Nozzle h-s Diagram

the actual inlet pressure, Pi, and discharge pressure, Pd. A nozzle efficiency can be defined as the ratio of the actual increase in kinetic energy to the increase in kinetic energy corresponding to the isentropic process, i.e.,

ηnoz =

Cd2 − Ci2 2 Cad − Ci2

(2-64)

Occasionally, a simpler definition is used: It employs the velocity pressure, Pt – P, in place of C2 as the measure of kinetic energy and uses Pti – Pd as the ideal (no loss in total pressure) discharge kinetic energy. Simple substitution of these alternate kinetic energy terms into Eq. (2-64) yields

ηnoz = 1 −

Pti − Ptd Pi − Pd

(2-65)

The most fundamental measure of irreversibility is the increase in entropy. But an entropy increase is difficult to interpret, so total pressure loss is more commonly used. If Eq. (2-8) is applied to total conditions, with total enthalpy constant, ∆Pt Pti − Ptd = = T∆s ρ ρ

(2-66)



It is known from observation that total pressure loss is usually proportional to kinetic energy, so a logical definition of loss coefficient can be obtained by dividing by 1/2C2, i.e., the loss coefficient can be defined as

ω =

∆Pt 1 2

ρ C2

(2-67)

Experience has shown that use of velocity pressure as a measure of kinetic energy provides a better definition, i.e.,

ω =

∆Pt Pt − P

(2-68)

It is found that the second definition is much less sensitive to changes in the Mach number. This is very important in axial-flow compressor performance analysis. Typically, blade row performance is predicted with empirical loss coefficient models derived from low-speed cascade tests, but applied to blade rows with relatively high inlet Mach numbers. The flow is essentially incompressible (Mach number essentially zero) in the low-speed cascade tests, so the two definitions of loss coefficient are basically identical. But when empirical loss coefficient models are applied to predict losses in higher Mach number blade rows, Eq. (2-68) is found to yield substantially better results. Equations (2-67) and (2-68) are ambiguous with regard to what kinetic energy is to be used in the denominator. The usual practice is use of the largest kinetic energy relevant to the component. For diffusing components, such as diffusers and compressor blade rows, the inlet kinetic energy is usually employed. For accelerating components, such as turbine blade rows, the discharge kinetic energy is usually employed. The formulation for loss coefficient assumed constant total enthalpy, which is far from true for rotor blade rows. But chapter 3 will show that total enthalpy viewed in a frame of reference rotating with the blades (the relative total enthalpy) is constant for axial flow. So, the above definitions can be used for rotors in that context. There is a subtle thermodynamic effect present when applying empirical loss coefficient models to rotating blade rows. In fact, the flow is not usually precisely axial, i.e., there is usually some change in radius as a streamline passes through a rotor. Consequently, even the relative total enthalpy is not usually constant through the blade row. For axial-flow compressors, this effect is so small that it can be neglected in virtually all cases. The situation is quite different in centrifugal and mixed-flow compressors, where consideration of this thermodynamic effect is essential. Aungier (2000) develops a correction procedure to apply empirical loss coefficients to rotating blade rows with a significant change in radius from inlet to discharge. Since it is not necessary for axialflow compressors, it will not be included here. But investigators attempting to combine axial-flow compressor stages with mixed-flow or radial-flow stages should consult that reference.



2.11 GAS VISCOSITY Gas viscosity is often required for aerodynamic analysis. A generalized estimation method compatible with the previous equation-of-state models is provided by Dean and Stiel (1965), and is reviewed by Ried et al. (1966). It is necessary to depart from this book’s practice of maintaining independence from specific units in this case. The Dean and Stiel model predicts viscosity in centipoises and employs a dimensional parameter defined by 1 2 ξ = Tc 6 /  MPc 3 

(2-69)

where Tc is in degrees Kelvin and Pc is in atmospheres. The low-pressure fluid viscosity is given by 8

µ 0ξ = (3.4 ⋅10−4 )Tr 9 ; Tr ≤ 1.5

(2-70)

µ 0ξ = 0.001668(0.1338Tr − 0.0932) 9 ; Tr > 1.5

(2-71)

5

Then, the viscosity at any pressure is defined by a departure function ( µ − µ 0 )ξ = (1.08 ⋅10−4 )[exp(1.439ρ r ) − exp(−1.111ρ1r .858 )

(2-72)

Ried et al. (1966) indicate that accuracy within 5% can be expected for non-polar molecules, with somewhat larger errors possible for polar molecules. This viscosity estimation method is quite compatible with the equation-of-state models previously recommended in this chapter as long as the fluid is in the vapor phase. In principle, the real gas equations of state apply to the liquid or two-phase fluid also, but this viscosity model does not. The critical point data needed for Aungier’s modified Redlich-Kwong equation of state are sufficient for this gas viscosity model also, but appropriate care is required to adjust the units of Tc and Pc in Eq. (2-69) and the µ predicted.

2.12 A COMPUTERIZED EQUATION-OF-STATE PACKAGE One of the most useful tools for the developer of axial-flow compressor design and analysis software is a computerized equation-of-state package that can be used as a module in any other analysis to handle all equation-of-state input, output and thermodynamic calculations. Some useful features to include are the following. • Provide the non-ideal gas, ideal gas (with variable specific heats) and pseudo-perfect gas as options always available to the user. Be sure to take advantage of opportunities for reduced computation time for the simpler models. For a basic ideal gas, the pseudo-perfect gas model



•

•

•

•

•

•

functions directly as a thermally and calorically perfect gas model. Include the capability to specify pseudo-perfect gas data directly, or to have these data computed from the equation of state and the specified gas mixture. Maintain a database of gas property data for compounds you need. Establish the capability to set up any of the equation-of-state models from this data simply by selecting compounds in the mixture and supplying their mole fractions. For the models described in this chapter, the data needed are M, Pc, Tc, Vc (or zc), ω and cp(T). This writer uses a third-order polynomial fit of cp as a function of T, consistent with Ried et al. (1977), which is a good source of all the data required for a very wide range of compounds. Note that setup of the pseudo-perfect gas model from the equation of state must be requested by the main analysis when appropriate thermodynamic conditions are known, as described previously in this chapter. Include a routine to accomplish this on demand. Include logic to save and recover gas property data so that input files of other programs maintain a common format for all aerodynamic software. It is a good idea to include the logic for basic checks on data validity to be sure the equation of state is complete. Reserve a special code for the equation-of-state model to signal to the program using this module that data is lacking or invalid. It is useful to include a standard output routine that can be used to insert a description of the mixture and the equation of state in use in output files for other programs. Include routines for the basic thermodynamic calculations needed in aerodynamic analysis. Standard functions likely to be needed include V(T, P), P(T, V), T(V, P), a(T, P), h(T, P), s(T, P), ηad(Ti, Pi, Td, Pd) and ηp(Ti, Pi, Td, Pd). A routine to compute the isentropic change in P and T for a specified change in h is essential for conversion between static and total thermodynamic conditions and between rotating and stationary coordinate systems. A routine to compute the speed of sound at sonic flow conditions for known total conditions is often useful. Include the capability to edit the thermodynamic model to modify the compounds in the mixture, their mole fractions and the equation-ofstate model to be used. The Dean and Stiel (1965) viscosity estimation model can be included to treat all gases and gas mixtures formed from the gas property database. If pseudo-perfect gas data are specified rather than calculated from the equation of state, viscosity data will also need to be specified.

Using this approach, development of aerodynamic design and analysis software will involve minimal logic related to the equation of state. In general, axial-flow compressor state calculations should be limited to the vapor phase using the constraint expressed in Eq. (2-42). If special consideration involving liquid phases is necessary, the approximate liquid phase models described in Aungier (1998, 2000) and the equilibrium flash calculation described in Aungier (2000) may be useful.



EXERCISES 2.1 Consider uniform axial flow at the inlet of a compressor with no inlet guide vanes (i.e., stations 0 and 1 are identical) with velocity Cz1 and Cθ1 = 0. For a thermally and calorically perfect gas, show that ratios ρ1/ρt1 and a1/at1 are functions of Mach number M1 = Cz1 / a1 (these results were used in Chapter 1 without derivation to develop the equivalent flow and speed parameters). 2.2 The flow at station 1 of Exercise 2.1 passes through a rotor exiting at station 2 with Cz2 = Cz1 and with a swirl velocity, Cθ2, at the mean radius where the blade speed is ωr = U. The total pressure loss coefficient across the blade row, based on the inlet velocity pressure, is –. For a thermally and calorically perfect gas, derive expressions for ω the ideal (no loss) discharge total temperature and pressure, Tt2id and Pt2id, and the actual values, Tt2 and Pt2, at the mean radius in terms of the inlet parameters Pt1, Tt1, H1 and P1. Use the Euler turbine equation, Eq. (1-17), to compute the change in total enthalpy across the rotor. 2.3 For the rotor row of Exercise 2.2, extend the Mach number equivalence parameters of Chapter 1 to include an expression for the equivalent total enthalpy rise across the rotor. Show that if the flow is – = 0), Mach number equivalence at the inlet of isentropic (i.e., that ω the blade row will produce Mach number equivalence at the discharge. 2.4 The rotor row of Exercises 2.1 and 2.2 is to be operated with a new working fluid. The operation with the new working fluid will start with the same inlet conditions (Pt1, Tt1, ρt1) as for the original working fluid. Its operating speed and mass flow rate will be determined by requiring Mach number equivalence at station 1. The two fluids have the same gas constant, R, but the original working fluid has k = 1.4 and the new working fluid has k = 1.38. Assuming that the flow is isentropic, show that both the equivalent speed and flow parameters cannot be satisfied at station 2 (the rotor exit). Is complete Mach number equivalence achieved in this case? 2.5 An axial-flow compressor is to be operated with propane as the working fluid with an inlet pressure of 200 kPa. Fluid property data for propane are M = 44.1, Tc = 369.83° K, Pc = 4249.6 kPa and ω = 0.152. Estimate the lowest inlet temperature that can be used to avoid the risk of liquid erosion due to two-phase flow. 2.6 Two axial-flow compressors operate on a thermally and calorically perfect gas with k = 1.4. Both compressors have an adiabatic efficiency of 85%. The pressure ratios of the two compressors are 3.0 and 5.0. Calculate the polytropic efficiencies of the two compressors. 2.7 An axial-flow compressor consists of three repeating stages, each with a stage pressure ratio of 1.1 and a stage adiabatic efficiency of 85%. The working fluid is a thermally and calorically perfect gas with k = 1.4. Compute the overall adiabatic efficiency of the three-stage compressor. Repeat this problem assuming all efficiencies are polytropic.



2.8 Consider adiabatic flow of a calorically and thermally perfect gas with k = 1.4 through a diffuser. The inlet total temperature is 300° K, the inlet total pressure is 230 kPa, the inlet static pressure is 200 kPa, the static pressure recovery coefficient is 0.6 and the total pressure loss coefficient based on the inlet velocity pressure, (Pti – Pi), is 0.1. Compute the diffuser efficiency, ηdif.


Chapter 3

FLUID MECHANICS

Fluid mechanics and thermodynamics are the fundamental sciences used for the aerodynamic design and analysis of axial-flow compressors. This chapter highlights some fundamental concepts from fluid mechanics to complement the concepts from thermodynamics covered in Chapter 2. The governing equations will be developed in forms suitable for the various aerodynamic analyses commonly employed for axial-flow compressors. Detailed solution procedures will be covered in subsequent chapters. Several types of fluid dynamic analysis are useful for this purpose. The through-flow analysis is widely used in both design and performance analysis. This involves solving the governing equations in the hub-to-shroud plane at stations located between blade rows. The flow is normally considered to be axisymmetric at these locations, but still three-dimensional because of the existence of a tangential velocity component. Empirical models are employed to account for the fluid turning and losses that occur when the flow passes through the blade rows. A simplification of this analysis is the “pitch-line” or “mean-line” onedimensional flow model, which ignores the hub-to-shroud variations. These were very common for many years, but are no longer particularly relevant to the problem. Computers are sufficiently powerful today that there is really no need to simplify the problem that much. The through flow in an axial-flow compressor is strongly influenced by viscous effects near the end walls. The primary influence from these end-wall boundary layers is commonly described as end-wall blockage. An inviscid through-flow analysis ignores the low momentum fluid in the boundary layers and will overestimate the mass flow that the passage can accommodate for a given flow field solution. To compensate the common practice is to impose a blockage factor to effectively reduce the passage area. This requires consideration of boundary layer analysis to estimate the appropriate blockage factors to be used. More fundamental internal flow analyses are often useful for specific components, particularly blade rows. These include two-dimensional flow analyses in either the blade-to-blade or hub-to-shroud direction, and quasithree-dimensional flow analyses developed by combining and interacting these two-dimensional analyses. Again, wall boundary layer analysis is often used to evaluate viscous effects. Any of these analyses may be used in a design mode as well as an analysis mode. A design mode seeks to define the gas path geometry (end-wall contours and blades) to produce the desired flow field, while an analysis mode seeks to predict the flow field from specified geometry.



Viscous computational fluid dynamics (CFD) solutions are also in use for axialflow compressors. These are typically three-dimensional flow analyses, which consider the effects of viscosity, thermal conductivity and turbulence. In most cases, commercial viscous CFD codes are used although some in-house codes are in use within the larger companies. Most design organizations cannot commit the dedicated effort required to develop these highly sophisticated codes, particularly since viscous CFD technology is changing so rapidly that any code developed will soon be obsolete unless its development continues as an ongoing activity. Consequently, viscous CFD is not covered in this book beyond recognizing it as an essential technology and pointing out some applications for which it can be effectively used to supplement conventional aerodynamic analysis techniques.

NOMENCLATURE a = b = C = E = → e = f = H = h = I = m = ˙ = m n = P = r = → r = s = T = u = → V = v = W = ˙ = w x = y = z = δ = δ* = θ = κ = ν = ρ = τ = φ = ω =

sound speed stream sheet thickness absolute velocity entrainment function unit vector body force total enthalpy static enthalpy rothalpy meridional coordinate mass flow rate normal coordinate pressure radius position vector in space entropy temperature velocity in x direction general vector velocity in y direction relative velocity power coordinate along a wall coordinate normal to a wall axial coordinate boundary layer thickness displacement thickness tangential coordinate and momentum thickness curvature force defect thickness gas density torque and shear stress streamline slope with axis and a general function rotation speed


Fluid Mechanics • 43

Subscripts e m n r t w 1 11 12 2 22 θ

= = = = = = = = = = = =

boundary layer edge condition meridional component normal component radial component total thermodynamic condition parameter at a wall blade inlet parameter and meridional defect parameter meridional defect parameter tangential flux defect parameter blade exit parameter and tangential defect parameter tangential defect parameter tangential component

Superscripts ′ = a relative value in the rotating coordinate system

3.1 FLOW IN A ROTATING COORDINATE SYSTEM The analysis of the flow in rotor blade rows is accomplished in a coordinate system, which rotates with the blade. The flow conditions in a rotating coordinate system are referred to as the relative conditions. If a blade row is rotating with an angular velocity, ω, the relative tangential velocity in a coordinate system rotating with the blade, Wθ, is related to the absolute tangential velocity, Cθ, by Wθ = Cθ − ω r

(3-1)

The axial and radial velocity components are independent of the rotation, i.e., Wz = Cz

(3-2)

Wr = Cr

(3-3)

It will be more convenient to work with the meridional velocity component, Wm, defined as Wm = Wz2 + Wr2 = Cm

(3-4)

Wm is the velocity component lying in the meridional (constant θ) plane and in a stream surface. A stream surface is defined as a surface having no fluid velocity component normal to it and, therefore, no mass flow across it. Thus, the defining characteristic of a stream surface is that the mass flow rate between it and the hub contour surface is constant everywhere. The meridional coordinate, m, is measured along the stream surface and in a meridional plane, i.e.,



( dm)2 = ( dr )2 + ( dz)2

(3-5)

For axial-flow compressors, it is almost always reasonable to assume stream surfaces are axisymmetric. That assumption will be used throughout this book. Figure 3-1 illustrates a schematic of a stream surface and unit vectors for the meridional and polar coordinates. Normal to these unit vectors, and to the stream surface, is the third coordinate of interest, the normal coordinate, n, as shown in Fig. 3-2. It is convenient to develop the governing equations of fluid mechanics in this “natural” coordinate system (θ, m, n), where by definition Wn = Cn = 0

(3-6)

Now consider the flow through a thin stream sheet, i.e., a thin annular passage bounded by two stream surfaces. The torque, τ, acting on the fluid between meridional stations 1 and 2 is given by conservation of angular momentum. ˙ ( r2Cθ 2 − r1Cθ1) τ =m

(3-7)

This torque must balance the power input, i.e., ˙ = ωτ = ω m ˙ ( r2Cθ 2 − r1Cθ1) w

(3-8)

Combining Eq. (3-8) with Eq. (2-4) yields the well-known Euler turbine equation

FIGURE 3-1 Schematic of a Stream Surface



FIGURE 3-2 Natural Coordinate System H2 − H1 = ω ( r2Cθ 2 − r1Cθ1)

(3-9)

This is the general energy equation relating the total enthalpy change produced by a transfer of mechanical energy between the fluid and a rotating blade row. It is convenient to introduce the rothalpy, I, defined by I = H − ω rCθ

(3-10)

On introducing Eq. (3-10) into Eq. (3-9), it can be seen that rothalpy is constant on a stream surface. Hence, rothalpy is the basic parameter expressing energy conservation for a rotating blade row. It is also valid for a stationary blade row, since I = H in that case, and Eq. (3-9) requires that H be constant in the absence of energy transfer with a rotating blade row. Aerodynamic analysis of axial-flow compressors involves alternately solving the governing equations in rotating coordinates (rotors) and stationary coordinates (stators). Hence, we need to relate the relative total enthalpy, H′, in a rotating coordinate system to the absolute total enthalpy, H, in a stationary coordinate system. Noting that static thermodynamic conditions are identical for either coordinate system, h = H ′ − 12 W 2 = H − 12 C 2

(3-11)

The relative velocity, W, follows from Eqs. (3-1), (3-4) and (3-6)



W = Wm2 + Wθ2

(3-12)

Equations (3-1), (3-10), (3-11) and (3-12) combine to yield H ′ = H − ω rCθ + 12 (ω r )2 = I + 12 (ω r )2

(3-13)

Noting that entropy is constant between total and static conditions, and therefore between a rotating and a stationary coordinate system, Eq. (3-13) can be used to relate the two coordinate systems. For example, the change in all other relative total thermodynamic conditions between the two coordinate systems can be calculated from an appropriate equation of state as a function of (H, H′, s). This requires calculation of the isentropic change in the parameter of interest for a specified change in enthalpy. Hence, Eq. (3-13) is an important relation that allows us to relate all thermodynamic parameters between the stationary and the rotating coordinate systems. Also, since I is constant on the stream surface, Eq. (3-13) allows calculation of H′ at all points on a stream surface when one value is known, e.g., at the inlet.

3.2 ADIABATIC INVISCID COMPRESSIBLE FLOW Adiabatic compressible inviscid flow analysis is commonly used in turbomachinery. This flow model assumes that fluid viscosity and thermal conductivity can be neglected. Basic conservation of mass, momentum and energy, supported by a suitable equation of state, govern the flow. It is useful to derive the governing equations in a rotating coordinate system, noting that these equations will be valid for a stationary coordinate system if ω is set to zero. The vector form of the momentum equation can be written (Novak, 1967; Vavra, 1960; Wu, 1952) r r r r r r r dC 1 r dW = − ∇P = + 2(ω × W ) + ω × (ω × r ) ρ dt dt

(3-14)

where the last two terms in Eq. (3-14) are the Coriolis and centrifugal accelerations imposed by the rotating coordinate system, and the time derivative is the substantial derivative, i.e., r r r r r dW ∂W = + (W ⋅ ∇)W dt ∂t

(3-15)

Hence, the momentum equation in rotating coordinates is r r r r r r r r r r ∂W ∇P + (W ⋅ ∇) W + 2(ω × W ) + ω × (ω × r ) = − ρ ∂t

(3-16)

Using standard vector identities, this equation can also be written as



r r r r r r r ∂W ∇P 2r 2 1 − W × (∇ × W + 2ω ) − rω er + 2 ∇W = − ρ ∂t

(3-17)

where → er is a unit vector in the radial direction. On introducing Eqs. (2-3), (2-8), (3-10) and (3-11), an alternate form of Eq. (3-17) is obtained. r r r r r r r ∂W − W × (∇ × W + 2ω ) = T ∇s −∇I ∂t

(3-18)

The continuity and energy equations in vector form are r ∂ρ r + ∇ ⋅ (ρ W ) = 0 ∂t r r ∂I 1 ∂P − + (W ⋅ ∇)I = 0 ∂t ρ ∂t

(3-19) (3-20)

Equations (3-16) through (3-20) are vector equations, which are valid in any coordinate system. To express the equations in the natural coordinates (θ, m, n), standard curvilinear coordinate transformations are used. These can be found in most advanced calculus books, which cover vector field theory. Vavra (1960, Appendix A) provides specific and detailed derivations of the vector operators and governing equations in natural coordinates. For general reference, the important vector operators are provided without derivation at the end of this chapter. The resulting governing equations are ∂ρ 1  ∂ rρ Wm ∂ρ Wθ  + + + κ n ρ Wm = 0 ∂t r  ∂m ∂θ 

(3-21)

∂Wm ∂Wm Wθ ∂Wm sin φ 1 ∂P + Wm + − [Wθ + ω r]2 = − ρ ∂m r ∂θ r ∂t ∂m

(3-22)

∂Wθ ∂Wθ Wθ ∂Wθ Wm sin φ 1 ∂P + Wm + + [Wθ + 2ω r] = − ∂t ∂m r ∂θ r rρ ∂θ

(3-23)

cos φ 1 ∂P [Wθ + ω r]2 = r ρ ∂n

(3-24)

∂I 1 ∂P ∂I Wθ ∂I − + Wm + =0 ∂m r ∂θ ∂t ρ ∂t

(3-25)

κ mWm2 +

The curvature of the stream sheet, κm, and of the normal surface, κn, are related to the angle φ shown in Fig. 3-2.

κm = − κn =

∂φ ∂m

∂φ 1 ∂b = ∂n b ∂m

(3-26) (3-27)



Parameter b in Eq. (3-27) is the thickness of a stream sheet bounded by two stream surfaces, as shown in Fig. 3-2. Hence, the curvature κn is related to the meridional divergence or convergence of the stream surfaces. That form is useful in some applications of these governing equations, particularly when analyzing the two-dimensional flow in a blade-to-blade stream surface. Equations (3-22) through (3-24) can be expressed differently using Eq. (3-18). ∂s ∂I ∂Wm Wθ  ∂Wm ∂( rWθ + ω r 2 )  − + −  =T r  ∂θ ∂ m ∂ m ∂t ∂m 

(3-28)

∂s ∂I ∂Wθ Wm  ∂Wm ∂( rWθ + ω r 2 )  − − −  =T r  ∂θ ∂θ ∂θ ∂t ∂m 

(3-29)

κ mWm2 +

Wθ ∂( rWθ + ω r 2 ) ∂Wm ∂I ∂s + Wm = −T ∂n ∂n r ∂n ∂n

(3-30)

Since there are only two velocity components (i.e., Wn = 0), one of the three momentum equations is redundant. The redundant equation has been replaced by the assumption that the stream surfaces are known or can somehow be determined as part of the solution. If the meridional surfaces are not stream surfaces, the governing equations must be modified to include a normal velocity component, Wn. This will not be required for analyses described in this book, although there is no reason why a flow analysis could not be accomplished in an arbitrary (θ, m, n) coordinate system. Aungier (2000) includes the more general form of the governing equations appropriate for that type of analysis.

3.3 ADIABATIC INVISCID COMPRESSIBLE FLOW APPLICATIONS The governing equations are applied in a variety of analyses in the aerodynamic design and analysis of axial-flow compressors. Most analyses employ the timesteady form of the governing equations, although the unsteady form does find application when the fluid velocity exceeds the sonic flow velocity. One of the most common applications is to determine the flow in the meridional plane. This application normally restricts the solution to stations outside of the blade rows, using empirical models to impose the influence of blade rows between stations in terms of fluid turning and total pressure loss correlations. These analyses treat the flow as axisymmetric to require conservation of mass, normal momentum and energy. Early practice was to develop these analyses in a simple cylindrical coordinate system (θ, m, r) such that the normal coordinate is replaced by the radial coordinate. Hence, the normal momentum equation was commonly referred to as the radial equilibrium equation in that context. Various simplifications may be employed in these analyses. If stream surface curvature is neglected (κm = 0) and the gradient of entropy across the passage is assumed to be zero, the term “simple radial equilibrium” has been used to describe the analysis. If the gradient of entropy is included, the solution is often referred to as “simple non-isentropic



radial equilibrium.” When curvature and entropy are both included, the term “full radial equilibrium” is often used. Advances in computer technology and numerical analysis techniques have reduced the role of simple radial equilibrium solutions to cases where the entropy gradient cannot be properly defined, such as the general-purpose stage design described in Chapter 10. Simple non-isentropic radial equilibrium continues to be useful in basic blade row or stage design, where the streamline curvatures to be encountered in the actual compressor are not known. Indeed, simple non-isentropic radial equilibrium analysis is often quite sufficient for actual axial-flow compressors, where stream surface curvatures may be negligible. This is often true for industrial axial-flow compressors. When the stream surface curvature can be ignored, a dramatic reduction in computation time is realized, since the analysis becomes a simple marching solution. This follows from the fact that the flow at any axial station is not dependent on the flow at downstream stations. A simple variant used by this writer is to approximate stream surface curvatures from end-wall contours by simple linear interpolation. This allows the advantages of a simple non-isentropic radial equilibrium analysis, yet can approximate curvatures imposed by end-wall contour design. Solutions for the two-dimensional flow in the meridional plane within blade passages are also fairly common. These usually seek to predict the average flow in the passage from the hub to the shroud as a two-dimensional flow problem. In the more general case, these hub-to-shroud analyses may solve for the twodimensional flow on specific stream surfaces from hub to shroud. In both cases, either the flow angle or Wθ distributions throughout the passage must be supplied to replace solution of the tangential momentum equation. Analysis of the flow passing through a blade row and lying on a stream surface is also common in axial-flow compressor design and analysis. These two-dimensional flow analyses are commonly called blade-to-blade flow analyses. Typically, the stream surface geometry is specified along with the distribution of the stream sheet thickness, b. Then, conservation of mass, energy, tangential momentum and meridional momentum can describe the flow. If the flow is assumed to be isentropic, Eqs. (3-28) and (3-29) show that one of the momentum equations is redundant, resulting in a simpler problem. This is fairly common practice for subsonic flow problems and is referred to as potential flow or irrotational flow. This results in a classical boundary value problem of an elliptic equation. Indeed, the governing equations for inviscid flow are elliptic in form as long as W < a throughout the flow field. When supersonic flow is encountered (W > a), the governing equations become hyperbolic in mathematical form, which requires a marching type solution—such as the method of characteristics—rather than a boundary value problem solution. Cases where the flow is supersonic throughout are rare. Usually mixed subsonic-supersonic flow is involved. Then the timesteady governing equations are elliptic in some regions and hyperbolic in others, requiring two different solution techniques that must be matched together in some fashion. It is now fairly common practice to employ the time-unsteady equations for these cases. The advantage of that approach is that the unsteady equations of motion are hyperbolic in form for both subsonic and supersonic flow. This allows a single numerical method to be used for the mixed subsonicsupersonic flow case. This approach is commonly called the “time-dependent” or “time-marching” method of solution.



Hub-to-shroud and blade-to-blade flow analyses may also be combined to form a “quasi-three-dimensional” flow analysis within blade passages. A hubto-shroud analysis can supply the stream surface geometry and stream sheet thickness distribution required for a blade-to-blade flow analysis. Similarly, a blade-to-blade flow analysis can supply the distribution of flow angle or tangential velocity required for a hub-to-shroud flow analysis. Thus an iterative solution solving these two two-dimensional flow problems with interaction between them can provide an approximate three-dimensional flow analysis. This approach was originally suggested by Wu (1952) and is a commonly used analysis technique. Analysis of the complete three-dimensional inviscid flow problem is seldom used today. The additional information supplied relative to the simpler and faster quasi-three-dimensional flow analysis is relatively minor. Also, the evolution of three-dimensional viscous CFD analysis techniques has discouraged use of a three-dimensional inviscid flow analysis. As discussed at the beginning of this chapter, most turbomachinery design groups use one of the many excellent commercially available viscous CFD codes when a more detailed and fundamental analysis is needed.

3.4 BOUNDARY LAYER ANALYSIS Adiabatic inviscid flow analyses, such as those described in the previous sections, are commonly augmented by boundary layer analysis techniques to evaluate viscous effects that are not considered by those analyses. The basic premise of boundary layer theory is that viscous effects are confined to a thin layer close to the physical surfaces bounding the flow passages (Schlichting, 1968, 1979). This is by no means always the case in axial-flow compressors, but selective use of boundary layer analysis has been found to be very effective in many applications. As with the discussion of three-dimensional inviscid flow analysis in the previous section, practical trade-offs with fully viscous flow analyses must be considered. Boundary layers in axial-flow compressors always involve significant threedimensional character. Yet, there is little merit to a fully three-dimensional boundary layer analysis today, when commercially available viscous CFD codes can treat the problem much more accurately. Rather, it is the simplified boundary layer analysis techniques that are most effective in augmenting the inviscid flow analyses discussed in the previous section. Indeed, it is little short of remarkable that boundary layer analysis has been used so effectively in axial-flow compressor design and analysis, considering the fact that the fundamental boundary layer approximations are almost always violated to some degree and very often to a substantial degree. There are two important types of boundary layer analysis commonly used in axial-flow compressor aerodynamics. These involve boundary layers on the blade surfaces and those on the compressor endwall contours. Blade surface boundary layers are of interest since they play a key role in viscous losses and stall or boundary layer separation. End-wall boundary layers are extremely important in performance analysis, since they can produce substantial viscous blockage effects that have significant impact on a compressor’s performance.



Two-dimensional boundary layer analysis is a useful approximation for blade surface boundary layers, particularly in blade design. The primary goals of blade design are really governed by viscous effects, namely minimizing viscous losses and avoiding or delaying flow separation. Two-dimensional boundary layer analysis provides at least a qualitative assessment of these effects, yet adds very little complexity or computational time to the overall analysis. Indeed, conventional practice for blade design is to design blade sections in the context of simple two-dimensional cascades, where two-dimensional boundary layer analysis is directly applicable. These blade sections are then “stacked” to create the actual three-dimensional compressor blade. When applied to the actual compressor blade surface boundary layers, two-dimensional boundary layer analysis provides only qualitative results, since three-dimensional effects that are not considered by the analysis often become significant. By contrast, end-wall boundary layers are necessarily three-dimensional, due to the presence of the swirl velocity component and tangential blade forces. Similar to inviscid flow analysis in the meridional plane, it is fairly common practice to conduct the boundary layer analysis for stations between blades—where an axisymmetric, three-dimensional boundary layer approximation can be used— while relying on empirical models to impose the blade row influence. This approach is a common basis for end-wall boundary layer blockage calculations for aerodynamic performance analysis (e.g., Balsa and Mellor, 1975). This axisymmetric three-dimensional boundary layer model has also been used within blade passages to provide an approximation to the boundary layer averaged between blade passages (Horlock, 1970, Aungier, 2000). Again, empirical models are required to model the influence of blade forces. The basic boundary layer equations relevant to these two types of boundary layers will be developed in the remainder of this chapter. Specific applications of these governing equations will be discussed, as required, in subsequent chapters to support the various inviscid flow analyses.

3.5 TWO-DIMENSIONAL BOUNDARY LAYER ANALYSIS Basic conservation of mass and momentum provide the governing equations for two-dimensional boundary layer flow over an adiabatic wall.

u

∂ρ bu ∂ρ bv + =0 ∂x ∂y

(3-31)

∂u ∂u 1 ∂P 1 ∂τ +v + = ∂x ∂y ρ ∂x ρ ∂y

(3-32)

where τ is the shear stress. The coordinates (x, y) and velocity components (u, v) are illustrated in Fig. 3-3, along with a typical boundary layer velocity profile. The stream sheet thickness, b, has been included in Eq. (3-31) since it is often a function of x in turbomachinery applications, i.e., streamlines often converge or diverge. This directly affects conservation of mass. The basic assumption of boundary layer theory is that the pressure is constant across the



FIGURE 3-3 Boundary Layer Nomenclature

boundary layer, i.e., P is a function of x only. Boundary layer analysis in turbomachinery is most conveniently accomplished by applying the governing equations in integral form. Equation (3-31) can be integrated across the boundary layer, using the Liebnitz rule to interchange the order of integration and differentiation, to yield δ

∂ ∂δ ∂ bρ udy = bρ eue [bρ eue (δ − δ * )] − bρ e ve = ∂x ∫ ∂x ∂x

(3-33)

0

The subscript e denotes inviscid flow conditions at the boundary layer edge, δ is the boundary layer thickness and δ* is called the displacement thickness or mass defect thickness, defined as δ

ρ eueδ * = ∫ [ρ eue − ρ u]dy

(3-34)

0

Equation (3-34) can be rewritten as δ

∫ ρ udy = ρeue [δ − δ

*

]

(3-35)

0



The displacement thickness is a fictitious thickness that can be used to correct the mass balance relative to the inviscid flow solution. As seen from Eq. (3-35), if the inviscid boundary layer edge or “free stream” conditions are applied within the boundary layer and the thickness δ* is assumed to have zero mass flow, mass conservation will be corrected for viscous effects. Equations (3-31) and (3-32) can be combined to express the momentum equation in conservation form. This yields 1 ∂bρ u2 ∂ρ uv ∂P ∂τ + + = b ∂x ∂y ∂x ∂y

(3-36)

Analogous to the displacement thickness, the momentum thickness or momentum defect thickness is defined as

ρ eue2θ = ∫ ρ u[ue − u]dy

(3-37)

Equations (3-35) and (3-37) combine to yield δ

∫ ρ u dy = ρeue [δ − δ 2

2

*

− θ]

(3-38)

0

If the free stream conditions are applied within the boundary layer with no flow in the thickness δ* and, in addition, no momentum in the thickness θ, momentum conservation will be corrected for viscous effects. Hence, if δ*and θ can be predicted, we have a simple method to correct the known inviscid free stream mass and momentum flux for viscous effects. This is really the basis of integral boundary layer analysis methods. Integrating Eq. (3-36) across the boundary layer, again using the Liebnitz rule, and noting that P = Pe is constant across the boundary layer, yields δ

∂ ∂δ ∂P bρ u2dy − ρ eue2 + ρ eue ve + δ e = −τ w ∂x ∂x ∫ ∂x

(3-39)

0

Combining Eqs. (3-33), (3-38) and (3-39) yields u ∂ 1 ∂ ∂P [bρ eue2 (δ − δ * − θ )] − e [bρ eue (δ − δ * )] + δ e = −τ w b ∂x b ∂x ∂x

(3-40)

Equation (3-40) can be rearranged to yield  ∂P 1 ∂bρ eue2θ ∂u ∂u  + δ * ρ eue e − τ w = δ  e + ρ eue e  b x ∂x ∂x ∂ ∂x  

(3-41)

By applying Eq. (3-32) at the boundary layer edge, where the gradients of u and τ in the y direction are zero, it is easily seen that the right-hand side of Eq. (3-41) is



identically zero. Hence, Eq. (3-41) reduces to the well-known momentum integral equation. 1 ∂bρ eue2θ ∂u + δ * ρ eue e = τ w b ∂x ∂x

(3-42)

In the special case of two-dimensional, axisymmetric flow, it can be noted that b is proportional to radius, r, and the momentum integral equation becomes 1 ∂rρ eue2θ ∂u + δ * ρ eue e = τ w r ∂x ∂x

(3-43)

The momentum integral equation is valid for both laminar and turbulent boundary layers. Laminar boundary layer analysis usually employs specific boundary layer flow profile assumptions to permit direct integration of the momentum integral equation. Turbulent boundary layer analysis usually employs several empirical models for solution, which may include specific boundary layer flow profile assumptions. Usually, turbulent boundary layer analysis employs a second conservation equation, such as conservation of mass, energy or moment of momentum (Rotta, 1966). This writer prefers conservation of mass as the second equation, commonly called the entrainment equation. In this case, Eq. (3-33) is written in the form ∂ [bρ eue (δ − δ * )] = bρ eue E ∂x

(3-44)

The parameter E is called the entrainment function, which specifies the rate at which free stream fluid is entrained into the boundary layer at the boundary layer edge. To employ this model, an empirical correlation for E is required, which must be derived from experiment. Combining Eqs. (3-33) and (3-44), it is seen that the entrainment function is given by E=

∂δ ve − ∂x ue

(3-45)

Indeed, entrainment is governed by the gradient of the shear stress at the boundary layer edge. Hence, entrainment should depend on the shape of the boundary layer profiles, which is the usual basis for empirical models.

3.6 AXISYMMETRIC THREE-DIMENSIONAL BOUNDARY LAYER ANALYSIS Equation (3-43) describes axisymmetric boundary layers where the flow field is two-dimensional, i.e., there is no tangential velocity component. When a



tangential velocity component is present in an otherwise axisymmetric flow field, the meridional and tangential components of the boundary layer profiles develop differently, resulting in a three-dimensional flow problem. The analysis of these problems is referred to as axisymmetric three-dimensional boundary layer analysis. This type of analysis is directly applicable to swirling flows in annular passages with no blades or vanes present, such as inlets and diffusers [e.g., Aungier 1988(b); Davis, 1976; Senoo et al., 1977]. This model has also been used effectively in axial-flow compressor performance analysis with application to stations between blade rows (e.g., Balsa and Mellor, 1975). When a blade row lies between successive computing stations, the axisymmetric assumption will have been violated within the blade passage. This requires use of empirical models to address the influence of the blade rows. Horlock (1970) also reports some success while applying this model within blade row passages. While these flows are far from axisymmetric, this model is used to provide an evaluation of the average or mean boundary layer behavior between the blades on the end-wall contours. Aungier (2000) uses this approach for quasi-threedimensional flow analysis, a practice also followed in this book. The governing equations for axisymmetric three-dimensional boundary layer flow in a rotating coordinate system in natural coordinates (θ, m, y) are 1 ∂ρ Wm ∂ρ Wy + =0 r ∂m ∂y Wm

(3-46)

∂Wm ∂Wm sin φ 1 ∂P ∂τ  + Wy − (Wθ + ω r )2 =  fm − e − m  r ∂m ∂y ρ ∂m ∂y 

(3-47)

∂Wθ ∂Wθ sin φ 1 ∂τ  Wm (Wθ + 2ω r ) =  fθ − θ  + Wy + r ∂m ∂y ρ ∂y 

(3-48)

Wm

A rotating coordinate system is needed for turbomachinery applications, since end-walls may be either rotating or stationary. The coordinate system should rotate with the end-wall to simplify imposing the boundary condition that all velocity components are identically zero at the wall. The terms fm and fθ in Eqs. (3-47) and (3-48) are body force terms used to account for blade forces acting on the flow. The blade forces at the boundary layer edge can be evaluated directly by applying Eqs. (3-47) and (3-48) to the inviscid flow at the boundary layer edge, where all inviscid free stream conditions are known and where Wn and the shear stress terms are identically zero. Hence, fme = ρ eWme fθe = ρ eWme

∂Wme ∂Pe sin φ + − ρ e (Wθe + ω r )2 ∂m ∂m r

∂Wθe sin φ ρ W ∂rCθe + ρ eWme (Wθe + 2ω r ) = e me ∂m r r ∂m

(3-49) (3-50)

The boundary layer equations are converted to integral form in the same fashion as described earlier. The algebra is more tedious and several additional defect thicknesses are required. The resulting integral equations are



∂ [rρ eWme (δ − δ1* )] = rρ eWe E ∂m

(3-51)

∂ ∂Wme 2 [rρ eWme θ11] + δ1* rρ eWme − ρ eWθe sin φ [Wθe (δ 2* + θ 22 ) + 2ω rδ 2* ] ∂m ∂m (3-52) = r[τ mw + fmeν m ]  ∂Wθe  ∂ 2 [r ρ eWmeWθeθ12 ] + rδ1* ρ eWme  r + sin φ (Wθe + 2rω ) m ∂m ∂  

(3-53)

= r 2 [τ θw + fθeνθ ]

The various mass, momentum and force defects used in these equations are defined as δ

ρ eWmeδ1* = ∫ ( ρ eWme − ρ Wm )dy

(3-54)

0

δ

2 ρ eWme θ11 = ∫ ρ Wm (Wme − Wm )dy

(3-55)

0

δ

ρ eWmeWθeθ12 = ∫ ρ Wm (Wθe − Wθ )dy

(3-56)

0

δ

ρ eWθeδ 2* = ∫ ( ρ eWθe − ρ Wθ )dy

(3-57)

0

δ

ρ eWθ2eθ 22 = ∫ ρ Wθ (Wθe − Wθ )dy

(3-58)

0

δ

ν m fme = ∫ ( fme − fm )dy

(3-59)

0

δ

νθ fθe = ∫ ( fθe − fθ )dy

(3-60)

0

The momentum integral equations can be simplified by assuming that the blade forces are constant through the boundary layer. This would certainly be consistent with the usual boundary layer approximations, but it is now accepted that this is often not true for end-wall boundary layers of axial-flow compressors. Mellor and Wood (1971) advanced compelling arguments for the existence of force defects. Smith (1970) and Hunter and Cumpsty (1982) measured force defects experimentally. For vaneless annular passages, these body force terms will normally vanish, but they can be used to advantage when boundary layers merge to form fully developed viscous flow profiles [Aungier, 1988(b)].



3.7 Vector Operators in Natural Coordinates The development of the governing equations in an axisymmetric natural coordinate system presented in this chapter requires use of several standard vector operators. Appendix A of Vavra (1960) provides detailed derivations of these operators. Here, the various operators are presented for reference purposes, without derivation. The gradient of any function φ is given by r ∂φ r ∂φ r 1 ∂φ r ∇φ = em + en + eθ ∂m ∂n r ∂θ

(3-61)

→

where → e is a unit vector. The divergence of any vector V is given by r 1  ∂rV ∂rVn ∂Vθ  m ∇ ⋅V =  + + + κ nVm + κ mVn r  ∂m ∂θ  ∂n

(3-62)

→

The curl of any vector V is given by r r  ∂V r ∂V ∇ × V =  n − m + κ nVn − κ mVm  eθ ∂ m ∂ n   1  ∂rVθ ∂Vn  r 1  ∂Vm ∂rVθ  r +  − em +  − en r  ∂n ∂θ  r  ∂θ ∂m 

(3-63)

The Laplacian of any function φ is given by ∇2φ =

1 ∂2φ 2

r ∂θ

2

+

∂2φ ∂m

2

+

∂2φ

 1 ∂r  ∂φ  1 ∂r  ∂φ + +κ n + +κm  r ∂m  ∂m  r ∂n  ∂n

∂n2

(3-64)

In evaluating the convective derivative in Eq (3-15), the following vector identity is useful. r r r (V ⋅ ∇)V =

r r r r r r ⋅ V ) − V × (∇ × V )

1 ∇(V 2

(3-65)

Equation (3-65) is usually written as r r r (V ⋅ ∇)V =

r

1 ∇V 2 2

r r r − V × (∇ × V )

(3-66)

→

where V is the magnitude of the vector V as given by the dot product.

EXERCISES 3.1 Consider time-steady flow in a rotating coordinate system. Use Eqs. (3-28) through (3-30) to analyze the flow at stations outside of the



3.2

3.3

3.4

3.5

3.6

blade row passages where the flow can be considered to be axisymmetric, but may have a tangential velocity component. Develop equations governing the variation of Cθ, s and I on stream surfaces. For the flow analysis in Exercise 3.1, modify Eq. (3-30) to consider cases where the relative flow angle, β′, is known for all stream surfaces, where tanβ′ = Wθ / Wm. Repeat for cases where the absolute flow angle, β, is known for all stream surfaces where tanβ = Cθ / Cm. For Eq. (3-66), express V in terms of its components in the three coor→ dinate directions. Derive an equation for –21 ∇V2 in terms of V and its derivatives. Consider one-dimensional, time-steady flow in a simple annular passage, i.e., κm = 0 and all gradients with respect to n and θ are identically zero. The passage width, b(m), is a function of the meridional coordinate. Derive a set of governing equations for this problem from Eqs. (3-21) through (3-27). Consider time-steady one-dimensional flow at the exit of a simple annular passage, with two identical boundary layers on the end-walls. The boundary layer parameters θ, δ* and δ and the inviscid core flow data ρ, u and P at the passage exit are known. The flow is incompressible, i.e., ρ is constant and Pt = P + 1/2ρu2. Develop expressions for the exit mass and momentum flow in terms of the boundary layer and inviscid core flow parameters. Assume that the boundary layer flow and inviscid core flow in Exercise 3.5 mix instantaneously into a uniform flow with no change in static pressure. By requiring conservation of mass and momentum, show that the total pressure loss between the inviscid core flow and the fully mixed flow is given by ∆Pt =

1 2

ρ eue2 [(2δ * / b)2 + 4θ / b]


Chapter 4

AXIAL-FLOW COMPRESSOR BLADE PROFILES

The traditional approach to axial-flow compressor aerodynamic design was to use various families of airfoils as the basis for blade design. American practice was based on various families defined by the National Advisory Committee for Aeronautics (NACA), the most popular being the 65-series family. British practice often centered about the C-series families, using circular-arc or parabolic-arc camberlines. As design requirements began to favor transonic operation, doublecircular-arc blades became popular. The performance characteristics of these airfoil families are well understood due to extensive experimental cascade testing, much of which is available in the literature. In recent years, use of blades designed for a prescribed surface velocity distribution or blade loading style, instead of for predefined airfoil families has become popular. Often, inverse design methods that predict the blade shape required for the desired blade loading are used. As the relation between blade shape and preferred loading styles became better understood, it also became common to use conventional or direct analysis methods in a trial-and-error mode to arrive at the same result. These airfoils have been referred to as prescribed velocity distribution (PDF) blades (Cumpsty, 1989), even though the term controlled diffusion airfoils is probably more common today. Although the literature offers general guidelines for these designs, the actual airfoil designs in use are proprietary. In general, the performance characteristics of these airfoils are well known only to the organizations that developed them. As discussed in the preface to this book, this situation posed a significant complication to the goal of providing a complete description of the working design and analysis system. It was quickly recognized that it is no longer possible to write a book that can be directly applied to all of the many proprietary designs in use today. But this is not considered to be a serious limitation. In this writer’s experience, the process of adapting classical blade performance prediction models to a more modern controlled diffusion airfoil design is not particularly difficult, assuming the performance characteristics of the airfoil are known. This chapter provides a complete description of the more commonly used traditional airfoil families, and Chapter 6 provides a detailed description of the performance modeling for these same airfoil families. This ensures that this book is at least complete in the context of classical axial-flow compressor technology.



General concepts from the literature used to guide the development of controlled diffusion airfoils are also briefly reviewed in this chapter.

NOMENCLATURE a = distance along chord to the point of maximum camber b = distance normal to the chord line to the point of maximum camber Cl0 = isolated airfoil lift coefficient c = chord length d = length defined in Eq. (4-24) i = incidence angle o = blade throat opening R = circular-arc radius of curvature s = blade pitch (spacing) tb = blade maximum thickness x = coordinate along the chord y = coordinate normal to the chord yC = y coordinate at the origin of RC α = angle of attack β = flow angle relative to axial direction χ = blade angle relative to the chord line δ = deviation angle γ = stagger (setting) angle κ = blade angle relative to the axial direction θ = camber angle σ = solidity φ = parameter defined in Eq. (4-30)

Subscripts C L U 1 2

= = = = =

camberline parameter blade lower or pressure surface parameter blade upper or suction surface parameter blade leading edge parameter blade trailing edge parameter

4.1 CASCADE NOMENCLATURE Figures 4-1 and 4-2 illustrate the basic parameters used to describe axial-flow compressor blades and cascades. Blades are defined by a mean camberline, y(x), upon which a profile or thickness distribution, tb(x), is imposed. The angles between slopes to the camberline and the chord line at the leading and the trailing edges are designated as χ1 and χ2, respectively. The blade camber angle is defined as

θ = χ1 + χ 2

(4-1)


Axial-Flow Compressor Blade Profiles • 61

FIGURE 4-1 Basic Airfoil Geometry

FIGURE 4-2 Basic Cascade Geometry



The pitch or the spacing between adjacent blades, s, and the chord length, c, define the cascade solidity, σ, by

σ = c/ s

(4-2)

The angle between the chord line and the axial direction is referred to as the stagger angle, or setting angle, γ. The angle between the inlet velocity vector, W1, and the chord line of the staggered blade is called the angle of attack, α. The flow angle with respect to the axial direction will be designated as β1. The angles between slopes to the camberline and the axial direction at the leading and the trailing edges will be designated as κ1 and κ2, respectively. Similarly, the flow angle at the blade trailing edge will be designated as β2. The flow incidence angle, i, the deviation angle, δ, and the angle of attack, α, are defined as i = β1 − κ1

(4-3)

δ = β2 − κ 2

(4-4)

α = β1 − γ

(4-5)

This nomenclature is directly applicable to blades based on well-defined camberlines such as the circular-arc and parabolic-arc camberlines typical of British practice. American practice has often been based on blades derived from NACA aircraft wing airfoils, which typically have infinite camberline slopes at the leading and trailing edges. In those cases, parameters such as κ, χ, θ, i and δ lose significance unless a suitable approximate or reference camberline is used to define them. Common practice has been to use an equivalent circular-arc camberline as a reference for the NACA 65-series blades (Johnsen and Bullock, 1965). This writer has made similar use of an equivalent parabolic-arc camberline for the NACA A4K6 63-series guide vanes (Dunavant, 1957), where the point of maximum camber is not at mid-chord. Construction of blades from the base camberline and profile has occasionally been a source of confusion. Profile thickness distributions are normally supplied for zero-camber blades as a function of distance along the chord, which also is the camberline for that case. When imposing a profile on a blade with camber, the thickness distribution data should be interpreted in terms of dimensionless distance along the camberline rather than along the chord line.

4.2 NACA 65-SERIES PROFILE The NACA 65-series blades are derived from NACA aircraft wing airfoils designed for approximately uniform loading. The original aircraft wing airfoil was not structurally suitable for the compressor cascade application. There are a number of adaptations of the original profile thickness distribution in use. The profile reported by Emery et al. (1958) is representative of the basic NACA 65-series cascade profile. Table 4-1 provides the thickness distribution from that reference. Even that profile has been structurally suspect due to its sharp trailing edge, as


Axial-Flow Compressor Blade Profiles • 63 Table 4-1 Dimensionless Data for Standard Axial-Flow Compressor Blade Types NACA 65-(10)10 Series x/c ~ % 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 RLE/c ~ % RTE/c ~ %

y/c ~ % 0 0.250 0.350 0.535 0.930 1.580 2.120 2.585 3.365 3.980 4.475 4.860 5.150 5.355 5.475 5.515 5.475 5.355 5.150 4.860 4.475 3.980 3.365 2.585 1.580 0 —— ——

tb/c ~ % 0 1.544 1.864 2.338 3.480 4.354 5.294 6.080 7.332 8.286 9.006 9.520 9.848 9.992 9.926 9.624 9.060 8.292 7.364 6.312 5.168 3.974 2.770 1.620 0.612 0 0.687 0

NACA A4K6 63 Series y/c ~% 0 0.376 —— 0.792 1.357 2.248 —— 3.531 4.420 5.040 5.458 5.710 5.824 5.820 5.713 5.516 5.239 4.891 4.479 4.011 3.492 2.922 2.308 1.642 0.912 0 —— ——

tb/c ~ % 0 —— —— 1.542 2.114 2.924 —— 4.020 4.772 5.312 5.682 5.908 6.000 5.942 5.754 5.446 5.034 4.602 4.170 3.740 3.308 2.876 2.444 2.014 1.582 0 0.297 0.600

C.4 Series tb/c ~ % 0 —— —— 3.30 4.54 6.16 7.24 8.04 9.10 9.66 —— 10.0 —— 9.78 —— 9.14 —— 8.10 —— 6.74 —— 5.08 —— 3.20 2.12 0 1.20 0.60

well as it being very thin toward the trailing edge. Kovach and Sandercock (1961) describe a more satisfactory modification for use in compressors. They use the basic distribution from Table 4-1 up to 60% of the chord. Then the thickness is varied linearly to match a trailing edge radius equal to 0.8% of chord. It is likely there are many other variants on the 65-series profile that are in use for reasons of structural integrity. The base profile has its maximum thickness at 40% of chord. NACA 65-series airfoils are designated by their lift coefficient and maximum thickness-to-chord ratio. The lift coefficient in tenths appears first in parentheses, followed by the thickness-to-chord ratio as a percentage. Hence a 10% thick airfoil with a lift coefficient of 1.5 is designated as NACA 65-(15)10. For lift coefficients less than one, the parentheses may be omitted. The base camberline is defined for a lift coefficient of 1.0 and is supplied in Table 4-1 (Emery et al., 1958). Simply multiply these coordinates by the lift coefficient to create other camberlines. Hence, the base airfoil for the NACA 65-series is the NACA 65-(10)10. Figure 4-3 illustrates this airfoil, using the base thickness of the NACA 65-010 from Table 4-1.



FIGURE 4-3 The NACA 65-(10)10 Airfoil

As noted, the slope of the NACA 65-series camberline becomes infinite at the leading and trailing edges. For this reason, experimental data from cascade testing are normally expressed in terms of angle of attack and fluid turning instead of incidence angle and deviation angle. It is now accepted practice to define effective inlet and discharge blade angles using an effective circular-arc camberline. The circulararc is defined as that which passes through the end points and the point of maximum camber at mid-chord (Johnsen and Bullock, 1965). Figure 4-4 shows a comparison of the equivalent circular-arc camberline with the NACA 65-(12) camberline. The construction of the circular-arc camberline reviewed in the next section can be used for that purpose. Figure 126 of Johnsen and Bullock (1965) provides a

FIGURE 4-4 Equivalent Circular-Arc Camberline



graphical relation between the effective camber angle and the lift coefficient. As will be seen in the next section, Clo and the equivalent θ can be related analytically by tan(θ / 4) = 0.1103 Clo

(4-6)

4.3 CIRCULAR-ARC CAMBERLINE The circular-arc camberline is commonly used in conjunction with the British C.4 series blade profile. It is also the camberline used for the double-circular-arc profile, and is reported to be used in place of the NACA 65-series camberline, when using the NACA 65-series profile (Cumpsty, 1989). As mentioned in the previous section, it is commonly used as an effective camberline for the NACA 65series blades to provide a meaningful definition of the leading and trailing edge blade angles. This camberline is completely defined by the camber angle, θ, and chord length, c. Figure 4-5 illustrates its construction. The radius of curvature, RC, of the camberline is given by

FIGURE 4-5 Circular-Arc Camberline Construction



c / 2 = RC sin(θ / 2)

(4-7)

The coordinates of the origin of the radius of curvature are (0, yC), where yC = − RC cos(θ / 2)

(4-8)

Then, for any x from – c / 2 to c / 2, y = yC + RC2 − x2

(4-9)

Using Eqs. (4-7) and (4-8), the camberline coordinate, y(0), at mid-chord can be expressed as 2 y(0) / c = [1 − cos(θ / 2)] / sin(θ / 2) = tan(θ / 4)

(4-10)

which is the basis for Eq. (4-6), where y(0) / c = 0.05515 is given in Table 4-1 and multiplied by Cl0 to obtain the value for any lift coefficient.

4.4 PARABOLIC-ARC CAMBERLINE The parabolic-arc camberline is also used with the British C.4 profile and can be used with other profiles as well. This writer has used it as an equivalent camberline to define effective blade angles for the NACA A4K6 63-series guide vane camberline. The parabolic-arc allows a more general blade loading style than the circular-arc. Front-loaded, mid-loaded and rear-loaded blades are all possible, depending on where the point of maximum camber is located. Figure 4-6 illustrates this blade style. The point of maximum camber is located at x = a and y = b, which provides the basic definition of the camberline. The basic constraints to be satisfied are y(0) = 0

(4-11)

y( c) = 0

(4-12)

y( a) = b

(4-13)

y'( a) = 0

(4-14)

The camberline is generated using the general second-order equation for a parabola that is given in many standard mathematics references. It can be written as Ax2 + 2 AE xy + By2 + Cx + Dy + E = 0

(4-15)

There appears to be a problem since we have five coefficients but only four constraints. However, one of the coefficients is arbitrary, e.g., we can divide through



FIGURE 4-6 The Parabolic-Arc Camberline

the equation by B and eliminate it. Hence Eqs. (4-11) through (4-14) are sufficient to determine all coefficients in Eq. (4-15). The algebra is very tedious, but the result can be shown to be x2 +

c − 2a c2 − 4ac ( c − 2a)2 2 xy + y − cx − y=0 2 b 4b 4b

(4-16)

Normally, specification of the blade camber angle, θ, or the blade angles, χ1 and χ2, is preferred. Differentiating Eq. (4-16) and evaluating the derivatives at x = 0 and x = c yields tanχ1 = 4b / (4a − c)

(4-17)

tanχ 2 = 4b / (3c − 4a)

(4-18)

Equations (4-1) and (4-16) through (4-18) can be combined to yield. b / c = { 1 + (4 tanθ )2 [a / c − ( a / c)2 − 3 / 16] − 1} / (4 tanθ )

(4-19)

This defines the parabolic-arc camberline in terms of camber and the ratio, a/c. For compressor blades, it is reasonable to restrict the leading and trailing edge angles to be less than 90°, i.e.,



FIGURE 4-7 The C.4 and 65-Series Profiles

0.25 < a / c < 0.75s

(4-20)

For any value of x, Eq. (4-16) can be solved for y as a standard quadratic equation. This approach will be singular for a/c = 0.5, since two of the terms drop out. But that special case is a simple and direct solution. Both cases can be treated by a numerical recursion equation of the form y = x( c − x) / [

( c − 2a)2 4 b2

y+

c − 2a c2 − 4ac x− ] 4b b

(4-21)

Simply start with y = 0 and repeatedly solve this equation to converge on the correct value of y.

4.5 BRITISH C.4 PROFILE The C.4 profile is one of several profiles in the British C-series (Howell, 1942). The C.4 series received the most attention in the literature relative to its performance characteristics (e.g., Johnsen and Bullock, 1965). Its base thickness distribution is tabulated in Table 4-1. Figure 4-7 shows an overlay of the C.4 and the NACA 65-series profiles. The C.4 profile is thicker toward the leading edge and has its maximum thickness at 30% of chord, compared to 40% for the 65-series profile. This would be expected to make it less effective for higher Mach number applications, but it would normally offer advantages relative to structural integrity. Similar comments apply to the trailing edge region. But, as noted, the 65-series profile is usually modified in that region. This profile is normally applied to circular-arc or parabolic-arc camberlines. It is reported that a later series, the C.7 profile, has seen more use in compressors and has many features in common with the 65-series profile (Cumpsty, 1989). C-series profiles are designated by a code giving the tb, profile, θ, camberline and a/c. Hence, a 10C4/20P40 blade is a 10% thick C.4 profile with a 20° camber angle using a parabolic-arc camberline with a/c = 0.4. A 10C4/20C50 blade would be similar, but with a circular-arc camberline. This writer has not had direct experience with this profile,



but it is well-established and empirical performance prediction models do exist for its application. On the other hand, there really is no reason why the 65-series profile cannot be applied to the circular-arc and parabolic-arc camberlines. That approach would be expected to yield better performance, particularly at higher Mach numbers.

4.6 DOUBLE-CIRCULAR-ARC PROFILE The double-circular-arc profile is constructed with both surfaces formed by circular-arcs, which blend with a nose radius, r0, applied at both the leading and trailing edges. Designate the lower and upper surface arc radii of curvature as RL and RU, respectively, as illustrated in Fig. 4-8. The construction of the camberline, with a radius of curvature, RC, has been described in Section 4.3. Here, construction of the upper surface of the profile will be illustrated. Construction of the lower surface is quite similar, except that certain parameters, such as tb and r0, are assigned negative values. The distance, ∆xU, from mid-chord to the center of the nose radius at the trailing edge is given by ∆xU = ( RU − r0 ) sin(θU / 2) = c / 2 − r0 cos(θ / 2)

(4-22)

where θ is the blade camber angle and θU is shown in Fig. 4-8. The distance, ∆yU, is given by

FIGURE 4-8 Double-Circular-Arc Profile



∆yU = RU − y(0) − tb / 2 + r0 sin(θ / 2) = RU − d

(4-23)

where the camberline coordinate, y(0), is given by Eq. (4-10) and d is defined as d = y(0) + tb / 2 − r0 sin(θ / 2)

(4-24)

The Pythagorean theorem applied to the right triangle in Fig. 4-8 requires [ RU − r0 ]2 = [ RU − d]2 + [c / 2 − r0 cos(θ / 2)]2

(4-25)

After some tedious algebra, this yields RU =

d2 − r02 + [c / 2 − r0 cos(θ / 2)]2 2( d − r0 )

(4-26)

The upper surface circular-arc extends through polar angles from -θU / 2 to θU / 2, constructed using the radius of curvature, RU, and the location of its origin at x = 0 and y = y(0) + tb / 2 – RU. The leading and trailing edge radii are constructed about their centers at y = r0 sin(θ / 2) and x = ± [c / 2 – r0 cos(θ / 2)] to blend with the circular-arc.

4.7 NACA A4K6 63-SERIES GUIDE VANE PROFILE Dunavant (1957) provides design and application data for a very effective vane profile for use as inlet guide vanes. This vane has excellent flow guidance and a wide incidence operating range. The camberline is developed by combining a frontloaded (A) profile with Cl0 = 0.4 and a uniform-loaded (K) profile with Cl0 = 0.6, which is designated as the A4K6 camberline corresponding to Cl0 = 1. This is combined with the 6% thick NACA 63-series profile as the base guide vane geometry. The base camberline coordinates and thickness distribution are listed in Table 4-1 and illustrated in Fig. 4-9. Similar to the 65-series blades, the camberline coordinates can be scaled directly by lift coefficient to alternate camberlines. Similarly, the thickness distribution can be scaled to other values from the base 6% thick profile. The general vane designation is 63-(Cl0 A4K6)nn, where nn is the maximum thickness as percent of chord. The maximum thickness of this vane is at 35% of chord. The location of the point of maximum camber can be estimated by interpolation to be at approximately a / c = 0.375 with b / c = 0.0583 for Cl0 = 1. As is the case with the NACA 65-series camberline, the leading and trailing edge camberline slopes are infinite. Here, an equivalent parabolic-arc camberline can be used to allow viable definitions of leading and trailing edge blade angles, incidence angle and deviation angle. Equation (4-19) can be solved for camber angle to yield tan θ =

b/c 2

2[a / c − ( a / c) − 3 / 16 − ( b / c)2 ]

(4-27)



FIGURE 4-9 NACA 63-(10A4K6)06

Hence, for the A4K6 camberline, tan θ =

291.5Cl0 468.75 − (5.83Cl0 )2

(4-28)

which yields θ = 33.8° at Cl0 = 1. A simpler approximation is obtained by generalizing the equivalent circular-arc camberline conversion of Eq. (4-6) by dividing the right-hand side by 2a / c to yield. tan(θ / 4) = 0.05515Cl0 / ( a / c)

(4-29)

which yields θ = 33.5° at Cl0 = 1. As long as only the NACA 65-series and A4K6 camberlines require conversion between θ and Cl0, this equation can be applied to either one. It also can be easily inverted for circular-arc and parabolic-arc camberlines to permit application of empirical blade performance correlations given as a function of Cl0 to those camberline types.

4.8 CONTROLLED-DIFFUSION AIRFOILS The standard blade profiles described in this chapter have been used extensively for axial-flow compressors. They are well understood, reliable and can yield excellent performance when properly applied. But, in recent years, many investigators have explored alternatives offering better Mach number range and higher efficiency. These are often referred to as controlled diffusion airfoils, since the design of the profiles is based on producing carefully controlled blade surface Mach number distributions. Hobbs and Weingold (1984) and Dunker et al. (1984) have reviewed the basic design strategy. They indicate that the key features are: • A continuous acceleration along the suction surface and near the leading edge to avoid laminar boundary layer separation or premature separation.



• The peak Mach number should not exceed 1.3 to avoid shock-waveinduced separation. • Carefully controlled deceleration along the suction surface from the peak Mach number to avoid turbulent boundary separation ahead of the trailing edge. • A nearly constant subsonic Mach number distribution on the pressure surface. Figure 4-10 is a qualitative schematic of the type of Mach number distribution employed. A key feature is to avoid shock wave–boundary layer interaction, such that boundary layer analysis can be used effectively in establishing the desired controlled diffusion characteristic along the suction surface. Initially, inverse techniques, which compute blade geometry from specified Mach number distributions, were used. Once the basic concept was clarified, it became possible to use direct methods to iteratively refine the geometry to achieve the desired aerodynamic characteristics. Typical controlled diffusion profiles (Hobbs and Weingold, 1984) appear to be more robust than the 65-series in the forward portion of the profile, followed by a relatively thin aft region of almost constant thickness. But that is not necessarily a general conclusion. This is really a design concept to produce specific, proprietary profile designs. It has not resulted in a standard airfoil family that can be employed and analyzed in a general sense. Some standardization is clearly possible as evidenced by the fact that controlled diffusion

FIGURE 4-10 Controlled Diffusion Airfoil



airfoils have found favor for industrial axial-flow compressors, where unique profile development for each compressor is not economically feasible. But to employ this concept, specific profiles of the desired characteristics must be at least initially designed. And performance prediction models must also be established, since those used for standard profiles are unlikely to be adequate.

4.9 BLADE THROAT OPENING The blade throat opening, o, is the minimum distance between adjacent blades, as illustrated in Fig. 4-11. It is an important parameter when conducting an aerodynamic performance analysis. The throat opening governs the onset of local flow choking within the blade passage. At sufficiently high inlet Mach number levels, this will define the maximum flow capacity that the compressor can pass. The best approach to determine the throat opening is to define adjacent blades using the stagger angle, camberline and profile coordinates to locate the minimum distance between the blades. For that purpose, this writer uses a computer database containing the information in Table 4-1. Then, before an actual performance analysis, the throat openings along the span of all blade rows are computed by a simple trial-and-error process. Typically, about 50 points are distributed along both blade surfaces and the minimum distance is easily located. This process is carried out in the context of a two-dimensional cascade with the correct solidity. The ratio of throat-opening-to-pitch, o / s, is computed. Conformal mapping shows that this ratio is applicable to the annular cascade of a compressor (e.g., Aungier, 2000, Chapter 7). There is no need to attempt something more sophisticated, since this is a rather trivial problem for a computer.

FIGURE 4-11 Throat Geometry



Plus, these data are added to the basic problem input file, so throat openings are only computed once for each compressor to be analyzed. The same process can be used for any blade type, such as the controlled diffusion airfoils discussed in the previous section. It is only necessary to have the camberline and profile coordinates available from a database. It is also possible to approximate throat openings with reasonable accuracy using an empirical correlation. This writer developed a throat-opening correlation for NACA 65-series blades from the carpet plots of Dunavant et al. (1955). A modified stagger angle parameter, φ, is defined as

φ = γ (1 − 0.05C1l0.5 ) + 5C1l0.5 − 2

(4-30)

and the ratio of throat opening-to-pitch, o / s, is estimated from o / s = [(1 − tb σ / c) cos φ ]

σ

(4-31)

Results from this correlation are compared to actual values of o/s for typical NACA 65-series blades in Fig. 4-12. This correlation has been found to be reasonably accurate for other blade types also. As a somewhat extreme example, it is compared to actual throat openings for a C4-series profile that is imposed on a parabolic-arc camberline with a/c = 0.4 in Fig. 4-13. Equation (4-29) was used to

FIGURE 4-12 Throat-Opening Correlation



FIGURE 4-13 Extended Throat-Opening Correlation

relate lift coefficient and camber angle for this comparison. While both the profile and camberline are quite different from the NACA 65-series blades for which the correlation was developed, accuracy is still quite good except at very low stagger angles in combination with high camber angles. That small region of inaccuracy is unlikely to be encountered in a real compressor application. Nevertheless, it is prudent to use the actual throat openings for performance analysis applications. This removes the uncertainty as to applicability of the empirical correlation to the specific blades used. It also ensures accurate treatment of special blade types for which the correlation is not likely to apply. A common example is an inlet guide vane row, where the blade throat will normally be located at the trailing edge, similar to a turbine blade row. A compressor blade throat correlation is unlikely to handle this situation accurately. While inlet guide vane choking is not common, it could be a factor if stagger angles are set too high. This could produce an unexpected limit on the compressor’s flow capacity unless the problem is handled correctly.

4.10 STAGGERED BLADE GEOMETRY Once the camberline and profile coordinates for any of the proceeding blade types have been generated along the chord, the geometry of the staggered blade in the cascade is obtained by a simple rotation of coordinates to the stagger angle, γ. The staggered blade inlet and discharge angles are given by



κ1 = χ1 + γ κ 2 = γ − χ2

(4-32) (4-33)

From Eqs. (4-1), (4-32) and (4-33), it follows that

θ = κ1 − κ 2

(4-34)

For the circular-arc or the NACA 65-series equivalent circular-arc approximation, it is easily shown that

χ1 = χ 2 = θ / 2

(4-35)

γ = (κ1 + κ 2 ) / 2

(4-36)

For the parabolic-arc or equivalent parabolic-arc camberlines, Eqs. (4-17) through (4-19) can be used to compute χ1 and χ2 as a function of θ and a/c for use in Eqs. (4-32) through (4-34). Indeed, it is easily shown that the same approach can be used for circular-arc and equivalent circular-arc camberlines, since Eqs. (4-17) through (4-19) are equivalent to Eqs. (4-35) and (4-36) when a/c = 0.5.


Chapter 5

TWO-DIMENSIONAL BLADE-TO-BLADE FLOW THROUGH CASCADES OF BLADES

Prediction of the flow through cascades of blades is fundamental to all aspects of axial-flow compressor aerodynamic design and analysis. Although the flow through the annular cascades of blades in a compressor is really a three-dimensional flow problem, there are many advantages to considering the simpler problem of two-dimensional flow in cascades. It offers a very natural view of cascade fluid dynamics to make it easier for designers to develop an understanding of the basic flow processes involved. Indeed, very simple two-dimensional cascade flow models were used in this educational role long before computational methods and computers had evolved enough to produce useful design results. Today, blade-to-blade flow analysis is a useful design and analysis tool that provides reasonable approximations to many problems of interest. Inviscid blade-to-blade flow analysis addresses the general problem of two-dimensional flow on a stream surface in an annular cascade, as discussed in Chapter 3. Two-dimensional boundary layer analysis can be used to approximate viscous effects. Although very useful, this approach does have limitations. It ignores secondary flows that develop from the migration of low momentum boundary layer fluid across the stream sheet. It also loses accuracy when significant flow separation is present. This chapter considers theoretical methods to model two-dimensional blade-toblade flow. The next chapter considers a different approach to the problem, where empirical models are derived from two-dimensional cascade test data. The methods presented in this chapter are basically the same as those presented in Aungier (2000), but are adapted to conventions more commonly used for axialflow compressors. Aungier (2000) provides detailed guidance relative to implementation of the methods in numerical analyses that is not repeated here. Readers interested in those details should consult the original reference.

NOMENCLATURE A = area and a function used in Eq. (5-22) a = sound speed and a parameter in Eq. (5-52)



B b C cf cp D E → n H Hk H1 h I K M m ˙ m → r P Q Reθ r S T t u V v W x y z α β γ δ δ′ δ* δE δh δu η θ Λ µ ξ ρ Φ

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

function in Eq. (5-23) stream sheet thickness absolute velocity and a function in Eq. (5-24) skin friction coefficient specific heat at constant pressure function in Eq. (5-25) function in Eq. (5-26) and the entrainment function a unit vector total enthalpy and shape factor of Eq. (5-129) kinematic shape factor, Eq. (5-130) shape factor, Eq. (5-128) enthalpy rothalpy shape factor of Eq. (5-122) Mach number = C/a meridional coordinate mass flow rate unit vector normal to an area element pressure velocity component of Eq. (5-76) momentum thickness Reynolds number radius blade pitch = r (θ1 – θ0) temperature time general function volume general function relative velocity distance along a wall distance normal to a wall axial coordinate angle of attack = βin – γ flow angle and coordinate angle (Fig 5-4) blade stagger angle boundary layer thickness density thickness displacement thickness energy thickness enthalpy thickness velocity thickness dimensionless tangential coordinate (Fig. 5-3) polar angle and momentum thickness shape factor of Eq. (5-113) stabilizing term and viscosity streamwise coordinate (Fig. 5-3) density stabilizing term


Two-Dimensional Blade-to-Blade Flow Through Cascades of Blades • 79

φ ψ τw ω – ω

= = = = =

stream surface slope angle with axis stream function wall shear stress rotation speed total pressure loss coefficient

Subscripts in m out q t θ ξ 0 1

= = = = = = = = =

cascade inlet condition meridional component cascade discharge condition component normal to ξ coordinate total thermodynamic condition tangential component component tangent to ξ coordinate parameter on blade surface θ0 (Fig. 5-3) parameter on blade surface θ1 (Fig. 5-3)

Superscripts (η) (ξ) ′ ″

= = = =

relative to the η direction relative to the ξ direction relative condition and first derivative second derivative

5.1 THE BLADE-TO-BLADE FLOW PROBLEM Figures 5-1 and 5-2 illustrate the basic problem to be considered. Figure 5-1 shows a schematic of a streamline pattern on a stream sheet between adjacent blades in a cascade of blades. It will be sufficient to consider a single passage between two blades, since the flow in all blade passages will be assumed to be identical. As discussed in Chapter 3, a stream sheet is a thin annular passage bounded by two stream surfaces, where a stream surface has no velocity component normal to it, i.e., it has no mass flow across it. A schematic of a stream sheet that might be used is illustrated in Fig. 5-2. Stream sheets will be assumed to be axisymmetric. Although that is not a necessary assumption, it does greatly simplify the problem. This chapter considers the stream sheet geometry to be known. In later chapters, techniques to define the stream sheet geometry to support these blade-to-blade flow methods will be covered, including generalization of the process into a quasi-three-dimensional flow analysis. The approach to be used is to generate a time-steady inviscid flow analysis followed by a blade surface boundary layer analysis to approximate viscous effects. It is assumed that the rothalpy and entropy are both constant at the upstream boundary. From the time-steady form of Eqs. (3-25), (3-28) and (3-29), it can be



FIGURE 5-1 Blade-to-Blade Plane Flow

FIGURE 5-2 Blade-to-Blade Stream Sheet



seen that these assumptions require rothalpy and entropy to be constant over the entire stream sheet. The steady form of Eqs. (3-28) and (3-29) governs conservation of momentum in the stream sheet. It is seen that these two equations are identical for the present problem. Indeed, from Eqs. (3-1), (3-29) and (3-63) it is easily shown that the component of the absolute vorticity normal to the stream sheet must be zero. This reduces the problem to a classical potential flow problem governed by conservation of mass, Eq. (3-21), and the irrotational flow condition, Eq. (3-29). But a major complication encountered in blade-to-blade flow analysis arises from the mathematical character of the governing equations of motion for timesteady inviscid flow. When the flow is everywhere subsonic (W < a), the governing equations are elliptic in form. This presents a classical boundary value problem, where the solution is completely determined by conditions imposed on the boundaries of the solution domain. But when the flow is supersonic (W > a), the governing equations are hyperbolic in form. This type of problem requires some type of marching solution, such as the method of characteristics. When supersonic flow is present, there is usually subsonic flow present in the solution domain. The mixed subsonic-supersonic flow problem requires two solution techniques that must be matched in some fashion. It is now common practice to consider time-unsteady flow in those cases, since the governing equations for unsteady inviscid flow are hyperbolic for both subsonic and supersonic flow. This solution technique is commonly called the time-marching or time-dependent technique. The solution is simply advanced in time until it has reached essentially a steady flow prediction. When the time-marching approach is used, the simplifications that lead to the potential flow model are no longer present. For example, Eq. (3-25) must be solved, since rothalpy can no longer be treated as a constant on the stream sheet. Nor are the two momentum equations for the flow in the stream sheet identical for the time-unsteady case. Thus, a potential flow analysis can be used for subsonic flow, but a more general analysis will be needed for a mixed subsonic-supersonic flow case. These more general solutions are commonly referred to as Euler techniques, which include the time-marching method. The basic characteristic of an Euler method is that conservation of mass, energy and all relevant momentum equations are solved without simplification, such as assuming isentropic flow.

5.2 COORDINATE SYSTEM AND VELOCITY COMPONENTS Figure 5-3 illustrates the solution domain to be considered for the blade-to-blade flow solution. It is convenient to use a coordinate transformation to define new coordinates (ξ, η) such that the blade surfaces correspond to lines of constant η. The new coordinates are given by m

ξ=

dm

∫ cos β

(5-1)

0

η = [θ − θ0 ] / [θ1 − θ0 ]

(5-2)



FIGURE 5-3 Blade-to-Blade Solution Domain

This transforms the complex solution domain of Fig. 5-3 into a simple rectangular domain in (ξ, η) space where η varies from 0 to 1. The blade surface tangential coordinates, θ0 and θ1, are illustrated in Fig. 5-3, and β is the angle between a tangent to a constant η line and the meridional direction, i.e.,  r∂θ  tan β =   = tan β0 + [tan β1 − tan β0 ] η  ∂m η

(5-3)

Outside the blade passage, θ0 and θ1 are somewhat arbitrary, except that θ1 = θ0 + 2π / N, where N is the number of blades. Figure 5-4 shows an expanded view of a basic control volume within the solution domain from Fig. 5-3. It also illustrates special velocity components useful for developing the governing equations in (ξ, η) space. Wξ and Wq are simply the velocity components parallel to and normal to a constant η line, respectively. The following equations relate these velocity components to the usual Wm and Wθ velocity components. Wξ = Wm cos β + Wθ sin β

(5-4)

Wq = Wθ cos β − Wm sin β

(5-5)

Wm = Wξ cos β − Wq sin β

(5-6)

Wθ = Wq cos β + Wξ sin β

(5-7)



FIGURE 5-4 A Control Volume

5.3 POTENTIAL FLOW IN THE BLADE-TO-BLADE PLANE As discussed previously, potential flow analysis on a blade-to-blade stream surface involves solution under the assumption that the absolute flow is time-steady, inviscid and adiabatic, with rothalpy and entropy both constant on this surface. No restrictions are imposed relative to variation of rothalpy and entropy normal to the stream surface. In general, rothalpy and entropy will vary from stream surface to stream surface, which is quite compatible with the present analysis. The governing equations are conservation of mass, Eq. (3-21), and the irrotationality condition, Eq. (3-29). The steady form of these equations could be transformed directly into the (ξ, η) space by developing transformations for the derivatives from Eqs. (5-1) and (5-2). But a more accurate numerical analysis is obtained by a more fundamental development using the basic control volume shown in Fig. 5-4. Conservation of mass for this control volume can be stated as  ρ bW    ρ bWq  q  2∆m  −    cos β   cos β  m,η + ∆η  m , η − ∆ η   +2∆η ( Sρ bWm )m − ∆m,η − ( Sρ bWm )m + ∆m,η = 0

[

]

(5-8)

where the subscripts identify specific grid points on the control volume of Fig. 5-4, and the tangential spacing, S(m), is defined by



S = r (θ1 − θ0 )

(5-9)

Taking the limit as ∆m and ∆η approach zero, Eq. (5-8) reduces to the following continuity equation. ∂  ρ bWq  ∂( Sρ bWm ) =0  + ∂η  cos β  ∂m

(5-10)

Developed in this fashion, it can be noted that the continuity equation contains coordinates m and η and velocity components Wq and Wm from two different coordinate systems. While perhaps a little unusual, it does result in a more precise statement of conservation of mass for use in a numerical analysis. Numerical approximations to the governing equations will apply them to a finite control volume. More accurate numerical approximations will result from using the control volume approach to develop the equations instead of using a mathematical transformation of the derivatives in Eq. (3-21). The condition for irrotational absolute flow in the stream surface requires that the component of the absolute vorticity normal to the stream sheet be zero, i.e., r r r r r r r en ⋅ (∇ × C ) = en ⋅ [∇ × (W + rω eθ )] = 0

(5-11)

Stokes theorem is a convenient method to impose this condition on the control volume. Stokes theorem relates the line integral of the velocity about any closed path to the integral of the normal component vorticity over the area enclosed by the path. In the rotating coordinate system, Stokes theorem can be applied as r

r

r

r

r

r

r

r

∫ W ⋅ dr = ∫ [en ⋅ (∇ × W )da = − ∫ en ⋅ (∇ × rω eθ )da

C

A

(5-12)

A

When applied to the control volume, this yields  W    Wξ  ξ  2∆m  −    cos β   cos β  m,η + ∆η  η η m , − ∆  

[

+2∆η ( SWθ )m − ∆m,η − ( SWθ )m + ∆m,η

]

S ∂r 2ω = 4∆η ∆m r ∂r

(5-13)

Taking the limit as ∆m and ∆η approach zero, Eq. (5-13) reduces to ∂  Wξ  ∂( SWθ ) + 2Sω sin φ  = ∂η  cos β  ∂m

(5-14)

where φ is the stream sheet angle with the axial direction as shown in Fig. 5-2 Hence,



sin φ =

∂r ∂m

(5-15)

A stream function, ψ, is defined by ˙ m

∂ψ = − ρ b(Wθ − Wm tan β ) ∂m

(5-16)

∂ψ = Sρ bWm ∂η

(5-17)

˙ m

· where m is the stream sheet mass flow rate. Hence, the velocity components are given by Wm = Wθ =

˙ m bρ

˙ ∂ψ m Sbρ ∂η

 tan β ∂ψ ∂ψ   S ∂η − ∂m   

(5-18) (5-19)

By substituting Eqs. (5-5), (5-18) and (5-19) into Eq. (5-10), it is easily shown that the definition of ψ identically satisfies the continuity equation. Note that ψ varies from 0 to 1 as θ varies from θ0 to θ1 or as η varies from 0 to 1. Thus, both conservation of mass and energy are satisfied, requiring solution of the irrotationality condition only to predict the inviscid flow field. Introducing the stream function into Eq. (5-14) yields the required equation. ˙ (1 + tan 2 β ) ∂ψ m ˙ tan β ∂ψ  ∂ m −   Sbρ bρ ∂m  ∂η  ∂η

=

˙ tan β ∂ψ mS ˙ ∂ψ  ∂ m −   + 2Sω sin φ ∂m  bρ ∂η bρ ∂m 

(5-20)

This equation can be simplified to the form A

∂2ψ ∂η 2

− 2B

∂2ψ ∂2ψ ∂ψ ∂ψ = 2Sω sin φ +C +D +E 2 ∂η∂m ∂η ∂m ∂m

(5-21)

where ˙ / ( Sbρ cos 2β ) A(m, η) = m

(5-22)

˙ tan β / ( bρ ) B(m, η) = m

(5-23)

˙ / ( bρ ) C(m, η) = mS

(5-24)



D(m, η) =

∂A ∂B − ∂η ∂m

(5-25)

E(m, η) =

∂C ∂B − ∂m ∂η

(5-26)

Boundary conditions for ψ(m, η) on the solution domain shown in Fig. 5-3 are reasonably straightforward. On the blade surfaces,

ψ (m, 0) = 0

(5-27)

ψ (m,1) = 1

(5-28)

For the side boundaries outside of the blade passage, the periodicity condition is used. Since the flow is identical in all blade passages, the flow field must repeat in the tangential direction with a period of ∆η = 1. The periodic nature of the flow can be used to extend the solution into adjacent passages such that points on these side boundaries can be treated the same as interior points in the solution domain.

ψ (m, η + 1) = ψ (m, η) + 1

(5-29)

ρ(m, η + 1) = ρ(m, η)

(5-30)

Wm (m, η + 1) = Wm (m, η)

(5-31)

Wθ (m, η + 1) = Wθ (m, η)

(5-32)

Conditions on the upstream and downstream boundaries may be assigned as uniform flow, basically requiring that ψ vary linearly in the tangential direction. But reliability of the numerical analysis is improved if a less stringent boundary condition is imposed. A good choice is to require that the flow angle be constant on these boundaries. Then, if the geometry of the side boundaries is defined such that β equals the local flow angle on the upstream and downstream boundaries, the appropriate upstream and downstream boundary condition is ∂ψ ∂ψ = cos β =0 ∂ξ ∂m

(5-33)

as can be seen from Eq. (5-16). Typically, β will be assigned to vary uniformly along the side boundary from the upstream boundary flow angle to the blade leading edge blade angle, and analogously for the downstream boundary. Once this distribution of β along the side boundaries is specified, simple integration of Eq. (5-3) yields θ0(m) and θ1(m), noting that β0 = β1 outside the blade passage. A potential flow analysis can be accomplished for virtually any specified flow angles at the upstream and downstream boundary. But, in reality, these two flow angles are not independent. A prediction of the downstream flow angle for any



upstream flow conditions is to be preferred. To accomplish this, some additional constraint is required. Typically, the well-known Kutta condition is applied at the trailing edge. This simply requires that the pressures on the two sides of the blades must be equal at the trailing edge. This follows from the fact that there is no longer a tangential blade force to sustain a pressure difference. It is equivalent to imposing the following trailing edge condition. [W (m, 0)]te = [W (m,1)]te

(5-34)

Typically, iterative adjustment of the discharge flow angle is accomplished as part of the solution process until a value that satisfies Eq. (5-34) is found. Equation (5-21) is solved subject to the above boundary conditions while treating the gas density field throughout the solution domain as constant, starting with an initial guess for the density field. After the stream function is predicted, Eqs. (5-18) and (5-19) are used to compute the velocity field. Then, the density field is updated using an appropriate equation of state from Chapter 2. The stream function is then recalculated, and the process continued until convergence is achieved. A grid structure is developed over the solution domain, as illustrated in Fig. 5-3. The spacing between nodes ∆m and ∆η is constant in each direction. Unequal spacing can be used, but this writer’s experience has shown that the benefits of unequal spacing do not justify the added complexity and reduced computational speed. Equation (5-21) is reduced to linear form using standard finite–difference approximations for the derivatives. If subscripts i and j are used to designate the ith meridional node and the jth tangential node, the relevant finite-difference approximations for interior points are ∂ψ ψ i +1, j − ψ i −1, j = 2∆m ∂m ∂ψ ψ i , j +1 − ψ i , j −1 = 2∆η ∂η 2

∂ψ ∂m2 2

∂ψ ∂η 2

= =

(5-35) (5-36)

ψ i +1, j − 2ψ i , j + ψ i −1, j ( ∆m)2

(5-37)

ψ i , j +1 − 2ψ i , j + ψ i , j −1 ( ∆η)2

ψ i +1, j +1 − ψ i −1, j +1 − ψ i +1, j −1 + ψ i −1, j −1 ∂ψ = 4∆m∆η ∂m∂η

(5-38)

2

(5-39)

Equations (5-35) through (5-39) are easily derived using truncated Taylor series expansions of ψ as functions of m and η (Aungier, 2000). The terms D and E in Eqs. (5-25) and (5-26) require approximations for the first derivatives at nodes on the boundary. These can be developed in the same way as for interior points. They are



∂ψ 4ψ i +1, j − 3ψ i , j − ψ i + 2 , j = 2∆m ∂m

(5-40)

∂ψ 4ψ i , j +1 − 3ψ i , j − ψ i , j + 2 = 2∆η ∂η

(5-41)

∂ψ 3ψ i , j − 4ψ i −1, j + ψ i − 2 , j = 2∆m ∂m

(5-42)

∂ψ 3ψ i , j − 4ψ i , j −1 + ψ i , j − 2 = 2∆η ∂η

(5-43)

Substitution of the finite-difference approximations into Eq. (5-21) for any interior node (i, j) yields a simple linear equation for ψ

ψ i , j + A˜ i , jψ i −1, j + B˜ i , jψ i +1, j + C˜ i , jψ i , j −1 + D˜ i , jψ i , j +1 + E˜ i , j [ψ i +1, j +1 − ψ i +1, j −1 − ψ i −1. j +1 + ψ i −1, j −1] = Q˜ i , j

(5-44)

The coefficients in Eq. (5-44) are given by  Ci , j Ei , j  A˜ i , j = −  −  2 2 ∆m   ( ∆m)

 2Ai , j 2Ci , j  +   2 ( ∆m)2   ( ∆η)

(5-45)

 Ci , j Ei , j  B˜ i , j = −  +  2 2∆m   ( ∆m)

 2Ai , j 2Ci , j  +   2 ( ∆m)2   ( ∆η)

(5-46)

 Ai , j Di , j  C˜ i , j = −  −  2 2 ∆η  η ( ) ∆ 

 2Ai , j 2Ci , j  +   2 η ( ) ( ∆ ∆m)2  

(5-47)

 Ai , j Di , j  D˜ i , j = −  +  2 2∆η   ( ∆η)

 2Ai , j 2Ci , j  +   2 ( ∆m)2   ( ∆η)

(5-48)

 2Ai , j 2Ci , j  +   2 ( ∆m)2   ( ∆η)

 Bi , j  E˜ i , j =    2∆m∆η 

[

Q˜ i , j = − 2Sω sin φ

]

 2Ai , j 2Ci , j  +   2 ( ∆m)2   ( ∆η)

(5-49)

(5-50)

Numerical solution of the linearized stream function equation can be accomplished with a relaxation technique (Katsanis 1968, 1969) or a matrix method (Smith and Frost, 1969; Aungier, 2000). This writer has used both methods and has found the matrix method to be superior in both computational speed and reliability. In the matrix method, Eq. (5-44) is used to develop equations for all nodes, resulting in a matrix equation, where the major matrix is a square matrix with the number of rows and columns equal to the number of nodes. It is a very sparse matrix, having non-zero values only in a band about its diagonal, so that a



rather small number of values actually need to be stored in the computer’s memory. Aungier (2000) provides a rather detailed description of a very efficient matrix solution for this problem. After each solution for the stream function, the density field must be updated using a new velocity field estimate from Eqs. (5-18) and (5-19). Since rothalpy is constant on the stream sheet, Eqs. (3-11) and (3-13) require h = H ′ − 12 W 2 = I + 12 ( rω )2 − 12 W 2

(5-51)

And, since entropy is also constant on the stream sheet, all thermodynamic properties can be calculated as a function of (h, s), using an appropriate equation of state from Chapter 2. While conducting iterations to converge on a density field, convergence on a discharge flow angle to satisfy the Kutta condition, Eq. (5-34), is also accomplished. As long as the flow is subsonic, this procedure of lagging the density solution one iteration behind the stream function solution offers excellent numerical stability and rapid convergence. Once velocities greater than sonic velocity are encountered, the solution will become unstable, and will almost always diverge. Blade-to-blade flow problems involving local patches of supersonic flow are often encountered in axial-flow compressor analysis. The usefulness of a potential flow analysis is greatly increased if it is extended to be capable of addressing these transonic flow cases. This can be accomplished by readjusting the mass flow rate and speed, or the inlet total sound speed, such that the inlet velocity triangle is the same as the actual problem but all velocities are subsonic. After solving the subsonic flow problem, the streamline pattern is assumed to be correct, and some type of streamline curvature numerical technique can be used to calculate the flow for the actual inlet conditions. Katsanis (1969) is a good example of this type of extension of a potential flow solution. The weakness in this approach is that the resulting flow field will no longer satisfy the irrotationality condition. Aungier (2000) presents a better technique, which can predict an irrotational flow field that conserves mass and involves local patches of supersonic flow. The inlet total sound speed is adjusted to reduce all velocities to subsonic values. At the same time, the stream sheet thickness distribution is adjusted such that the resident velocity field, which conserves mass for the fictitious subsonic problem, will also conserve mass for the actual transonic flow problem when the actual inlet conditions and steam sheet thickness distribution are used. Thus, when the subsonic flow solution is obtained, it is only necessary to accomplish a final update of all thermodynamic data with the predicted velocity field to satisfy all governing equations for the transonic flow case. Implementation of this extension to the analysis is straightforward, but depends to some degree on the equation of state used in the analysis. Aungier (2000) provides an illustration of its implementation for the special case of a calorically and thermally perfect gas. The stability of a numerical solution of the blade-to-blade flow problem can be significantly influenced by the manner in which the grid structure is established near the leading and trailing edges of the blades. Figure 5-5 illustrates two methods of locating the grid near the leading edge. In one case, the first nodes on the blade surface are outside the passage, touching the blade at a single point. The other case locates the first nodes on the blade surfaces inside the passage with a



FIGURE 5-5 Leading Edge Grid Structure

node on each blade surface. Experience has shown that the first method can result in local numerical instability, while the second method almost never experiences that problem. The cause of instability has been traced to the behavior of the stagnation streamline, coupled with the finite-difference approximations for derivatives with respect to η used at blade surface points. In the illustration in Fig. 5-5, it is seen that the stagnation streamline passes between the blade surface and the node next to the surface when the leading edge nodes are outside the passage. Since ψ has the same value on the stagnation streamline and the blade surface, a difference approximation to the tangential derivative will provide a poor estimate. And, without special logic in the solution, it is not obvious which direction the tangential difference approximation for the node on the blade should use, i.e., Eq. (5-41) or (5-43). In the case illustrated, nodes to the right of the leading edge node should be used, i.e., Eq. (5-41). Minor changes in gas density at the leading edge can induce the stagnation streamline to move, possibly even making it shift to the opposite blade surface. It is not uncommon for this to result in an oscillation on successive iterations, all occurring very local to this region, such that a converged solution is never realized. Numerical damping, refining the grid near the leading edge, etc., can alleviate this problem. But a simpler and more effective approach is to move the leading edge nodes into the passage as illustrated in Fig. 5-5. This removes the ambiguity regarding the direction to be used for the surface derivatives, and the local oscillation problem almost never occurs when this is done. It is a very simple method to avoid the tendency toward local instability near the leading and trailing edges.



Figure 5-6 illustrates typical results from this potential flow solution. Predicted blade surface Mach numbers are compared to experimental results reported by Dunavant et al. (1955). For the purpose of a blade loading diagram comparison, the experimental surface pressure coefficient data were converted to Mach numbers using standard compressible flow relations. This is really the same example used in Aungier (2000), but the analysis is now capable of a more precise treatment of the blade geometry. Basically, the methods described in Chapter 4 are now included in the analysis to very precisely define the blade geometry for standard axial-flow compressor blade camberlines and profiles. Overall, rather good agreement is achieved between predictions and experiment. Near the trailing edge, the boundary layer analysis described later in this chapter predicts boundary layer separation as noted in Fig. 5-6. This would be expected to suppress further diffusion of the velocity, much as seems to have occurred in the experiment. Some additional insight into the quality of the predicted results is provided from empirical blade performance correlations that are described in the next chapter. For this particular cascade test, those correlations indicate that the angle of attack is within 0.5° of the optimum value. Noting that the predicted Mach numbers on the two sides of the blade are nearly equal at the leading edge, the analysis indicates that the flow enters the blade smoothly, which is often used as an indication of an optimum inlet angle. Indeed, these predicted Mach numbers should not be expected to be exactly equal under optimum inlet conditions, since the leading edge node structure lies inside the passage as described previously. In effect, the leading edge predicted values are slightly downstream of the true entrance point for the flow. The potential flow analysis predicts a discharge

FIGURE 5-6 Potential Flow Results



flow angle of 24.4°, which is in reasonably good agreement with the experimental value of 25.0°. Hence, this potential flow analysis provides very useful information about the performance of this cascade that is consistent with the cascade test results and with empirical cascade performance models from Chapter 6. Figure 5-7 illustrates the transonic capability of this extended potential flow analysis. The case shown is the same as that used for Fig. 5-6, except that the higher inlet Mach number results in a substantial supersonic patch within the flow field. The agreement between the predictions and the experimental data of Dunavant et al. (1955) is reasonably good, both in terms of the blade loading diagram and the discharge flow angle. The procedure used is very robust, providing rapid and reliable convergence on these transonic flow problems. It is a very valuable extension for axial-flow compressor analysis, permitting much wider use of the potential flow method and resulting in far fewer solution failures. As long as Mach numbers do not become so large that imbedded shock waves significantly influence the flow, the predictions are generally quite accurate as well, as evidenced by this comparison with experimental data.

5.4 LINEARIZED POTENTIAL FLOW ANALYSIS The two-dimensional potential flow analysis of the previous section can be simplified to provide an exceptionally fast blade-to-blade flow analysis, yet provide surprisingly good prediction accuracy [Aungier, 1988(a), 2000]. If the sole

FIGURE 5-7 Transonic Potential Flow Solution



purpose of an analysis is to accomplish stand-alone blade-to-blade flow prediction, there is really little reason to consider this simplified model. The twodimensional potential flow analysis described above is so fast that it is not necessary to accept any compromise in accuracy for increased computation speed. The real purpose of the linearized method is its use in a quasi-threedimensional flow analysis, where blade-to-blade analyses are conducted on several stream sheets and must be repeated many times. It is described here to prepare for its later use in that more general application. The linearized blade-to-blade model is nothing more than a simplification of the potential flow model of the previous section. The stream function is approximated by

ψ (m, η) = a(m)[η − η 2 ] + η 2

(5-52)

From Eq. (5-17) it is easily shown that this is equivalent to assuming that the quantity ρbWm varies linearly with η. Equation (5-14) will be solved in integral form, noting that W = Wξ on the blade surfaces, i.e., 1

∂SWθ W1 W0 − = dη + 2Sω sin φ cos β1 cos β0 ∫ ∂m

(5-53)

0

The velocity normal to the blade surfaces must be zero, which requires that W=

˙ ∂ψ Wm m 1 = cos β cos β Sbρ ∂η

(5-54)

Combining Eqs. (5-52), (5-53) and (5-54) yields ˙ (2 − a) W1 m = cos β1 Sbρ cos2β1

(5-55)

˙ W0 ma = cos β0 Sbρ cos2β 0

(5-56)

which supply the terms on the left-hand side of Eq. (5-53). From Eqs. (5-18), (5-19) and (5-52) ˙ [tan β ( a − 2aη + 2η) − a′S(η − η 2 )] / ( bρ ) SWθ = m

(5-57)

where the prime notation denotes the total derivative with respect to m. As was done with the two-dimensional potential flow analysis, the stream function solution will be accomplished with the density held constant. To simplify the equations to follow, define



˙ tan β / ( bρ ) u(m, η) = m

(5-58)

˙ / ( bρ ) v(m, η) = mS

(5-59)

Differentiating Eq. (5-57) and introducing Eqs. (5-58) and (5-59) yields ∂SWθ ∂u ∂v ( a − 2aη + 2η) + (1 − 2η)ua′ − ( va′′ + = a′ )(η − η 2 ) ∂m ∂m ∂m

(5-60)

Using truncated Taylor series expansions for any function, F(η), for values at η = 0, 0.5 and 1, a three-point difference approximation to the integral is obtained. 1

∫ F(η)dη = (F0 + 4F + F1) / 6

(5-61)

0

where the overbar designates a value at η = 0.5. With the above equations and some tedious algebra, the integral term in Eq. (5-53) is given by 1

∫

0

∂SWθ dη = [au0′ + u0a′ + 4u ′ − va′′ − v ′a′ + u1′(2 − a) − u1a′] / 6 ∂m

(5-62)

Combining Eqs. (5-53), (5-55), (5-56) and (5-62) yields a simple linear differential equation. a′′ + Aa′ + Ba = C

(5-63)

where A, B and C are functions of m only A (m) = [ v ′ − u0 + u1] / v u′ − u′ v0 6  v1 B(m) = 1 0 − +  v vS2  cos2β1 cos2β0 C ( m) =

(5-64)   

(5-65)

2u1′ + 4u ′ + 12ω sin φ 12v1 − v vS2 cos2β1

(5-66)

Equation (5-63) is solved from the blade leading edge to the trailing edge. The leading edge boundary condition follows from the known inlet angular momentum supplied by the upstream flow, Wθ,in. Integrating Wθ across the passage at the leading edge using Eq. (5-61) yields the following leading edge boundary condition. a′ + a[u1 − u0 ] / v = [4u + 2u1 − 6SWθ ,in ] / v

(5-67)

The Kutta condition is again used as the trailing edge boundary condition, i.e., W0 = W1 at the trailing edge. From Eqs. (5-55) and (5-56), the trailing edge boundary condition is



a = 2 cos β0 / [cos β1 + cos β0 ]

(5-68)

The governing equation is cast in finite-difference form using the difference approximations previously introduced to develop a matrix equation for solution. In this case, a simple tri-diagonal matrix is obtained, except for the equation at the leading edge point, which contains one extra term. Inversion of this matrix is a rather trivial problem for the numerical analysis, which results in an exceptionally efficient blade-to-blade flow analysis. The iteration process to update the gas density field follows the same process as that for the two-dimensional potential flow, and the same transonic flow extension is incorporated. Aungier (2000) provides a fairly detailed description of the numerical analysis used to implement this model. Figure 5-8 compares results from the linearized and two-dimensional potential flow analyses for the same test case used earlier. The linearized method does not resolve the blade loading too well near the leading edge, but does reasonably well over most of the blade. The linearized method predicts a discharge flow angle for this case of 25.8°, which is in reasonably good agreement with the experimental value of 25.0°. The blade surface flow predictions at the leading edge are generally ignored, since the solution imposes the upstream boundary condition at that point. It will, by definition, correctly match the average angular momentum at the leading edge, but not necessarily the blade surface data, particularly on relatively blunt blades such as this one. Later in this book, it will be seen that the role of this

FIGURE 5-8 Linearized Potential Flow Results



analysis in a quasi-three-dimensional flow analysis is to define the streamwise variation of the average angular momentum, which it does rather well. Typical practice is to provide a direct and automated interface to the two-dimensional method for a detailed blade loading analysis after the quasi-three-dimensional flow analysis is completed. At that point, the stream sheet geometry and flow conditions are well defined for input to the two-dimensional method. The advantage offered by the linearized method for the quasi-three-dimensional flow application is exceptional computation speed while accurately estimating the average angular momentum distribution through the blade passage.

5.5 THE TIME-MARCHING METHOD The potential flow method can be applied to a very wide range of axial-flow compressor blade-to-blade flow problems, particularly with the transonic flow extension. But when Mach numbers become too high, a more general analysis technique is needed. The time-marching method provides a more general solution capability that is suitable for subsonic, supersonic or mixed subsonic-supersonic flow problems. Von Neumann and Richtmyer (1950) suggested this method for treating flows with imbedded shock waves. Except for some interest from mathematicians (Lax, 1954; Lax and Wendroff, 1964), the method received little attention until computers evolved enough for it to be used on practical problems. This writer participated in the development of this technique for application to hypersonic reentry vehicles in the late 1960s [Aungier, 1968, 1970, 1971(a), 1971(b)]. Soon after its successful application to the reentry problem, the method became popular for the blade-to-blade flow problem (e.g., Gopolakrishnan and Bozzola, 1973; Denton, 1982). The time-marching technique of Aungier [1970, 1971(a), 1971(b)] has also been adapted to this application and later reported in Aungier (2000). An abbreviated description of this technique that is sufficient to understand the fluid dynamics of the problem is provided in this chapter. Readers interested in developing a numerical analysis to implement the procedure may find the expanded description in Aungier (2000) helpful. The time-marching solution will be accomplished using the same solution domain, velocity components and coordinate system as those used for the potential flow analysis, i.e., Fig. 5-3 and Eqs. (5-1) through (5-7). The governing equations are Eqs. (3-21), (3-22), (3-23) and (3-25), which are to be solved over the stream sheet in their full time-unsteady form. The solution is advanced in time until the flow becomes approximately steady with time. The solution approaches the steady state asymptotically, so a true steady-state solution is never actually achieved. Rather, the solution is advanced in time until variations with time are considered negligible. As was discussed previously for the potential flow analysis, it is not advisable to simply transform the derivatives in the governing equations to solve them in the (ξ, η) coordinate system. A more precise numerical analysis will be achieved by developing the equations for the control volume to be used in the solution. This is accomplished by applying the integral form of the equations of motion to the control volume in Fig. 5-4. The integral form of the continuity, momentum and energy equations for inviscid, time-unsteady flow are



∫

V

∫

V

r r ∂ρ dV + ∫ ρ (W ⋅ n)dA = 0 ∂t

(5-69)

A

r r r r r r r ∂ρ W dV + ∫ ρ W (W ⋅ n)dA + ∫ P e ( e ⋅ n)dA = ∂t

r

∫ fdV

(5-70)

r r r r  ∂H ′ ∂P  − dV + ∫ ρ H ′( W ⋅ n)dA = ∫ ρ ( f ⋅ W )dV  ∂t ∂t  A V

(5-71)

A

A

V

∫ ρ

V

where V and A denote volume and area integrals, respectively, → n is a unit vector normal to the area and directed out of the control volume, → e is a unit vector along → → w and f is a body force. The body force is used to account for the Coriolis and centrifugal acceleration terms in the rotating curvilinear coordinate system. After some tedious algebra, application of these integral equations to the control volume yield: ∂ρ ∂ ∂ [Sbρ Wm ] + [bρ Q] = 0 + ∂t ∂m ∂η

(5-72)

∂ρ Wm ∂ ∂bP ∂ [Sb( ρ Wm2 + P )] − tan β [bρ QWm ] + + ∂t ∂m ∂η ∂η 1 ∂Sb = SBρ sin φ (Wθ + rω )2 + P r ∂m

(5-73)

Sb Sb

Sb

∂ρ Wθ 1 ∂ ∂ ∂ + [rSbρ Wm (Wθ + rω )] + [b( ρ QWθ + P )] = rω [Sbρ Wm ] (5-74) r ∂m ∂t ∂η ∂m Sb

∂( ρ I − P ) ∂ ∂ + [Sbρ Wm I ] + [bρ QI ] = 0 ∂t ∂m ∂η

(5-75)

These equations have been written in conservation form, such that they will be valid when applied across a shock wave as suggested by Lax (1954). Since shock waves can form when the flow is supersonic, this is an important consideration in the time-marching method. The parameter Q is a special velocity component to conserve properties at the constant η boundaries of the control cell, defined by Q = Wq / cos β = Wθ − Wm tan β

(5-76)

Note that Q = Wq = 0 for points on the blade surface. For these points, there is only one velocity component, Wξ. Applying the integral momentum equation in the ξ direction yields a special momentum equation for points on the blade surfaces. Sb

∂ρ Wξ

∂ ∂ [Sb( ρ WmWξ + P cos β )] + [bρ QWξ ] ∂t ∂m ∂η ∂ [Sb cos β ] + Sbρ sin φ cos β rω 2 =P ∂m +

(5-77)



Wm = Wξ cos β

(5-78)

Wθ = Wξ sin β

(5-79)

Equations (5-77) through (5-79) replace Eqs. (5-73) and (5-74) for points on the blade surfaces. The boundary condition for blade surface points is that the velocity normal to the surface is zero, which is satisfied by Eqs. (5-78) and (5-79). For the side boundaries outside of the blade passage, the procedure used is the same as that for the potential flow problem, i.e., the solution is extended into adjacent passages using the periodicity condition so these points can be treated in the same fashion as any interior point. The upstream and downstream boundaries require more care. The number and type of boundary conditions depend on how these boundaries are influenced by the flow inside the solution domain. A fundamental property of hyperbolic differential equations is that there are certain characteristic directions along which derivatives of the dependent variables normal to these “characteristics” can be discontinuous. For each characteristic direction, certain dependent variables can be determined by integration along them. Aungier (2000) derives the characteristic directions for the unsteady flow problem for the upstream and downstream boundaries. Since Wm is normal to these boundaries, these unsteady characteristics are defined by dm = Wm + a dt

(5-80)

dm = Wm − a dt

(5-81)

dm = Wm dt

(5-82)

This simply shows that information can be transmitted within the flow field by waves traveling at the fluid velocity and waves where the fluid velocity is augmented or opposed by the local acoustic velocity. Aungier (2000) also shows that the characteristics of Eqs. (5-80) and (5-81) determine Wm and P, while the characteristic of Eq. (5-82) determines Wθ and I. These characteristics can be used to define the number and type of boundary conditions needed for the upstream and downstream boundaries. Figure 5-9 shows these three characteristics drawn on an m-t diagram for an upstream boundary with a subsonic meridional velocity component. Since one of the characteristics that determines the flow at time t + ∆t lies within the solution domain, one dependent variable on the boundary can be computed as part of the solution, while the other three must be assigned as boundary conditions. The characteristic for Wθ and I lies outside the solution domain, so both must be assigned as boundary conditions, with one more boundary condition to be supplied. A logical choice for the computed dependent variable is density. Then, P and Wm follow directly from the equation of state and the definition of rothalpy if entropy is known. The most logical upstream boundary conditions for this case are Wθ, Pt, and Tt. They are usually known conditions for



FIGURE 5-9 Subsonic Upstream Boundary

a blade-to-blade flow analysis and they specify rothalpy and entropy through the equation of state and Eq. (3-10). So, for an upstream boundary with Wm < a, the continuity equation will be solved for density, with all other dependent variables computed from the boundary conditions. Figure 5-10 shows the m-t diagram for an upstream boundary where the meridional velocity component is supersonic. Here, all of the characteristics that determine the flow at t + ∆t lie outside of the solution domain. Consequently, none of the dependent variables on this boundary can be computed from the solution. All dependent variables must be assigned as boundary conditions when Wm > a on an upstream boundary. Figure 5-11 shows the m-t diagram for a downstream boundary where the meridional velocity component is subsonic. Here, one of the characteristics that determine the flow at t + ∆t lies outside the solution domain. This means that one dependent variable must be specified as a boundary condition, while the other three are computed as part of the solution. The usual practice is to specify the discharge static pressure as the boundary condition. It can be convenient to specify the mass flow rate instead, but then the solution procedure must compute the discharge pressure needed to produce that mass flow rate. Figure 5-12 shows the mt diagram for a downstream boundary where the meridional velocity component is supersonic. Here, all of the characteristics that determine the flow at t + ∆t lie inside the solution domain. This means that all dependent variables can be computed from the solution, with no boundary conditions required. Usually, it is not possible to state that Wm is supersonic. That is a very unique case for each set of



FIGURE 5-10 Supersonic Upstream Boundary

FIGURE 5-11 Subsonic Downstream Boundary



FIGURE 5-12 Supersonic Downstream Boundary

specified upstream boundary conditions. Rather, the discharge static pressure is normally specified in all cases. If the solution requires Wm > a to achieve that pressure, the solution procedure must ignore the boundary condition and compute all discharge conditions. In contrast to the potential flow analysis, it is noted that no empirical Kutta condition has been needed. Indeed, there would be no way to actually use it, since Wθ is always computed at the downstream boundary as part of the solution. If Wθ were specified as a boundary condition, a Kutta condition would be needed to select a value. Indirectly, the Kutta condition is applied by virtue of the periodicity condition for all solution points downstream of the trailing edge. It is important to note that the boundary condition requirements are not optional. For example, specifying more boundary conditions than outlined here will not produce a valid solution, and normally will cause the solution to diverge. Some investigators use the unsteady characteristics directly to solve for independent variables on the boundaries, e.g., Gopolakrishnan and Bozzola (1973). In most cases this just complicates the analysis for no good reason. The same thing is accomplished by simple upstream or downstream differences in the governing equations using Eqs. (5-40) and (5-42) with far less numerical logic. On occasion, unsteady characteristics can be used to advantage at boundary points. For example, this writer used them for the upstream bow shock boundary in the hypersonic reentry problem [Aungier, 1970, 1971(a), 1971(b)]. But there really is no merit to using them in the present application. The important role of unsteady characteristics is to identify the type of and the number of boundary conditions required.



A major issue to be addressed in a time-marching solution is numerical stability. It is well known that a finite-difference solution of the governing equations that is explicit in time is unconditionally unstable. The term explicit in time refers to predicting flow data at time t + ∆t from conditions at time t. In contrast, implicit solutions seek to use conditions at both t and t + ∆t for this purpose. Those solutions also have critical numerical stability issues, but this book will deal with only the explicit solution procedures. Von Neumann and Richtmyer (1950) developed a stability analysis procedure for this problem. They used it to achieve a stable numerical solution by including additional stabilizing terms in the governing equations similar in form to viscous terms. This approach is often referred to as including artificial viscosity terms. Lax (1954) used an averaging scheme for parameters at time t to project data at t + ∆t, which is equivalent to introducing artificial viscosity terms. Another approach that has been used is to apply a Taylor series expansion in time to the governing equations to extend them to second (or higher) order accuracy in time (e.g., Lax and Wendroff, 1964). Regardless of the approach used, stabilizing terms must always be added to the governing equations to achieve a stable explicit finite-difference solution. Unless extreme care is taken, these stabilizing terms can significantly influence the solution, possibly producing very unsatisfactory results. A method developed by this writer [Aungier 1970, 1971(a), 1971(b), 2000] has a definite advantage in that regard. It always permits the user of the analysis to reduce the influence of the stabilizing terms on the solution as much as necessary. But reduction of these effects is accomplished at the cost of requiring longer computation times for a steady-state solution. The method was developed by conducting a Von Neumann stability analysis on a simplified one-dimensional momentum equation to determine the minimum allowable magnitude of the stabilizing terms for a specified time step. Aungier (2000) describes this stability analysis in considerable detail. Here, only the results will be reviewed. It is convenient to represent any of the governing equations in the general form ut = v(ξ , η, t) + µ (ξ )uξξ + µ (η )uηη

(5-83)

where the subscript notation identifies first and second partial derivatives, and the last two terms are the stabilizing terms. For a stable solution for any time step, ∆t, the stability analysis shows that the coefficients of the stabilizing terms must satisfy the following conditions.

µ (ξ ) ≥ 12 ( Wm + a)2 ∆t

(5-84)

µ (η ) ≥ 12 ( Wθ + a)2 ∆t

(5-85)

µ (ξ ) ≥ 12 [( Wξ + a) cos β ]2 ∆t

(5-86)

µ (η ) ≥ 12 [( Wq + a) / cos β ]2 ∆t

(5-87)

This is satisfactory for nearly any blade-to-blade flow problem, but there are occasional exceptions when the grid structure is highly skewed and the node



spacing is much smaller in the tangential direction than in the meridional direction. The following empirical correction is applied after the meridional stabilizing term is established from Eqs. (5-84) and (5-86).

µ (ξ ) → µ (ξ ) + 12 [( Wθ + a) sin 2β ( ∆m) / ( S∆η)]2 ∆t

(5-88)

So far, ∆t has been treated as arbitrary, but that is really not the case. The stability analysis shows that the well-known Courant-Friedricks-Lewy (CFL) limit (Courant et al., 1928) must always be satisfied. This basically limits ∆t to the time it takes for the fastest relevant characteristic wave to travel between adjacent nodes in the solution field. Thus, the maximum value of the time step allowed by the CFL limit is given by ∆tmax ≤

∆m Wm + a

(5-89)

∆tmax ≤

S ∆η Wθ + a

(5-90)

The maximum time step is computed on all time iterations by applying Eqs. (589) and (5-90) at all nodes in the solution field. The actual time step used is some fraction of this maximum, specified by the user. ∆t = µ0∆tmax

(5-91)

Experience has shown that the following limits should be observed to avoid numerical stability problems. 0.1 ≤ µ0 ≤ 0.9

(5-92)

If the time derivative in Eq. (5-83) is approximated by a forward finite-difference approximation, the general governing equation is u(ξ , η, t + ∆t) = u(ξ , η, t) + [ v(ξ , η, t) + µ (ξ )uξξ + µ (η )uηη ]∆t

(5-93)

But since the coefficients of the stabilizing terms are also proportional to ∆t, as seen from Eqs. (5-84) through (5-88), the stabilizing terms are second order with respect to ∆t, while the dynamic terms represented by the general function v are first order with respect to ∆t. Hence, by simply using smaller values of ∆t the influence of the stabilizing terms can be reduced. Of course, that will mean more time iterations must be processed for the solution to approach a steady state. The approach used by this writer is to start the analysis with a fairly large value of µ0 (typically, 0.75) and steadily reduce it to some smaller value (typically, 0.25) as the solution approaches a steady state. This allows fairly large time steps to be used in the early iterations to accelerate the approach to a



steady state, but relatively small time steps when the solution is close to a steady state to reduce the effect of the stabilizing terms. The stability analysis of Aungier (2000) produced other useful results. It shows that for points on the blade surfaces, no stabilizing term normal to the surface is required, i.e., the last term in Eq. (5-83) is omitted for blade surface points. It also shows that for Wm > a, and backward difference approximation for meridional partial derivatives ∂u = [ui, j − ui −1, j ] / ∆m ∂m

(5-94)

no stabilizing term in the ξ direction is required, i.e., the second term on the right-hand side of Eq. (5-83) can be omitted. A similar result is obtained for negative Wm, except a forward difference approximation is used ∂u = [ui +1, j − ui, j ] / ∆m ∂m

(5-95)

Hence, supersonic Wm with “upwind” differences requires no stabilizing term in the ξ direction. This permits special procedures to be used on partial derivatives with respect to m to improve accuracy for very high Mach number flows. This is not usually necessary for compressor applications, but can significantly improve results for the high Mach numbers often encountered in turbine blades. Basically a weighted average of forward and backward differences are employed, as follows: um = F[ui +1, j − ui, j ] / ∆m + (1 − F )[ui, j − ui −1, j ] / ∆m  2Wm Wn 1 F= − 2 ( Wm + a + Wm + a 

  )2 

[Wm (m + ∆m,η) + 2 Wm (m,η) + Wm (m − ∆m,η)] am = 14 [ am (m + ∆m, η) + 2 am (m, η) + am (m − ∆m, η)]

Wm =

1 4

µ (ξ ) → 4µ (ξ )F(1 − F )

(5-96) (5-97) (5-98) (5-99) (5-100)

This procedure uses basic central-difference approximations and the basic stabilizing term form outlined previously when Wm = 0. As Wm→ a, F → 0. Hence, for Wm ≥ a, the solution will use an upwind difference approximation for the partial derivative, and the ξ stabilizing term will be zero. The result is that the minimum magnitude in the stabilizing terms allowed for stability is always used. It also results in faster convergence and sharper ‘shock capturing” when imbedded shock waves form in the flow field. For a period of time, a similar procedure was used relative to partial derivatives and stabilizing terms relative to the η direction. It was found that no significant benefit resulted for high Mach number



turbine blade problems, and the procedure occasionally slowed convergence and sometimes produced mild numerical instability. Interaction with the side boundaries and the higher probability that Wq and Q at a given node may change sign during the solution are believed to be the source of the problem. Presently, this writer uses standard central difference approximations and the basic stabilizing terms relative to the η direction. The basic form of the stabilizing terms significantly affects their influence on the solution. The stabilizing terms should be formulated such that their magnitude is expected to be small when the flow approaches a steady state. The recommended stabilizing terms to be added to the right-hand side of Eqs. (5-72) through (5-75) and Eq. (5-77), in that order, are Φ ρ = Sb[µ (ξ )ρ mm + µ (η )ρηη ]

(5-101)

Φ m = µ (ξ ) ( Sbρ Wm )mm + µ (η ) ( Sbρ Wm )ηη + Φ ρ Wm

(5-102)

Φη = Sbρ [µ (ξ ) ( rCθ )mm + µ (η ) ( rCθ )ηη ] / r + Φ ρ Wθ

(5-103)

Φ I = Sbρ [µ (ξ )Imm + µ (η )Iηη ] + Φ ρ I

(5-104)

Φξ = µ (ξ ) ( SbρWξ )mm + Φ ρ Wξ

(5-105)

The terms involving Φρ in Eqs. (5-102) through (5-105) can be shown to be direct corrections for the known error in mass conservation caused by Φρ when solving Eq. (5-72). The numerical analysis used is a fairly straightforward finite-difference analysis applied to the grid structure illustrated in Fig. 5-3. The only subtle feature required relates to defining the side boundaries outside of the blade passages. When Eq. (5-77) is solved at the blade leading and trailing edges, the finite-difference approximation to the ξ derivative will involve the closest node outside the blade passage. Equation (5-77) is derived for nodes where Wq = 0. As the solution proceeds, the side boundaries outside the blade are continually readjusted such that β is equal to the local flow angle at the closest node outside the blade passage. This ensures that Wq will be zero at this node so that the difference approximation used in Eq. (5-77) will be valid. Aungier (2000) provides additional details on the numerical analysis that may be of interest to readers considering implementation of this method in a numerical analysis. Figures 5-13 and 5-14 illustrate typical results from this time-marching method for the problems discussed previously for the potential flow solution, including comparison with that method. For the subsonic case, the time-marching solution generally shows better agreement with the experiments than does the potential flow method, particularly near the leading and trailing edges. It does show somewhat higher pressure-surface Mach numbers than either the experiment or the potential flow analysis, but agreement is considered very satisfactory. The predicted discharge flow angle is 21.1°, which is not in as good agreement with the experimental value of 25.0° as was achieved by the potential flow prediction of 24.4°. The transonic flow case shows similar trends, except that the potential flow analysis more accurately locates the point of maximum



FIGURE 5-13 Time-Marching Solution Results

FIGURE 5-14 Transonic Time-Marching Results



Mach number. Comparison of the time-marching method with this experimental data is particularly difficult and a little misleading. This case is very close to the passage choke flow limit. The time-marching method predicts choke at a slightly lower mass flow rate than indicated by the potential flow analysis and the experiment, no doubt due to the stabilizing term influence. Indeed, the mass flow rate used in the time-marching solution is slightly lower than for the potential flow solution and the experiment. The time-marching results correspond to the lowest discharge pressure that could be assigned without causing the flow to choke, expand to supersonic velocities near the discharge and then form shock waves to adjust to the discharge pressure. Both the potential flow and time-marching blade-to-blade flow analyses are seen to provide very useful guidance as aerodynamic design and analysis tools. For the sample problems considered, the potential flow method would be preferred due to the absence of stabilizing term effects and due to the much shorter computation time required for a solution. But the time-marching method provides a capability for solution at Mach number levels that are beyond the capability of potential flow methods.

5.6 BLADE SURFACE BOUNDARY LAYER ANALYSIS Two-dimensional boundary layer analysis is a valuable supplement to the inviscid blade-to-blade flow analyses discussed in this chapter. This type of analysis can be added to an inviscid flow analysis with very little increase in computation time. The basic assumption of boundary layer theory is that viscous effects are confined to a thin layer close to the walls. This is usually a valid assumption for the flow in a blade-to-blade stream sheet, where the boundary layers form on the blade surfaces. However, there are usually significant pressure gradients normal to the stream sheet. Unlike the inviscid flow, the low momentum of the boundary layer fluid will be unable to balance these normal pressure gradients, causing the boundary layer fluid to migrate across stream sheets. Two-dimensional boundary layer analyses must usually be considered to be an approximation, since they neglect this effect. Yet they provide a practical and useful method to quantify viscous effects and to identify likely boundary layer separation problems. These phenomena are very important since they govern the level of total pressure loss that will occur as the flow passes through the blade row. Boundary layer analysis in turbomachinery is most conveniently accomplished by applying the equations in integral form. The basic governing equation is the momentum-integral equation derived in Chapter 3 as Eq. (3-42). In the context of the present problem, this equation can be written 1 ∂bρ eWe2θ ∂We + δ * ρ eWe = τw b ∂x ∂x

(5-106)

The subscript e designates inviscid, boundary layer edge parameters, x is the distance along the blade surfaces and τw is the wall shear stress. The momentum and displacement thicknesses are defined as


108 • AXIAL-FLOW COMPRESSORS δ

ρ eWe2θ = ∫ ρ W (We − W )dy

(5-107)

0

δ

ρ eWeδ * = ∫ ( ρ eWe − ρ W )dy

(5-108)

0

where y is the distance normal to the wall and δ is the boundary layer thickness. Usually the wall shear stress is expressed in terms of the skin friction coefficient, cf, where cf =

τw 1

2 2 ρ eWe

(5-109)

When the boundary layer initially forms on the blade, the flow will be laminar. Typically, the boundary layers on axial-flow compressor blades soon transitions to turbulent flow. Hence, both laminar and turbulent boundary layer analyses are needed. This writer prefers the laminar boundary layer analysis of Gruschwitz (1950) as reviewed by Schlichting (1968). It is a generalization of the classical incompressible Karmen-Pohlhausen solution (Pohlhausen, 1921) to compressible flow. Among other advantages, this solution allows a very direct treatment of transition to turbulence by simple application of conservation of mass and momentum. This method employs a universal boundary layer velocity profile in the form W

We

= C1η + C2η 2 + C3η3 + C4η 4

(5-110)

y

η=

1 ρ dy δ ′ ∫ ρe

(5-111)

0

δ

δ′ = ∫

0

ρ dy ρe

(5-112)

Denoting the fluid viscosity by µ, a shape factor, Λ, is defined as Λ=

ρ e2 (δ ′ )2 dWe ρ w µ dx

(5-113)

Then, by matching the boundary layer edge conditions, the coefficients in Eq. (5-110) are C1 = 2 + Λ / 6, C2 = − Λ / 2, C3 = Λ / 2 − 2, C4 = 1 − Λ / 6

(5-114)

and the momentum thickness is given by



θ Λ Λ2 37 = − − δ ′ 315 945 9072

(5-115)

The boundary layer energy thickness, δE, and velocity thickness, δW, are given by δ 798048 − 4656Λ − 758Λ2 − 7 Λ3 δE ρ W  W2  =∫ 1 − 2  dy = 4324320 δ′ ρ W  We  0 e e 

(5-116)

δ

 3 δW W Λ FWe2 = ∫ 1 − − +  dy = 10 120 2cpTe δ′ We  0

(5-117)

where the function F is given by 2

F = 0.232912 − 0.831483

 Λ   Λ  Λ + 0.650584 + 17.8063   100 100  100 

3

(5-118)

For the case of adiabatic walls with the Prandtl number equal unity, the boundary layer enthalpy thickness is δ

 δh ρW  h We2δ E =∫  − 1 dy = 2cpTeδ ′ δ′ ρW h  0 e e  e

(5-119)

and the displacement thickness is given by

δ * = δ h + δW

(5-120)

Again considering adiabatic walls with the Prandtl number equal unity, the following parameters are introduced for convenience b0 =

T′ ρe = t ρ w Te

(5-121)

2

θ  ρ θ 2 dWe K = Λ   = b0 e µ dx δ ′ 

(5-122)

and Gruschweitz (1950) shows that 1c 2 f

=

µ  Λ 1+ ρ eWeδ ′  6 

(5-123) 2

 37 Λ Λ2  K= − −  Λ  315 945 9072 

(5-124)



The above equations are sufficient to permit numerical integration of Eq. (5-106) along x, starting at the leading edge, where θ = 0, to predict θ, δ* and δ at all other stations. In the process Gruschwitz limits the shape factor, Λ to lie in the range −12 ≤ Λ ≤ 12

(5-125)

where Λ = -12 corresponds to boundary layer separation. Normally, the boundary layer can be expected to undergo transition to turbulent flow. Many criteria for boundary layer transition have been proposed. This writer recommends use of the momentum thickness Reynolds number in the following transition criterion: Reθ =

ρ eWeθ > 250 µ

(5-126)

Once the boundary layer undergoes transition to turbulent flow, different empirical relations are required to integrate Eq. (5-106). This writer prefers the entrainment method of Head (1958, 1968) as adapted by Green (1968) to compressible flows. This requires integration of the entrainment equation, presented previously as Eq. (3-44). In context of the present application, this is written as ∂bρ eWe (δ − δ * ) = bρ eWe E ∂x

(5-127)

The entrainment function, E, specifies the rate at which mass is entrained into the boundary layer at the boundary layer edge. To integrate Eqs. (5-106) and (5-127), empirical relations are required for E, cf and for the various boundary layer thicknesses as a function of θ and (δ – δ*). Head’s entrainment method was developed for incompressible boundary layers using the following shape factors as its basis: H1 ≡ (δ − δ * ) / θ *

H = δ /θ

(5-128) (5-129)

Green (1968) recommends using a kinematic shape factor Hk in place of H when generalizing an incompressible boundary layer model to compressible flow. δ

Hk =

W 1 ρ  dy 1−  ∫ θ ρ e  We  0

(5-130)

Green shows that for adiabatic walls, with the Prandtl number equal unity, Hk can be related to H by H = ( Hk + 1)Tt′ / Te − 1

(5-131)



Solution of the governing equations requires empirical models for E and cf and to relate Hk and H1. Many alternate empirical models have been suggested for this purpose. This writer has found the following relations suitable for the present application: Hk = 1 + [0.9 / ( H1 − 3.3)]0.75

(5-132)

E = 0.025( Hk − 1)

(5-133)

The skin friction coefficient correlation of Ludwieg and Tillmann (1950) is commonly used for incompressible turbulent boundary layer analysis. cf , inc = 0.246 exp(−1.561Hk ) Reθ−0.268

(5-134)

Green (1968) recommends correcting this incompressible flow correlation to compressible flow by cf = cf , incHk ( Hk + 1) / [2Hk + H ( Hk − 1)]

(5-135)

These empirical relations are sufficient to integrate Eqs. (5-106) and (5-127), starting at the transition point. At the transition point, the mass and momentum flow in the turbulent boundary layer must match the values computed for the laminar boundary layer. From Eqs. (3-35) and (3-38), this requires (δ − δ * )turb = (δ − δ * )lam

(5-136)

θ turb = θ lam

(5-137)

From the Gruschwitz (1950) profiles, it can easily be shown that 7 Λ  δ −δ* = δ′ +  10 120 

(5-138)

Equation (5-137) is normally applied at the transition point, along with some assumption on how H changes during transition from laminar to turbulent flow. Use of the Gruschwitz laminar boundary layer model permits a more fundamental method based on conservation of mass and momentum. This writer uses Hk ≥ 2.4 as a separation criterion for turbulent boundary layers. During the analysis, the kinematic shape factor is limited to this value to avoid solution divergence and permit the analysis to continue through a separation zone. This is necessary, since it is not uncommon for the boundary layer to reattach, particularly on the pressure surface of the blade. When boundary layers on the blade surfaces have been computed, it is useful to predict the total pressure loss coefficient for the cascade from the boundary layer data at the trailing edge. Following Lieblein and Roudebush (1956), the



total pressure loss coefficient based on the cascade inlet velocity pressure can be approximated by 2

ω =

 cos βin  2Θ + ( ∆* )2 ∆Pt = ( Pt − P )in  cos β out  (1 − ∆* )2 Θ= ∆* =

(5-139)

∑θ

(5-140)

∑δ *

(5-141)

S cos β out S cos β out

where the summations are carried out for the boundary layers on both blade surfaces at the blade trailing edge. In the case of rotor blade rows, this total pressure loss coefficient can also be used to estimate the rotor efficiency via the methods described in Chapter 2. Equation (5-139) is derived directly from conservation of mass and momentum, at the blade trailing edge, while assuming that the low momentum boundary layer fluid instantly mixes with the inviscid free stream fluid while the static pressure remains constant. Those readers who completed Exercise 3.6 have, in fact, already derived a simple incompressible form of this equation under the same assumptions. Loss coefficients computed from two-dimensional boundary layer predictions should be regarded as quite approximate. The analysis ignores secondary flow effects associated with boundary layer migration normal to the stream sheets, which are often quite significant in an annular cascade within an axial flow compressor. Also, the boundary layers are often predicted to separate at some point along the blade surface. The basic assumptions of boundary layer theory are not satisfied in separation zones, causing the predictions to be of questionable accuracy. It can be expected that the loss coefficient in an annular cascade of an axialflow compressor will be significantly higher than is predicted by Eq. (5-139). Loss coefficients calculated in this fashion do have qualitative significance to guide the designer in evaluation of relative differences in loss for alternative cascade designs. Figure 5-15 shows predicted boundary layer shape factors, H, of Eq. (5-129) that are generated by a two-dimensional boundary layer analysis conducted using results from the inviscid potential flow analysis results as the boundary layer edge conditions. The inviscid flow blade-loading diagram for the case considered is shown in Fig. 5-6. Transition from laminar to turbulent flow occurs close to the leading edge for both blade surfaces. The deceleration of the inviscid velocity on the suction surface is severe enough to lead to a significant boundary layer separation zone starting well upstream of the blade trailing edge. As noted in the discussion of Fig. 5-6, the experimental data appears consistent with the premise that boundary layer separation may have occurred in this region. Numerical analysis for the two-dimensional boundary layer is relatively straightforward. The governing equations are parabolic in mathematical form. Hence, a simple marching type solution is needed, since the solution at each streamwise station depends only on the upstream boundary layer parameters.



FIGURE 5-15 Boundary Layer Analysis Result

Streamwise derivatives in the governing equations are replaced with backwarddifference approximations of the form of Eq. (5-94). An iterative solution procedure is used to complete the integration of the governing equations at each streamwise station before proceeding to the next station. Aungier (2000) outlines the procedure used by this writer. In this book, the outline of a numerical procedure is left as an exercise for the interested reader, with appropriate guidance provided in the exercises.

5.7 SUMMARY The analysis methods described in this chapter are representative of available theoretical methods for predicting the two-dimensional flow in cascades. There are many alternate solution techniques in use, both for the inviscid flow and the boundary layer. But the precise technique used is less significant than the approximations that are inherent in the two-dimensional flow model. It has been seen that reasonable estimates of the fluid turning can be achieved, but reasonable estimates are not good enough when errors are compounded in a multistage compressor analysis. The limitations relative to loss prediction have been discussed in the previous section. These theoretical methods suffer from a more fundamental limitation that has not yet been discussed. Both the inviscid flow and the boundary layer analyses suffer a dramatic loss of accuracy when applied under far off-design conditions, where severe flow separation may exist. Indeed,



there is very little merit in even applying them to such cases, yet these situations must be addressed when analyzing a compressor. Faced with this problem, design engineers normally turn to empirical models or seek a more fundamental analysis technique. Empirical modeling of cascade performance has a very long history in axial-flow compressor aerodynamic technology. Supported by extensive cascade testing, particularly by the NACA, rather accurate empirical models are available. These empirical models play such an essential role in axial-flow compressor aerodynamic design and analysis that they are covered in some detail in Chapter 6. Use of more fundamental theoretical methods is also receiving much attention. One well-established extension of the methods in this chapter is to combine them with a hub-to-shroud flow twodimensional flow analysis to form a quasi-three-dimensional flow analysis. Through interaction between the two analyses, it is possible to identify the stream sheet geometry, which has simply been treated as a known quantity in this chapter. In the case of centrifugal compressor aerodynamic design and analysis, the quasi-three-dimensional flow model plays an essential role (Aungier, 2000). Its advantages for axial-flow compressors are far less dramatic, but sufficient to warrant presenting the technique in Chapter 12 of this book. The more fundamental advance in theoretical analysis is the use of viscous flow computational fluid dynamics (CFD) codes. The design engineer will typically employ one of several commercially available CFD codes that are well suited to turbomachinery applications. Viscous CFD offers considerable promise for removing most of the limitations of the methods described in this chapter. Indeed, it has already greatly alleviated many of them, although much remains to be done in the areas of turbulence modeling, numerical methods and computational speed before these methods can be considered exact. Viscous CFD is occasionally applied to the two-dimensional blade-to-blade flow problem, but its real merit lies in treating the fully three-dimensional flow problem, where the important secondary flow patterns are also modeled. At present, the primary role of viscous CFD is in the area of advanced blade or stage design where its more fundamental fluid dynamics models can be used to advantage for reducing losses and increasing the operating range within practical computer running times.

EXERCISES 5.1 Derive an alternate expression for Eq. (5-10) by expressing the partial derivatives with respect to m and θ in terms partial derivatives with respect to ξ and η and substituting them into Eq. (3-21). Reduce this alternate continuity equation to a finite-difference form using centraldifference approximations similar in form to Eqs. (5-40) and (5-41). Simplify this difference equation to a mass balance equation for the control volume of Fig. 5-4, and show that the mass balance achieved for a numerical analysis by this alternate derivation is less accurate than that given by Eq. (5-8). 5.2 Develop Taylor series expansions about the meridional coordinate position, m, for ψ(m+∆m) and ψ(m-∆m) in terms of ψ(m) and its derivatives. Use these relations to derive Eqs. (5-35) and (5-37).



5.3

5.4

5.5

5.6

Express the order of accuracy of these finite-difference approximations in terms of ∆m. Develop Taylor series expansions about the meridional coordinate position, m, for ψ(m+∆m) and ψ(m+2∆m) in terms of ψ(m) and its derivatives. Use these relations to derive Eq. (5-40). Express the order of accuracy of this finite-difference approximation in terms of ∆m. From a truncated Taylor series expansion about the meridional coordinate position, m, for u(m+∆m), derive Eq. (5-95). Express the order of accuracy of this finite-difference approximation in terms of ∆m. The paragraph preceding Eq. (5-101) notes that the form of the stabilizing terms used in a time-marching solution is an important consideration. Consider flow in the portion of the solution domain upstream of the blade where the flow is uniform on the upstream boundary. Comment on the expected magnitudes of the stabilizing terms associated with µ(ξ) in Eqs. (5-102) through (5-104) when the flow is approximately steady state. Why is the term Sbρ outside of the partial derivative in Eqs. (5-103) and (5-104) but inside the partial derivative in Eq. (5-102)? If the form of the stabilizing terms causes them to be very small in magnitude near a steady state, how can they stabilize the numerical analysis? Consider the numerical solution of the laminar boundary layer equations at a specific streamwise station where all boundary layer edge conditions and all boundary layer parameters at the upstream station are known. (a) Define a procedure to estimate a “safe” initial guess for θ at the current station from the upstream station, which accounts for change in just boundary layer edge conditions. Suggest a “safe” procedure for the case where the upstream station is the leading edge. (b) Outline a procedure that might be followed to predict the laminar boundary at this station in terms of specific steps [parameter to predict and equation number(s) to be used]. It is understood that these steps will be repeated, in order, until the process converges.

5.7 Repeat Exercise 5.6 for turbulent boundary layers, except that in this case the upstream station cannot be the leading edge, so that issue is not relevant.


Chapter 6

EMPIRICAL PERFORMANCE MODELS BASED ON TWO-DIMENSIONAL CASCADE TESTS

The theoretical methods to analyze the flow in cascades presented in Chapter 5 are useful design and analysis methods, but they are not sufficient for day-to-day axial-flow compressor design activity. The many limitations restricting the accuracy and range of application of those methods have been discussed in Chapter 5. Consequently, empirical models are commonly used as a means of predicting the basic performance of cascades in axial-flow compressor design and analysis. These empirical models are derived from experimental data obtained from twodimensional cascade testing. Extensive testing of this type has been accomplished, particularly by the NACA. Rather sophisticated empirical methods are available for the standard blade profiles discussed in Chapter 4. In the case of advanced blade types, such as the controlled diffusion airfoils discussed in Chapter 4, alternate empirical models may be required. Typically these will be an adaptation of the methods used for the standard airfoil profiles, specifically addressing the improved performance characteristics achieved by the new profile design. The basic objective of the empirical modeling process is to predict the fluid turning and total pressure loss for a cascade under fairly general operating conditions. Also, operating conditions where near-optimum performance can be expected need to be identified by the empirical models. These are often referred to as design conditions, since they are appropriate for use under the compressor’s design operating condition where optimum performance is usually desired. Indeed, it will be seen in Chapters 10 and 11 that these can be used as a basis for selecting blade geometry to match the desired flow field through the compressor. Special empirical models to address blade tip leakage and seal leakage through stator shroud seals will also be covered in this chapter.

NOMENCLATURE a = distance to point of maximum camber along chord Bwake = blade wake blockage



b = exponent factor in Eq. (6-21); also distance from chord line to the point of maximum camber C = absolute velocity Cc = seal carryover coefficient Cr = seal contraction ratio Ct = seal throttling coefficient Cl0 = lift coefficient c = blade chord D = diffusion factor in Eq. (6-29) Deq = equivalent diffusion factor h = blade height i = incidence angle Ksh = blade shape parameter Kt,i = design incidence angle thickness correction factor Kt,δ = design deviation angle thickness correction factor M = Mach number Mc = critical Mach number m = slope parameter in Eq. (6-19) ˙ = mass flow rate m N = number of seal fins Nrow = blade row number (sequential through the compressor) n = slope parameter in Eq. (6-12) o = throat opening P = pressure p = seal pitch, Fig. 6-22 R = range from design inlet angle to stall angle, gas constant or radius of curvature Rec = blade chord Reynolds number r = radius s = blade pitch T = temperature t = seal point thickness, figure 6-22 tb = blade maximum thickness Uc = leakage velocity W = relative velocity Z = number of blades in a blade row α = angle of attack; also parameter in Eq. (6-38) β = flow angle χ = blade angle with chord δ = deviation angle δc = blade tip or seal clearance γ = stagger angle κ = blade angle with the meridional direction θ = camber angle θw = blade wake momentum thickness ρ = gas density φ = angle defined in Fig. 6-17 ψ = angle defined in Fig. 6-17


Empirical Performance Models Based On Two-Dimensional Cascade Tests • 119

σ τ ω – ω

= = = =

solidity torque rotation speed total pressure loss coefficient

Subscripts c m max s t u 0 1 2 10 θ *

= = = = = = = = = = = =

negative stall angle parameter or leakage parameter meridional component; also minimum loss condition maximum value positive stall angle or bow shock wave condition total thermodynamic condition parameter on upper (suction) blade surface zero-camber condition parameter at blade inlet parameter at blade exit parameter for 10% thick profile tangential component sonic flow condition

Superscripts * = design condition ′ = relative condition

6.1 CASCADE GEOMETRY AND PERFORMANCE PARAMETERS Figure 6-1 illustrates the basic parameters and nomenclature used to describe the cascade flow. Subscripts 1 and 2 are used to designate conditions at the inlet and discharge of the blade, respectively. Blades have a chord length, c, and a tangential spacing between adjacent blade camberlines or pitch, s. The stagger or setting angle, γ, is the angle between the chord line and the axial direction. The flow enters the cascade with velocity, W1, and the flow angle with the axial direction is β1. The angle between the inlet velocity vector and the chord line is called the angle of attack, α, i.e.,

α = β1 − γ

(6-1)

The flow exits the cascade with velocity, W2, and the flow angle with the axial direction is β2. The blade angles κ1 and κ2 are the angles between the camberline and the axial direction at the leading and trailing edges, respectively. The incidence angle, i, and the deviation angle, δ, are defined as



FIGURE 6-1 Cascade Nomenclature

i = β1 − κ1

(6-2)

δ = β2 − κ 2

(6-3)

As discussed in Chapter 4, the angles κ1 and κ2 are not well defined for the NACA families of blade camberlines. For this reason, NACA cascade data is usually presented in terms of the angle of attack and the fluid turning, ε.

ε = β1 − β2

(6-4)

It has become fairly common practice to use an equivalent circular-arc camberline for the NACA 65-series camberline as a basis for defining i, δ, κ1 and κ2, following the procedure described in Chapter 4. Several other parameters will be used in the empirical correlations reviewed in this chapter. The minimum passage opening or throat width, o, is illustrated in Fig. 6-1. The location of the point of maximum camber, a / c, is illustrated in Fig. 4-6. The thickness-to-chord ratio, tb / c, is illustrated in Fig. 4-1. The solidity, σ, and the camber angle, θ, are defined by

σ = c/ s θ = κ1 − κ 2

(6-5) (6-6)

Equation (4-29) will be used to relate lift coefficient, Cl0, of the NACA camberlines to camber angle, i.e., tan(θ / 4) = 0.05515Cl0 / ( a / c)

(6-7)



6.2 DESIGN ANGLE OF ATTACK OR INCIDENCE ANGLE The design incidence angle, i*, or design angle of attack, α*, define a near-optimum or minimum-loss inlet angle for the cascade. Figure 6-2 shows a chart of the design angle attack by Herrig et al. (1957) for NACA 65-series blades with solidity and lift coefficient as parameters. The selection of α* was based on achieving smooth blade surface pressure distributions, particularly on the suction surface. This writer has formulated the following empirical model based on that design chart.

α * = [3.6K sh Kt,i + 0.3532θ ( a / c)0.25 ]σ e

(6-8)

e = 0.65 − 0.002θ

(6-9)

The data in Fig. 6-2 applies to NACA 65-series blades with tb / c = 0.1, so Kt, i = Ksh = 1 and a / c = 0.5. The parameters Ksh and Kt,i are adapted from the design incidence correlation of Lieblein (1960), which is also presented in NASA SP-36 (Johnsen and Bullock, 1965). Lieblein shows that design incidence angle correlations for NACA 65-series blades with tb / c = 0.1 can be extended to other profile types and other thickness-to-chord ratios by applying the correction factors to the design incidence (or design angle of attack) for camber angle equal zero. Ksh assumes values of 1.0 for NACA profiles, 1.1 for the C4-series profile and 0.7 for the double-circular-arc profile. Figure 6-3 shows the correction term Kt, i

FIGURE 6-2 The Design Angle of Attack



FIGURE 6-3 Thickness Correction for α*

provided graphically in Johnsen and Bullock (1965), compared to the following empirical equation: Kt, i = (10tb / c)q

(6-10) 0.3

q = 0.28 / [0.1 + (tb / c)

]

(6-11)

The parameter a/c introduced into Eq. (6-8) serves to extend the correlation to the NACA A4K6 inlet guide vane camberline, and is presumed to be applicable to the parabolic-arc camberline also, although experimental data isn’t available for the writer to actually confirm that. Lieblein’s (1960) design incidence angle correlation is developed from basically the same data as that in Fig. 6-3, but with the intent to identify the minimum loss incidence angle. The equivalent circular-arc camberline is used for the NACA 65-series blades as the basis for defining the incidence angle. The form of Lieblein’s correlation is i* = K sh Kt, i (i0* )10 + nθ

(6-12)

The first term on the right-hand side of Eq. (6-12) is the design incidence angle for a camber angle of zero. It is computed from a correlation for NACA 65-series blades with tb / c = 0.1, corrected by Ksh and Kt,i. The base zero-camber incidence



angle graphical correlation from Johnsen and Bullock (1965) is shown in Fig. 6-4, along with predictions from the following empirical equations developed from the graphical correlation. (i0* )10 =

β1p − 0.1σ 3 exp[(β1 − 70) / 4] 5 + 46 exp(−2.3σ ) p = 0.914 + σ 3 / 160

(6-13) (6-14)

Figure 6-5 shows the graphical correlation for the slope factor, n, from Johnsen and Bullock (1965) along with predictions from the following empirical equation: n = 0.025σ − 0.06 −

(β1 / 90)(1+1.2σ ) 1.5 + 0.43σ

(6-15)

It can be noted that both Eqs. (6-13) and (6-15) contain the flow angle, which is, itself, a function of i*. Hence, an iterative solution is needed to compute the design incidence angle using Lieblein’s (1960) correlation. Liebleins’s model applies only to blades where a/c = 0.5. To treat the NACA A4K6 guide vanes and parabolic-arc blades, this writer has used the following procedure. A pseudoblade inlet angle is computed as

FIGURE 6-4 Zero-Camber Design Incidence Angle



FIGURE 6-5 Design Incidence Angle Slope Factor

κ1 = γ +

1

2θ

(6-16)

This is used to compute a pseudo-design incidence angle using Lieblein’s correlations. Then, the design angle of attack for the actual blade is adjusted by

α * = κ1 + i * − γ + ( a / c − 0.5)θ

(6-17)

The actual design incidence angle is given by i* = α * + γ − κ1

(6-18)

This writer has used both the design angle of attack correlation of Eq. (6-8) and Lieblein’s design incidence models rather extensively in axial-flow compressor aerodynamic performance analysis. Clearly, Lieblein’s model is based on a less subjective criterion than α* and includes more parameters in the correlation. But, in practice, the design angle of attack model has consistently resulted in more accurate compressor performance predictions. That is not necessarily a significant evaluation of the two models. A performance analysis uses many other empirical models, which are discussed in the remainder of this chapter. Performance prediction accuracy depends on the complete set of models, more than on



any single model. So, all that really can be said is that this writer has had better success using the design angle of attack model.

6.3 DESIGN DEVIATION ANGLE Lieblein (1960) also supplies an empirical model for the design deviation angle, δ*, corresponding to operation at the design incidence angle. This model is reviewed in somewhat greater detail in NASA SP-36 (Johnsen and Bullock, 1965). The model is similar in form to Lieblein’s design incidence angle model, i.e.,

δ * = K sh Kt,δ (δ0* )10 + mθ

(6-19)

Ksh is the same as for the design incidence angle model. Figure 6-6 shows the graphical form of the base zero-camber deviation angle from Johnsen and Bullock (1965) along with predictions from the following empirical equation: (δ 0* )10 = 0.01σβ1 + [0.74σ 1.9 + 3σ ](β1 / 90)(1.67 +1.09σ )

(6-20)

The slope parameter, m, is expressed as a function of its value for a solidity of 1.0, m1.0, corrected for other values of solidity in the following form.

FIGURE 6-6 Zero-Camber Deviation Angle



m = m1.0 / σ b

(6-21)

The graphical form of the parameters m1.0, b and Kt, δ from NASA SP-36 (Johnsen and Bullock, 1965) are shown in Fig. 6-7, 6-8 and 6-9, along with predictions from empirical equations used by this writer. Defining x = β1 / 100, the slope factor for the NACA 65-series camberline is modeled as m1.0 = 0.17 − 0.0333 x + 0.333 x2

(6-22)

For circular-arc camberlines, m1.0 = 0.249 + 0.074 x − 0.132 x2 + 0.316 x3

(6-23)

The exponent, b, is modeled by b = 0.9625 − 0.17 x − 0.85 x3

(6-24)

Kt,δ = 6.25(tb / c) + 37.5(tb / c)2

(6-25)

It can be seen that Lieblein’s design deviation angle model does not apply to the NACA A4K6 inlet guide vane or parabolic-arc camberlines. For those blade types,

FIGURE 6-7 Design Deviation Angle Slope Factor



FIGURE 6-8 Design Deviation Angle Exponent

FIGURE 6-9 Thickness Correction for *



a modified design deviation angle correlation by Howell (1942, 1945) can be used. Howell’s correlation is expressed in terms of the discharge flow angle, which is a function of the deviation angle, but that problem is easily remedied by substituting for β2 using Eq. (6-3). Howell also does not account for blade thickness-to-chord and profile shape effects, so it is adjusted using Lieblein’s corrections for those effects. The modified form used is

δ* =

0.92( a / c)2 + 0.002κ 2 θ 1 − 0.002θ / σ

σ

+ ( K sh Kt,δ − 1)(δ 0* )10

(6-26)

Equation (6-26) is also used for all inlet guide vanes, regardless of camberline or profile type. Inlet guide vanes are unique in the sense that they are normally the only cascades in an axial-flow compressor that accelerate the flow. Indeed, at sufficiently large stagger angles, such that the blade passage throat is located at the discharge, they function just like turbine nozzle blade rows. In those cases, an axial-flow turbine deviation angle correlation would be more appropriate. But that is not really a viable approach, since the throat of inlet guide vanes also may be located at the inlet. In the case of adjustable inlet guide vanes, both situations may be encountered in the same compressor. Basically, the function of inlet guide vanes is to turn the flow from an inlet flow angle of zero to some larger flow angle. In the context of a compressor analysis, this means an inlet guide vane will have a negative camber angle. The preceding empirical equations are all applicable to positive camber angles only. They can be applied to a cascade with negative camber simply by changing the signs of all blade angles (θ, γ, κ1 and κ2) and flow angles (β1 and β2), applying the correlations, and then changing all signs again to cast the results in the sign convention used for the compressor analysis. When these sign corrections are applied, inlet guide vanes have the unique feature that the corrected γ is negative. Also, the corrected κ2 will have a relatively large negative value. Howell’s design deviation angle correlation is the only one known to this writer that can properly handle this situation. Indeed, deviation angles predicted by Eq. (6-26) for the NACA A4K6 inlet guide vane camberline are in rather good agreement with the design charts of Dunavant (1957).

6.4 DESIGN LOSS COEFFICIENT AND DIFFUSION FACTORS Now that a reference or design incidence angle and the corresponding fluid turning or deviation angle have been established, to completely characterize the performance of the cascade at the design operating condition, the corresponding design total pressure loss coefficient has to be predicted. In the context of twodimensional cascade test results, the loss coefficient involved is referred to as the profile loss coefficient. It is approximately related to the wake momentum thickness, θw (Lieblein, 1959) by

ω =2

θ w σ  W2  c cos β2  W1 

2

(6-27)



Equation (6-27) is similar to Eq. (5-139), except that wake properties rather than boundary layer properties are used. Terms involving the wake shape factor, H, are neglected, since it is close to unity at typical measuring stations in a cascade test. Velocity relative to the blade, W, is used for generality, although W = C for a stationary cascade test. The wake momentum thickness is primarily a function of blade surface skin friction and the blade surface velocity distributions. In particular, the amount of deceleration or diffusion of the surface velocity is a primary factor in determining the wake momentum thickness. In the context of a general blade-loading diagram, this is characterized by the magnitude of the maximum velocity, Wmax, relative to the discharge velocity, W2. With reference to Fig. 6-10, it can be seen that Wmax is primarily a function of W1, W2 and the blade loading distribution, ∆W. For irrotational flow, Stokes’ theorem of vector field theory, Eq. (5-12), can be used to show that ∆W is a function of the change in the tangential velocity across the blade row, i.e., ∆W ∝ W1 + f [(Wθ 2 − Wθ1) / σ ]

(6-28)

This fact has been employed to develop various diffusion factors for use as correlating factors for loss coefficient and to estimate the diffusion or loading limits where boundary layer separation leads to an abrupt increase in loss. Two popular diffusion factors are the so-called D-factor, D, of Lieblein et al. (1953) and the equivalent diffusion factor, Deq, of Lieblein (1959).

FIGURE 6-10 Typical Blade Loading Diagram



D ≈ (Wmax − W2 ) / W1

(6-29)

Deq ≈ Wmax / W2

(6-30)

Equation (6-29) is replaced with a very specific approximation for use as a correlating parameter. D = 1−

W2 Wθ1 − Wθ 2 + W1 2σW1

(6-31)

Johnsen and Bullock (1965) provide a good review of the development of the Dfactor as a correlating parameter. Through comparison with extensive twodimensional cascade data, it is shown that θw / c can be well represented by a single curve as a function of D. From Eq. (6-27) it is expected that loss coefficient should also correlate with D in the form 2

ω cos β2  W1   W  = f ( D) 2σ  2

(6-32)

That has been confirmed by comparison with cascade test data. It is also shown that a correlation can be developed in the form

ω cos β2 = f ( D) 2σ

(6-33)

The correlation based on Eq. (6-33) offers less resolution than a correlation based on Eq. (6-32), and is far less useful for identifying the loading limit where an abrupt increase in loss is observed. Despite these weaknesses, the correlation form of Eq. (6-33) was adopted. After a correlation of loss coefficient as a function of D was developed from two-dimensional cascade data, it was evaluated against loss data from annular compressor cascade data. It is found that losses in a compressor cascade are significantly higher than for a simple two-dimensional cascade test. Even at mid-span, where end-wall boundary layer and tip clearance effects should be minimal, three-dimensional effects were found to be significant. Based on the annular cascade data, the two-dimensional cascade correlation was revised to the form shown in Fig. 6-11, which is well approximated by the following empirical equation:

ω * cos β2* = 0.0035[1 + 3.5D* + 37( D* )4 ] 2σ

(6-34)

D* rather than D is used to emphasize that the correlation only applies for operation at the design incidence angle. For values of D* > 0.6, an abrupt increase in loss is observed, so D* = 0.6 is adopted as the diffusion limit. Equation (6-34) should not be applied in those cases. The premise of this design loss coefficient



FIGURE 6-11 Loss Coefficient as a Function of D*

correlation is that well-designed compressor cascades will not operate beyond the diffusion limit at the design incidence condition. Lieblein (1959) develops the equivalent diffusion factor of Eq. (6-30). From experimental cascade data for NACA 65-series and C.4 circular-arc blades, Lieblein developed the following empirical equation for Wmax / W1 for operation at minimum loss, i.e., for operation at the design incidence angle: *

 Wmax  cos2 β1* [tan β1* − tan β2* ]   = 1.12 + 0.61 σ  W1 

(6-35)

Hence, the design equivalent diffusion factor is *

*

W  W1*  Wmax  cos β2* Wm1 * Deq =  max  =   W1  W2*  W1  cos β1* Wm2

(6-36)

Lieblein considered only two-dimensional cascades where the axial velocity is constant through the cascade, so he could express Eq. (6-36) in terms of inlet and discharge flow angles only. From two-dimensional cascade data, Lieblein developed an empirical correlation for θw / c as a function of the equivalent diffusion factor. This can be converted to a correlation of design loss coefficient by applying Eq. (6-27). The result is shown in Fig. 6-12, along with predictions from the following empirical equation.



FIGURE 6-12 Loss Coefficient as a Function of Deq

2

ω * cos β2*  W1*  2 8 * *  *  = 0.004[1 + 3.1( Deq − 1) + 0.4( Deq − 1) ] 2σ  W2 

(6-37)

It is seen that losses increase rather abruptly for equivalent diffusion factors greater than about 2. Lieblein suggests that an equivalent diffusion factor of 2.0 at design incidence should be considered a diffusion limit, beyond which an abrupt increase in loss can be expected. To permit use of the equivalent diffusion factor as an indicator of the off-design diffusion limit, Lieblein extended Eq. (6-35) to include operation at incidence angles greater than the design incidence angle. The off-design equivalent diffusion factor for i ≥ i* is computed from Eq. (6-30), using cos2 β1 Wmax = 1.12 + 0.61 [tan β1 − tan β2 ] + α (i − i* )1.43 W1 σ

(6-38)

The parameter α = 0.0117 for NACA 65-series blades and α = 0.007 for C4 circular-arc blades. J. Klapproth, in a discussion included in Lieblein (1959), extended these results to include general annular compressor cascades where the axial velocity and radius are not constant, and the cascade may be rotating. The more general form is cos2 β1 r1Cθ 1 − r2Cθ 2 Wmax = 1.12 + 0.61 + α (i − i* )1.43 W1 σ r1Cm1

(6-39)



Equation (6-39) accounts for changes in the meridional velocity and in the blade circulation due to a change in radius and due to blade row rotation. By using this form, the equivalent diffusion factor can be applied to the more typical cases encountered in axial-flow compressor cascades for both design and offdesign operation. In principle, Fig. 6-11 or Fig. 6-12 can be used to estimate the loss coefficient at the design incidence angle. But, in practice, the blade profile loss is only part of the loss that occurs in an axial-flow compressor blade row. Other important sources of the overall total pressure loss include effects due to tip clearance, stator shroud leakage, end-wall boundary layers, Mach number, Reynolds number and secondary flows. Howell (1942, 1945) recognized this fact many years ago and included additional loss models to account for the effects of end-wall and secondary flow losses. Howell expresses these additional losses in terms of drag coefficient, CD, where

ω = CDσ cos2 β1 / cos3 β

(6-40)

tan β = (tan β1 + tan β2 ) / 2

(6-41)

The end-wall drag coefficient used by Howell is CDa = 0.02s / h

(6-42)

where h is the blade height. Howell’s secondary flow drag coefficient is based on the blade lift coefficient, CL CDs = 0.18CL2

(6-43)

CL = 2 cos β (tan β1 − tan β2 ) / σ

(6-44)

The loss coefficient correlation of Fig. 6-11 presumably includes at least the secondary flow loss, since it is adjusted to reflect the higher losses seen in compressor cascades relative to simple two-dimensional cascades. That is not the case for the correlation in Fig. 6-12. Neither correlation is expected to account for other loss sources listed above. Where possible, these additional losses should be approximated by specific empirical models. Approximate methods to model Mach number effects and tip clearance effects are discussed later in this chapter. Some loss sources can often be neglected, such as Reynolds number effects. Attempts to model end-wall boundary layer losses have been reported in the literature, but none that appear to be particularly general or reliable. It is often more effective to adjust the profile loss models to reflect the higher losses expected in a compressor cascade. Equations (6-34) and (6-37) can be written in the more general form of

ω * cos β2* = K1[K2 + 3.5D* + 37( D* )4 ] 2σ ω * cos β2* 2σ

(6-45)

2

 W*  2 8 * * 1  *  = K1[K2 + 3.1( Deq − 1) + 0.4( Deq − 1) ] W  2

(6-46)



The parameters, K1 and K2, are empirical constants or functions to be determined by comparing performance predictions with experiment for a suitable number and variety of axial-flow compressors. Their precise form depends on how the performance analysis is formulated, particularly with respect to which loss sources are modeled explicitly and which sources are lumped into K1 and K2. The approach adopted for this writer’s performance analysis is discussed at the end of this chapter, along with some variations and alternatives from the literature. The more general loss coefficient model of Fig. 6-12 or Eq. (6-46) appears to be the better choice. There seems to be some resistance to that choice due to its development from specific blade loading distribution information. It is sometimes argued that its application to other blade camberlines and profiles requires specific blade loading information for development of a new empirical correlation. But that really doesn’t justify choosing the model of Fig. 6-11 and Eq. (6-45) instead. In truth, the definition of D given in Eq. (6-31) has an inherent assumption regarding the blade loading distribution, even though specific blade loading information was not used to define it. There is no reason to believe that D is a more general parameter than Deq, when it is applied to blade types different from those for which its loss coefficient empirical model was developed.

6.5 POSITIVE AND NEGATIVE STALL INCIDENCE ANGLES Figure 6-13 is a schematic showing the typical variation of loss coefficient with incidence angle. The loss coefficient is fairly constant over a range of incidence

FIGURE 6-13 Off-Design Loss Coefficient



angles near the design incidence angle, but increases rapidly when the cascade is operated too far from the design incidence angle. It is conventional practice to define the limits of low-loss operation by the positive and negative stall incidence angles, is and ic, where the loss coefficient becomes twice the minimum loss coefficient, as shown in Fig. 6-13. Herrig et al. (1957) developed an approximate correlation of the positive and negative stall angles of attack for NACA 65series blades from low Mach number two-dimensional cascade test data. Figure 6-14 shows their correlation along with data computed from the following empirical equations: 0.48    30   θ α c − α * = −9 + 1 −    β1c   4.176  

(6-47)

 β1 s  θ α s − α * = 10.3 + 2.92 −  15.6  8.2 

(6-48)

where α, β1 and θ are all expressed in degrees. Note that Eq. (6-47) is singular if β1c = 0. This writer’s practice is to limit β1c ≥ 20°. Since α is a function of β1, these equations are not directly usable in a performance analysis. But since β1 = α + γ, they can be applied by a simple iterative solution. Since α – α* is simply the incidence angle range to stall, these empirical models would be expected to be rea-

FIGURE 6-14 Stall Angles of Attack



sonable approximations for blade types other than the NACA 65-series. Indeed, they have proven to be effective when used for performance analysis of doublecircular-arc blades, which suggests they should be applicable to the C4 circulararc blades as well. One exception is the NACA A4K6 camberline where cascade test data shows a nearly constant range from the design angle of attack, given approximately by αs – α* = 10° and αc – α* = –10° (Dunavant, 1957). This may suggest that a correction is needed for blades in which the point of maximum camber is not at mid-chord, but it is more likely that the constant incidence ranges to stall is due to the fact that the inlet flow angle is constant for inlet guide vanes. Nevertheless, it should be recognized that the applicability of these empirical models to the parabolic-arc camberline has not been established.

6.6 MACH NUMBER EFFECTS When applying the low-speed cascade empirical correlations to an actual compressor blade row, it is necessary to apply corrections to account for Mach number effects. As Mach number increases, the low-loss working range for the cascade, αs – α* and α* – αc, is reduced relative to the low-speed cascade correlations of Eqs. (6-47) and (6-48). These low-loss working ranges for low-speed cascades will be designated as Rs and Rc, i.e., R c = α * − α c = i* − ic

(6-49)

R s = α s − α * = is − i*

(6-50)

Johnsen and Bullock (1965) note that as Mach number increases (i* – ic) and (is – i*) are reduced by approximately the same amount for moderate Mach number blade profiles such as the NACA 65-series and C4-series blades. But for high Mach number blade profiles such as the double-circular-arc profile, (i* – ic) is reduced much faster than (is – i*) as Mach number increases. This writer computes the negative and positive stall incidence angles for all blade sections by ic = i* − R c/ [1 + 0.5M1′3 ]

(6-51)

is = i* + R s/ [1 + 0.5( K sh M1′)3 ]

(6-52)

but with the constraint that Ksh ≤ 1. The negative stall incidence angle is also required to be no less than the value corresponding to an inlet flow angle for which the mass flow rate is 2% from the blade choke mass flow. An appropriate equation of state from Chapter 2 is used to compute the sonic flow gas density, ρ*, and velocity, W*, for the local inlet relative total thermodynamic conditions. Assuming the stream sheet thickness is constant between the inlet and the throat, basic conservation of mass yields the inlet flow angle corresponding to choke:



ρ1W1s cos β1Choke = oρ*W*

(6-53)

This imposes a lower limit on the negative stall incidence angle given by ic ≥ β1Choke − κ1 + 1o

(6-54)

Hence, the approach to choked flow, where i will be less than ic , will be associated with large and rapidly increasing loss coefficients as illustrated in Fig. 6-13. In principle, the loss should become infinite when the passage is choked. But the milder choke condition used here is more appropriate for an axial-flow compressor performance analysis. Blade passage choke is often a local condition along the blade height, which causes a redistribution of the mass flow toward unchoked sections of the blade. If the loss increase near choke is too severe, a numerical solution will often diverge before this redistribution can occur. Beyond that, this simple onedimensional flow choke calculation is not precise enough to treat it as an absolute limit. The procedure suggested here has been found to be an effective compromise for use in an axial-flow compressor aerodynamic performance analysis. When these Mach number adjustments are imposed, it may also be necessary to readjust the minimum loss incidence angle and the minimum loss coefficient. Indeed, it is quite possible for the design incidence angle, i*, to be less than ic. Figure 6-15 illustrates the type of corrections that may be imposed on the off-design loss coefficient at elevated Mach number levels. A minimum loss incidence angle, im, is defined by

FIGURE 6-15 Mach Number Corrections



im = ic + (is − ic ) R c / ( R c + R s )

(6-55)

For moderate Mach number blade profiles far from the choked flow limit, it can be seen that im = i*. But for higher Mach number profiles, or cases where limit in Eq. (6-54) becomes active, Eq. (6-55) results in im > i*, similar to the schematic shown in Fig. 6-15. As a minimum, the minimum loss coefficient must be adjusted for the difference between im and i*, using the low Mach number loss curve directly. For incidence angles between ic and is, this writer uses a simple second-order power law relation for the off-design loss coefficient, i.e.,

ω = ω m + ω m [(i − im ) / (ic − im )]2 ; i ≤ im 2

ω = ω m + ω m [(i − im ) / (is − im )] ; i ≥ im

(6-56) (6-57)

It follows directly that the minimum correction required for the minimum loss coefficient is

ω m = ω * [1 + (im − i* )2 / R2s ]

(6-58)

Except for this fairly minor correction, loss coefficients at the minimum loss incidence show little variation with Mach number until the fluid velocities on the blade surfaces become supersonic. At that point, shock waves can form locally, eventually causing boundary layer separation to significantly increase the minimum loss coefficient. Equation (6-39), applied with i = i* and δ = δ*, defines the maximum velocity, Wmax, as illustrated in Fig. 6-10. It follows that the critical Mach number, M′c, where the flow first becomes supersonic on the blade surfaces, is given by Mc′ = M1′ W* / Wmax

(6-59)

When the inlet Mach number exceeds the critical Mach number, the minimum loss coefficient is estimated from

ω m = ω * [1 + (im − i* )2 / R2s ] + K sh [( M′ / Mc′ − 1)W* / W1]2

(6-60)

When applying Eq. (6-60), the limit M′ ≤ 1 is imposed since bow shock wave losses are handled separately, as will be seen in the next section.

6.7 SHOCK WAVE LOSS FOR SUPERSONIC CASCADES The Mach number effects described in the previous section do not account for the additional loss caused by the upstream shock wave when the flow entering the blade row is supersonic. Semi-empirical models to correct for bow shock losses have been reported by Swan (1961) and Miller et al. (1961), both of which



employ essentially identical models. Swan’s description is perhaps a little more complete and will be used as the basis for the present discussion. The physics of the problem considered is illustrated in Fig. 6-16. The flow approaches the blade row at a supersonic velocity, passes through an oblique bow shock wave and undergoes a supersonic expansion over the suction surface of the blade. Finally, the expanded, higher Mach number flow passes through the stronger normal shock wave denoted as the passage shock wave in Fig. 6-16. The approach used is to compute the blade surface Mach number modeled as a simple PrandtlMeyer isentropic expansion. The passage shock loss is then calculated using an upstream Mach number that is the average of the blade inlet Mach number and the expanded flow blade surface Mach number. Here, Swan’s (1961) ideal gas model will be generalized somewhat so that it can be used for non-ideal gas equations of state as well. The Prandtl-Meyer expansion is accomplished for a basic double-circular-arc blade profile. The flow is assumed to approach the blade row at a flow angle tangent to the blade suction surface. This is considered fairly representative of the geometry for any profile likely to be used for supersonic inlet flow. And the rather narrow incidence angle operating range expected for supersonic inlet flow also makes the suction surface tangency assumption reasonable. From Figs. 6-16 and 6-17, it is seen that the flow must expand through an angle φ along the suction surface. If the blade profile is assumed to have a sharp leading edge, Eqs. (4-7) and (4-10) can be used to define the upper or suction surface of the blade as 2Ru / c = sin(θ u / 2)

(6-61)

2bu / c = tan(θ u / 4)

(6-62)

FIGURE 6-16 Shock Wave Loss Model



FIGURE 6-17 Expansion Angle Geometry

where θu is the upper surface “camber angle” and Ru is the upper surface radius of curvature. Similarly, the mean camberline circular-arc can be expressed by 2R / c = sin(θ / 2)

(6-63)

2b / c = tan(θ / 4)

(6-64)

where θ is the true camber angle, R is the camberline radius of curvature, and b is analogous to bu of Fig. 6-17, but for the mean camberline. Since bu – b = tb / 2, as can be seen in Fig. 4-8, it follows that tan(θ u / 4) = tan(θ / 4) + tb / c

(6-65)

For a circular-arc, the angle of the arc with the chord line at the inlet is given by Eq. (4-35), i.e.,

χ u1 = θ u / 2

(6-66)

Since the flow angle is assumed to be equal to the suction surface inlet angle, Eqs. (4-32) and (6-66) yield

β1 = 90o − ψ = θ u / 2 + γ

(6-67)

ψ = 90o − θ u / 2 − γ = 90o − θ u / 2 − κ1 + θ / 2

(6-68)

where γ is the blade stagger angle and κ1 is the blade inlet angle, as shown in Fig. 6-1. The law of sines and basic trigonometry applied to the triangle shown in Fig. 6-17 yields tan φ =

s cos ψ s sin ψ + Ru

(6-69)



Swan (1961) develops Eq. (6-69) in a more general form by including the leading edge nose radius. That is a relatively insignificant effect, considering the approximate nature of the model. Hence, the assumption of a sharp leading edge is used here to avoid the need to specify the blade nose radius for an aerodynamic performance analysis. The blade surface Mach number, M′S, and velocity, WS, entering the passage shock wave is given by the well-known Prandtl-Meyer isentropic expansion through the expansion angle φ. In general form, this can be expressed as WS

φ=

∫

M′ 2 − 1

W1

dW W

(6-70)

For thermally and calorically perfect gases, this equation can be integrated analytically and expressed in terms of the Prandtl-Meyer angle, which is usually designated as ν(M). To permit use of the model for any appropriate equation of state, Eq. (6-70) must be integrated numerically, noting that the Prandtl-Meyer expansion is an isentropic process. The shock wave total pressure loss is calculated for a normal shock wave with an inlet Mach number of Min ′ = M1′MS′

(6-71)

Again, for thermally and calorically perfect gases, the total pressure loss across a normal shock wave can be expressed analytically. To generalize the calculation for any appropriate equation of state, a numerical solution is required. This involves simple conservation of mass, momentum and energy across the shock wave in the form ( ρ W )in = ( ρ W )out 2

(6-72) 2

( P + ρ W )in = ( P + ρ W )out

(6-73)

Hin = Hout

(6-74)

sout ≥ sin

(6-75)

Equations (6-72) through (6-74) can be solved by simple iteration, noting that the flow downstream of the shock wave is subsonic. Equation (6-75) requires that the second law of thermodynamics be satisfied as stated in Eq. (2-6). Then the downstream total pressure is calculated to yield the desired total pressure loss across the shock wave.

6.8 OFF-DESIGN CASCADE PERFORMANCE CORRELATIONS The calculation of loss and fluid turning at off-design incidence angles is more complex than for the design incidence angle. Although two-dimensional cascade



test data provide some useful guidance, the off-design performance of an annular cascade in an axial-flow compressor is far more complex. Successive blade rows in a compressor are normally closely spaced, resulting in significant interaction between them. Two-dimensional cascade test data is based on measurements rather far downstream where blade wakes and flow distortion is minimal. That is far from the case encountered in a compressor. Rotating stall can significantly influence the flow in a compressor cascade, particularly when the compressor operates at rotation speeds well below its design speed. Two-dimensional cascade test data provides no insight into this phenomenon. A very fundamental difference between compressor cascades and cascade testing is the fact that the flow is far from two-dimensional in the compressor. In a compressor, significant changes in axial velocity and radius across the cascade are common and end-wall boundary layers often influence a substantial portion of the flow field. By contrast, substantial effort is made to minimize these effects in two-dimensional cascade tests, typically by use of boundary layer suction. Consequently, off-design blade row performance empirical model development requires calibration against axial-flow compressor performance data, including a range of compressor types and operating conditions. It follows that these empirical models are substantially influenced by the overall strategy used for the compressor performance analysis. In effect, it soon becomes impossible to separate the parts from the whole. Numerous approximations are required to model the through-flow, end-wall boundary layers, tip clearance, shroud leakage, etc., in a performance analysis. It is important to recognize that the off-design empirical correlations suggested here are simply methods that have been found effective in the context of this writer’s performance analysis as it is described in this book. The blade row incidence angle and the axial velocity ratio across the cascade significantly influence the off-design deviation angle. Johnsen and Bullock (1965) provide an empirical model for variation of the deviation angle with incidence angle at the design incidence angle. This graphical correlation is shown in Fig. 6-18, along with predictions from the following empirical equation: *

 ∂δ  4 2.5  ∂i  = [1 + (σ + 0.25σ )(β1 / 53) ] / exp(3.1σ )  

(6-76)

G. Mellor has developed plots of β2 as a function of β1 from NACA cascade data (see Fig. 3.9, Horlock, 1958). These are equivalent to plots of δ versus i. A schematic typical of Mellor’s constant-stagger angle characteristics is shown in Fig. 6-19. It can generally be concluded that the deviation angle approaches a constant value near the negative stall incidence angle. At the positive stall incidence angle, the slope of the β2 versus β1 curves approaches unity. For some time, this writer used these two observations and Eq. (6-76) in several unsuccessful attempts to construct a general deviation angle correlation as a function of incidence angle to match the observed behavior at ic, i* and is, as suggested by Novak (1973). While this produced reasonable results for certain applications and operating conditions, definite exceptions seemed to always exist where this type of model could not match actual compressor performance data. The requirement



FIGURE 6-18 Off-Design Deviation Slope

FIGURE 6-19 Schematic of 2 Versus 1 in Cascades



that the fluid turning does not increase with incidence beyond the positive stall incidence angle is particularly troublesome when analyzing the performance of compressors. While this appears to be the case in cascade test results, it does not permit predicting compressor performance at some of the very severe incidence angles at which they are often capable of operating. After experimenting with various modifications and limits, a somewhat arbitrary model was tried, assuming a linear variation of deviation angle with incidence angle with a slope given by Eq. (6-76). This produced such a dramatic improvement in the accuracy and versatility of the compressor performance analysis that the attempt to define a more general empirical correlation was abandoned. The axial velocity ratio across the cascade also has a definite influence on the deviation angle. The correction developed by Pollard and Gostelow (1967) from cascade test results is about as good as any, although it is based on rather limited evidence. Combining their axial velocity correction with the assumed linear variation of deviation angle with incidence angle results in the expression used by this writer for offdesign deviation angle prediction. *

 ∂δ  δ = δ * +   (i − i* ) + 10(1 − Wm2 / Wm1)  ∂i 

(6-77)

where i and δ are expressed in degrees. Inlet guide vanes require special treatment. It was noted previously in this chapter that inlet guide vanes usually function like turbine nozzle blade rows, where the throat area at the blade discharge largely dictates the discharge flow angle. Hence, the second term on the righthand side of Eq. (6-77) is omitted for inlet guide vanes, such that deviation angle is not influenced by incidence angle. This is an important feature when compressors employ adjustable inlet guide vanes. In those cases, the vanes may be operated at extremely large incidence angles. Comparison of predicted and measured performance of those types of axial-flow compressors shows no evidence of any incidence angle influence on the deviation angle of the inlet guide vanes. Figure 6-20 illustrates the model used for off-design loss coefficient. Define a normalized incidence angle parameter as

ξ = (i − im ) / (is − im ); i ≥ im

(6-78)

ξ = (i − im ) / (im − ic ); i < im

(6-79)

– , the offDesignating the upstream shock wave loss coefficient (if any) by ω s design loss coefficient is given by

ω = ω s + ω m [1 + ξ 2 ]; − 2 ≤ ξ ≤ 1

(6-80)

ω = ω s + ω m [5 − 4(ξ + 2)]; ξ < −2

(6-81)

ω = ω s + ω m [2 + 2(ξ − 1)]; ξ > 1

(6-82)

As noted in Fig. 6-20, Eqs. (6-81) and (6-82) are simple linear extrapolations of Eq. (6-80) outside of its designated range of application. The values of the loss



FIGURE 6-20 Off-Design Loss Coefficient

coefficients at ic, im and is are consistent with the definitions of those incidence angles established earlier in this chapter. The remainder of the correlation is completely empirical, based on optimizing the performance prediction accuracy for a wide range of axial-flow compressor types and operating conditions. The lower limit imposed on Eq. (6-80) is largely a safety feature, since operation at such large negative incidence angles is almost never encountered. By contrast, the upper limit imposed on Eq. (6-80) and the extrapolation defined by Eq. (6-82) are essential, and are commonly encountered, particularly when an axial-flow compressor operates at speeds well below its design speed. Limiting the rate of increase in loss coefficient with incidence angle in these conditions really compensates for the fact that the compressor is likely to operate in rotating stall. These stall zones effectively block some of the blade passages such that the incidence angles are not as large as those indicated by an ideal, axisymmetric flow analysis. This writer’s performance analysis makes further provision for this type of operation by imposing an area blockage, Bwake, due to the blade wakes. This is based on the blade loading level indicated by Lieblein’s equivalent diffusion factor evaluated at off-design conditions, i.e., using the off-design incidence and deviation angles in Eq. (6-39) to compute the off-design equivalent diffusion factor using Eq. (6-30). The wake blockage is given by Bwake = 0; Deq ≤ 2

(6-83)

Bwake = 1 − (2 / Deq )0.9 ; Deq > 2

(6-84)



It will be seen later in this book that the through-flow analysis interprets Bwake as the fraction of the stream sheet area that is unavailable for through flow.

6.9 BLADE TIP CLEARANCE LOSS The calculation of the blade tip clearance loss is based on the same semi-empirical model used for centrifugal compressor impellers in Aungier (2000). Figure 6-21 shows the tip clearance geometry for a typical rotor blade. The situation for unshrouded stator blades is similar, except that the clearance lies along the hub contour. The pressure difference on the two sides of the blade produces a leakage flow through the clearance gap, basically dissipating the pressure difference. The pressure difference across the blade must balance the blade torque as given in Eq. (3-7). For the clearance gap, this can be expressed as

τ = πδ c [( rρ Cm )1 + ( rρ Cm )2 ][r2Cθ 2 − r1Cθ1]

(6-85)

The average pressure difference across each blade in the blade row is ∆P = τ / ( Zrtipδ cc cos γ )

(6-86)

where Z is the number of blades in the blade row. The fluid velocity of the leakage flow is estimated from ∆P and the assumed throttling coefficient of Aungier

FIGURE 6-21 Blade Tip Clearance Geometry



(2000) for the first blade row, but is reduced as the blade row number, Nrow, increases, i.e., 0.2 Uc = 0.816 2∆P / ρ / Nrow

(6-87)

As will be seen in Chapter 8, end-wall boundary layer growth in multistage axial-flow compressors results in a substantial tangential blade force defect, which is expected to reduce the tip leakage flow. The dependence on Nrow was determined empirically from comparison of predicted and measured performance of several multistage axial-flow compressors. The leakage mass flow rate is given by ˙ c = ρ UcZδ cc cos γ m

(6-88)

The clearance gap total pressure loss for the entire blade row is ˙c/m ˙ ∆Pt = ∆P m

(6-89)

This total pressure loss is clearly concentrated in the clearance gap region of the flow field. But if it is applied in that fashion, losses along the end-wall will accumulate from each blade row, eventually causing the through-flow analysis to diverge. The basic problem is that the flow in a multistage compressor undergoes considerable fluid mixing at each blade row due to secondary flows. But a conventional through-flow analysis does not account for this. Hence, once a loss is imposed on a stream sheet in the analysis, it stays on that stream sheet through the remainder of the compressor. To avoid solution divergence, this writer imposes the leakage total pressure loss as a linear distribution across the annulus, such that the integrated ∆Pt is equal to the value given by Eq. (6-89), but the total pressure loss is zero at the wall opposite from the clearance gap. Since successive blades encounter clearance (or shroud seal leakage) losses on alternate end-wall contours, this produces a mild concentration of these losses near the end-walls, but with about half of the loss applied at mid-passage. Experience has shown that this yields a stable performance analysis that correlates well with overall compressor experimental performance data.

6.10 SHROUD SEAL LEAKAGE LOSS Figure 6-22 illustrates an alternate style of blade tip geometry that is often used on stator blade rows. Here a shroud band is attached integral with the blades. Typically seal fins are attached to the shroud band to reduce the clearance gap leakage flow, yet protect the compressor from serious damage in the event that a rotor excursion should cause the shaft to come in contact with the shroud. The seal allows larger clearances between the shroud band and the shaft to be used without causing excessive leakage. If contact occurs, the seal strips will be sacrificed, but the more expensive components should be undamaged.



FIGURE 6-22 Shroud Seal Geometry

The total pressure loss due to shroud leakage is that given by the leakage mass flow and the pressure difference across the blade row, i.e., ˙ leak / m ˙ ∆Pt = ∆P m

(6-90)

This total pressure loss is distributed across the annulus in exactly the same fashion as the tip clearance loss. The leakage mass flow through the labyrinth seal can be computed by the method of Egli (1935), who expresses the leakage mass flow in the form ˙ leak = 2π rsealδ cCtCcCr ρ RT m

(6-91)

Aungier (2000) provides general empirical equations to approximate the coefficients in Egli’s model. Here, a simplified model can be used, since the number of fins, N, in a shroud seal is always relatively small. The contraction ratio is approximated by Cr = 1 −

1   54.3 3+   1 + 100δ c / t 

3.45

(6-92)



If PR designates the ratio of the lower pressure to higher pressure across the seal, the throttling coefficient is approximated by Ct =

2.143 ln( N ) − 1.464 [1 − PR ](0.375PR ) N − 4.322

(6-93)

If N ≤ 12, the carryover coefficient is approximated as Cc = 1 +

X1[δ c / p − X2 ln(1 + δ c / p)] 1 − X2

(6-94)

X1 = 15.1 − 0.05255 exp[0.507(12 − N )]

(6-95)

X2 = 1.058 + 0.0218N

(6-96)

Equation (6-94) yields a maximum when δc / p = X2 – 1. This represents a poor seal design, which exceeds the range of Egli’s model. It is rarely encountered, but it is wise to require δc / p ≤ X2 – 1 when applying Eq. (6-94). Aungier (2000) provides a more detailed discussion and the extension of the empirical equations to N > 12.

6.11 IMPLEMENTATION, EXTENSIONS AND ALTERNATE METHODS Integration of empirical performance models into an axial-flow compressor performance analysis is by no means a trivial process. Chapters 7 and 8 discuss the through-flow analysis and end-wall boundary layer analysis, which are the other two important components of an aerodynamic performance analysis. It is convenient to discuss the component parts of the performance analysis individually, but the parts cannot really be separated from the whole. An objective of this book is to provide a complete description of the methods used by the author. Completion of the description of the empirical loss models requires a definite qualification. The final adjustments suggested here depend on the other components of the performance analysis to be discussed in the next two chapters, as well as on the use of the complete set of empirical models reviewed in this chapter. The final adjustments may be ineffective if used in a different context. To complete this discussion, the parameter, K1 and K2, in Eq. (6-46) must be provided. These parameters adjust the profile loss model to compensate for loss sources not specifically modeled here, such as end-wall and secondary flow losses. In the simplest form, K1 and K2 may be considered as empirical constants. That is actually not a bad assumption. Rather good performance prediction accuracy has been achieved simply using K1 = 0.004 and K2 ≈ 4. That is possible because extremely low values of the aspect ratio are not normally encountered in typical compressors. It is more prudent to include specific provision for aspect ratio effects as suggested by Howell (1942, 1945) in Eq. (6-42). Consider a simple model for skin friction loss in a cascade. If cf is the skin friction coefficient, this can be represented by



ω SF ∝ cf c / ( s cos β2 )

(6-97)

This can be extended to include an approximate end-wall loss estimate by adding the blade chord-to-height ratio, i.e.,

ω SF ∝ cf [c / ( s cos β2 ) + c / h]

(6-98)

This simple argument leads to the following expression for K2: K2 = 1 + ( s / h) cos β2

(6-99)

The correction is applied to K2 rather than K1, since blade loading is not expected to significantly influence the end-wall loss. By comparison of performance predictions with experiment for a several axial-flow compressors, K1 = 0.0073, combined with Eq. (6-99), was found to yield good results, as will be demonstrated in Chapter 9. It is also useful to include an approximate correction for Reynolds number. Although not normally used in the author’s performance analysis, there are extreme cases where a correction may be necessary. The Reynolds number correction is applied only to the skin friction portion of the simple cascade profile loss, as represented by Eq. (6-37). For application to Eq. (6-46), Eq. (6-99) is generalized to the form K2 = 1 + ( s / h ) cos β2 + 0.004KRe / K1

(6-100)

For laminar flow (Rec < 2.5 × 105), KRe = 2.5 x 105 / Rec − 1

(6-101)

For turbulent flow (Rec > 2.5 × 105),

[

]

KRe = log(2.5 x 105 ) / log(Rec )

2.58

−1

(6-102)

These corrections are based on the blade chord Reynolds number, Rec, derived from classical Reynolds number formulations for boundary layer skin friction coefficients. The skin friction models used can be found in a variety of books on boundary layer theory, e.g., Schlichting (1968, 1979). These corrections are consistent with the correlation in Fig. 6-12, which is based on data with Rec ≈ 2.5 × 105. Fig. 6-23 illustrates the basis of this Reynolds number correction. Cascade loss coefficient data from Johnsen and Bullock (1965) are shown as a function of blade chord Reynolds number. The empirical correction curves shown are obtained by normalizing Eqs. (6-101) and (6-102) to the data point closest to Rec = 2.5 × 105 for each data set.



FIGURE 6-23 Reynolds Number Effect

It is appropriate to close this chapter with a reference to an excellent technical paper by Koch and Smith (1976). They present a very careful development of a significant alternate approach to the present methods. Koch and Smith rely on an independent empirical model for end-wall loss and end-wall work formulated from compressor test data. They developed a profile loss model using a compressible boundary layer analysis, including corrections for Mach number, Reynolds number and stream sheet contraction. An alternate model for bow shock loss is also included. Attempts by this author to employ their approach many years ago were unsuccessful, due to the fact that their end-wall loss model simply did not correlate with experimental data for the specific class of axial-flow compressors of interest at the time. But Koch and Smith have shown good agreement with experiment for a significant number of cases, suggesting that this author’s experience may not be typical. Indeed, this reference has substantially influenced this author’s performance analysis, even though its empirical performance models are not used directly. This will become apparent in Chapter 8 of this book. It is a very significant reference that is highly recommended.

EXERCISES 6.1 Consider NACA 65-series blades with tb / c = 0.1. Hence, a / c = 0.5 and Ksh = Kt,i = 1. Based on Eq. (6-8), what are the independent variables defining the design incidence angle? Repeat for Eq. (6-12).



6.2 Repeat Exercise 6.1, but consider the design inlet angle, β*1, as the dependent variable, instead of i*. 6.3 A performance analysis has been developed following the procedures of this chapter, but using Eq. (6-12) as the basis for the design incidence angle. Is there any reason to compute α* using Eq. (6-8)?


Chapter 7

MERIDIONAL THROUGH-FLOW ANALYSIS

The term meridional through-flow analysis refers to an analysis of the flow in the meridional plane, i.e., a plane defined by a constant polar angle of cylindrical coordinates. A solution in the meridional plane can completely characterize the flow field if the flow is locally axisymmetric. This is usually considered to be a reasonable approximation for hub-to-shroud computing stations located outside of the blade rows. Hence, common practice is to locate all hub-to-shroud computing stations in a meridional through-flow analysis before, between or after the blade rows. This requires a means to define the influence of the blade rows in a form that can be imposed on the solution. Typically, this is accomplished by specifying the flow angle or swirl velocity and the entropy rise or total pressure loss associated with flow passing through the blade row. In the case of performance analysis of an existing axial-flow compressor design, the empirical models of Chapter 6 can be used. When designing an axial-flow compressor, the influence of the blade rows is specified directly. When the meridional through-flow analysis is completed, the geometry of the blade rows is selected to produce the specified influence. A properly formulated meridional through-flow analysis is a very powerful technique that can be used to support a variety of axial-flow compressor aerodynamic design and analysis functions. In this book, this technique will be used for aerodynamic performance analysis, general stage design and complete axial-flow compressor design. This chapter develops the governing equations and describes methods of solution appropriate to these various applications.

NOMENCLATURE A = annulus area a = sound speed Bwake = wake blockage B* = stream surface repositioning damping factor C = absolute velocity F = general function f = general function



H h I KB KW M m ˙ m n P r s T W y z β ε κm λ φ θ ρ ω

= = = = = = = = = = = = = = = = = = = = = = = =

total enthalpy static enthalpy rothalpy boundary layer blockage factor wake blockage factor Mach number meridional coordinate mass flow rate normal coordinate pressure radius entropy temperature relative velocity coordinate along a quasi-normal axial coordinate flow angle with m-direction φ – λ; deviation of quasi-normal from a true stream surface normal stream surface curvature quasi-normal angle, Eq. (7-1) stream surface angle with axial direction polar angle gas density rotation speed (radians/second)

Subscripts h m s t θ 1 2 3

= = = = = = = =

hub parameter meridional component shroud parameter total thermodynamic condition tangential component condition at point preceding point being considered condition at point being considered condition at point following point being considered

Superscripts ′ = relative condition * = parameter on stream surface where Wm is specified for annulus sizing

7.1 Meridional Coordinate System Figure 7-1 illustrates the basic meridional coordinate system used in a meridional through-flow analysis. A series of meridional computing stations or


Meridional Through-Flow Analysis • 155

FIGURE 7-1 Definition of the Problem

straight-line quasi-normals are distributed through the compressor, all of which lie outside of the blade rows. The annulus is divided into a series of stream tubes separated by stream surfaces. A stream surface is defined as a surface having no mass flow across it, or equivalently, no velocity component normal to it. Hence, the hub and shroud contours always define stream surfaces. The intermediate stream surfaces are normally defined by requiring that the mass flow between each stream surface and the hub stream surface be constant through the solution domain. Quasi-normals in axial-flow compressors are often simple radial lines, although provision should be made for non-radial quasi-normals to treat cases such as that of the inlet section illustrated in Fig. 7-1. Figure 7-2 illustrates the quasi-normal construction in further detail. As the name implies, the intention is to construct the quasi-normal to be approximately normal to the stream surfaces. The quasi-normal is defined by the (z, r) coordinates of both end points or by one end point plus the quasi-normal angle, λ. As illustrated in Fig. 7-2, the quasi-normal angle is defined by tan λ =

∆z zh − zs = ∆r rs − rh

(7-1)

The stream surface slope angle, φ, is given by sin φ =

∂r ∂m

(7-2)



FIGURE 7-2 Quasi-Normal Construction

where m is the meridional coordinate, measured along a stream surface. Noting that λ is also the angle between a normal to the quasi-normal and the axial direction, the angle, ε, between a quasi-normal and a true stream-surface normal is given by

ε =φ−λ

(7-3)

Quasi-normals do not have to be defined with extreme precision, but it is preferred that ε be relatively small if possible to achieve better numerical accuracy. This means that ε on the hub-and-shroud contours should be approximately equal in magnitude, but opposite in sign. In axial-flow compressors, this is usually not an important consideration since a simple radial quasi-normal normally results in very small values of ε. But for cases like that of the inlet passage illustrated in Fig. 7-1, a little more care may be needed. In Chapter 3, the governing equations for adiabatic inviscid flow were developed in the natural coordinate system (θ, m, n). This coordinate system is convenient for derivation of the equations, but less convenient for solution. The reason is that two of the coordinates, m and n, must be determined as part of the solution. It is advantageous to maintain the stream surface coordinate, m, despite the need to compute it in the solution, since many of the basic conservation equations explicitly apply along m. But the determination of n is an unnecessary complication offering no benefit to a numerical analysis. Katsanis (1964) suggested the use



of the quasi-normal coordinate, y, shown in Fig. 7-2, to avoid that complication. It is easily shown that derivatives with respect to n can be expressed in the form 1 ∂ ∂ ∂  =  − sin ε  ∂n cos ε  ∂y ∂m 

(7-4)

Since y is a fixed coordinate at all computing stations, the numerical analysis is greatly simplified.

7.2 INVISCID ADIABATIC FLOW ON A QUASI-NORMAL The relevant equations for adiabatic inviscid flow are developed in general form in Chapter 3. For the present application, Eqs. (3-21), (3-25), (3-29) and (3-30) can be used after they are simplified to their axisymmetric, time-steady form. It will be convenient to satisfy conservation of mass in integral form instead of using Eq. (3-21). Conservation of mass along a quasi-normal can be expressed in the form ys

˙ = 2πK B ∫ KW rρ Wm cos ε dy m

(7-5)

0

KB is the end-wall boundary layer blockage factor, which corrects the area available for through-flow for viscous blockage effects. It is the fraction of the total area available for through-flow after subtracting the hub-and-shroud boundary layer displacement thicknesses from the overall quasi-normal length. Methods to estimate KB will be described in Chapter 8. For now, it is simply recognized that it must be specified in some manner. KW is the blade wake blockage factor, which serves a similar purpose in correcting for local wake blockage. From Eqs. (6-83) and (6-84), it is given by KW = 1 − Bwake

(7-6)

The axisymmetric, time-steady tangential and normal momentum equations follow directly from Eqs. (3-29) and (3-30). ∂( rWθ + ω r 2 ) ∂rCθ = =0 ∂m ∂m

κ mWm2 +

∂Wm ∂I ∂s Wθ ∂( rWθ + ω r 2 ) + Wm = −T r ∂n ∂n ∂n ∂n

(7-7) (7-8)

where κm is the stream surface curvature given by Eq. (3-26), i.e.,

κm = −

∂φ ∂m

(7-9)



The axisymmetric, time-steady energy equation follows directly from Eq. (3-25). ∂I =0 ∂m

(7-10)

Substitution of Eqs. (7-7) and (7-10) into Eq. (3-28) shows that for axisymmetric flow, entropy must also be conserved in the meridional direction, i.e., ∂s =0 ∂m

(7-11)

At first glance, Eqs. (7-7) and (7-11) would appear to preclude use of this model for axial-flow compressors. It is known that entropy and angular momentum change along stream surfaces when the flow passes through blade rows. But these equations do not preclude changes between quasi-normals. They only require that the local gradients be zero for the flow to be locally axisymmetric. Introducing Eq. (7-4) into Eq. (7-8), and simplifying the result with Eqs. (7-7), (7-10) and (7-11), yields the following expression for the normal momentum equation. Wm

∂Wm ∂Wm Wθ ∂( rCθ ) ∂I ∂s + κ mWm2 cos ε − Wm sin ε + = −T r ∂y ∂m ∂y ∂y ∂y

(7-12)

Equation (7-12) can also be expressed in terms of the relative flow angle, noting that Wθ = Wm tanβ′.

Wm

∂Wm ∂Wm Wm tan β ′ ∂( rWm tan β ′ + ω r 2 ) + κ mWm2 cos ε − Wm sin ε + r ∂y ∂m ∂y (7-13) ∂I ∂s = −T ∂y ∂y

After some basic algebra and trigonometry, Eq. (7-13) simplifies to the form ∂Wm ∂Wm Wm2 tan β ′ ∂( r tan β ′ ) 2 W W κ cos ε sin ε + − + m m m r ∂m ∂y cos2 β ′ ∂y ∂I ∂s +2Wmω tan β ′ cos λ = −T ∂y ∂y Wm

(7-14)

It is convenient to express the normal momentum equation in the following general form: ∂Wm f ( y) = f1( y)Wm + f2 ( y) + 3 ∂y Wm

(7-15)



When Cθ and Wθ are known, the functions in Eq. (7-15) are f1( y) = −κ m cos ε +

sin ε ∂Wm Wm ∂m

f2 ( y) = 0 f3 ( y) =

∂I ∂s Wθ ∂( rCθ ) −T − ∂y ∂y ∂y r

(7-16) (7-17) (7-18)

When β′ is known, the functions are  tan β ′ ∂( r tan β ′ ) sin ε ∂Wm  f1( y) = cos2 β ′ −κ m cos ε − +  r ∂y Wm ∂m  

(7-19)

f2 ( y) = −2ω cos β ′ sin β ′ cos λ

(7-20)

 ∂I ∂s  f3 ( y) = cos2 β ′  − T  ∂ y ∂ y 

(7-21)

Numerical integration of Eq. (7-15) is straightforward if the functions f1, f2 and f3 are truly functions of y only. But it can be seen that f3 includes a term involving the meridional gradient of Wm, which appears to depend on the flow solution on other quasi-normals. Novak (1967) shows that this difficulty can be removed by use of the differential form of the continuity equation, i.e., Eq. (3-21). For timesteady, axisymmetric flow, this reduces to ∂( rρ Wm ) + κ n rρ Wm = 0 ∂m

(7-22)

Combining Eqs. (3-27) and (7-21) and expanding the result yields 1 ∂Wm 1 ∂ρ sin φ ∂φ + + + =0 ρ ∂m ∂n Wm ∂m r

(7-23)

Since entropy is conserved along stream surfaces, the term involving the gradient of ρ can be expanded using Eq. (2-26) to yield 1 ∂ρ 1 ∂P  ∂ρ  1 ∂P =   = ρ ∂m ρ ∂m  ∂P  s ρ a2 ∂m

(7-24)

Finally, introducing Eq. (3-22) for time-steady axisymmetric flow yields 1 ∂ρ 1  C 2 sin φ ∂Wm  = 2 θ − Wm  ρ ∂m a  r ∂m 

(7-25)



Combining Eqs. (7-23) and (7-25) yields 1 ∂Wm sin φ ∂φ 2 (1 − Mm ) = −(1 + Mθ2 ) − Wm ∂m r ∂n

(7-26)

where the meridional and tangential Mach numbers are given by Mm = Wm / a

(7-27)

Mθ = Cθ / a

(7-28)

Introducing Eqs. (7-4) and (7-9) into Eq. (7-26) yields 1 ∂Wm sin φ 1 ∂φ 2 (1 − Mm ) = −(1 + Mθ2 ) − − κ m tan ε Wm ∂m r cos ε ∂y

(7-29)

Hence, Eq. (7-29) can be used to evaluate the meridional gradient of Wm on any quasi-normal, independent of the solution on other quasi-normals. It is necessary to take the precaution of avoiding a singularity only if Mm = 1 should occur. This writer imposes the following constraint when applying Eq. (7-29). 2 1 − Mm ≥ 0.1

(7-30)

Equation (7-15) can also become singular if Wm = 0. Aungier (2000) avoids that problem by using conservation of mass in a stream tube in the form ˙ = ρ Wm ∆A ∆m

(7-31)

The stream tube area term is given by ∆A = 2π rKW cos ε ∆y

(7-32)

Typically, all stream tubes are assumed to contain equal mass flows, although alternate definitions can certainly be used. Now introduce the function, f4, given by f4 ( y) = f2 ( y) + f3 ( y)

ρ ∆A ˙ ∆m

(7-33)

Then, Eq. (7-15) can be written as ∂Wm = f1( y)Wm + f4 ( y) ∂y

(7-34)

The solution of this linear differential equation can be found in almost any textbook on differential equations as


Meridional Through-Flow Analysis • 161 y

Wm ( y) = Wm (0)F( y) + F( y)∫

0

f4 ( y) dy F ( y)

(7-35)

where the function F(y) is given by y  F( y) = exp  ∫ f1( y)dy   0 

(7-36)

The meridional velocity on the hub contour, Wm(0), is the constant of integration for Eq. (7-34). It is determined from conservation of mass through Eq. (7-5). Equations (7-34) and (7-5) are solved in an iterative numerical scheme, successively improving the estimate of Wm(0) until mass is conserved and the normal momentum equation is satisfied. This requires calculation of thermodynamic properties such as ρ and a, using an appropriate equation of state from Chapter 2. At any point, the relative total enthalpy is given by Eq. (3-13), i.e., H ′ = I + 12 ( rω )2

(7-37)

The local static enthalpy is given by h = H ′ − 12 W 2

(7-38)

Then static thermodynamic conditions are computed from relative total thermodynamic conditions for the change in enthalpy, (h – H′), while holding entropy constant. When computing the flow profile on a quasi-normal, it is also necessary that the numerical analysis be able to recognize choked flow. The choke condition corresponds to the maximum mass flow rate that can pass through the annulus for the specified total thermodynamic conditions and swirl velocity or flow angle. One way to identify choke is to compare mass flow rates calculated from Eq. (75) for two different values of Wm(0). If the calculated mass flow and Wm(0) vary in opposite directions, the higher value of Wm(0) is beyond the choke limit. An iteration scheme can be used to converge on the actual choking value of Wm(0). It has been found to be simpler, and equally effective, to monitor the average meridional Mach number, Mm, of Eq. (7-27). For uniform, swirling flow in an annulus, it can be shown that the condition for choke is Mm = 1. For the more general case considered here, a reasonable criterion for choke is 1 ys

ys

∫ Mmdy ≥ 1

(7-39)

0

Indeed, this parameter is easily employed to limit the value of Wm(0) used while seeking to converge on the mass flow. The existence of a choke condition at



a quasi-normal is not necessarily an indication of choke for the compressor. It can be caused by incorrect stream sheet geometry in the early iterations. Hence, the overall numerical scheme must include provision to continue the analysis until it is clear that the choke condition is real. But during the process of calculating the flow profiles on a specific quasi-normal, it does represent the maximum mass flow rate that can pass through the annulus. The procedure and equations outlined in this section can be used to compute the flow along a quasi-normal under the following conditions: • The stream surface coordinates, slopes and curvatures along the quasinormal are specified. • The total thermodynamic conditions along the quasi-normal are specified. • The flow angle or the swirl velocity distribution along the quasi-normal is specified. What remains is description of how these conditions are to be established in the overall numerical analysis. For generality, the procedures described are presented for the rotating frame of reference. They are equally valid for the stationary frame of reference if ω = 0 and C, β and H are substituted for W, β′ and I, respectively.

7.3 LINKING QUASI-NORMALS Next, the determination of the total thermodynamic conditions and swirl or flow angle profiles on a quasi-normal will be considered. It will continue to be assumed that the stream surface coordinates, slopes and curvatures are all known. This section describes techniques needed to link successive quasi-normals together so that the quasi-normal flow analysis of the previous section can be conducted on all quasi-normals for a specified stream surface pattern. To start the process, boundary conditions are needed for the first quasi-normal. The usual process is to supply specifications of the distributions of the inlet flow angle or absolute tangential velocity and inlet total thermodynamic conditions (e.g., Pt and Tt) along the first quasi-normal. Values of these parameters on the stream surfaces of the first quasi-normal are determined by interpolation from these profile specifications. Then the procedures of the previous section can be used to compute the inlet flow profiles. For all quasi-normals after the first one, the process required is to link the quasi-normal being analyzed to the upstream quasi-normal, where all flow data are known. Subscript 1 will be used to designate known data on the upstream quasi-normal, and subscript 2 will be used to designate conditions on the quasinormal being analyzed, where both apply to the same stream surface. The simplest case is successive quasi-normals in a simple annular passage with no blade row between them. In this case, Eqs. (7-7), (7-10) and (7-11) provide the linking relations, i.e., ( rCθ )2 = ( rCθ )1

(7-40)



I2 = H2 − ω ( rCθ )2 = I1 = H1 − ω ( rCθ )1

(7-41)

s2 = s1

(7-42)

From Eqs. (7-40) and (7-41), it is seen that H2 = H1 is the true linking condition implied by Eq. (7-41). Hence, all data needed for the quasi-normal flow analysis of the previous section are available. When a blade row lies between the two quasi-normals, the empirical models of Chapter 6 are used to estimate the influence of the blade row. The process will be described in the rotating frame of reference, recognizing that it is applied to a stationary blade by simply setting ω = 0. The empirical models of Chapter 6 supply the blade row total pressure loss coefficient and discharge relative flow angle. But the empirical models require knowledge of the discharge meridional velocity, which is not yet known. Hence, an iterative solution procedure is required, typically starting with the assumption that Wm2 = Wm1 on all stream surfaces. The estimate of the discharge meridional velocity profile is improved by successive application of the empirical models of Chapter 6 and the quasinormal flow analysis of the previous section until the process converges. First, the inlet relative conditions are computed from the known upstream absolute flow conditions. I1 = H1 − ω ( rCθ )1

(7-43)

Wθ1 = Cθ1 − r1ω

(7-44)

W1 = Wm21 + Wθ21

(7-45)

H1′ = h1 + 12 W12

(7-46)

Other relative total conditions (e.g., P′t1 and T′t1) can be computed from the equation of state and the known values of entropy and relative total enthalpy. At the discharge station, conservation of rothalpy requires I2 = I1

(7-47)

H2′ = I2 + 12 ( r2ω )2

(7-48)

The ideal (no loss) discharge total pressure is computed from the equation of state, using the known discharge relative total enthalpy and the inlet entropy. Then the actual discharge relative total pressure is computed from the total pressure loss coefficient. Pt′2 = Pt′2id − ω ( Pt1′ − P1)

(7-49)

All other relative total thermodynamic conditions and the entropy at the discharge are computed using the equation of state and the known relative total pressure and relative total enthalpy. Hence, all data required for the quasi-normal



flow analysis of the previous section are known. When the blade performance and discharge flow profile iteration process is converged, absolute discharge conditions are computed as Cθ 2 = Wθ 2 + r2ω = Wm2 tan β2′ + r2ω

(7-50)

H2 = I2 + ω ( rCθ )2

(7-51)

Cm2 = Wm2

(7-52)

2 2 C = Cm 2 + Cθ 2

(7-53)

7.4 REPOSITIONING THE STREAM SURFACES After solving the equations for conservation of mass and momentum to determine all flow field data throughout the solution domain, the new data will generally not be consistent with the stream surface geometry used in the process. The variation of mass flow along any quasi-normal can be easily determined in functional form using a modified form of Eq. (7-5). y

˙ ( y) = 2π ∫ KW rρ Wm cos ε dy m

(7-54)

0

Note that the boundary layer blockage factor, KB, has been omitted in Eq. (7-54). This writer prefers to treat the blockage factor as a simple area correction applied to conservation of mass. In this approach, the hub-and-shroud stream surfaces are always positioned on the corresponding end-wall contours. An alternate approach is to reposition the hub-and-shroud stream surfaces from the end-wall contours by the end-wall boundary layer displacement thicknesses described in Chapter 8. In that case, the lower limit of integration in Eq. (7-54) will be yh rather than zero. That added sophistication has not resulted in any observable improvement to performance prediction accuracy, yet it can often complicate convergence and numerical stability. In either case, the interior stream surfaces are to be repositioned to yield the correct fraction of the mass flow calculated for the shroud stream surface using Eq. (7-54). The actual mass flow is not used so as to avoid any influence from numerical errors in conservation of mass or boundary layer analysis. Both of these numerical calculations are governed by specified convergence tolerances, but neither will yield an exact result. The correct locations of the interior stream surfaces for the computed flow field can be obtained by interpolation from this function to yield the values of y that correspond to the correct fraction of the mass flow function at ys. These computed values of y yield the stream surface coordinates (z, r) for all stream surfaces on all quasi-normals. z( y) = zh + ( zs − zh )( y − yh ) / ( ys − yh )

(7-55)

r( y) = rh + ( rs − rh )( y − yh ) / ( ys − yh )

(7-56)



These equations are written in general form, but are simplified in this writer’s approach, where yh = 0. If the stream surfaces are simply repositioned to these new positions, the streamline curvature numerical technique is known to be numerically unstable. Normal practice is to reposition the stream surfaces to locations that are some fraction, F, of the distance between the old positions and the new positions calculated for the current flow field data. This writer uses the numerical damping procedure suggested by Novak (1973). For quasi-normals outside of the blade passages, such as the present application, Novak recommends (1 − M2 ) y2 1 = 1 + * m 2s F B ( ∆m)

(7-57)

B* is an empirical constant, typically about 8, and ∆m is the minimum meridional spacing with the adjacent quasi-normals. When applying Eq. (7-57), this writer imposes the constraint that Mm ≤ 0.95. Once the stream surfaces have been repositioned, new meridional coordinates are computed for all stream surfaces.   ∂r  2  ∫ 1 +  ∂z   dz z1   z

m=

(7-58)

where the partial derivative in Eq. (7-58) is evaluated numerically from the stream surface (z, r) coordinates and z1 is the value of z at the first quasi-normal. Other stream surface geometry data follow directly from Eqs. (7-1), (7-2), (7-3) and (7-9). Equation (7-58) sets m = 0 at the first quasi-normal, but that is arbitrary, since only relative values of m along a stream surface are significant for the present solution procedure.

7.5 FULL NORMAL EQUILIBRIUM SOLUTION There are several useful solution procedures that can be applied for a throughflow analysis. The most general method is to solve the complete normal momentum equation as given in Eqs. (7-12) and (7-14). From the common practice of using radial lines as quasi-normals, this is often referred to as a full radial equilibrium solution. In the more general form used in this chapter, full normal equilibrium solution is a more appropriate term. Figure 7-3 shows a flow chart of a typical full normal-equilibrium solution procedure referenced to the methods outlined in Sections 7.2 through 7.4. The process starts by initializing the stream surfaces throughout the solution domain. This is usually accomplished by applying Eqs. (7-54) through (7-56), while assuming that the flow is uniform from the hub to the shroud and ε = 0. The inlet boundary conditions are imposed, and the flow is computed for the first quasi-normal using the iterative procedure described in Section 7.2. The process then involves solving of the flow field at all other quasi-normals. In this case, it is necessary to impose the linking calculations of Section 7.3 as well as the iterative procedure of Section 7.2. In



FIGURE 7-3 Full Normal Equilibrium Solution Flow Chart

general, the linking calculations depend on the flow field being calculated, so a second iteration loop is needed to converge on those calculations. When flow field on all quasi-normals has been treated, the stream surfaces can be repositioned using the procedures of Section 7.4. This requires a third or outer iteration loop to converge on the stream surface positions. When the flow field, linking calculations and stream surface positions are all self-consistent within an acceptable tolerance, the through-flow solution is complete. As shown in the flow chart, the end-wall boundary layer calculations described in Chapter 8 are normally carried out during this process, to include the end-wall boundary layer blockage effect. The procedure is quite simple in concept, but a number of complications may be encountered. One common problem is that the flow is choked at a quasi-normal. It is not immediately obvious whether the choke condition is real, since incorrect stream surface positions might cause a false choke indication. The flow chart shows logic to require at least one completed stream surface reposition operation before the choke is considered valid. It may be desirable to require more than one completed reposition option. When the through-flow analysis is conducted interactively on personal computers, the simplest approach is to switch from an automatic iteration procedure to a manual one when choke occurs, such that the user can decide whether to continue for another outer iteration or terminate the solution. Another common complication is convergence problems with the outer iteration loop on stream surface positions. The damping procedure described in Section 7.4 is quite effective, but there is some uncertainty with regard to B* in Eq. (7-57). B* = 8 is a good choice for most problems,



but exceptions do occur. It is good practice to monitor the maximum stream surface position errors on successive iterations. If that error is increasing, B* can be decreased to impose more damping on the numerical analysis. A good way to make this adjustment is to impose the following correction: B* → ( B* + 1) / 2

(7-59)

7.6 SIMPLIFIED FORMS OF THE THROUGH-FLOW ANALYSIS The full normal equilibrium through-flow analysis outlined in the previous sections is commonly referred to as the streamline curvature technique. It is wellsuited for implementation in a relatively robust and reliable numerical analysis. But the streamline curvature technique is by no means totally free of numerical stability and convergence problems. The process of repositioning stream surfaces is, by far, the major source of these problems. It is also the process responsible for most of the computer time required for a through-flow analysis. The main purpose served by this relatively complex process is to determine φ, κm and ε. If these terms are neglected or approximated in some fashion, the entire outer iteration loop of the streamline curvature technique can be eliminated. The solution then becomes a simple marching solution, where the analysis proceeds through the solution domain, treating the quasi-normals in sequence. This is possible because the flow analysis on a quasi-normal becomes totally independent of conditions on downstream quasi-normals. The locations of the interior stream surfaces must still be established using Eqs. (7-54) through (7-56), but now the process is numerically stable and requires no numerical damping procedures. The process of relocating the stream surfaces can then be accomplished as a normal part of the process of solution at each quasi-normal. In principle, this can be accomplished by solving Eqs. (7-2) and (7-9) using upstream finite-difference approximations. However, this really provides an estimate of the stream surface curvature within the blade passage rather than at the blade passage exit where the solution is to be accomplished. It is not at all uncommon for the stream surface curvature to be dramatically different at these two locations, often even having opposite signs. The difficulty arises from the fact that Eq. (7-9) is really equivalent to determining the second derivative of r as a function of z along the stream surface. Upstream finite-difference approximations to second derivatives are often seriously in error. At best, this approach may be capable of accounting for large curvature effects associated with passages having large end-wall contour curvatures. But in the majority of situations encountered in axial-flow compressors, it is better to ignore stream surface curvature entirely than to use upstream finite-difference approximations. There certainly is merit to providing a through-flow analysis with the capability to approximate large stream surface curvatures present when passage curvatures are large. The inlet portion of the flow passage illustrated in Fig. 7-1 is a good example of a situation where stream surface curvature cannot be ignored. Since the end-wall contours are normally completely specified in advance, the



hub-and-shroud stream surface curvatures can be accurately estimated. A good way to extend a simplified through-flow analysis for it to be applicable to problems having significant passage curvature effects, is to assume that φ and κm vary linearly with y from the hub to the shroud. This approach can recognize significant passage-induced stream surface curvatures without the risk of producing the highly erroneous curvatures that can follow from an upstream finitedifference approach. In the context of axial-flow compressor performance analysis, performance predictions using this simplified model and the full normal equilibrium model are consistently found to be virtually identical. And the computer time required for solution is dramatically reduced when the simplified model is used. The situation is quite different when a through-flow analysis is used as part of an aerodynamic design procedure. In these cases, typical practice is to specify one of the end-wall contours and to calculate the other from conservation of mass. This process of sizing the annulus may also include design of the blade rows, or it may employ a standard stage design. In either case, the slopes and curvatures of one of the end-walls cannot be well approximated in a marching type solution. In these cases, it is best to simply neglect stream surface curvature effects entirely. Since design-mode applications are normally restricted to stations before and after blade rows, this is generally a reasonable approximation. Usually any fine-tuning of the design that may be required is easily accomplished with a normal performance analysis. To apply this approximation to the throughflow analysis procedure outlined in the previous sections, it is only necessary to set ε = κm = 0 in Eqs. (7-12) through (7-19). This type of analysis has commonly been called a simple non-isentropic radial equilibrium solution, although in the more general quasi-normal structure used here, it is better described as a simple non-isentropic normal equilibrium solution. There are also applications where the through-flow analysis must be further simplified by ignoring the entropy gradients along the quasi-normal. This occurs in industrial axial-flow compressors when designing a standard stage to be used throughout the compressor. This is often done as a means of minimizing the cost of manufacturing the compressor. Typically, the stagger angles of the blades in this standard stage will be adjusted somewhat through the machine to fine-tune the performance to the customer’s requirements. Since the standard stage may be used in a variety of applications, it is not possible to compute the blade row performance in any general context. Nor do entropy gradient effects have much significance in this case, since they depend on all of the blade rows preceding the location of a specific stage. In this case, it is best to simply ignore entropy gradients as well as the curvature effects. This type of analysis is commonly referred to as a simple radial equilibrium solution. The term simple normal equilibrium is perhaps more appropriate, although generally this type of analysis will employ radial quasi-normals in any case. Figure 7-4 illustrates a typical flow chart for any of these simplified forms of the through-flow analysis. On comparing this flow chart with the one shown in Fig. 7-3, it is seen that the entire outer iteration loop has been eliminated. In this case, repositioning of stream surfaces will be accomplished while computing the flow field at each quasi-normal. The procedure of Section 7.4 is used for this purpose, but without any numerical damping. This type of analysis is extremely fast



FIGURE 7-4 Simplified Through-Flow Analysis Flow Chart

and numerically very stable. The full normal equilibrium model certainly should be included for generality. But it is truly remarkable how seldom it is really required for typical axial-flow compressor applications. In addition, the simplified forms allow application of the through-flow analysis to a number of aerodynamic design functions where use of the full normal equilibrium model would be totally impractical. In summary, a through-flow analysis should normally be developed in a fairly general form to include the capability of employing any of the following aerodynamic models. • • • •

Full normal equilibrium. Approximate (linear, hub to shroud) stream surface slope and curvature. Simple non-isentropic normal equilibrium. Simple (isentropic) normal equilibrium.

In this way, the same through-flow analysis can be applied to a variety of axialflow compressor aerodynamic design and analysis functions.

7.7 ANNULUS SIZING The annulus sizing process mentioned in the previous section is one useful aerodynamic design function that is conveniently incorporated directly into the



through-flow analysis. In this mode of solution, only one of the end-wall contours is specified. The analysis is used to determine the other end-wall contour, while conserving mass and matching a desired distribution of Wm through the compressor. Typically, the Wm distribution is specified for the mean stream surface, although any other stream surface can be used as well. For generality, a superscript, *, will be used to designate parameters on the stream surface for which Wm will be specified. The special requirements for annulus sizing include • Specify z and r for one end-wall contour and the angle, λ, of Fig. 7-2 for all quasi-normals. • Specify values of Wm for the selected stream surface for all quasi-normals. • Neglect stream surface curvature effects, typically using the simple nonisentropic normal equilibrium model. The annulus area, A, at any quasi-normal is given by A = π ( rs + rh ) ys

(7-60)

As an initial estimate for the annulus area, use the specified meridional velocity, the total density and conservation of mass ˙ / ( ρt*Wm* ) A=m

(7-61)

where the inlet total density is used for the first quasi-normal, and its value at the upstream quasi-normal is used for all others. From Eq. (7-60) and the specified value of λ, it is easily shown that rs2 = rh2 + ( A cos λ ) / π

(7-62)

ys = A / [π ( rs + rh )]

(7-63)

zs = zh − ys sin λ

(7-64)

Equations (7-62) through (7-64) yield the coordinates of the unknown end-wall contour from those of the known contour and the passage area. Next, the usual through-flow analysis is conducted while using the specified meridional velocity as a constant of integration in Eq. (7-35). Although Eq. (7-35) is written with the hub meridional velocity (at y = 0) as the constant of integration, the value at y* can be used with a simple substitution. From Eq. (7-35) it is easily shown that

* / F( y* ) − Wm (0) = Wm

y*

∫

0

f4 ( y) dy F ( y)

(7-65)

The annulus sizing differs from the procedures presented for the analysis mode only with regard to the application of Eq. (7-5). Here, it is used to compute the



mass flow rate, m˙ c, for the estimated annulus area. Then the annulus area estimate is improved by ˙ /m ˙c A → Am

(7-66)

and the process is repeated until acceptable convergence on the mass flow rate is achieved.

7.8 NUMERICAL APPROXIMATIONS A numerical analysis based on the procedures described in this chapter is relatively straightforward, except as it relates to approximations for the stream surface curvature terms. When the streamline curvature technique was introduced, common practice was to employ spline-connected cubic approximations for this purpose (e.g, see Walsh et al., 1962). Although the spline fit seems almost ideal for the smooth curves expected, experience eventually convinced most investigators that it is not a particularly good choice. The spline fit has a definite tendency to destabilize the analysis, to increase the demands on the numerical damping procedures. The simple three-point finite-difference approximation for the partial derivatives in Eqs. (7-2) and (7-9) is a much better choice for this application. These are derived from truncated Taylor series approximations for the central point in a series of three points, similar to the derivation of Eqs. (5-35) and (5-36). In the present case, the points are not likely to have equally spacing, as was the case in Chapter 5. A more general approximation can be shown to be  (m3 − m2 )( f2 − f1) (m2 − m1)( f3 − f2 )   ∂f  1 +   ∂m  = m − m  m2 − m1 m3 − m2  2 3 1 

(7-67)

where the subscripts 1, 2 and 3 designate any three successive points along the curve. Similarly, three-point difference approximations can also be derived for end points on the curve to yield  (m3 − m1)( f2 − f1) (m2 − m1)( f3 − f1)  −   m2 − m1 m3 − m1  

(7-68)

 (m3 − m1)( f3 − f2 ) (m3 − m2 )( f3 − f1)   ∂f  1 −   ∂m  = m − m  m3 − m2 m3 − m1  3 2 1 

(7-69)

 ∂f  1  ∂m  = m − m  1 3 2

For this application, which can often induce numerical stability problems, a simple two-point difference approximation is a better choice for end points, i.e.,  ∂f   ∂f  f2 − f1  ∂m  =  ∂m  = m − m  1  2 2 1

(7-70)



Equation (7-70) will also be required for all quasi-normals when the simplified through-flow models of Section 7.6, since data at m = m3 will not be known when solving at m = m2, in those simple marching type analyses. Solution of Eq. (7-35) requires numerical approximations for partial derivatives with respect to y and for integrals with respect to y. Equations (7-67) through (7-69) have been found to be good choices for the derivative approximations at interior and end points, where y is substituted for m. For numerical integration, an approximation for the second derivative at interior values of y is also required. A suitable three-point approximation can be derived from truncated Taylor series to yield  ∂2 f  2  f3 − f2 f −f  − 2 1  2 =  − − y y y y y  ∂y 2 3 1 3 2 2 − y1 

(7-71)

A Taylor series approximation to the integral between points 1 and 2, where point 2 is an interior point, is easily shown to be y2

  ∂f  ( y2 − y1)  ∂2 f  ( y2 − y1)2  f ( y)dy =  f ( y2 ) −   + 2  ( y2 − y1)  2 6  ∂y  2  ∂y 2   y1

∫

(7-72)

Similarly, the integral between points 2 and 3, where point 3 is the last point, is given by y3

  ∂f  ( y3 − y2 )  ∂2 f  ( y3 − y2 )2  f ( y)dy =  f ( y2 ) +   + 2  ( y3 − y2 )   2 6  ∂y  2  ∂y 2   y2

∫

(7-73)

These two equations can be used to compute all integrals with respect to y from 0 to ys for any number of stream surfaces by simple summation of results between successive points. Equations (7-67) and (7-71) provide the approximations for the derivative terms in Eqs. (7-72) and (7-73).

EXERCISES 7.1 Geometry data for an axial-flow compressor blade is normally specified as a function of radius. Data sufficient to define the blade might include the chord, c, the thickness-to-chord ratio, tb / c, and any two of the following: camber angle, θ, stagger angle, γ, inlet angle, κ1, or exit angle, κ2. The number of blades, Z, and the location of maximum camber, a/c, will be constant for each blade row. A through-flow analysis is to be applied in an aerodynamic performance analysis such that the blade geometry on each stream surface is required so that the empirical models of Chapter 6 can be used to



estimate the blade row loss and fluid turning. Consider a stream surface passing through a blade row with inlet coordinates (z1, r1) and exit coordinates (z2, r2). Develop a method to estimate the blade geometry on the stream surface from the known blade geometry as a function of radius. 7.2 Equation (7-29) is recommended for evaluating the meridional gradient of the meridional velocity component. Alternatively, this gradient might be evaluated using the finite-difference approximations of Section 7.8. Discuss the relative merits of these two alternative approximations. 7.3 Equation (7-39) has been recommended as an approximate criterion for choked flow in the annular passage, outside of the blade rows. A rigorous calculation of the choked flow limit could be accomplished by determining the constant of integration, Wm(0), for Eq. (7-35) that yields the maximum mass flow rate. Give two reasons why the approximate criterion should be adequate for a through-flow analysis in an axial-flow compressor. Under what conditions might the more rigorous method be preferred? 7.4 Discuss the advantages and disadvantages of using Eqs. (7-31) through (7-33) as a means of avoiding a singularity in Eq. (7-34). Consider the accuracy of the approximation used for both interior and end-wall stream surfaces. For a fully converged inviscid through-flow solution, where can such a singularity occur? Is the approximation more acceptable if an end-wall boundary layer analysis is conducted as part of the overall solution?


Chapter 8

END-WALL BOUNDARY LAYER ANALYSIS

The flow in the end-wall regions has a substantial influence on the aerodynamic performance of axial-flow compressors. As noted in Chapter 6, a significant portion of the losses in axial-flow compressors is directly associated with the endwall flow. The through-flow analysis of Chapter 7 requires some external specification of the viscous end-wall blockage factor for solution of Eq. (7-5). In addition, individual stage loading limits and the compressor surge flow limit are often associated with end-wall stall. Unfortunately, there is no available theoretical aerodynamic model capable of predicting the detailed behavior of these highly complex end-wall flows. Indeed, even modern computational fluid dynamics (CFD) viscous flow solvers are found to be incapable of resolving many of the important flow patterns that are observed in the end-wall regions of axial-flow compressors. When fundamental analysis techniques are not sufficient to treat a problem of interest, engineers commonly resort to a combination of theoretical and empirical models. That approach is always used when formulating an aerodynamic performance analysis for axial-flow compressors. The role of end-wall boundary layer models used within specific performance analyses varies considerably. It is always necessary to address the problem of end-wall blockage effects to effectively apply an inviscid through-flow analysis to the problem. Attempts to model end-wall work and loss effects from boundary layer analysis results will be briefly discussed in this chapter. But, in this writer’s experience, none of the available end-wall boundary layer models is sufficiently accurate and reliable for that purpose. Chapter 6 has already described empirical models used to extend cascade loss models to account for clearance and end-wall loss effects. This chapter presents an end-wall boundary layer analysis used to account for end-wall boundary layer blockage effects. The blade row performance models of Chapter 6, the through-flow analysis of Chapter 7 and this end-wall boundary layer analysis are the basic components of an aerodynamic performance analysis. Chapter 9 describes the performance analysis and qualifies it by comparing performance predictions with experimental data. In keeping with the stated objective of this book, Chapters 6 through 9 provide a detailed description of the aerodynamic performance analysis. But it should be emphasized that qualification of the performance analysis evaluates its basic components in combination. That type of qualification does not separate the parts from the whole. In Chapters 6 and 7, it was possible to discuss the assumptions, approximations and limitations of the



models. That is not the case in the present chapter. The merits of the present endwall boundary layer analysis cannot be established beyond demonstrating its effectiveness in supporting the methods of Chapters 6 and 7 to predict the overall performance of axial-flow compressors.

NOMENCLATURE B C cf E f g H H1 H2 KB m ˙ m n P q Reθ r s Uleak V W y β γ ∆ δ δc δ* δ 1* δ*2 θ θ11 θ12 θ22 µ ν ρ τ φ ψ ω

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

fractional area blockage absolute velocity skin friction coefficient entrainment function blade force blade row staggered spacing boundary layer streamwise shape factor boundary layer meridional shape factor boundary layer tangential shape factor blockage factor meridional coordinate and tangential velocity profile exponent mass flow rate meridional velocity profile exponent pressure inlet dynamic head momentum thickness Reynolds number radius blade pitch leakage flow tangential velocity velocity relative to the wall velocity relative to the blade row distance normal to the wall flow angle blade stagger angle leakage flow correction parameter boundary layer thickness blade clearance boundary layer streamwise displacement thickness boundary layer meridional displacement thickness boundary layer tangential displacement thickness tangential coordinate and streamwise momentum thickness meridional momentum thickness tangential momentum flux thickness tangential momentum thickness fluid viscosity blade force defect thickness fluid density shear stress contour angle with the axial direction pressure coefficient rotation speed


End-Wall Boundary Layer Analysis • 177

Subscripts e in leak m out s w θ

= = = = = = = =

boundary layer edge condition blade inlet condition seal leakage parameter meridional component blade discharge condition streamwise component wall condition tangential component

Superscripts ′ = = + = - =



parameter in rotating coordinates average of blade inlet and discharge values upstream leakage flow correction downstream leakage flow correction

8.1 HISTORICAL DEVELOPMENT OF END-WALL BOUNDARY LAYER THEORY Early attempts to account for the effects of end-wall boundary layer blockage relied on assigned blockage factors. Typical practice was to assign the fractional area blockage to vary linearly through the compressor (e.g., Sandercock et al., 1954). Alternatively, the boundary layer displacement thickness or the blockage was assumed to vary linearly through the front stages and remain constant in the rear stages (e.g., Voit, 1953; Jansen and Moffet, 1967). These blockage allowances were rather arbitrary, based largely on the investigator’s experience from previous compressors. Indeed, they really were rather arbitrary corrections that appeared to explain differences between calculated and measured compressor performance. Considering the relatively crude calculation methods used, these corrections probably compensated for many weaknesses and omissions in the analyses in addition to the end-wall boundary layer blockage effects. Early attempts to compute end-wall boundary layer blockage in multistage compressors using boundary layer analysis techniques were published simultaneously by Stratford (1967) and Jansen (1967). Both methods seek to predict the average boundary layer growth assuming that the blade forces are conserved inside the boundary layer and that the stream surface slope can be neglected. This reduces the problem to consideration of a simplified axial momentum-integral equation similar to Eq. (3-52) but with sinφ = νm = 0. These assumptions uncouple the axial momentum-integral equation from the tangential momentum-integral equation, so that Eq. (3-53) is unnecessary. Stratford employed flat-plate approximations for the boundary layer shape factor, H, and the wall shear stress. Jansen used an approximate integral solution (Schlicting, 1968 and 1979). Subsequent investigations showed that these early analyses are overly simplified. Nevertheless, they introduced the important concept of analyzing the gap-averaged or pitchaveraged boundary layer flow using integral boundary layer analysis techniques.



Smith (1970) reported detailed end-wall boundary layer data measured in a multistage axial-flow compressor. His measurements clearly demonstrate that blade forces are not constant across the boundary layer. He also provides experimental evidence to support the “repeating stage model.” Basically this model assumes that the end-wall boundary layers achieve an equilibrium condition after passing through several stages, such that the blade row exit and inlet flow profiles are essentially identical. Once the equilibrium condition is reached, boundary layer growth is viewed as primarily a function of aerodynamic loading. Smith suggests that the repeating stage concept has been recognized for many years (e.g., Howell, 1947), but has not been used effectively. Smith provides experimental boundary layer data that may provide at least a preliminary basis for using this model. In a subsequent publication (Koch and Smith, 1976), the same basic boundary layer data was reworked as a sum of the hub-and-shroud boundary layers, as shown in Figs. 8-1 and 8-2. The combined data show less data scatter as compared to Smith (1970). The pressure coefficient, ψ, is a sum of the rotor and stator pressure coefficients for the stage, specifically defined by

ψ =

∆protor + ∆pstator qrotor + qstator

(8-1)

where q is the inlet dynamic head. The meridional displacement thickness, δ 1*, and the tangential force defect thickness, νθ, are defined in Eqs. (3-54) and (3-60),

FIGURE 8-1 Displacement Thickness Data



FIGURE 8-2 Tangential Force Defect Data

and δc is the blade tip clearance. The staggered spacing, g, is a function of the blade pitch, s, and stagger angle, γ, where average values are used, g = s cos γ

(8-2)

Hunter and Cumpsty (1982) report similar results obtained with an isolated rotor. Koch and Smith use the empirical curves shown in Fig. 8-1 to estimate the sum of the displacement thicknesses or blockage. This requires that ψmax values be supplied by some unspecified stall criterion. The displacement thicknesses plus the tangential force defect are then used to estimate the efficiency reduction due to end-wall losses. There is no doubt that Smith made a substantial contribution to our knowledge of end-wall boundary layers. But there is little reason to believe that the empirical models outlined can be used for general application. To recognize that the empirical curves shown are far from correlations of experimental results, one need only note that the tip clearance-to-staggered spacing ratios for all experimental data in Fig. 8-1 lie between 0.028 and 0.062. Careful study of the original reference shows that the experimental data trends contradict the empirical curves about as often as they are in agreement. And the excessive data scatter in Fig. 8-2 provides no real basis for any empirical correlation of the tangential force defect. Although far from a complete end-wall flow model, these references provide important insight into the end-wall boundary layer problem. It is clear that any end-wall boundary layer theory must address the



problem of blade force defects. Substantial tangential force defects clearly do exist. It must be presumed that the same is likely to be true for meridional force defects. But the repeating stage model poses special constraints. For an equilibrium blockage to be established, it is clear that meridional blade force defects must become small, or possibly even negative. Unfortunately, none of the above experimental investigations provided any direct insight relative to the meridional force defect behavior. It is apparent that our knowledge of the mechanisms behind the blade force defects is far from adequate to support an accurate endwall loss calculation method. There have been numerous attempts to formulate end-wall boundary layer analyses since the original attempts by Stratford and Jansen. One of most significant was the work of Mellor and Wood (1971), who provided a rather rigorous development of the basic gap-averaged governing equations. Their development includes effects previously neglected, such as blade force defects and “jump” conditions, to treat boundary layers moving between rotating and stationary frames of reference. They conducted numerous analytical studies to show how these phenomena can be used to explain observed flow behavior not modeled by the simpler methods. Like Smith (1970), Mellor and Wood considered the need to explain the existence of a repeating stage condition to be important. Balsa and Mellor (1975) continued this development by attempting to formulate a usable end-wall boundary layer analysis. In a serious attempt to address the blade force defect problem, the difference in the tangential and meridional momentum defect thicknesses was approximated with a secondary flow model. It was also assumed that the overall blade force remains approximately normal to the mean mainstream velocity vector. These conditions are sufficient to solve the governing equations and do offer a mechanism that can produce the repeating stage condition. This development includes techniques to convert boundary layer data into an end-wall loss prediction. Although undoubtedly the most complete theoretical development up to that time, several important features were largely ignored. The critically important boundary layer shape factor relating the displacement and momentum thicknesses was somewhat arbitrarily assigned to match predictions to experiment in sample overall compressor performance predictions. The boundary layer skin friction coefficient was treated in a similar fashion. The coupling between the tangential and meridional momentum-integral equations was ignored, even though it can be very significant when φ is not zero in Eqs. (3-52) and (3-53). An important series of publications on this subject by Professor Hirsch and associates also should be mentioned. These include Hirsch (1974 and 1976) and De Ruyck et al. (1979 and 1980). These references provide a very detailed discussion of the important force defect terms and investigate various alternative blade force defect models derived from secondary flow theory. They also attempt to model some of the features that were handled rather arbitrarily by Balsa and Mellor. Although the results fall short of a complete end-wall boundary layer theory, these references provide considerable insight into the problem and are definitely recommended. Basically, the current state of the art of end-wall boundary layer theory is insufficient to analyze the complete end-wall flow problem, including the associated end-wall losses. The prediction of end-wall boundary layer blockage to



support the aerodynamic performance analysis of axial-flow compressors is probably the best that can be expected. Nevertheless, the experimental and analytical studies discussed above have clearly identified several important features that strongly influence the end-wall flow problem. Even the more modest goal of predicting end-wall blockage effects requires consideration of the blade force defect terms and should include a mechanism that can produce the repeating stage condition.

8.2 THE END-WALL BOUNDARY LAYER EQUATIONS End-wall boundary layer analysis in axial-flow compressors considers equations for the flow averaged over the pitch between adjacent blades. The governing equations are basically identical to the axisymmetric three-dimensional boundary layer equations presented in Chapter 3. Most investigators have ignored the streamline slope terms in the governing equations, basically assuming that φ = 0. Also, it has been common practice to develop the governing equations in either stationary coordinates or in a coordinate system fixed to the blade rows. In the real problem, the wall and blades rotate at different speeds at the blade tips, unless an attached shroud is present. This effect is usually ignored, except perhaps as a correction to the wall shear stress. Aungier (2000) develops the end-wall boundary layer equations for use in centrifugal compressors, where these effects cannot be neglected. This more general form of the governing equations will be used here also. The absolute velocity will continue to be designated as C and the velocity relative to the blade by W. But the velocity relative to the wall is also needed, and will be designated by V. If the rotation speed of the blade is ω and the rotation speed of the wall is ωw, the three velocities are related by Cm = Wm = Vm

(8-3)

Cθ = Wθ + rω = Vθ + rω w

(8-4)

The no-slip condition requires that the fluid velocity at the wall must vanish in the coordinate system fixed to the wall. So the governing equations should be written in that coordinate system. The axisymmetric three-dimensional boundary layer equations presented in Chapter 3 are valid for any rotating coordinate system. So they are easily transformed to the coordinate system fixed to the wall by substituting V for W and ωw for ω. Consequently, the governing equations for the end-wall boundary layer flow problem are easily shown to be ∂ [rρ eVme (δ − δ1* )] = rρ eVe E ∂m

(8-5)

∂ ∂V 2 [rρ eVme θ11] + δ1* rρ eVme me − ρ eVθe sin φ [Vθe (δ 2* + θ 22 ) + 2ω w rδ 2* ] ∂m ∂m = r[τ mw + fmeν m ]

(8-6)



 ∂V  ∂ 2 [r ρ eVmeVθeθ12 ] + rδ1* ρ eVme  r θe + sin φ (Vθe + 2rω w ) = r 2 [τ θw + fθeνθ ] (8-7) ∂m ∂ m   The blade forces at the boundary layer edge are fme = ρ eVme fθe =

∂Vme ∂Pe sin φ + − ρ e (Vθe + ω w r )2 ∂m ∂m r

 ρ V ∂rCθe ρ eVme  ∂Vθe r + sin φ (Vθe + 2rω w ) = e me r r  ∂m ∂m 

(8-8) (8-9)

Equations (8-7) and (8-9) can be combined to yield ∂ 2 [r ρ eVmeVθeθ12 ] + r 2δ1* fθe = r 2 [τ θw + fθeνθ ] ∂m

(8-10)

If y is the distance normal to the wall and δ is the boundary layer thickness, the various defect thicknesses in the boundary layer equations are defined as δ

ρ eVmeδ1* = ∫ ( ρ eVme − ρ Vm )dy

(8-11)

0

δ

2 ρ eVme θ11 = ∫ ρ Vm (Vme − Vm )dy

(8-12)

0

δ

ρ eVmeVθeθ12 = ∫ ρ Vm (Vθe − Vθ )dy

(8-13)

0

δ

ρ eVθeδ 2* = ∫ ( ρ eVθe − ρ Vθ )dy

(8-14)

0

δ

ρ eVθ2eθ 22 = ∫ ρ Vθ (Vθe − Vθ )dy

(8-15)

0

δ

ν m fme = ∫ ( fme − fm )dy

(8-16)

0

δ

νθ fθe = ∫ ( fθe − fθ )dy

(8-17)

0

Solution of this set of equations requires empirical correlations for the entrainment function, E, the wall shear stresses, τwm and τwθ, and the force defect thicknesses, νm and νθ. It also requires empirical relationships between the mass and momentum defect thicknesses, typically derived from some assumed form of the boundary layer velocity profiles.



8.3 THE BOUNDARY LAYER VELOCITY PROFILE ASSUMPTIONS Aungier [1988(b)] obtained good agreement with experimental data for axisymmetric swirling flow in vaneless annular passages by solving the axisymmetric three-dimensional boundary layer equations using simple power-law velocity profile assumptions. Vm  y  =  Vme  δ  Vθ  y  =  Vθe  δ 

n

(8-18)

m

(8-19)

Substitution of these profile assumptions into the definitions of the various mass and momentum defect thicknesses, assuming density can be regarded as essentially constant, yields the following relationships. n = θ11 / (δ − δ1* − 2θ11)

(8-20)

m = θ12 ( n + 1)2 / [δ − θ12 ( n + 1)]

(8-21)

H1 = δ1* / θ11 = 2n + 1

(8-22)

δ − δ1* = 2H1θ11 / ( H1 − 1)

(8-23)

δ1*

/ δ = n / ( n + 1)

H2 = δ 2* / θ 22 = 2m + 1

δ 2*

/ δ = m / (m + 1)

(8-24) (8-25) (8-26)

These profile assumptions and empirical relations require some modification when applied to flows through blade rows. This can be illustrated by considering the flow through an axial-flow compressor stage on the hub contour. For the rotor, the blades and the hub end-wall rotate at the same speed, so V = W in the boundary layer equations. If the downstream stator is shrouded, both the hub end-wall and the blades are stationary, so V = C. From Eqs. (8-4) and (8-13), it is easily shown that the jump condition between the rotor exit and the stator inlet is Cθ eθ12 = Wθ eθ12 ′

(8-27)

The prime designates the tangential momentum defect thickness viewed in the rotating coordinate system. Since Wθ and Cθ normally have opposite signs, it can be seen that the tangential momentum defect thickness will change sign at any meridional station where the wall rotation speed changes from rotating to stationary, or inversely. Hence, the boundary layer velocity profile assumption must include cases where the θ12, θ22 and δ 2* are negative and H2 < 1. In this simple



illustration, a deficit in tangential momentum in the rotating coordinate system corresponds to excess tangential momentum in the stationary coordinate system. The boundary layer equations can accommodate this jump condition, but the power-law profile assumption of Eq. (8-19) cannot. Aungier (2000) extended the power-law profile model to accommodate this situation. When m < 0.05, Eq. (819) is replaced by Vθ  y  =  Vθe  δ 

0.05

2

 y  y + 0.1705(1 − 20m)1 −     δ  δ 

0.1

(8-28)

Figure 8-3 shows typical velocity profiles from Eqs. (8-19) and (8-28). Both equations yield identical results at m = 0.05. Equation (8-28) simply extends the power-law profiles in a plausible fashion to accommodate cases where the integrated tangential momentum in the boundary layer exceeds the boundary layer edge value, i.e., where m becomes negative. From Eqs. (8-18) and (8-28) and the various boundary layer defect thickness definitions,

δ 2* 20m = δ 21

(8-29)

θ 22 = 0.95m − 1.684m2 δ

(8-30)

θ  ( n + 1.1)( n + 2.1)( n + 3.1) 0.05 m = 0.05 +  12 − ( n + 1)( n + 1.05)  6.82  δ

(8-31)

H2 =

δ 2* 20 = θ 22 21(0.95 − 1.684m)

(8-32)

Figure 8-4 presents the functional relationship between m and H2 from Eqs. (8-25) and (8-32). If the entrainment equation is solved for (δ – δ 1*) and the momentum integral equations are solved for θ11 and θ12, n and m can be computed from Eqs. (8-20) and (8-21) or (8-31). Then the boundary layer profile assumptions provide all other boundary layer data. This writer’s analysis limits H1 and H2 to a maximum of 2.4, which is used as the boundary layer profile separation limit. Empirical models to compute the entrainment function, wall shear stresses and blade force defect thicknesses are required to complete the formulation of the analysis.

8.4 EMPIRICAL MODELS FOR ENTRAINMENT AND WALL SHEAR STRESS The wall shear stress terms can be approximated using a suitable skin friction coefficient model. Assume that the shear stress is directed along the boundary layer edge streamline. Then the shear stress components are related to the skin friction coefficient, cf.



FIGURE 8-3 Extended Boundary Velocity Profiles

FIGURE 8-4 The Boundary Layer Shape Factor



cf =

τ mw = τ θw =

τw 1 2

(8-33)

ρ eVe2

1c ρ VV 2 f e e me

(8-34)

1c ρ VV 2 f e e θe

(8-35)

The well-known skin friction coefficient model by Ludwieg and Tillmann (1950) is one of the most accurate methods available for turbulent boundary layers. It can be expressed as cf = 0.246 exp(−1.561H )( ρ eVeθ / µ )−0.268

(8-36)

where θ and H are the momentum thickness and shape factor in the free stream direction, and µ is the fluid viscosity. Designate the free stream component of the fluid velocity within the boundary layer by Vs. Vs = Vm cos β e + Vθ sin β e

(8-37)

Noting that Vse = Ve, the defect thicknesses in the free stream direction are given by δ

ρ eVe2θ = ∫ ρVs (Ve − Vs )dy

(8-38)

0

δ

ρ eVeδ * = ∫ ( ρ eVe − ρVs )dy

(8-39)

0

and H = δ* / θ. Substituting Eq. (8-37) into Eqs. (8-38) and (8-39) yields expressions for these defect thicknesses in terms of the axisymmetric three-dimensional boundary layer defect thicknesses.

δ * = δ1* cos2 β e + δ 2* sin 2 β e θ = (θ11 +

δ1* ) cos4

β e + (θ 22 + δ 2* ) sin 4

β e + 2(θ12 + δ1* ) sin 2 β e cos2 β e

(8-40) −δ

*

(8-41)

Some care is required to avoid values of these defect thicknesses that will invalidate Eq. (8-36). It is recommended that all defect thicknesses on the right-hand side of Eqs. (8-40) and (8-41) be limited to be no less than zero. In addition, when solving Eq. (8-36), it is recommended that H ≤ 2.4 be required, where H = 2.4 is regarded as a boundary layer separation limit. The momentum thickness Reynolds number in Eq. (8-36) should be limited to a value to ensure that transition to turbulent flow has occurred. A reasonable transition limit for this purpose is to require ρeVeθ / µ ≥ 250.



When solving the boundary layer equations in blade free passages, this writer has observed that the entrainment function is best approximated by E = 0.025( H1 − 1)

(8-42)

Since gradients in the tangential direction all vanish for these cases, it is not too surprising that the shape of the meridional flow profile controls the fluid entrainment at the boundary layer edge. This approach was adopted by Davis (1976) at this writer’s suggestion and has also been shown to be effective by Aungier [1988(b)] and Schumann (1985). When solving the boundary layer equations within a blade passage, where the flow is guided by the blades, the free stream shape factor, H, is more relevant, much like the case in classical two-dimensional boundary layer theory, i.e., E = 0.025( H − 1)

(8-43)

For the present application to axial-flow compressor performance analysis, the boundary layer equations are normally solved across blade rows. Equation (8-43) is also used in this case, even though the solution is carried out using data outside the blade passage, where the flow is considered to be axisymmetric. In this case, the boundary layer development occurs primarily within the blade passage where the flow is guided by the blades. Hence the entrainment process should be governed in the same way as when solving within a blade passage.

8.5 THE BLADE FORCE DEFECT THICKNESSES The important blade force defect thicknesses need to be specified. The best available experimental data is the tangential blade force defect data by Koch and Smith (1976) shown in Fig. 8-2, along with an empirical curve used by them. Koch and Smith used the empirical curve to estimate end-wall losses. They proposed a separate correlation for end-wall blockage as illustrated in Fig. 8-1. It is evident that the empirical curve in Fig. 8-2 does not correlate to the experimental data with sufficient accuracy for it to be used in predicting end-wall blockage. This writer has reworked the data of Koch and Smith in the form shown in Fig. 8-5. Staggered spacing is used to normalize νθ instead of the displacement thickness. Since νθ is known to increase with blade clearance (Hunter and Cumpsty, 1982), it was somewhat arbitrarily corrected by subtracting half of the average blade clearance. Data scatter remains significant, but the result does appear to suggest trends that might be used in an empirical correlation, such as the empirical curve shown in Fig. 8-5. In the rare cases where the boundary layer thickness is less than the blade clearance, this model yields very questionable results. It is unlikely that νθ can become significant until the flow about the blade profiles comes under the influence of the distorted boundary layer flow profiles. To avoid that weakness, the data from Fig. 8-5 was expressed in the form shown in Fig. 8-6, which seems equally satisfactory as the basis for an empirical correlation.



FIGURE 8-5 Tangential Force Defect Model

FIGURE 8-6 Adjusted Tangential Force Defect Model



For the purpose of end-wall blockage prediction, the more critical parameter is the meridional blade force defect thickness, νm. There are diverse and contradictory opinions concerning νm and no experimental data available for guidance. Smith (1970) and Koch and Smith (1976) argue that νm must be small based on the existence of the repeating stage condition. Balsa and Mellor (1975) argue that the blade force should remain approximately normal to the free streamline, similar to the blade force components at the boundary layer edge. This assumes that νm = νθ, since both are governed by the same fundamental fluid dynamics. They also consider the need to explain the repeating stage condition to be a basic requirement. They assume that the blades guide the flow toward a collateral flow condition such that θ11 and θ12 approach a common value at the blade discharge, but with a correction for blade clearance effects. In this way, small or even negative values of νm are possible to provide the mechanism for the repeating stage condition. De Ruyck and Hirsch (1980) propose a correlation for νm with secondary flow parameters, but impose a transverse force defect correction similar to that of Balsa and Mellor. None of these models has been found to be particularly effective for end-wall blockage prediction, although the model of Balsa and Mellor provided the most promising results. Hence, this writer’s blade force defect model is an adaptation of that method. Following Balsa and Mellor (1975) it is assumed that the blade force remains oriented in the same direction as the free stream blade force such that νm = νθ. Hence the subscript can be omitted and the blade force defect simply designated as ν. A base defect thickness is estimated as

ν0 =

(0.12 g + δ c / 2)(8θ11 / g)3 1 + (8θ11 / g)3

(8-44)

This equation is similar to the empirical curve shown in Fig. 8-6. The average meridional momentum thickness is used instead of the displacement thickness because it is less likely to be subject to abrupt changes during the analysis. It had been expected that the equation would require adjustment by some multiplying factor to compensate for the change in the independent variable relative to the correlation in Fig. 8-6. It was somewhat of a surprise to find that qualification of the performance analysis against experimental performance indicated that no adjustment is necessary. Indeed, for most of the axial-flow compressors analyzed in the qualification study, Eq. (8-44) was quite sufficient to provide very good performance prediction accuracy using the models presented in this Chapter and in chapters 6 and 7. But a few cases were encountered where the blade force defects from Eq. (8-44) appeared to overestimate the blockage. Based on the previous discussions, this is to be expected. It is clear that the blade force defect must become small or even negative if the flow in the compressor approaches the repeating stage condition. A correction procedure not unlike that of Balsa and Mellor was found effective in correcting the blade force defect from Eq. (8-44) in those cases. If the blades provide sufficient guidance to the flow to force the boundary layer to be completely collateral at the discharge, θ11 and θ12 will approach a common value when viewed in a coordinate system fixed to the blades. Regarding W as the velocity relative to the blade row being considered, this requires that the boundary layer at the blade discharge satisfy the following condition:



Vθθ12 − Wθθ11 = 0

(8-45)

As noted in Eq. (8-27), the first term in Eq. (8-45) is identical to the tangential momentum defect in the coordinate system fixed to the blade row. If we use a prime to designate the tangential momentum thickness in the coordinate system fixed to the blade row, Eq. (8-45) can be written as Wθθ12 ′ − Wθθ11 = 0

(8-46)

Hence Eq. (8-45) is the condition for collateral flow. If Eq. (8-10) is integrated across the blade row of interest, the meridional momentum thickness at the discharge can be represented by Vθeθ12 = A + ν

r 2 fθe∆m r 2ρ eVme

(8-47)

Similarly, integrating Eq. (8-6) across the blade row and multiplying by Wθe yields Wθeθ11 = B + ν Wθe

rfme∆m 2 rρ eVme

(8-48)

For convenience, assume that the blade force is normal to the mean free stream velocity, i.e., fme = − Wθe fθe / Vme Wθeθ11 = B − ν Wθe

rfθe∆m Wθe 2 V rρ eVme me

(8-49) (8-50)

If a blade force defect increment ∆ν is imposed, Eqs. (8-47) and (8-50) yield  r W W  rf ∆m ∆(Vθeθ12 − Wθeθ11) = ∆ν  + θe θe  θe  r VmeVme  rρ eVme

(8-51)

This can be used to calculate the increment in the blade force defect thickness needed to impose the collateral flow condition, i.e., ∆(Vθeθ12 − Wθeθ11) = ξ∆ν = − (Vθeθ12 − Wθeθ11)

(8-52)

 r W W  rf ∆m ξ =  + θe θe  θe  r VmeVme  rρ eVme

(8-53)

where



In practice, the solidity would have to be infinite for the blade guidance to be sufficient to force collateral flow in the boundary layer. Hence the condition imposed in the analysis is ∆ν = − (Vθeθ12 − Wθeθ11)F / ξ

(8-54)

Since a correction appeared to be required only in cases where the blockage and defect force thicknesses become relatively large, it was reasonable to expect F to be similar in form to the correlation for ν0, which proved to be true. F=

1 (8θ11 / g)3 2 1 + (8θ11 / g)3

(8-55)

Note that F is similar in form to the function illustrated in Fig. 8-5, but approaches one-half as the quantity (8θ11/g) becomes large. The overall blade force defect thickness is given by

ν = ν0 + ∆ν

(8-56)

Equation (8-54) is to be solved using boundary layer data obtained with ∆ν = 0. In practice, it is preferable to correct the blade force defect thickness continually as the governing equations are being solved in an iterative fashion. That can be done as long as the correction process accounts for any non-zero value of ∆ν used in the previous iteration. It can easily be shown that updates for ∆ν on all iterations can be imposed as ∆ν → [∆ν − (Vθeθ12 − Wθeθ11) / ξ ]F

(8-57)

8.6 SEAL LEAKAGE EFFECTS FOR SHROUDED BLADES When the blade row is shrouded at the end-wall being analyzed, the boundary layer data must be corrected for the shroud seal leakage. The shroud seal leakage mass flow calculation has been described in Chapter 6. The sign convention used for the leakage mass flow is positive when the leakage is directed from the blade row discharge to the inlet, as illustrated in Fig. 8-7. The shroud seal leakage will change the mass flow rate in the boundary layer. The leakage flow is expected to have no meridional velocity component, so the boundary layer meridional momentum flow rate should be unchanged. The absolute tangential velocity of the leakage flow entering the boundary layer is assumed to be half of the local speed of the rotating wall. If the leakage flow is out of the boundary layer, its tangential velocity relative to the wall is assumed to be essentially zero, due to the no-slip condition. The details of the mass and momentum balances to account for the leakage flow will be left to the exercises. Here, only the results will be given. Using the nomenclature defined in Fig. 8-7, the blade row inlet boundary layer mass flow is corrected by



FIGURE 8-7 A Shrouded Blade Row

(δ − δ1* )+ = (δ − δ1* )in + ∆+ ∆+ =

˙ leak m (2π rρ eVme )in

(8-58) (8-59)

The meridional momentum balance yields + θ11 = θ11 + ∆+ in

(8-60)

If ωc is the compressor rotation speed, the tangential velocity of leakage flow entering the boundary layer, relative to the blade shroud, is Uleak =

1 2

rω c + Vθ − Cθ

(8-61)

The tangential momentum balance, for either positive or negative leakage mass flow rate, yields (Vθθ12 )+ = (Vθθ12 )in + ∆+Vθ

in

− 12 ( ∆+ + | ∆+ |)Uleak

(8-62)



These leakage corrected boundary layer data are sufficient to define the powerlaw profile and correct all other boundary layer parameters using Eqs. (8-20) through (8-26). The boundary layer analysis is conducted across the blade row, with the wall rotation speed set equal to the blade rotation speed. Then mass and momentum balances and Eqs. (8-20) through (8-26) are used to correct the calculated discharge boundary layer data for the seal leakage to obtain the final blade discharge boundary layer data. ∆− =

˙ leak m (2π rρ eVme )out

(8-63)

(δ − δ1* )out = (δ − δ1* )− − ∆− − θ11 = θ11 − out −

−

(Vθθ12 )out = (Vθθ12 ) − ∆ Vθ

∆

(8-65) −

out

(8-64)

− −

+ 12 ( ∆ − | ∆ |)Uleak

(8-66)

8.7 BOUNDARY LAYER JUMP CONDITIONS The boundary layer analysis has been developed in terms of the fluid velocity relative to the wall, where the wall may be stationary or rotating. It is necessary to consider jump conditions for cases where the boundary layer moves from a stationary wall to a rotating wall, or inversely. Indeed, the boundary layers along the hub contour in axial-flow compressors with shrouded stator blades must make this transition at every blade row. For clarity, designate the velocity relative to the wall rotating at rotation speed ω as W and C to be the velocity relative to the stationary wall. Boundary layer parameters relative to the rotating wall will be designated with a prime. Hence, Wθ = Cθ − ω r

(8-67)

Substituting Eq. (8-67) into the definitions of the various boundary layer defect thicknesses yields Cθeθ12 = Wθeθ12 ′

(8-68)

Cθeδ 2* = Wθeδ 2′ *

(8-69)

Cθ2eθ 22 = Wθ2eθ 22 ′ + ω rWθeδ 2′ *

(8-70)

Cθ2eθ 22 (1 + H2 ) = Wθ2eθ 22 ′ (1 + H2′ ) + 2ω rWθeδ 2′ *

(8-71)

Equations (8-68) and (8-71) contain basic terms in the momentum-integral equations. Note that they have identical values in either rotating or stationary coordinates, so no special logic is required to convert them at a transition between a rotating and stationary wall. Consequently, integration of the governing



equations can proceed through the compressor without any complex logic at transitions between rotating and stationary walls.

8.8 SOLUTION PROCEDURE Since the end-wall boundary layer analysis presented in this chapter considers only turbulent boundary layers, boundary layer data must be specified at the compressor inlet. The data needed are θ11, θ12, H1 and H2. It is reasonable to start the analysis with H1 = 1.4, which is a typical value for simple flat-plate boundary layer flow. The classical one-seventh power-law profile is a reasonable choice for the tangential velocity profile, i.e., m = 1 / 7 or H2 = 1.286. The boundary layer fractional area blockage, B, is assigned at the inlet. Assume the blockage is equally split between the hub-and-shroud contours and compute the meridional displacement thickness for either end-wall by 2π rρ eVmeδ1* =

1 2

˙ Bm

(8-72)

Equations (8-22) through (8-26) provide all other initial boundary layer data. If the analysis is to be started immediately upstream of the first blade row, experience has shown that B = 0.02 is a good choice. If the analysis can be started well upstream of the first blade row, a very small value can be entered for B so that the boundary layer development ahead of the first blade row is predicted by the analysis. Throughout the analysis, the boundary layer is always constrained to be turbulent by imposing a limit on the Reynolds number based on θ11, i.e., Reθ = ρ eVeθ11 / µ ≥ 250

(8-73)

Hence, specifying a very small value of B at the inlet station forces the analysis to start with a boundary layer essentially at the transition point from laminar to turbulent flow. The analysis on each end-wall can be accomplished with a simple marching technique, starting at the second meridional station, and completing the solution at each station before proceeding to the next. The basic procedure is 1. Impose the inlet shroud seal leakage corrections of Section 8.6, if applicable. Estimate initial values of the momentum thicknesses from the upstream station using simplified forms of Eqs. (8-6) and (8-7). ∂ 2 [rρ eVme θ11] = 0 ∂m

(8-74)

∂ 2 [r ρ eVmeVθeθ12 ] = 0 ∂m

(8-75)

Assume H1 and H2 are equal to the upstream values and initialize all other parameters using Eqs. (8-22) through (8-26).



2. Compute the entrainment function and the wall shear stresses as described in Section 8.4. If applicable, compute the blade force defect using the methods described in Section 8.5. 3. Integrate Eq. (8-5) for (δ – δ 1*), Eq. (8-6) for θ11, and Eq. (8-7) for θ12. 4. Compute n and δ from Eqs. (8-20) and (8-24). As a precaution, require θ12 ≤ 0.99δ / (n + 1). Compute m from Eq. (8-21). 5. Compute the other boundary layer parameters from Eqs. (8-22) through (8-26). 6. Check for convergence on θ11, H1 and ν. If not converged, return to Step 2 and repeat. 7. Impose the discharge shroud seal leakage corrections of Section 8.6, if applicable. 8. Return to Step 1 and treat the next meridional station until all stations have been analyzed. When the boundary layer analysis is completed at all meridional stations on both end-walls, the end-wall boundary layer blockage and the blockage factor can be computed at all meridional stations for use in the meridional through-flow ˙ is the compressor mass flow rate, they are given by analysis of Chapter 7. If m B=

Shroud

∑

˙ 2π rρ eVmeδ1* / m

(8-76)

Hub

KB = 1− B

(8-77)

Since the boundary layer and through-flow analyses are conducted sequentially in an iterative fashion, some limits and numerical damping are normally required. The boundary layer edge parameters may be quite unrealistic in the early iterations, resulting in rapid changes in the predicted blockage values.

8.9 TYPICAL RESULTS Figures 8-8 through 8-10 illustrate end-wall blockage predictions from this endwall boundary layer analysis for three different styles of axial-flow compressors. The NACA 8-stage compressor (Voit, 1953) is designed with two inlet transonic stages. The NACA 10-stage compressor (Johnsen, 1952) is a conservative design using all subsonic stages. The NACA 5-Stage compressor (Kovach and Sandercock, 1961) is designed with all transonic stages. The analysis of the NACA 10stage compressor had to be started at the exit of an undefined inlet guide vane using measured vane discharge flow angles. The inlet blockage is set to 2% of the annulus area as recommended above. The analyses of the other two compressors started well up in the inlet passage with minimum (transition point) blockage levels to let the boundary layer develop naturally based on the analysis. It is seen that blockage levels and blockage distributions predicted for these three compressors are quite different, and show considerable variation of blockage level with overall pressure ratio, PR. It will be seen in Chapter 9 that the overall



FIGURE 8-8 NACA 8-Stage Compressor Blockage





performance predictions obtained using these end-wall boundary layer blockage predictions are quite accurate for all of these compressors. Hence, the present end-wall boundary layer analysis is quite effective in its intended role of supporting the aerodynamic performance analysis.

EXERCISES 8.1 Consider the inlet of the shrouded stator blade shown in Fig. 8-7. From conservation of mass for the upstream boundary layer and the leakage mass flow, derive Eq. (8-58) for the boundary layer downstream of the injected leakage flow. The velocity, V, is relative to the blade shroud, 8.2 The leakage mass flow in Exercise 8.1 enters with Vm = 0. From conservation of meridional momentum for the upstream boundary layer and the leakage mass flow, derive Eq. (8-60) for the boundary layer downstream of the injected leakage flow. 8.3 The leakage mass flow in Exercise 8.1 enters with tangential velocity Uleak, relative to the blade shroud. From conservation of tangential momentum for the upstream boundary layer and the leakage mass flow, derive Eq. (8-62) for the boundary layer downstream of the injected leakage flow.



8.4 Repeat Exercises 8.1 through 8.3 for the discharge of the stator blade, noting that the tangential velocity of the leakage flow leaving the boundary layer, relative to the blade shroud, is zero. 8.5 (δ – δ 1*), θ11 and θ12 are the boundary layer parameters obtained from the basic conservation equations. Develop expressions for these parameters as functions of n, m and δ, using Eqs. (8-11), (8-12), (8-13), (8-18) and (8-19). Use the results to derive Eqs. (8-20) and (8-21).


Chapter 9

AERODYNAMIC PERFORMANCE ANALYSIS

An aerodynamic performance analysis is essential for nearly all aspects of axialflow compressor aerodynamic design and application. There are basically two types of performance analysis techniques in common use: One-dimensional or mean-line methods analyze the performance along a mean stream surface. If applied to well-designed stages, the mean-line performance may be considered to be representative of the overall performance, at least near the compressor’s design operating conditions. This approach is more questionable for off-design performance prediction. In those cases, blade incidence angle matching, blade loading levels, etc., may vary dramatically at different locations along the blade span, such that the mean-line performance is no longer representative of the overall performance. This is particularly true with respect to blade loading limits, blade stall and end-wall stall, which establish the compressor’s surge limit. In off-design operation, the extremes in incidence angle and flow diffusion will almost always occur on the hub or the shroud contours. Some one-dimensional methods include approximate calculations at the hub-and-shroud contours to provide additional guidance to the user and to better evaluate tip clearance losses, shroud leakage effects and end-wall boundary layer blockage. The more general approach is to conduct the performance analysis for a series of stream surfaces from hub to shroud. These methods are referred to by various names, such as streamline methods, through-flow methods, streamline-curvature methods or three-dimensional methods. Hub-to-shroud performance analysis is a more accurate term, which is used in this book. It has been common practice to conduct hub-toshroud performance analysis using the full normal equilibrium equation as described in Chapter 7. The longer computer times required and reduced reliability due to numerical instability for these techniques is probably the main reason that one-dimensional methods have continued to be used. But if the through-flow analysis offers the approximate normal equilibrium models described in section 7.6, the advantages of a hub-to-shroud performance analysis can be obtained with computation speed and reliability comparable to those of a mean-line method. Consequently, there is really no longer a need for one-dimensional performance analysis methods for axial-flow compressors. The component parts of a hub-to-shroud performance analysis have been presented in Chapters 6 to 8. This chapter describes methods to integrate those component analyses into a hub-to-shroud performance analysis and suggests some



useful features that can make it more effective. It also compares performance predictions from this performance analysis against experimental data for axialflow compressors to demonstrate the merits of the procedures presented in Chapters 6 to 8.

NOMENCLATURE AR b c Deq g i L N P r s tb VR W WRE Z z γ θ κ σ φ – ω

= = = = = = = = = = = = = = = = = = = = = = =

diffuser area ratio diffuser passage width blade chord length equivalent diffusion factor staggered spacing stream surface number diffuser or camberline length number of stream surfaces pressure radius cascade pitch blade maximum thickness diffuser velocity ratio relative velocity equivalent velocity ratio number of blades axial coordinate stagger angle camber angle blade camberline angle solidity stream surface slope angle total pressure loss coefficient

Subscripts c t 1 2

= = = =

corrected (smoothed) parameter total thermodynamic condition inlet condition discharge condition

Superscripts ′ = relative condition

9.1 GEOMETRY CONSIDERATIONS Application of procedures from previous chapters requires specific procedures for specifying the geometry of the end-wall contours, quasi-normals and blades


Aerodynamic Performance Analysis • 201

FIGURE 9-1 Basic Gas Path Geometry for an Axial-Flow Compressor

in a form suitable for the performance analysis. Figure 9-1 illustrates the basic gas path geometry for a typical axial-flow compressor. The hub-and-shroud contour coordinates are entered at the end points of a series of quasi-normals that are used to conduct the analysis. In this example, all quasi-normals are simple radial lines, although that is certainly not a requirement. This analysis used a single quasi-normal between successive blade rows, although more could have been used if they were considered necessary. For each quasi-normal after the first one, it is necessary to specify what type of blade row lies upstream. Some choices for this specification are rotor, stator, guide vane or none. It is useful to distinguish between stators and guide vanes so that the analysis can correctly distinguish stages by stage number to permit individual stage performance data to be output. The end-wall contour specification should also identify the first and last quasinormals for which the hub wall is rotating, so that the end-wall boundary layer analysis can use the correct wall rotation speed. In this case, that is simply the first and last quasi-normal. It is also useful to provide for alternate input to size the annulus. In that case, coordinates are specified for only one end-wall contour. The angle between the quasi-normal and a radial line and the meridional velocity for the mean streamline are the other data required. The program will then compute the annulus area required and the coordinates of the other end-wall as outlined in Chapter 7. The blade geometry also needs to be specified. The standard blade profile types described in Chapter 4 should be available to make full use of the empirical blade performance models of Chapter 6. The blade construction logic of Chapter 4 is recommended so that the blade throat openings can be computed accurately. Alternatively, the empirical approximation of Eq. (4-31) can be used, although the more precise calculation is preferred. When the latter approach is used, provision should be made to compute the throat openings one time and save the computation in the program’s input file once all other blade geometry is available. It is inefficient to have the program perform these calculations every time a performance analysis is conducted. If a special blade profile such as a controlled diffusion airfoil is in use, it will be necessary to extend the methods of Chapters 4 and 6, or to confirm that use of one of the standard profiles can adequately approximate the performance of the profile. Usual practice is to specify the geometry of blade sections at a series of radii that extend over a range at least as large as the range of end-wall radii at the quasi-normals before and after the blade row. Usually it is reasonable to specify a single value of the location of the point of maximum camber, a / c, for each blade row. Similarly, a single specification per blade row can be



supplied for the number of blades in the blade row, Z, and the blade tip clearance, δc. The most useful data to supply for the radial sections are the chord, c, the thickness-to-chord ratio, tb / c, and the blade inlet and discharge angles, κ1 and κ2. For double-circular-arc blades, a specification for the leading and trailing edge radii may also be necessary, unless a standard specification is considered acceptable. The blade angles are the most useful specifications, but may not be the most convenient, so there is provision to opt for choices such as camber angle, θ, stagger angle, γ or lift coefficient, Cl0. The program can then compute κ1 and κ2 from the data actually supplied, as described in Chapter 4. In the case of shrouded stator blades, the shroud seal clearance and the number of seal fins will also be required instead of the blade tip clearance. There are a number of features that can easily be incorporated to greatly simplify the blade geometry specification process, particularly for industrial axial-flow compressors. These include: • • • • •

The capability to copy the geometry of an already specified blade row. The capability to apply a geometrical scale factor to the data. The capability to impose a radial shift on the data. The capability to impose a change in stagger angle, i.e., to rotate the blade. The capability to import geometry from a file for a commonly used standard profile. • The capability to export geometry to a file for use as a standard profile for future analyses. • The capability to import geometry designed by the stage design procedure described in Chapter 10. Often application of one or more of these procedures can greatly simplify the blade geometry input process so it requires at most, only minor editing, avoids the need to enter all data. When conducting the actual performance analysis, the blade geometry on a stream surface is required. If the stream surface coordinates at the blade row inlet and discharge are (z1, r1) and (z2, r2), respectively, the stream surface angle, φ, is estimated from tan φ =

r2 − r1 z2 − z1

(9-1)

Then the corrected blade geometry on the stream surface is given by c → c / cos φ

(9-2)

tb / c → cos φ tb / c

(9-3)

tan κ1 → cos φ tan κ1

(9-4)

tan κ 2 → cos φ tan κ 2

(9-5)

s = π ( r1 + r2 ) / Z

(9-6)

Where base values are obtained from the input data by interpolation with respect to radius to obtain κ1 at r1, κ2 at r2, and other data at the mean radius. Then all other blade geometry data can be computed as described in Chapter 4.



9.2 CASCADE PERFORMANCE CONSIDERATIONS Some care is required in applying and interpreting the empirical cascade performance models of Chapter 6 when predicting the performance of an axial-flow compressor. When analyzing a compressor at far off-design conditions, specific blade rows may operate locally in deep stall or in choke while other blade rows operate under near optimum conditions. If the empirical models of Chapter 6 are applied directly, the overall performance analysis will be very unreliable and incapable of analyzing many cases where the compressor is capable of operation. The major reason for this is that the meridional through-flow analysis of Chapter 7 does not permit any fluid mixing between stream sheets, such that entropy can build up locally on a stream surface to cause the solution to diverge. In the actual compressor blade rows, there are substantial secondary flow patterns or boundary layer migration, which does result in fluid mixing between stream sheets. Thus, it is necessary to impose limits on the loss coefficient models and apply some artificial smoothing procedures to simulate the fluid mixing processes. This writer imposes the following limit on all loss coefficients:

ω ≤ 0.5

(9-7)

This will rarely be required, except when analyzing flow points where local choke is encountered in the blade passages. This can be encountered when the compressor is operating close to its choke limit and flow profiles are highly distorted. Smoothing of the total pressure loss is a more important consideration. Without it, the performance analysis can be very unreliable for predicting performance close to the surge line. This is particularly true at low off-design speeds, where severe local blade stall is commonly encountered. If N stream surfaces are used from hub to shroud, the total pressure loss on stream surface number i is given by ( ∆Pt′)i = ω i ( Pti′ − Pi )in

(9-8)

The prime designates the total pressure in a frame of reference relative to the blade row, and the subscript i designates the stream surface, numbered sequentially from the hub contour. Smoothed total pressure loss values are computed from

[

]

( ∆Pt′)i, c = ( ∆Pt′)i −1 + 2( ∆Pt′)i + ( ∆Pt′)i +1 / 4 ; 1 < i < N

(9-9)

( ∆Pt′)1, c = 2( ∆Pt′)2, c − ( ∆Pt′)3, c

(9-10)

( ∆Pt′)N , c = 2( ∆Pt′)N −1, c − ( ∆Pt′)N −3, c

(9-11)

Equations (9-10) and (9-11) are derived from a simple trapezoidal-rule numerical approximation for the integral of the total pressure loss over the two stream sheets adjacent to the walls. They result in those integrals being identical when approximated using either the smoothed or the uncorrected total pressure loss values at those three points.



9.3 STALL AND COMPRESSOR SURGE CONSIDERATIONS An important consideration in performance analysis is estimating the compressor’s surge limit from the detailed performance data. This is rather subjective and imprecise, but there are some useful indicators that have been found to be helpful in this process. It is well known that a pressure versus flow characteristic at constant speed, which has a positive slope, is theoretically unstable. Hence, a predicted characteristic that approaches a slope of zero can be considered to be an obvious indication of probable surge. But in most cases, the performance analysis will not be capable of resolving the characteristic’s slope accurately enough to rely on this. Indeed, the occurrence of a positive slope is often not even obvious from experimental performance data. In the presence of blade or end-wall boundary layer stall, there may be a very abrupt drop in compressor discharge pressure that is not apparent from either the predictions or experiment. Of course, in the case of the predictions, this lack of resolution is really deliberate. Practices such as the loss smoothing previously discussed are employed to achieve acceptable reliability, which will preclude resolving any abrupt changes in performance due to the onset of local stall. The best we can do is attempt to identify stall parameters that may indicate such an abrupt change is likely to occur. Blade stall is expected to be a function of the limit loading diffusion factors, such as D or Deq of Chapter 6. In this writer’s experience, those parameters are of limited value as a means of estimating stall limits likely to be associated with the compressor’s surge limit. It is quite common to observe values of these diffusion factors well in excess of typical recommended loading limits at operating points well removed from the observed surge line. A stall criterion proposed by de Haller (1953) is given by W2 / W1 < 0.72

(9-12)

The generally accepted opinion today is that de Haller’s criterion is associated with an end-wall stall rather than a blade stall. In any event, this criterion has been found to be more effective than the diffusion factors as an indicator of the onset of surge. But, again, a significant number of exceptions are encountered where velocity ratios well below this limit are predicted at operating points that are quite far from a compressor’s surge limit. Koch (1981) improved on the simple de Haller criterion by including blade geometry parameters. Koch approximated the blade passages, averaged over a stage, as a simple diffuser. He was able to achieve reasonable agreement between observed stage pressure coefficients at stall with the two-dimensional diffuser data of Sovran and Klomp (1967) for a specific inlet blockage level. To obtain this correlation, the compressor stage data was extensively corrected to adjust for differences in blade clearance, Reynolds number and axial spacing between blade rows. This correlation is intended as a means to estimate the maximum achievable stage pressure coefficient. It is of limited value as a stall criterion for use in an axial-flow compressor performance analysis. Here, a stall criterion for a blade row instead of a stage is needed. The use of pressure coefficient is also undesirable, since it results in a stall criterion that is a strong function of the blade performance models, which are by no



FIGURE 9-2 Two-Dimensional Diffuser Correlation

means universal methods. Finally, the Koch correlation requires a number of corrections for it to be applicable to a specific axial-flow compressor. This writer has developed a correlation for the onset of stall that is more appropriate to the present application. Reneau et al. (1967) developed a graphical correlation for the geometry of simple two-dimensional diffusers operating at their peak static pressure recovery. Aungier (2000) correlated their data in the form AR = 1 + 0.4( L / b1)0.65

(9-13)

Where AR is the diffuser area ratio, L is the diffuser length and b1 is the diffuser width at the inlet. Since the diffuser test data correspond to basically incompressible flow, this can also be related to the ideal diffuser velocity ratio, VR. VR = 1 / [1 + 0.4( L / b1)0.65 ]

(9-14)

Equations (9-13) and (9-14) are presented in Fig. 9-2. Note that these equations identify peak pressure recovery condition, but are not dependent on the value of the peak pressure recovery. It is reasonable to expect the onset of stall in any diffusing passage to closely correspond to this peak pressure recovery condition. To apply this concept to a blade passage, it is assumed that L should be the camberline



length and b1 should be the staggered spacing. If the camberline is approximated as a circular-arc, this can be written as L / b1 → L / g = L / ( s cos γ ) = θσ / [2 sin(θ / 2) cos γ ]

(9-15)

When applying an incompressible flow model to compressible flow through a blade row, it is always preferable to employ velocity pressures rather than velocity to minimize the influence of Mach number effects, i.e., VR → WRE =

Pt′2 − P2 Pt1′ − P1

(9-16)

WRE will be called the equivalent relative velocity ratio across the blade row. Hence the expected stall criterion is WRE <

(0.15 + 11tb / c) / (0.25 + 10tb / c) 1 + 0.4[θσ / {2 sin(θ / 2) cos γ }]0.65

(9-17)

The numerator on the right-hand side of this equation reflects a very weak dependence on the ratio of tb/c observed while comparing stall estimates from Eq. (9-17) with observed axial-flow compressor surge limits. These comparisons have also suggested two other modifications: First, the application to blade passages requires extrapolating Eq. (9-14) to values of L / b1 that are considerably lower than those covered by the test data and correlation of Reneau et al. (1967). Thus, a limit is imposed on the effective value of L/b1 used on the right-hand side of Eq. (9-17) by requiring

θσ / [2 sin(θ / 2) cos γ ] ≥ 1.1

(9-18)

Second, Eq. (9-17) can be too pessimistic for highly diffusing blades, where blade wake blockage can be expected to be significant. It is rather subjective and very tedious to attempt to estimate stall in those cases. It has been observed that Eq. (917) becomes pessimistic whenever Deq is greater than about 2.2. When Deq > 2.2, an empirical correction to Eq. (9-17) has been found to be reasonably effective, i.e., WRE <

[(2.2 / Deq )0.6 ](0.15 + 11tb / c) / (0.25 + 10tb / c) 1 + 0.4[θσ / {2 sin(θ / 2) cos γ }]0.65

(9-19)

The stall criterion for Deq< 2.2 and tb / c = 0.1 is illustrated in Fig. 9-3. In summary, this writer employs the following three basic criteria as a basis for estimating the onset of compressor surge: 1. When the discharge pressure versus flow characteristic approaches a slope of zero. 2. When end-wall boundary layer stall is predicted (H1 = 2.4). 3. When stall is indicated by Eqs. (9-17) through (9-19).



FIGURE 9-3 The Blade Row Stall Criterion

The first two criteria are employed at all speeds, while the third criterion is most reliable for speeds greater than about 85% of design speed. At lower speeds, the flow profiles typically become so distorted that the accuracy of the profile predictions with the inviscid through-flow analysis of Chapter 7 becomes too questionable for it to rely heavily on the third criterion. This will be illustrated in the presentation of results from the performance analysis.

9.4 APPROXIMATE NORMAL EQUILIBRIUM RESULTS The approximate normal equilibrium model described in Section 7.6 offers substantial advantages in computation speed and reliability over the full normal equilibrium method. This approximation assumes that the stream surface slope and curvature vary linearly between the two known end-wall contours. In this section, typical results from the performance prediction procedures described in this book using the approximate normal equilibrium model will be reviewed. In the next section, results for the same problems using the full normal equilibrium model will be reviewed for comparison. Figure 9-4 compares performance predictions with experiment for the NACA 10-stage subsonic axial-flow compressor. The compressor design is described in Johnsen (1952). The performance data are provided in Budinger and Thomson



FIGURE 9-4 The NACA 10-Stage Compressor

(1952). All blade rows in this compressor use fairly conventional NACA 65-series blades. Design data for the inlet guide vane are not provided in the references, so the performance analysis was started at a station downstream of the inlet guide vane using experimental measurements of the flow angle distribution, with an assumed end-wall boundary layer blockage of 2%. Typical end-wall boundary layer blockage predictions for this case were shown in Fig. 8-8. The estimated stall line based on stall criterion #3 is also shown in Fig. 9-4. The experimental compressor surge limit is well approximated by the lowest flow data point for each speed line. It is seen that stall criterion #3 is a good indicator of surge at the design speed, but is pessimistic at lower speeds, where stall criterion #1 is the best indicator. This compressor has the unusual characteristic that stall criterion #3 is first encountered in the front stages at the rotor tips and that the limit expressed in Eq. (9-18) is active at these locations. Under these conditions, stall criterion #3 is of questionable validity and it is far too insensitive to variations in mass flow rate to be very useful. At the tips of the front stage rotors of an axialflow compressor, the variation of WRE with mass flow rate is far too weak to resolve a meaningful stall limit. For example, the difference in WRE between the experimental and predicted surge limits at 90% speed is just 0.015. This is the only axial-flow compressor encountered by this writer with this unusual characteristic. It is normally the case that blade diffusion limits are first encountered along the hub contour, where WRE shows the strongest variation with mass flow rate. Flow diffusion limits are easier to avoid at the rotor tips, and designers



have strong incentive to do so due to the higher Mach number levels encountered there. Figure 9-5 compares performance predictions with experiment for the NACA 5-stage transonic axial-flow compressor (Kovach and Sandercock, 1961). Sandercock et al. (1954) describes the detailed design. Kovach and Sandercock (1954) provide the experimental performance data. The first two stator rows and all rotor rows in this compressor use double-circular-arc blades, while the last three stator rows use NACA 65-series blades. This case provides a fairly significant test of all high Mach number performance models used, including bow-shock losses, supercritical Mach number effects and blade passage choking. This analysis was started well up in the inlet passage with negligible inlet end-wall boundary layer blockage. Figure 9-1 illustrates the compressor cross-section and the computational stations used in this analysis. Typical end-wall boundary layer blockage predictions for this case were shown in Fig. 8-9. The estimated stall line from stall criterion #3 is also shown in Fig. 9-5. The experimental compressor surge limit is not well defined in the references, but presumably can be approximated by the lowest flow data point on a speed line. At 90% and 100% speed, there is substantial scatter in the experimental data, so the surge limit is less obvious and possibly better indicated by the highest pressure-ratio point achieved on the speed line. It is seen that stall criterion #3 is reasonably significant as an indicator of surge at all speeds. This case provides a fairly dramatic illustration of the merits of stall criterion #3, as shown in Fig. 9-6. The predicted equivalent velocity




FIGURE 9-6 Illustration of Stall Criterion 3

ratios defined by Eq. (9-16) along the hub contour are shown for all blade rows, along with the stall limit values computed from Eqs. (9-17) through (9-19). The simple de Haller limit of Eq. (9-12) is also shown for reference. It is apparent that the simple de Haller limit would predict a substantially lower pressure ratio at stall. Indeed, the de Haller criterion predicts that blade stall is present for the entire predicted characteristic shown in Fig. 9-5. For the specific blade geometry used in this compressor, Eqs. (9-17) through (9-19) predict a stall limit well below the de Haller limit. Although an unusually extreme example, this case serves to demonstrate the importance of a more fundamental stall criterion than that proposed by de Haller. Figure 9-7 compares performance predictions with experiment for the NACA 8-stage transonic axial-flow compressor (Voit, 1953; Geye et al., 1953). The first two rotor blade rows use double-circular-arc blades. All other blade rows use NACA 65-series blades. The performance analysis for this case is a little less precise than the previous examples. The hub contour around the first stator row is not well defined in Voit (1953) and the exit guide vane row geometry is not supplied. The exit guide vane is not expected to have a significant influence on the total-to-total pressure ratio prediction. So an exit guide vane was simply designed for this compressor for use in the performance analysis. The exit guide vane design used a blade configuration similar to that of the last stator row, with camber and stagger angles defined to effectively remove the swirl from the flow exiting the last stator row. The analysis was started well upstream in the inlet passage




with negligible inlet end-wall boundary layer blockage. Typical end-wall boundary layer blockage predictions for this case were shown in Fig. 8-7. The estimated stall line from stall criterion #3 is also shown in Fig. 9-7. The experimental compressor surge limit is well approximated by the lowest flow data point for each speed line (Geye et al., 1953). Stall criterion #3 is a good indicator of the surge limit at the 90% and 100% speed lines, but is somewhat conservative at the 70% and 80% speed lines, where stall criterion #1 is a better indicator.

9.5 FULL NORMAL EQUILIBRIUM RESULTS Figures 9-8 through 9-10 compare performance predictions using the full normal equilibrium model with the approximate normal equilibrium results from the previous section. Except for some minor differences in the stall lines estimated from the two models, it is seen that the performance predictions from the two models are essentially identical. It is quite evident that there is very little loss in performance prediction accuracy when the approximate normal equilibrium model is used for these three compressors. This is quite typical of this writer’s experience on other axial-flow compressor performance analyses. There is almost never a need to employ the full normal equilibrium model to obtain accurate overall compressor performance predictions.








9.6 CONCLUDING REMARKS This chapter provides supplemental procedures to apply the analyses described in Chapters 6 through 8 to the prediction of the aerodynamic performance of axialflow compressors. The resulting performance analysis has been applied to three NACA axial-flow compressors to demonstrate the level of prediction accuracy to be expected. The compressors considered cover a significant variety of design styles, ranging from the conservative subsonic NACA 10-stage compressor to the more aggressive transonic designs used in the NACA 5-stage and 8-stage compressors. But it must be recognized that the empirical models of Chapter 6 specifically cover only the convention blade profiles of Chapter 4. Application of these procedures to other blade profiles, such as proprietary controlled diffusion airfoils, may require supplemental empirical performance models to achieve the same level of prediction accuracy as demonstrated by the results provided in this chapter. The performance predictions reviewed in this chapter show that the approximate normal equilibrium model of Section 7.6 provides excellent prediction accuracy as well as dramatic improvements in computation speed and reliability. This supports an earlier comment in this chapter suggesting that one-dimensional mean-line performance analysis methods no longer offer any significant advantage over the more fundamental hub-to-shroud flow performance analyses. Some general guidelines have been suggested for estimating a compressor’s surge limit based on the performance analysis results. These guidelines can



normally be refined and improved through comparison with experimental results when the analysis is applied to user-specific design styles and blade profile types. Surge is a transient phenomenon dependent upon the complete system in which the compressor operates. Consequently, the surge limit cannot be predicted by a compressor performance analysis. A performance analysis can only provide fluid dynamic data that may be used as a guide to estimate the probable surge limit. It should be apparent from the results presented in this chapter that this process remains quite subjective, relying heavily on the experience and judgment of the aerodynamicist to interpret results from the performance analysis. At speeds close to the design speed, the stages are reasonably well matched. In these cases, stall criterion #3 is usually the best indicator. At speeds much less than the design speed, excessive front stage loading produces severe flow profile distortion, making the predicted values of WRE too inaccurate for stall criterion #3 to be very useful. There, stall criterion #1 is the most useful indicator. Although stall criterion #2 isn’t a factor in the cases considered in this chapter, it is occasionally encountered. There is evidence to suggest that stable operation is very unlikely when significant end-wall boundary layer separation is predicted. In the few cases encountered, stall criterion #2 provided a good estimate of the surge limit and none of those compressors actually operated beyond that limit.

EXERCISES 9.1 Show that the integration of the corrected total pressure loss data from Eqs. (9-9) and (9-10) with respect to mass flow rate between stream surfaces 1 and 3 yields the same result as integration of the uncorrected total pressure loss data. Assume the mass flow rate is identical in all stream ˙ . Use the simple trapezoidal rule approximation for sheets and equal to ∆m numerical integration, e.g., ˙ ∆m

∫

˙ ≈ 1 [( ∆Pt′ )1 + ( ∆Pt′ )2 ]∆m ˙ ∆Pt′dm 2

0

9.2 Derive an expression for the ratio of the blade camberline length-to-the staggered spacing, s cosγ, for a circular-arc camberline to confirm Eq. (9-15).


Chapter 10

COMPRESSOR STAGE AERODYNAMIC DESIGN

The aerodynamic design of an axial-flow compressor stage involves definition of the rotor and stator velocity triangles and blade geometry that will produce desired stage performance characteristics. When appropriate, this may include the design of inlet and exit guide vanes as well. In the context used in this book, there are subtle but important differences between stage design and compressor design. Stage design will involve the selection of basic dimensionless performance parameters and the design of appropriate blade rows to produce them. No specific reference is made to the precise application of the stage in an actual compressor. Without knowledge of the specific working fluid, Mach number levels, matching with adjacent stages, etc., stage design is a rather idealized process. It is not possible to establish precise end-wall contours and compute the losses in this context. The purpose of this chapter is to introduce basic dimensionless performance parameters and to discuss their application to the design process. Representative applications of the stage designs will be used to indicate some of the consequences of the choices made. Chapter 11 extends these basic design concepts to the design of blade rows for complete axial-flow compressors with specific operating conditions and working fluids. The stage-design procedures described in this chapter offer far more than abstract educational value. They certainly can be used to design stages for actual applications within a compressor, although the procedures in chapter 11 are more efficient. They often play a key role in the design of industrial axial-flow compressors. Each industrial axial-flow compressor usually has a unique design. Development and manufacturing costs become primary considerations when there are no duplicate machines to share them. It is fairly common practice to employ a standard repeating stage design for these compressors. Blade stagger angles are commonly adjusted to accommodate the specific applications. Scaling of the blades with corresponding modification of the number of blades per row may be used to satisfy mechanical strength requirements. Since the precise application is not predefined and unique designs for all stages are not acceptable, the design of a standard repeating stage will precisely follow the process described in this chapter. The specific axial-flow compressor design will involve application of this standard stage design with appropriate sizing of the annulus and various minor adjustments to accommodate the application. An example of this use of stage design procedures is included at the end of this chapter.



NOMENCLATURE C c H h i K m n P PR R RR r tb U W Z z β γ δ η θ κ ρ σ φ ψ ψlim ω

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

absolute velocity chord total enthalpy static enthalpy incidence angle ψ / ψlim vortex exponent vortex exponent pressure pressure ratio reaction recovery ratio radius maximum blade thickness local blade speed = ωr relative velocity number of blades axial coordinate flow angle stagger angle deviation angle efficiency camber angle blade camberline angle gas density solidity flow coefficient work coefficient value of ψ yielding W2 / W1 = 0.7 rotation speed, radians/sec.

Subscripts c h R S s t z θ 0 1 2 3 4

= = = = = = = = = = = = =

parameter on the reference radius hub condition rotor parameter stator parameter shroud condition total thermodynamic condition axial component tangential component inlet guide vane inlet condition rotor inlet condition stator inlet condition stator exit condition exit guide vane exit condition


Compressor Stage Aerodynamic Design • 217

Superscripts * = minimum loss condition ′ = relative condition

10.1 DIMENSIONLESS PERFORMANCE PARAMETERS In the design of an axial-flow compressor stage, the designer normally wants to control the flow capacity, the work done per stage and the distribution of the flow diffusion load between the rotor and stator. The specifications should be dimensionless to enable application to any specific stage inlet thermodynamic conditions, rotation speed and geometrical scale. This leads quite naturally to the dimensionless performance parameters introduced in Chapter 1, Section 1.5. If ω is the rotation speed, the local blade speed is given by U = ωr. The dimensionless flow capacity of the stage will be governed by the dimensionless axial velocity component or flow coefficient, defined by

φ = Cz1 / U

(10-1)

The flow coefficient has been defined at the rotor inlet, since an inlet guide vane is not necessarily defined as part of the stage design process. The dimensionless work per stage follows directly from the Euler turbine equation, Eq. (3-9). The work coefficient is defined as

ψ = ( H2 − H1) / U 2 = (Cθ 2 − Cθ1) / U

(10-2)

where the subscripts 1 and 2 refer to the rotor inlet and exit stations, respectively, using the station nomenclature illustrated in Fig. 10-1. The distribution of the flow diffusion between the rotor and stator can be expressed in terms of the fraction of the stage static enthalpy rise that occurs in the rotor, i.e., the stage reaction, defined as R = ( h2 − h1) / ( h3 − h1)

(10-3)

Alternatively, reaction may be defined in terms of static pressures, but that is less convenient for the design problem. Since Cr = 0 for this case, Eq. (3-11) and the basic kinematics of the velocity components yield, H = h + 12 C 2

(10-4)

C 2 = Cz2 + Cθ2

(10-5)

Noting that H3 = H2, Eqs. (10-2) and (10-4) yield h2 − h1 = U (Cθ 2 − Cθ1) − (C22 − C12 ) / 2

(10-6)



FIGURE 10-1 Stage Station Number Nomenclature

h3 − h1 = U (Cθ 2 − Cθ1) − (C32 − C12 ) / 2

(10-7)

Now consider a repeating stage design, i.e., a stage that is designed assuming it will be followed by another identical stage. This means that all velocity components at the stator exit and rotor inlet must be identical. Let us further require that Cz2 = Cz1. Then Eqs. (10-3) through (10-7) yield R = 1 − (Cθ 2 + Cθ1) / (2U )

(10-8)

It follows that for a repeating stage design with Cz2 = Cz1, all relevant stage velocity triangles can be specified in terms of the dimensionless performance parameters φ, ψ and R. For example, from Eqs. (3-1), (10-1), (10-2) and (10-8), it is easily shown that Cθ1 / U = Cθ 3 / U = 1 − R − ψ / 2

(10-9)

Cθ 2 / U = 1 − R + ψ / 2

(10-10)

Wθ1 / U = Cθ1 / U − 1 = − R − ψ / 2

(10-11)

Wθ 2 / U = Cθ 2 / U − 1 = ψ / 2 − R

(10-12)

tan β1 = (1 − R − ψ / 2) / φ

(10-13)



tan β2 = (1 − R + ψ / 2) / φ

(10-14)

tan β1′ = −( R + ψ / 2) / φ

(10-15)

tan β2′ = (ψ / 2 − R) / φ

(10-16)

Specifications for the inlet and exit guide vanes have not been discussed yet. The simplest specification is to require Cz to be constant across these vanes and Cθ0 = Cθ4 = 0.

10.2 APPLICATION TO STAGE DESIGN Development of the dimensionless performance parameters in the previous section assumed constant Cz (or φ) throughout the stage. But Cz is also governed by basic fluid dynamics, as has been described in Chapter 7. In particular, Cz must satisfy some form of the normal equilibrium equation. This means that the stage dimensionless performance can only be specified at one radius for any axial station. The radius at which the dimensionless performance is specified will be designated as rc, and a subscript, c, will be used to identify all parameters on that radius. Hence, the stage dimensionless performance will be specified by

φ c = Czc / Uc

(10-17)

ψ c = (Cθ 2c − Cθ1c ) / Uc

(10-18)

Rc = 1 − (Cθ1c + Cθ 2c ) / (2Uc )

(10-19)

In Chapter 7, normal equilibrium and conservation of mass were combined to complete the solution. That requires that the stage design be specific to the working fluid and Mach number level for which it is to be applied. In this chapter, it is useful to avoid imposing those constraints in the interest of a more general investigation of design alternatives. This will also accommodate the general industrial repeating stage design application discussed earlier in this chapter. The application-specific design problem will be considered in chapter 11. Hence the problem to be addressed in this chapter is illustrated in Fig.10-2. The hub, shroud and reference radii will all be held constant through the stage. The stage will be designed as a repeating stage with φc constant throughout the stage, i.e., Czc is constant on the reference radius. The absence of an equation of state and Mach number dependence precludes a meaningful calculation of losses from the models presented in Chapter 6, so the flow will be considered to be isentropic. Common design practice is to require constant total enthalpy from hub to shroud, which will be assumed to be the case in stage design studies considered in this chapter. It will be shown later in this chapter that it is relatively easy to include provisions to relax many of these constraints in an actual computerized stage design system to obtain a more general design capability. But that does not add any benefits to developing an understanding of the merits of alternate dimensionless performance specifications. The flow profiles at other radii will be supplied by applying the simple, isentropic normal equilibrium



FIGURE 10-2 The Stage Design Application

equation to a prescribed tangential velocity distribution. A general vortex type equation is applied at the rotor inlet. Cθ1 / Uc = (1 − Rc )( rc / r )n − (ψ c / 2)( rc / r )m

(10-20)

where the exponents n and m are specified. From Eq. (10-9), it can be seen that this vortex equation is consistent with the dimensionless performance specifications at r = rc. It will be seen later in this chapter that this vortex equation includes a wide variety of design styles, including the styles most commonly used. At the rotor exit, the constant total enthalpy assumption is used. Cθ 2 / Uc = Cθ1 / Uc + ψ c ( rc / r )

(10-21)

For our repeating stage design problem, the other tangential velocity distributions are given by Cθ 3 / Uc = Cθ1 / Uc

(10-22)

Cθ 0 / Uc = Cθ 4 / Uc = 0

(10-23)

The basic normal equilibrium equation is obtained from Eq. (3-30) by expressing it in stationary coordinates, neglecting streamline curvature and entropy gradients.



For the present case, the meridional coordinate is z, and the normal coordinate is r. This yields Cz

∂Cz Cθ ∂rCθ ∂H + = ∂r ∂r r ∂r

(10-24)

In cases to be considered in this chapter, the total enthalpy gradient will also be zero, but this term has been retained to accommodate provision for a more generalized work distribution. Equation (10-24) can be integrated numerically, with the constant of integration supplied by the known value of φc. Hence, the dimensionless velocity components and velocity triangles can be computed at all of the axial stations on Fig. 10-1 and at all radii once values for φc, ψc, Rc, n and m are supplied. Specified values of the hub, shroud and reference radii are also required. The design of blades to produce these velocity triangles is discussed in the following section. Practical application or evaluation of these ideal stage designs is provided through the performance analysis procedures described in Chapters 6 through 9. The ideal stage design is accomplished over a range of radii sufficient to pass the desired mass flow for the intended applications. The performance analysis then selects the portion of the blades designed that are actually required, using the annulus sizing capability described in Chapter 7, Section 7.7.

10.3 BLADE DESIGN Once the velocity component and velocity triangle distributions have been computed at the entrance and exit stations for all blade rows to be designed, the blade geometry to be used can be computed. Some basic geometry data must be specified to carry out the blade design process. Several alternative specifications are possible, but this writer finds the following specifications to be effective and relatively easy to supply: • • • •

Specify the blade camberline and profile type to be used. Specify the number of blades in the blade row, Z. Specify the chord, c, at the hub, reference and shroud radii. Specify the thickness-to-chord ratio, tb / c, at the hub, reference and shroud radii. • If required, specify the location of maximum camber, a / c. • For double-circular-arc blades specify the leading and trailing edge nose radii, r0. A convenient specification is the ratio, r0 / tb. • Specify the difference between the incidence angle and the design incidence angle, ∆i = i – i*, at the hub, reference and shroud radii. Values of c, tb / c and ∆i at other radii are obtained by interpolation from the three points supplied. The local solidity is given by

σ = cZ / (2πr )

(10-25)



The blade angle sign convention used in this book generally results in a negative turning angle for rotors and inlet guide vanes. To apply the blade geometry procedures of Chapter 4 and the empirical cascade performance models of Chapter 6 in these cases, it is necessary to change the signs on all flow and blade angles. After the blade design is complete, the signs of all angles are changed back to obtain blade geometry consistent with the sign convention used. The blade design process will be illustrated for the rotor blade, with the understanding that this sign adjustment is required. Design of other blade rows is handled in the same manner. The procedure is as follows: 1. Initialize the blade inlet and discharge angles equal to the relative flow angles, i.e., κ1 = β 1′ , κ2 = β 2′ . 2. Compute all other cascade geometry from κ1 and κ2 and the above geometry specifications, using the procedures of Chapter 4. 3. Compute i* and δ* using the procedures of Chapter 6. 4. Recompute the blade inlet and discharge angles from κ1 = β 1′ – i* – ∆i and κ2 = β 2′ – δ*. 5. Repeat Steps 2 through 4 until convergence on κ1 and κ2 is achieved. When the blades have been designed at all radii, all data required for the performance analysis are available. Detailed camberline and blade profile data can also be obtained as described in Chapter 4.

10.4 SELECTING THE STAGE PERFORMANCE PARAMETERS It may appear that the designer has a vast array of choices for stage design through the basic performance specifications of φc, ψc, Rc, n and m. This is certainly true to some degree, but practical considerations dramatically reduce the designer’s choices. When recommendations from the literature, gained from past stage design experience, are considered, the designer’s range of choices actually become quite limited. Figures 10-3 through 10-5 illustrate practical constraints. Here, it is assumed that the relative velocity ratio across any blade row should not be less than 0.7. This is an adjusted de Haller stall limit obtained from Eq. (9-17) using the limit expressed in Eq. (9-18) and with tb / c = 0.1. Noting that the minimum relative velocity ratio will almost always occur at either the hub or shroud radius, it is doubtful that designing for lower velocity ratios at the reference radius would be a wise choice. Eqs. (10-11) and (10-12) can be used to express the rotor blade relative velocity ratio as W2 = W1

φ c2 + (ψ c / 2 − Rc )2 φ c2 + (ψ c / 2 + Rc )2

(10-26)

Similarly, for the stator, Eqs. (10-9) and (10-10) yield C3 C1 = = C2 C2

φ c2 + (1 − Rc − ψ c / 2)2 φ c2 + (1 − Rc + ψ c / 2)2

(10-27)



FIGURE 10-3 ψ-φ Limits for 50% Reaction

FIGURE 10-4

φ-ψ Limits for 25% and 75% Reaction



FIGURE 10-5

φ-ψ Limits for 0% and 100% Reaction

Eqs. (10-26) and (10-27) yield identical results for 50% reaction, i.e., the velocity triangles for the rotor and stator are identical (except for the sign convention). If these equations are applied for 25% and 75% reaction, it can also be seen that the minimum velocity ratio will be the same for both cases. The minimum simply appears on different blade rows for these two cases. Thus, the restrictions imposed by the velocity-ratio limit are symmetrical about Rc = 0.5. It can be useful to specify the work coefficient as some fraction of the value at this velocityratio limit to more easily choose practical values. This is easily done using the following empirical correlation for ψc corresponding to the velocity ratio of 0.7.

ψ lim (φ , R) =

6 Rˆ  0.5  + 0.85   Rˆ  17

1.18

Rˆ = 0.5 + R − 0.5

φ

(2+0.1/ Rˆ )

(10-28) (10-29)

Results from this empirical equation are illustrated in Figs. (10-3) through (10-5). Another constraint used on Figs. 10-3 through 10-5 is that the absolute values of all flow angles should not exceed 70°. Larger flow angles are likely to be impractical with respect to maintaining an acceptable throat area. Beyond that, many of the empirical cascade performance correlations of Chapter 6 are limited to flow



angles less than 70°, which is sufficient reason to impose this limit. Noting that β3 = β1, Eqs. (10-14) and (10-15) can be used to express this limit as approximately

ψ c / 2 ≤ 2.75φ c − Rc

(10-30)

ψ c / 2 ≤ 2.75φ c + Rc − 1

(10-31)

Note that the restrictions imposed by this constraint are also symmetrical with respect to Rc = 0.5. Thus, these simple practical considerations have substantially reduced the range of choices for φc and ψc. It can also be noted that 50% reaction permits designing for the lowest values of φc and ψc. Indeed, as reaction increases or decreases from 50%, the major impact seen in Figs. 10-3 through 10-5 is that the acceptable range of design choices for φc moves toward higher values. There is also a reduction in the range of acceptable values of ψc for a given value of φc, but that effect is relatively minor. Another important consideration is obtaining an acceptable surge margin, i.e., an acceptable flow range between the flow rate at the design point and the flow rate at which surge occurs. Alternatively, surge margin may be expressed in terms of the difference in discharge pressure between the design point and the surge point. It is clear that improved surge margin can be obtained by designing for a steeper slope on the constant speed pressure-flow characteristics. This resists the trend for increased losses or abrupt stalls to force the characteristic toward an unstable positive slope as the flow rate is reduced. Cumpsty (1989) notes that it is useful to approximate the work input curve in the form

ψ = 1 − φ (tan β1 − tan β2′ )

(10-32)

This relation follows directly from Eqs. (10-13) and (10-16). The advantage of this form of the work input equation is that the flow angles involved are both discharge angles relative to the upstream blade row. It was shown in chapter 6 that the deviation angle is a very weak function of incidence angle. Hence, these flow angles can be considered to be approximately constant over a speed line. As illustrated in Fig. 10-6, this can be used to approximate Eq. (10-32) by

ψ ≈ 1 − (1 − ψ c )φ / φ c

(10-33)

It follows that the surge margin will be improved by choosing lower values of both φc and ψc. From Eqs. (10-13), (10-16) and (10-32) it is easily shown that the slope of ψ as a function of φ will be positive if ψc > 1 is selected. This is certainly inconsistent with a reasonable surge margin. Hence, ψlim > 1 is treated as a third constraint on Figs. 10-3 through 10-5. Cumpsty (1989) also notes that the slope of the work-input characteristic is important to resist the influence of local total pressure distortions, even at operating conditions far from surge. He illustrates this by approximating the flow as incompressible, such that Pt = P + 12 ρ W 2 = P + ρ U 2φ 2 / cos2 β ′

(10-34)



FIGURE 10-6 Approximate Work-Input Characteristic

Hence a local perturbation in total pressure, within an essentially constant static pressure flow field, can be approximated by

δPt = 2ρ U 2φδφ / cos2 β ′

(10-35)

It follows that a local deficit in total pressure will produce a local deficit in φ. If the work-input characteristic is sufficiently steep, the corresponding increase in ψ will reduce, and possibly eliminate, the total pressure deficit. Smith (1958) provides a more detailed analysis of the capacity of a stage design to resist local total pressure deficits. In a discussion at the end of the paper by Smith, Ashley reports that the NACA had independently arrived at almost exactly the same result. Using arguments analogous to Eq. (10-35), Smith developed an equation to express the magnitude of a pressure deficit, (δPt)out, at the stage exit caused by an local inlet deficit, (δPt)in. He expressed this in terms of a parameter, RR, which he called the recovery ratio. RR = 1 −

(δPt )out (δPt )in

(10-36)

It follows that for RR = 1, the inlet total pressure deficit has been completely removed at the stage exit. For the case corresponding to the present dimensionless



performance parameters at the reference radius, Smith’s recovery ratio can be expressed as RR = (cos2 β1 tan β1 − cos2 β2′ tan β2′ ) / φ + cos2 β1 cos2 β2′ tan β1 tan β2′ ) / φ 2 (10-37) Once again, it can be shown that RR is symmetrical about R = 0.5. Figs. 10-7 through 10-9 show typical results for various values of reaction. To maintain reasonable values of ψc in these predictions, φc was constrained to the limits shown in Figs. (10-3) through (10-5), and Eq. (10-28) was used in the form

ψ c = K cψ lim (φ c , Rc )

(10-38)

While values of RR = 1 are certainly not required, it is expected that small or negative values should be avoided where possible. Based on results similar to those shown in Figs. 10-7 through 10-9, Smith suggested that values of φc < 0.5 are preferred. Indeed, φ < 0.5 at all radii would be preferred, which is a much stronger constraint. Clearly, if φc = 0.5 were imposed as an upper limit on Figs. 10-3 through 10-5, the range of choices available to the designer would be narrow. This is not considered to be appropriate in light of the many successful axial-flow compressor designs that have used significantly higher values of φc. But there does appear to be a definite trend toward lower values of φc in more modern

FIGURE 10-7 RR for 50% Reaction



FIGURE 10-8 RR for 25% and 75% Reaction

FIGURE 10-9 RR for 0% and 100% Reaction



designs. Smith also concluded that the recovery ratio is not a strong function of reaction, which may be somewhat questionable. It is evident that RR < 1 is to be expected for any practical combination of φc and ψc when Rc is 0 or 1. Other guidance available in selection of φc, ψc and Rc are more in the form of opinions and rule-of-thumb practices. It has been common practice to favor 50% reaction designs, based on the intuitive benefit expected from distributing the flow diffusion load equally between the rotor and stator. Sometimes slightly higher reaction is recommended based on the assumption that rotors are slightly more efficient than stators. Similarly, designers seem to be most comfortable with choosing ψc ≤ 0.4 in most cases. None of these opinions have been well substantiated by either experiment or analysis. Indeed, such guidelines are probably rather academic, since the selection of φc, ψc and Rc is dependent on many other factors. The swirl vortex type discussed in the next section of this chapter is very important. That determines the stage performance at other radii, and may have a significant influence on the choice of the reference radius performance parameters. Also, the developments in this chapter do not consider the effect of the Mach number level, which can have important consequences and impose definite constraints. That can be illustrated by a design goal common to nearly all axial-flow compressor designs, i.e., achieving the maximum mass flow per unit frontal area and the minimum number of stages. For aircraft engine compressors this permits essential size and weight reductions, while for industrial compressors the important benefit is reduced size and cost. In either case, this immediately encourages selection of larger values of ψc and φc, although lower values of φc are generally preferred. This might be resolved by simply designing for a higher speed to satisfy both needs. But Mach number effects impose definite limits on that approach as well as result in performance penalties when the approach is used.

10.5 SELECTING THE SWIRL VORTEX TYPE In addition to the reference radius performance parameters, it is necessary to select the vortex type by specifying n and m in Eq. (10-20) to set the rotor inlet Cθ distribution. The Cθ distributions at other stations are given by Eqs. (10-21) through (10-23) for a constant-work, repeating stage. When the Cθ distributions are known, the Cz distributions at all stations are obtained by solving Eq. (10-24). Then the velocity triangles are known at all radii for all stations. If those velocity triangles are acceptable, the blades can be designed as described in Section 10.3. At this point in the design process, fluid dynamic data are available at all radii and must be evaluated. The same design preferences and practical limits considered relative to the reference radius certainly are relevant at other radii as well. It will be seen that problems associated with excessive flow diffusion or expected Mach number limits almost always occur first on the end-walls, usually at the rotor shroud and stator hub locations. It is not uncommon for an apparently very conservative selection of φc, ψc and Rc to produce unacceptable performance parameters at other radii. In the following sections, some commonly used vortex types will be reviewed to illustrate some of the important considerations in selecting the vortex type. To allow comparison of the various vortex types, a common design problem will be considered. All examples will be constant-work,



repeating stage designs. A hub-to-tip ratio of rh / rs = 0.5 will be considered, which is typical of the front stages in an axial-flow compressor. The reference radius will be set to rc / rs = 0.8, which approximately divides the annulus into two equal area passages. In principle, the reference radius is an arbitrary design parameter, but, in practice use of a typical mean-radius is usually necessary. Since Cz is constant on rc, use of a mean-radius ensures approximate conservation of mass, which will be required when the stage is applied in a compressor. If the stage design deviates too far from this, the velocity triangles produced after sizing of the compressor annulus area may be very different from those intended.

10.6 FREE VORTEX FLOW It is instructive to start with the popular free vortex design style, which is given by setting n = m = 1. From Eqs. (10-20) through (10-23) it can be shown that this yields constant angular momentum (rCθ) profiles at all stations. If rCθ and H are constant, Eq. (10-24) also yields constant Cz profiles. Indeed, since Cz is constant on the reference radius, this means that Cz is constant everywhere. Let us design a free vortex stage using very conservative dimensionless performance parameters. Following Smith, (1958) set φc = 0.5. To provide significant margin from the loading limit, set Kc = 0.8. Choose Rc = 0.5 to distribute the loading equally between the rotor and stator. It would be expected that such conservative design specifications and such a simple type of vortex would certainly result in an acceptable design. Figures 10-10 and 10-11 show some of the important results obtained with these design specifications. It is apparent from the flow angle distributions that this design will require a highly twisted rotor blade, which may pose structural and manufacturing problems. The stator blade will be highly staggered to match flow angles close to the 70° limit suggested earlier in this chapter. But more serious is the fact that the design yields a substantial region of negative reaction near the hub. This results in significant flow acceleration across the rotor, followed by significant deceleration across the stator. The stator is overloaded near the hub, and probably will stall. Since ψh > 1, stability is a concern, as discussed in Section 10.4. The unacceptable features of this example could have been anticipated. Since φ is constant across the blade rows at all radii, Eq. (10-8) can be applied at any radius to predict the reaction. From Eq. (10-20), free vortex flow yields Cθ1 / U = (1 − Rc − ψ c / 2)( rc / r )2

(10-39)

Equation (10-21) requires Cθ 2 / U = Cθ1 / U + ψ c ( rc / r )2

(10-40)

Substituting Eqs. (10-39) and (10-40) into Eq. (10-8) yields R = 1 + ( Rc − 1)( rc / r )2

(10-41)



FIGURE 10-10 Flow Angle Distributions

FIGURE 10-11 Blade Loading Distributions



Hence the condition to avoid negative reaction at the hub for free vortex flow is Rc ≥ 1 − ( rh / rc )2

(10-42)

For this example this requires Rc ≥ 0.609. Rc = 0.5 can be maintained by requiring rh / rc ≥ 0.7071. Since Cz is constant everywhere, the choice of rc is quite arbitrary for free vortex flow. It is likely that Rh significantly greater than zero would be preferred, which can be easily obtained using Eq. (10-41). This simple example illustrates that selection of the dimensionless stage performance parameters is strongly influenced by the vortex type to be used and the radius-ratio range required. Instead of refining the previous example, it is more useful to continue the discussion of free vortex flow with a more commonly used form obtained by adding the requirement that Cθ1 = Cθ3 = 0. The advantage of this form is that no inlet or exit guide vanes are required. From Eq. (10-9) this requires

ψ c = 2(1 − Rc )

(10-43)

To maintain a negative slope on the work-input characteristic, ψc < 1 is required. That requires Rc > 0.5, which means that the rotor is more highly loaded than the stator. To maintain a reasonable loading level in the rotor, it is desirable to specify the velocity ratio, W2c / W1c, as well. Equations (10-26) and (10-43) yield

φ c2 =

(W2c / W1c )2 − (1 − 2Rc )2 1 − (W2c / W1c )2

(10-44)

Figure 10-12 is a (Rc, φc) design chart with W2c / W1c as a parameter, with the usual practical limits imposed. Clearly, Smith’s recommendation of φc ≤ 0.5 requires rather high reaction. From Eq. (10-43) that also requires very low values of ψc. Thus, it may be difficult to follow Smith’s recommendation while maintaining a reasonable stage work input coefficient. For example, Fig. 10-12 includes the (Rc, φc) mean-radius design point for the first stage of the NACA 5stage axial-flow compressor (Sandercock et al., 1954). This is an actual example of a free vortex design with Cθ1 = 0, which illustrates the type of compromise that may be required in a practical stage design. To illustrate features of the free-vortex design, the basic specifications for the first stage of the NACA 5-stage axial-flow compressor were used to generate a stage design using the procedures presented in this chapter. Figure 10-13 shows the relevant flow angles for this design. A variable camber, highly twisted rotor blade will be required, although not as extreme as the earlier example. Figure 10-14 shows that the rotor tip inlet relative velocity is by far the largest velocity in the stage and will be the value to be evaluated in terms of Mach number levels. No general conclusion can be reached without specifying the stage inlet conditions and rotation speed, but the rotor tip will be the critical location. Indeed, as observed in chapter 9, the first stage of the NACA 5-stage compressor is a transonic design. The stator inlet and exit velocity levels are quite modest by comparison, so Mach number level should not be a concern. Figure 10-15 shows that the velocity ratios across the



FIGURE 10-12 Free Vortex Design Chart

FIGURE 10-13 Free Vortex Flow Angles



FIGURE 10-14 Free Vortex Velocities

FIGURE 10-15 Free Vortex Performance Parameters



blades are relatively close to the de Haller limit on both the rotor and the stator. As seen in Figure 10-12, the design specifications correspond to a rotor relative velocity ratio at the reference radius of only 0.75, so this is really to be expected. The reaction is significantly positive at the hub, which has avoided the velocity ratio problems seen in Fig. 10-11 for the simple example considered earlier. A smaller value of ψc probably should be used to avoid the ψh > 1 condition and to improve the velocity ratios across the blade rows. The blade design procedures described earlier in this chapter were used to design double-circular-arc blades to match the velocity triangles. The blades were designed with constant chord, and with tb / c = 0.1. The number of blades in each row was set to yield σ = 1 at the reference radius. Figure 10-16 shows the camber and stagger angle distributions for the rotor and the stator. As expected, a highly twisted, variable camber rotor blade is required. Variations of camber and stagger angles with radius on the stator are relatively modest. However, the radial variation in β2, as seen in Fig. 10-13, is probably too great to seriously consider using a constant camber, constant stagger stator blade for this stage.

10.7 CONSTANT REACTION VORTEX FLOW Another vortex type occasionally suggested is obtained by setting n = -1 and m = 1. Inserting these values into Eqs. (10-20) and (10-21) and substituting

FIGURE 10-16 Free-Vortex Blade Design



into Eq. (10-8) appears to yield a constant reaction design, which is the title commonly applied to this vortex type. It must be recognized that, normally, constant reaction will not be achieved, since φ is not generally constant across the rotor as is assumed by Eq. (10-8). This is easily illustrated by repeating the “conservative” design used as an example of free vortex flow in the previous section. Figures (10-17) and (10-18) show results from that design with the constant reaction type of vortex. It is seen that the design is by no means that of constant reaction. Indeed, reaction is close to zero at the hub radius. Velocity ratios across both the rotor and stator are well below the de Haller limit, which is not very encouraging for operation at the stage’s design point. But the major problem is that the axial velocity at the rotor discharge drops to zero below the tip radius. The stage design software used restricts the axial velocity from becoming negative to prevent fatal errors in the analysis. So the zone of zero axial velocity is actually a reverse flow zone. Data predicted in this zone are not accurate, as indicated by the dashed lines near the shroud radius. Once again, it is seen that the specific vortex style can produce an unacceptable design, even though the dimensionless performance parameters appear to be rather conservative. It is quite evident that the large variation in axial velocity between the rotor exit and inlet stations has completely nullified the intention of achieving an essentially constant reaction. One of the exercises at the end of this chapter will be to show that the radial gradient of reaction for any constant work, repeating stage is given by

FIGURE 10-17 Axial Velocity Distributions



FIGURE 10-18 Performance Parameter Distributions

rc

∂R = 2(1 − Rc )( rc / r )(2+ n) + ψ c [( rc / r )3 − ( rc / r )(2+ m) ] ∂r

(10-45)

With n = -1 and m = 1, this yields rc

∂R = 2(1 − Rc )rc / r ∂r

(10-46)

Equation (10-45) can be integrated to yield R = Rc + 2(1 − Rc ) ln( r / rc )

(10-47)

Clearly, the only case for which this vortex type will yield constant reaction is when Rc = 1. For other values of Rc, Eq. (10-47) can be used to compute a value of Rc that will yield a desired value of reaction at the hub. It is clear that reaction is completely independent of φc and ψc. However, it will be possible to eliminate the reverse flow zone by increasing φc. Indeed, another exercise at the end of this chapter will be to show that the condition to avoid reverse flow at the rotor exit for any radius, r, for the constant reaction vortex style is

φ c2 ≥ 2(1 − Rc )2 [( r / rc )2 − 1] + 2(1 − Rc )ψ c ln( r / rc )

(10-48)



Reverse flow will first occur at the rotor exit on the shroud. Hence Eq. (10-48) can be used to compute the minimum acceptable value of φc for any value of Rc using r = rs. Alternatively, the value of Rc required to avoid reverse flow for any φc can be estimated from Eq. (10-48) by trial and error. In this way, it can be shown that values of φc down to 0.5 will be achievable for this case with no reverse flow if Rc ≥ 0.6 is used. This limit was used to investigate the influence of φc and Rc on the stage design for this vortex type. The influence of φc is shown in Fig. 10-19. It is seen that φc has little influence on the velocity ratios across the blades. Increasing φc increases the inlet velocity for both blade rows. Since the loss is proportional to the inlet kinetic energy, this is expected to increase the losses. It may also lead to other problems due to the increased Mach number levels. The only real benefit from larger values of φc is the higher values of ψc obtained for the same value of Kc. The influence of Rc is shown in Fig. 10-20. Increasing Rc will increase the rotor inlet kinetic energy, raising concerns about loss and Mach number levels that are similar to the concerns regarding Fig. 10-19. The reduction of the stator inlet kinetic energy is some compensation for these concerns. The work coefficient is nearly independent of Rc for a specified Kc. Hub reaction and the minimum velocity ratio across the stator both benefit significantly from increased values of Rc. The minimum velocity ratio across the rotor appears a little erratic, which is caused by the location of the minimum value switching from the hub radius to the shroud radius. Probably the most notable observation is that there is an optimum value of Rc at about 0.79 that yields the highest minimum velocity ratio across

FIGURE 10-19 Influence of Flow Coefficient



FIGURE 10-20 Influence of Reaction

either blade row. This would be expected to provide the best stall margin, assuming the increased rotor inlet Mach number level does not become the limiting factor. To investigate this more closely, a constant reaction vortex stage was designed based on this optimum. For the purpose of comparison, a free vortex stage was designed for the same set of dimensionless performance parameters. Figures 10-21 through 10-23 compare some of the more significant results. It is seen that the constant reaction vortex flow yields the lowest peak rotor inlet velocity and offers greater margin from stall as indicated by the velocity ratios across the blade rows. Both vortex types require a twisted, variable-camber rotor blade, but the variations are less extreme for the constant reaction vortex style. Twisted, variable-camber stator blades are required for both vortex types, with slightly more extreme variations for the constant reaction type of vortex. The work input coefficient distributions for both vortex types are identical, with ψh = 0.913, so the work input characteristics at all radii should have the desired negative slope. It can be concluded that the constant reaction vortex style does offer some advantages over the free vortex style for this specific set of dimensionless performance parameters, particularly with regard to increased stall margin. The most important conclusion with regard to the constant reaction vortex style is that the usual claim that it yields blades of nearly constant reaction is simply not correct, unless Rc values that are very close to 1.0 are used. In other cases, it is necessary to choose a reaction at the reference radius that will yield an acceptable reaction at the hub, much like the case of the free vortex style. This is



FIGURE 10-21 Comparison of Velocity Ratios

FIGURE 10-22 Comparison of Rotor Blade Geometry



FIGURE 10-23 Comparison of Stator Blade Geometry

easily accomplished with the aid of Eq. (10-47), so effective use of the constant reaction vortex style is not too difficult. It is worth considering whether this vortex style can be modified so that it will more closely approximate a constant reaction stage. From Eq. (10-45) it is clear that there is really no alternate selections for m and n that will yield constant reaction. One of the exercises at the end of the chapter will be to show that the only possible constant reaction design is the 100% reaction case. A very modest reduction in the radial gradient of the reaction can be achieved by choosing n = -2 and m = 1, which yields rc

∂R = 2(1 − Rc ) ∂r

R = Rc + 2(1 − Rc )( r / rc − 1)

(10-49) (10-50)

This has the undesirable effect of producing steeper gradients near the shroud radius, to impose additional restrictions to avoid reverse flow. It is possible to more closely approximate constant reaction by choosing n = -1 or -2 and selecting m > 1. From Eq. (10-45) it can be seen that this introduces a negative contribution to the radial-gradient of R when r < rc to partially balance the usual positive gradient. However, it has the reverse and unfavorable effect when r > rc. This approach is generally not too effective, but it could be considered for very specific applications.



10.8 CONSTANT SWIRL AND EXPONENTIAL VORTEX FLOWS Horlock (1958) suggests that the exponential vortex style is often a good choice, and is obtained by setting n = 0 and m = -1. A very similar alternative is obtained by setting n = m = 0, which will be called the constant swirl vortex style. As seen from Eq. (10-20), the constant swirl vortex style yields constant Cθ1. With appropriate choices for φc, ψc, and Rc, these two vortex types may yield a constant-camber stator blade, possibly with a constant stagger angle as well. This is certainly attractive for reducing manufacturing cost and may offer other advantages, such as improved structural characteristics. From Eq. (10-45) it is easily shown that the reaction at any radius for exponential vortex flow is given by R = Rc − 2(1 − Rc )( rc / r − 1)

(10-51)

whereas for constant swirl vortex flow, R = Rc − [2(1 − Rc ) + ψ c ( rc / r − 1) / 2]( rc / r − 1)

(10-52)

Hence the exponential vortex style has the advantage that the value of Rc required to achieve any desired hub reaction is easily determined. It is also clear that the exponential vortex flow will yield the higher hub reaction for any value of Rc. While these advantages tend to favor the exponential vortex style, the constant swirl vortex flow also has a definite advantage, as illustrated by Figs. 10-24 and 10-25. These figures show the difference in the stator camber angle and stagger angle between the hub and the shroud for stages designed with these two types of vortex. If the difference in camber angle is small enough, a constant camber stator blade can probably be used. Similarly, if the difference in stagger angle is small enough, a constant stagger stator blade can probably be used. It is clear that both vortex styles can yield a constant-camber stator blade if an appropriate value of Rc is chosen. It can be seen that the precise value of Rc required varies with φc. Indeed, it will also depend on Kc to some degree. Clearly, the prospects for achieving a stator blade with both constant camber and constant stagger angles are substantially better if the constant swirl vortex flow is employed. To illustrate the advantages of constant swirl vortex flow, a stage was designed specifically to achieve a constant-camber, constant-stagger stator blade. The important stage performance data for this design are shown in Fig. 10-26. It will be noticed that this design used a lower value of Kc than has been used in previous examples. This was necessary to maintain a reasonable velocity ratio across the stator at the hub, which is typically the weakest area of a constant swirl or exponential vortex design. The reduced work input capability is not as pronounced as it may first appear. Indeed, Fig. 10-21 shows that, for similar reasons, a similar reduction in Kc would have been beneficial for the free vortex design as well. In comparison to the free vortex and constant reaction vortex designs in Fig. 10-21, it is seen that lower values of rotor-inlet relative velocity have been achieved. The basic limitations for this vortex style are associated with low values of R and C3 / C2 near the hub. Figure 10-27 shows the rotor and stator camber



FIGURE 10-24 Exponential Vortex Stators

FIGURE 10-25 Constant Swirl Vortex Stators



FIGURE 10-26 Constant Swirl Vortex Design Data

FIGURE 10-27 Constant Swirl Vortex Blade Angles



and stagger angles obtained using the blade design procedures of Section 10.3. In this case, NACA 65-series blades were designed with tb / c = 0.1 and σ = 1 at r = rc. The camber angles in Fig. 10-27 are the equivalent circular-arc camber angles defined in chapter 4. Note that the result is basically a constant-camber, constant-stagger stator blade. This is usually not too difficult to achieve. The constant swirl vortex style yields flow angles that essentially produce a linear variation of camber and stagger angle from hub to shroud. For any basic design parameters, constant camber is achievable by selecting the proper value of Rc, as illustrated in Fig. 10-25. By adjusting the other dimensionless performance parameters, the variation in stagger angle can usually be reduced to a point where constant stagger angles can also be used without significantly compromising performance. This would be a very cost-effective design for use as a standard repeating stage for industrial axial-flow compressors.

10.9 ASSIGNED FLOW ANGLE VORTEX FLOWS Occasionally, it is useful to assign the absolute flow angle distribution at the rotor inlet or discharge rather than the swirl velocity distribution. Noting that Cθ = Cz tanβ, it is easily shown that Eq. (10-24) can be replaced by cos2 β ∂( r tan β )2 cos2 β ∂H 1 ∂Cz + =− ∂r Cz ∂r Cz2 ∂r 2r 2

(10-53)

The specification of φc, ψc and Rc defines β1c and β2c. If it is assumed that β varies linearly with r, and the difference between hub and shroud, βs – βh, is specified at either the rotor inlet or discharge station, Eq. (10-53) can be solved for the corresponding distribution of Cz. Since Cθ = Cz tanβ, the swirl velocity distribution is also defined. For constant work stages, Eq. (10-21) defines the swirl distribution at the other station. This vortex specification is a useful variant on the constant swirl vortex type discussed in the previous section. Sometimes it is easier to finetune the blade angle distributions by refining the flow angle distribution rather than the swirl velocity distribution. Usually, it is more effective to specify the rotor exit flow angle distribution for this purpose. This vortex type can be quite useful, and it is easily incorporated into a computerized stage design system.

10.10 APPLICATION TO A PRACTICAL STAGE DESIGN The stage design examples provided in the previous sections of this chapter have been rather arbitrary in illustrating some of the features of the various types of vortex flow styles. In practice, stage design is far more focused on specific design objectives and constraints. Usually there is substantial conflict between various desired design objectives in terms of establishing the specific stage design parameters. In almost any axial-flow compressor design, minimum size, weight and cost will be important objectives. Clearly, these goals are best served by achieving a



large mass flow per unit frontal area, thus favoring a high flow coefficient stage. Similarly, reducing the number of stages required will be important, thus favoring a high work input coefficient design. As seen in Figs. 10-3 through 10-5, a high work input coefficient requires a high flow coefficient to avoid exceeding practical blade loading limits. So size, weight and cost considerations are definite incentives for the high flow coefficient and work input coefficient stages. Good aerodynamic performance will also be an important objective for any axial-flow compressor design. Usually this is expressed in terms of efficiency and surge margin requirements. Surge margin may be expressed in terms of flow range or pressure-rise range between the design point and the surge limit. Surge margin and efficiency are not really independent concepts, although specific applications may impose special surge margin requirements. Normally, the stage design point is expected to be approximately its best operating condition. It is quite obvious that there is little merit in achieving optimum performance too close to the surge limit, where the stage will almost never operate in an actual compressor. In Section 10.2 it was shown that improved surge margin can be expected to favor low flow coefficients and low work input coefficients. In Figs. 10-7 through 10-9, Smith’s (1958) recovery ratio parameter has strongly suggested that flow coefficients around 0.5 should be preferred. Smith (1958) seems to favor flow coefficient not exceeding 0.5 at any radius, although he notes that it may be difficult to achieve. Indeed, that seems a little impractical and perhaps not even desirable. Depending on the hub-to-shroud radius ratio, achieving a hub flow coefficient of 0.5 may require very low shroud flow coefficients. That may be as harmful as using flow coefficients that are too high, as can be seen in Fig. 10-7. Figs. 10-7 through 10-9 also show that low work coefficient (or K) is desirable and has the effect of reducing the significance of the flow coefficient selection. The role of stage efficiency in setting stage design parameters is less significant than whether the surge margin is adequate to make use of the efficiency achieved. Indeed, if maximum efficiency is used as the major design criterion, the design point is very likely to be very close to the surge limit. This follows from the fact that designing for high work input may be the best way to reduce the impact of the losses on efficiency. Excellent stage efficiencies have been achieved for a wide range of stage performance parameters and vortex types. There is little credible evidence to associate any set of performance parameters or vortex type to optimum and practical design-point efficiency levels. It is sometimes suggested that optimum efficiency is associated with 50% reaction designs, although that is of questionable validity and of very little practical value. The stage reaction, in terms of Rc, will usually be established by surge margin considerations. Specifically, achieving satisfactory hub reaction and velocity ratios across the blades at the hub and shroud will usually be the major factors influencing the selection of Rc. Design approaches directed specifically toward increasing the stage design-point efficiency are generally beyond the scope of the methods described in this chapter. For example, special blade designs such as the controlled diffusion airfoils mentioned in Chapter 4, may be used. Some designers favor special features in the end-wall boundary layer regions, such as increased end-wall work. With modern viscous computational fluid dynamics codes, special designs to minimize secondary flow effects may be considered.



Clearly the designer is faced with conflicting priorities to trade off performance against size, weight and cost. Usually a parametric study will be required to select the best compromise, including evaluation of performance with the methods of Chapter 9. Unless carefully structured, such a parametric study can be very confusing and time-consuming. Usually the most efficient approach is to investigate alternate values of φc, while specifying the minimum acceptable values of the blade row velocity ratios, W2/W1 and C3/C2. Generally, the designer will be able to select an acceptable range of values for φc, while the minimum velocity ratios ensure reasonable blade loading and surge margin capability. Other constraints will be specified and monitored. For example, reasonable limits on flow angle, flow coefficient and work coefficient at all radii will normally be required. To illustrate the process, a simple optimization study has been conducted for the following constraints: • • • • •

φc = 0.5 W2 / W1 and C3 / C2 ≥ 0.73 β and |β′| < 70° ψ<1 φ≤1

Designs were generated for constant swirl vortex, free vortex and constant reaction vortex stages. The free vortex stage was not required to have Cθ1 = 0 for this design. The design radius was chosen as the true mean stream surface, i.e., the root-mean-square average of the hub-and-shroud radii was used. The only parameters available to be selected are Rc and Kc (or ψc), which were chosen to best satisfy the constraints. In all cases, the velocity ratio across the stator proved to be controlling constraint, although the flow angle constraint was a significant factor for the constant reaction vortex. Figures 10-28 through 10-30 show the most important performance parameters to compare the three vortex types. It is seen that the free vortex design resulted in the highest rotor inlet relative velocities, making it the most susceptible to Mach number effects. The constant reaction vortex stage would appear to be the best choice for high Mach number applications, but that is a little misleading. That design is very close to the flow angle constraint near the shroud. It would be difficult to achieve adequate blade passage throat widths while using practical blade thicknesses, so blade passage choke could easily be a problem near the shroud. The constant reaction stage has the largest flow coefficient at the hub, but it is within the limit set for these sample designs. In other respects, the performance parameter distributions for the three designs are reasonably similar. Except for the rather large flow angles produced by the constant reaction vortex stage, the basic dimensionless performance and velocity triangle data provide little reason for preference of one design over the others. The more significant differences in these three designs are seen in Figs. 10-31 and 10-32, which compare the blade geometry required for the three stage designs. In all cases, NACA 65-series blades were designed with constant chord, constant tb / c = 0.1 and σ = 1 at r = rc. All rotor blades are variable camber, twisted blades, but the constant reaction vortex stage requires the most extreme variations. Indeed, a negative camber angle is required at the shroud for that



FIGURE 10-28 Constant Swirl Vortex Stage

FIGURE 10-29 Free Vortex Stage



FIGURE 10-30 Constant Reaction Vortex Stage

FIGURE 10-31 Rotor Blade Geometry



FIGURE 10-32 Stator Blade Geometry

design, with a very high stagger angle at the same location. The constant swirl vortex design results in very simple stator geometry, which is easily approximated by using constant camber and stagger angle blades. The free vortex stage requires variable camber, twisted stator blades, but with very reasonable variations. The constant reaction vortex stage requires highly twisted stator blades with a large variation in the camber angle. The constant reaction stage blades would need careful evaluation relative to its structural integrity and practicality for manufacturing. It is likely that application of this vortex style should be limited to higher hub-to-shroud radius ratios than those in this case The performance analysis described in Chapter 9 was applied to these three stage designs to evaluate their expected performance. The annulus geometry for all stages was obtained by assuming a constant hub radius and sizing the shroud radius based on the same volume flow rate and rotation speed. This design volume flow rate is easily computed from the rotation speed and the free vortex velocity triangles, where Cz is constant. The rotation speed and inlet thermodynamic conditions were chosen to limit the maximum rotor inlet relative Mach number to about 0.78 to avoid biasing the comparison with Mach number effects. Once the approximate annulus sizing was accomplished, the shroud radii were adjusted to produce a conical shroud wall to avoid the results being biased by local contour angle variations that are not representative of a stage’s true performance when used as a repeating stage in a multistage compressor. Similarly, end-wall boundary layer blockage and stream surface curvature



effects were ignored to obtain as fair a comparison as possible. This process produced the expected repeating stage for the free vortex and constant swirl vortex stage designs. That was not the case for the constant reaction vortex design, where a repeating stage configuration could not be achieved in practice. The annulus sizing to match the design velocity triangle for that case produced an increase in shroud radius across the rotor, followed by a large decrease across the stator. The most reasonable approximation to the annulus area schedule required was to simply assign a constant shroud radius. The source of the problem is the axial velocity ratios across the blade rows, which alter the predicted deviation angles as discussed in Chapter 6. As seen in Fig. 10-17, axial velocity ratios can be quite significant for this vortex style. The computerized stage design system used to generate these designs does not correct deviation angles for axial velocity ratio effects. To do so is usually misleading, since axial velocity ratios in an actual application may be quite different from those produced in the ideal stage design calculations. Application of a standard industrial compressor repeating stage involves use of various portions of the blade span in different stages as required to conserve mass flow rate. Adjustments in stagger angles and axial velocity levels through the multistage compressor are usually necessary to produce the required performance with a specific number of stages. If it is not possible to neglect the effect of axial velocity ratio on deviation angle in the basic stage design, the stage is simply not a good candidate for a standard repeating stage. That is really the case for this constant reaction vortex stage. The hub-to-shroud ratio used in this example is simply too low for a constant reaction vortex to produce a viable repeating stage. Of course, it is possible that this stage might be used in a single, specific application. In that case, the application-specific design procedures discussed in the next chapter should be used to design the blades for the stage. Figure 10-33 presents a comparison of the performance predictions for the three stage designs. The lower limit of volume flow rate for each stage corresponds to the predicted stall limit based on stall criterion #3 of Chapter 9. It can be seen that the free vortex and constant swirl vortex stages produce essentially the same performance characteristics. The free vortex stage has a slight advantage in stable operating range, while the constant swirl vortex stage has a slight advantage in efficiency. Neither of these differences can be considered significant within the uncertainty of the performance analysis procedures. The constant reaction vortex design produces lower pressure ratios and efficiencies than the other two stages, even though it had the highest design work input coefficient, ψc. Based on the expected stage performance, either the free vortex or constant swirl vortex design could be selected. The simpler stator geometry of the constant swirl vortex stage and the higher rotor inlet relative Mach number level of the free vortex design would favor selection of the constant swirl vortex design.

10.11 A REPEATING STAGE AXIAL-FLOW COMPRESSOR As mentioned at the beginning of this chapter, a repeating stage design can be used as a standard stage in industrial axial-flow compressors. Each industrial axial-flow compressor usually has a unique design. Pattern, tooling and development costs



FIGURE 10-33 Comparison of Predicted Performance

are critical issues, since there are seldom any duplicate machines to share those costs. Similarly, experimental confirmation of the performance of the finished compressor will be obtained before it is shipped to the customer. Failure to achieve the guaranteed performance may require very expensive rework of the compressor. Use of a standard repeating stage can alleviate these problems. Now pattern and tooling costs can be shared among many industrial compressors, even though each one has a unique design. The risk of a performance deficiency is also greatly reduced, since experience with previous compressors can be used for guidance. Application of a standard stage in a multistage compressor is reasonably straightforward using an aerodynamic performance analysis that is capable of sizing the annulus as described in Chapter 7. To illustrate the process, a ten-stage axial-flow compressor will be configured using the constant swirl vortex repeating stage designed in the previous section. The compressor will be designed with a constant hub radius, so that the same rotor blade can be used throughout, by simply trimming the blades to the proper shroud contour. Since constant-camber, constant-stagger stator blades can be used for the basic stage, a common stator blade is also usable. Under these constraints, it is easily seen that the basic blade geometry is identical for all stages. The simplest way to configure the basic compressor is to assign the mean stream surface meridional velocity through the compressor and size the annulus to define the approximate shroud contour. To simplify this process, the performance analysis is set up with one computing station between successive blade rows. Then the meridional velocity on the mean stream surface is assigned at specific computing stations and obtained at other



stations by linear interpolation with the station number. For this example, it will be assumed that the assigned meridional velocity must vary linearly with station number from the entrance to the first rotor to the exit of the last rotor. The compressor will be sized to use the full flow capacity of the standard stage, at the same rotation speed as that used in Fig. 10-33. So the meridional velocity at the entrance to the first rotor can be assigned immediately from the stage design data. Some fine-tuning is required to match the known shroud radius at this point due to the effect of gas density variations across the inlet guide vane and to the endwall boundary layer blockage assumed at the compressor inlet. This requires a minor adjustment to the design inlet mass flow until the correct shroud radius is obtained. The sizing of the annulus for the entrance to the inlet guide vane and the exit to the last stator and the exit guide vane can be rather arbitrary at this point. As long as the meridional velocities assigned at these stations are not unreasonable, they have little influence on the performance. Manual sizing of the annulus at those stations is rather trivial once the shroud contour is defined at the other stations. So the task to be accomplished in this simple example is selection of the meridional velocity to be assigned at the last rotor exit. The first priority is to obtain a reasonable shroud contour. For example, assigning the meridional velocity on the mean stream surface to be constant through the compressor resulted in the shroud radius increasing with axial distance in the rear stages. This could be anticipated since the mean stream surface radius decreases through the machine. Since the axial velocity for the constant swirl vortex increases as the radius is reduced, it could be expected that the assigned meridional velocity should increase through the compressor. The assigned meridional velocity to be used at the last rotor discharge was easily determined by trial and error, using the performance analysis. Figure 10-34 shows the Cm distribution used and Figure 10-35 shows the hub-and-shroud contours developed by the annulus sizing with that Cm distribution. Figure 10-36 shows the predicted performance using the methods of Chapter 9. The estimated surge limit is also shown, including the stall criterion from Chapter 9 that was first encountered for each speed line. It can be seen that a standard repeating stage designed by the methods of this chapter can be effectively employed in a multistage compressor. This application of a standard stage in a multistage compressor illustrates the conflicting priorities that the designer regularly encounters. The stage design in Section 10.10 followed practices expected to favor good efficiency and stable operating range. This was reasonably successful, but the cost-effectiveness of a ten-stage compressor that achieves a design pressure-ratio of about 2.2 is very questionable. The design constraints selected in Section 10.10 do not permit increasing the work coefficient to increase the pressure ratio. The compressor might be operated at a higher rotation speed to increase the pressure ratio, but only by exceeding the limit imposed on the rotor tip relative Mach number. It might be concluded that a standard repeating stage design is simply impractical for these design constraints. But that is really not the case. Rather, the stage hubto-shroud radius ratio of 0.5 used throughout this chapter is not a good choice for an industrial axial-flow compressor using this as a standard repeating stage. Figure 10-37 shows the important performance parameters for a constantswirl vortex stage designed to the same constraints as for the design in Fig. 10-28, but with a hub-to-shroud radius ratio of 0.7. Note that a substantial increase in



FIGURE 10-34 The Assigned Cm Distribution

FIGURE 10-35 Hub-and-Shroud Contours



FIGURE 10-36 Aerodynamic Performance Prediction

FIGURE 10-37 Alternate Constant-Swirl Vortex Stage



work coefficient is achieved while satisfying the same design constraints. An alternate 10-stage axial-flow compressor was configured using this alternate stage design as a standard repeating stage. This results in the predicted performance map shown in Fig. 10-38. This compressor achieves a pressure ratio of about 3.9 at the design mass flow rate. The larger hub-to-shroud radius ratio also results in higher efficiency for both the stage and the ten-stage compressor. Of course, the mass flow rate achieved is reduced relative to Fig. 10-36 due to the larger hub-to-shroud radius ratio. But scaling the geometry to a larger hub radius and reducing the rotation speed to obtain the desired mass flow rate, while maintaining the same dimensionless performance parameters and Mach number levels, could easily correct that. In practice, application of a standard stage is more complicated than this simple illustration. It is unlikely that a direct application of the repeating stage will produce the design point flow and pressure ratio required. For example, it might be concluded that the application requires a 91/2-stage compressor with that configuration. In that case, it will be necessary to modify the design to achieve the required performance in either a 9-stage or a 10-stage compressor. This may require adjusting some of the blade stagger angles and modifying the Cm distribution. The stagger angle adjustment modifies the velocity triangles, while the Cm distribution can be used to expand or contract them. Both approaches can change the work input per stage to refine the overall pressure ratio to a more appropriate value. Both approaches will probably be needed to modify the performance while maintaining a good match between stages. After some experience with a specific

FIGURE 10-38 Alternate Compressor Performance



standard stage, the designer usually develops a fairly systematic approach to achieving the desired modifications. The same methods can be used to improve the surge margin, to better optimize the efficiency at a specific operating point of special interest, or even to achieve a flow capacity somewhat greater than the value for which the stage was designed. Hence, there is considerable flexibility when applying a standard stage, even though it was designed to be a simple repeating stage.

10.12 A COMPUTERIZED STAGE DESIGN SYSTEM It is relatively simple to computerize the procedures described in this chapter into a systematic interactive design system. The most obvious use of this type of system is the design of standard industrial compressor stages such as those illustrated in the previous section. For that application, it is logical to ignore Mach number effects and losses, since they will vary with the specific application and the location of the stage in the multistage compressor. In Chapter 11, a more general approach is described where blades are designed specifically for each stage in a multistage compressor, including the effects of Mach number and losses. A computerized stage design system based on the methods from this chapter can be very useful to support that approach as well. It can provide a convenient method to investigate the alternate stage performance parameters and vortex types to be employed in the more general design approach. It can be rather cumbersome and confusing to attempt those investigations in the context of a multistage compressor design. Another application of a stage design system is for field-service repairs where a new stage design is required. This can occur if a stage is too damaged to determine its original geometry through reverse engineering. When a stage design system is available, it can often be a useful tool for a variety of simple and fast investigations to support other design activity. The effort required to develop a computerized system is so modest that it can be easily justified. It is advisable to include some additional flexibility, which is not covered in this chapter, when creating the computerized system. In particular, it is quite simple to provide for a variation of Czc through the stage. Similarly, provision for a radial variation in work input is easily included for cases where additional end-wall work or imposed radial energy gradients are considered desirable.

EXERCISES 10.1 Show that Eq. (10-45) is valid for any constant-work, repeating stage. 10.2 Show that Eq. (10-48) is the condition to avoid reverse flow at the rotor exit for a constant reaction vortex flow in a constant-work, repeating stage. 10.3 Show that a constant-work, repeating stage can have constant reaction only for 100% reaction.


Chapter 11

MULTISTAGE AXIAL-FLOW COMPRESSOR AERODYNAMIC DESIGN

The basic concepts introduced in chapter 10 to design an ideal axial-flow compressor stage are easily extended to the design of a complete axial-flow compressor. When the specific working fluid, rotation speed, mass flow rate and inlet thermodynamic conditions are known, the design process can include the influence of total pressure loss and Mach number levels. This is possible as long as the design is restricted to blade operating conditions reasonably close to the design incidence angles, i*, such that the blades operate in the low-loss incidence angle range where loss coefficients are reasonably constant. This is normal practice at the compressor’s design point, i.e., for the mass flow rate and speed condition at which the stages are to be well matched throughout the compressor. An axial-flow compressor design system of this type is unlikely to produce all of the aerodynamic and structural design features desired and all of the off-design operating characteristics required. But it can produce an excellent preliminary configuration to be fine-tuned using the performance analysis of Chapter 9. The same procedures can be used to design individual stages or blade rows intended for specific applications. For example, it might be used to design replacements for blade rows that are too badly damaged to be reverse-engineered. When this type of design system is available, the benefits of the ideal stage design system of Chapter 10 may seem questionable. Indeed, if sufficient care is taken, this more rigorous design approach may be used in place of the ideal stage design system to generate a standard industrial compressor stage design. It is necessary to avoid imposing restrictions that may preclude using the standard stage for intended applications different from the design conditions used. The procedures defined in this chapter size the compressor annulus to match the specific operating conditions. This imposes a specific stream surface pattern and limits the span of the blades designed to whatever is required to conserve mass. The influence of end-wall boundary layers is also significantly affected by the specific design conditions. If this system is applied at the lowest Mach number level expected and at a higher volume flow rate than is expected to be required, a stage design may be sufficient for general use as a standard stage.



But that typically reduces the design procedure to a more complicated version of the methods described in Chapter 10, effectively nullifying the benefits of the more rigorous models used. Indeed, the design methods of Chapter 10 are often useful for exploring alternative stage performance parameters before attempting a design using the methods of this chapter.

NOMENCLATURE C c Fg Fsh g H h i i* m n PR R r U W WRE Z z γ η θ σ φ ψ ω

= = = = = = = = = = = = = = = = = = = = = = = = = =

absolute velocity chord fractional axial gap between blades shroud-to-hub chord ratio axial gap between blade rows total enthalpy blade height incidence angle minimum loss incidence angle vortex exponent vortex exponent pressure ratio reaction radius blade speed = ωr relative velocity equivalent velocity ratio number of blades axial coordinate stagger angle efficiency camber angle solidity flow coefficient and stream surface slope angle work coefficient rotation speed, radians/sec.

Subscripts c h i m s θ 1 2 3

= = = = = = = = =

parameter on the mean stream surface parameter on hub contour stage number meridional component parameter on shroud contour tangential component rotor inlet condition rotor exit condition stator exit condition


Multistage Axial-Flow Compressor Aerodynamic Design • 261

11.1 THE BASIC COMPRESSOR DESIGN APPROACH Figure 11-1 illustrates the basic compressor design problem to be solved. The geometry of the end-wall contours and of all blade rows is to be generated, based on specified swirl vortex types and dimensionless performance parameters on the mean stream surface for all stages. As usual, the mean stream surface is defined by the requirement that equal mass flow rates are achieved on both sides of it. To accomplish the design, one of the end-wall contours is defined, while the other is computed from the annulus sizing procedure of Chapter 7. The case illustrated in Fig. 11-1 is typical of an assigned shroud contour design. Since all operating conditions are known, the hub contour can be computed as part of the design solution from simple conservation of mass. The design procedure to be used is a combination of the stage design method presented in Chapter 10 and a simplified version of the performance analysis of Chapter 9. The performance analysis is simplified by assuming that the loss coefficients are given by the design loss coefficient models of Chapter 6. This is a reasonable approximation as long as the incidence angle is not too different from the design incidence angle. Chapter 6 shows that design loss coefficient is primarily a function of the aerodynamic data. Indeed, from Eqs. (6-35) through (6-37), it is seen that the only geometrical parameter required is the solidity. The flow field throughout the compressor will be generated using specified swirl vortex types and dimensionless stage performance parameters, similar to the approach used in Chapter 10. The simple normal equilibrium equation, Eq. (10-24), is replaced by the meridional through-flow analysis of Chapter 7, using the annulus sizing

FIGURE 11-1 Compressor Geometry



option. By considering the calculation of solidity as part of the design calculations, the influence of total pressure loss can be included in the flow field calculations. End-wall boundary layer blockage effects are included using the boundary layer analysis of Chapter 8. As long as the design loss coefficient models are valid, this provides a rather accurate and complete prediction of the flow field throughout the compressor. The major approximation required is that streamline curvature effects must be neglected to size the annulus. But once the annulus has been sized, the design process can be repeated using the fixed end-wall geometry mode of the meridional through-flow analysis to allow curvature effects to be included. The development of this type of compressor aerodynamic design system is much simpler than it may appear. Nearly all of the computational methods required have been presented in previous chapters of this book. The design system used to generate results presented later in this chapter is a good example. It was rather easily formulated directly from the aerodynamic performance analysis. All that was needed was simplification of the blade performance models and revision of the meridional through-flow logic to include the stage performance specifications and airfoil design logic of Chapter 10. Indeed, nearly all of the computer codes needed for this design system already existed in the aerodynamic performance analysis and the ideal stage aerodynamic design system.

11.2 AERODYNAMIC PERFORMANCE SPECIFICATIONS The aerodynamic performance specifications used are similar to those employed in Chapter 10 for a stage design. For each stage in the compressor, the flow coefficient, work coefficient and reaction on the mean stream surface will be supplied. Parameters on the mean stream surface will be designated by a subscript c, and subscripts 1 through 3 will designate successive stations in the stage from rotor inlet to stator exit. Then the performance data for stage number i are given as

φ ci = (Cmc1 / Uc1)i ψ ci = ( Hc2 −

Hc1) / Uc21

(11-1)

= [( rc2Cθc2 / rc1 − Cθc1) / Uc1

Rci = 1 − (Cθc1 + Cθc2rc2 / rc1) / (2Uc1)

(11-2) (11-3)

The definitions of flow coefficient and work coefficient are direct generalizations of the definitions used in Chapter 10. The definition of reaction is chosen for convenience. It will be the true stage reaction only in the case of a repeating stage with a constant-radius mean stream surface. The vortex exponents, ni and mi, are also specified for each stage. Then the rotor inlet swirl velocity distribution for any stage is given by Cθ1 = Uc1[(1 − Rci )( rc1 / r1)

ni

− (ψ ci / 2)( rc1 / r1)

mi

]

(11-4)

The swirl velocity distribution at the rotor exit or stator inlet results from the assumption that the work input is constant from hub to shroud. From Eq. (11-2) this requires



Cθ 2 = [r1Cθ1 + ψ ciUc1rc1] / r2

(11-5)

If there is another stage following this one, the stator exit swirl velocity distribution is given by Eq. (11-4), using performance parameters for the next stage, i.e., Cθ 3 = Uc3 [(1 − Rc,i +1)( rc3 / r3 )

n i+1

− (ψ c,i +1 / 2)( rc3 / r3 )

m i+1

]

(11-6)

For the last stator, a special specification is required. A convenient choice is to specify the ratio of the exit to inlet swirl velocity as a constant for the last stator, i.e., Cθ 3 / Cθ 2 = constant

(11-7)

The meridional velocity on the mean stream surface must also be specified as the constant of integration for the normal momentum equation. The specifications used are Cmc1 = Uc1φ ci

(11-8)

Cmc3 = Uc3φ c,i +1

(11-9)

Cmc2 = (Cmc1 + Cmc3 ) / 2

(11-10)

If an inlet guide vane is used, its discharge conditions are identical to the inlet conditions for the first rotor. Similarly, the inlet conditions for an exit guide vane are identical to the discharge conditions for the last stator. At the stations before the inlet guide vane and after the exit guide vane, the swirl velocity is set to zero and it is assumed that the meridional velocity is constant across the adjacent blade row. If additional computing stations are inserted between blade rows, conservation of angular momentum supplies the required swirl velocity distribution from the adjacent station where data has been assigned, and the meridional velocity is set to the same value as that adjacent station. A useful variant on the above equations is to replace Uc and rc in Eqs. (11-4) through (11-9) by their values at the entrance to the first rotor. This often makes it easier to select appropriate values for the performance parameters by removing the influence of the unknown variation of the mean stream surface radius through the compressor. This is easily included as an option in the computerized design system. Hence for any assumed annulus and stream sheet geometry, the normal momentum equation, Eq. (7-12), can be integrated if the entropy distributions are known. As discussed in Chapter 7, Section 7.6, the stream surface curvature terms are neglected when sizing the annulus. Thus the compressor design solution is a simple marching process where the solution at each axial computing station is completed before proceeding to the next station. The entropy distributions are computed as part of the solution by computing the design total loss coefficients. As mentioned, this requires knowledge of the blade row solidity, which will be discussed in the next section. Then the annulus area is adjusted as required to balance the known mass flow rate. During this process, the stream sheet pattern is continually changing. Equations (11-4) through (11-10)



are continually applied to update the performance data as the stream sheet geometry changes.

11.3 BLADE DESIGN The basic process used to design the blades has been described in Chapter 10, Section 10.3. Some minor variations on that process are more appropriate for the present design problem. The following assumptions and specifications have been found to be effective and relatively simple to employ: • The basic airfoil camberline and profile types from Chapter 4 are specified. • If required, specify the location of the point of maximum camber, a / c. • Specify the ratio of the chord at the shroud to its value at the hub and require a linear variation from hub to shroud. • Specify the thickness-to-chord ratio on the hub and shroud and require a linear variation from hub to shroud. • Specify the solidity on the mean stream surface. • Specify the fraction of the axial spacing between computing stations, Fg, that is reserved for the axial gap, g, between blade rows. • Specify the difference between the actual incidence angle and the design incidence angle, (i – i*), on the hub and shroud and require a linear variation from hub to shroud. • Specify the blade tip clearance as a fraction of the shroud radius of the first rotor row. The blade design procedure is essentially the same as that described in Chapter 10, Section 10.3. The major difference is that the blades must be inserted within a predefined axial distance, since coordinates for one of the end-walls are specified. Here it is necessary to select the number of blades in the blade row and the chords such that the desired axial gap is maintained between blade rows. As illustrated in Figure 11-2, the chord at the hub, which provides the specified axial gap, is ch = ∆zh (1 − Fg ) / cos γ h

(11-11)

A similar calculation on the shroud contour yields cs = ∆zs (1 − Fg ) / cos γ s

(11-12)

But these two values are also constrained by the specified shroud to hub ratio, Fsh = cs / ch. This can be used to determine which of the previous two equations specifies a chord necessary to achieve the axial gap. Then the other chord is calculated from Fsh. Once the correct chord at the hub is known, the chord at any radius is given by



FIGURE 11-2 The Blade Row Axial Gap

c = ch [1 + ( r − rh )( Fsh − 1) / ( rs − rh )]

(11-13)

If Z is the number of blades in the blade row, the solidity is given by

σ = [ch Z / (2πr )][1 + ( r − rh )( Fsh − 1) / ( rs − rh )]

(11-14)

Solving Eq. (11-14) on the mean stream surface, and normalizing it by the result, yields

σ = σc

r [1 + ( r − rh )( Fsh − 1) / ( rs − rh )] r[1 + ( r − rh )( Fsh − 1) / ( rs − rh )]

(11-15)

Hence the solidity at any radius can be computed from the specified solidity at the mean stream surface before the blades are designed. Since that is the only blade geometrical parameter needed to compute the design loss coefficient, the design flow field analysis can be conducted prior to designing the blades. Then the precise chords and number of blades can be selected using Eqs. (11-11) through (11-14) and the known solidity at the mean stream surface. The largest chords and lowest number of blades permitted by the specified axial gap are selected. If tip clearance losses are included in the analysis, there is a weak dependence on chord and stagger angle, as can be seen from Eq. (6-86). The end-wall boundary layer analysis also depends on the stagger angle, as seen from Eqs. (8-44) and (8-55). Hence, initial guesses are made for the number of blades and the chord on



the mean stream surface that will yield the desired solidity. The stagger angles on the end-wall contours are approximated from the local flow angles until the first blade design is accomplished. The flow analysis and blade design is repeated until convergence on the chord (or number of blades) is achieved. One repetition is almost always sufficient. At least one repetition is necessary to adequately account for the effect of stagger angle on the end-wall boundary layers. Another minor difference relative to the procedure of Chapter 10 is that stream surfaces are normally not cylindrical surfaces. The blade geometry used in the flow field analysis needs to be adjusted for streamline slope, using the angle, φ, illustrated in Fig. 11-2 for the hub contour. The required corrections have been presented in Eqs. (9-1) through (9-6).

11.4 REFINING THE COMPRESSOR DESIGN In general, an axial-flow compressor design accomplished by the procedures outlined above will prove a little disappointing when analyzed in the performance analysis of Chapter 9. Even when directly derived from the performance analysis, this design procedure does not normally result in a design that is simply confirmed by it. Since the performance analysis works with assigned end-wall contours, it can directly estimate stream surface slopes and curvatures. In contrast, the design procedure outlined above has only a crude estimate of the sized contour’s slope and ignores curvature effects entirely. There are several simple steps that can be incorporated into the design procedure to minimize the differences seen when a performance analysis is conducted on the resulting design. First, it is very simple to include the capability to repeat the design using the sized contour on the previous attempt as the basis for contour slope calculation. This can be repeated until the contour slopes no longer show a significant change between successive attempts. When that has been achieved, it is very useful to be able to redesign the compressor with the resulting sized contour used directly, instead of sizing the annulus. This will also permit inclusion of stream surface curvature effects, while still generating a flow field consistent with the desired performance parameters and blade geometry consistent with that flow field. It is also useful to constrain the annulus sizing procedure to prohibit an increase in the annulus area in the streamwise direction. When a passage area increase is required, the annulus area can be set to the value at the upstream station and conservation of mass is used to override the local meridional velocity specification. Resolving inconsistencies with the performance analysis is usually not totally sufficient. When an end-wall contour is computed by the annulus sizing procedures, the resulting contour is likely to be unrealistic, often resulting in erratic variations in coordinates, slopes and curvatures. Refining the dimensionless performance parameters can reduce this erratic behavior. But that is a rather tedious process, and involves changes in the performance parameters that are really physically insignificant. It is much more effective to include procedures to permit smoothing and editing the contour coordinates. Then the design process can again be repeated with the revised contour to produce the desired performance. A simple smoothing procedure that has been found effective is illustrated in Fig. 11-3. In this case, the shroud has been sized with somewhat erratic results.



FIGURE 11-3 Contour Smoothing

The approach used is performance of a simple linear interpolation between adjacent axial computing stations to determine an alternate radius on the quasi-normal of interest. Then a smoothed point is defined as an average of the actual and interpolated points. The smoothed points can be used directly for all interior quasi-normals. Alternatively, smoothing can be omitted at certain quasi-normals, typically either at the rotor inlet or rotor discharge quasi-normals. That may be desirable to preserve key elements of the sizing process. It is best to implement this contour smoothing as an interactive process where the smoothed contour is displayed graphically, with the original contour points in the background. Then the designer can try alternate smoothing strategies to select the most promising method. This can also include provision to directly edit some coordinates that do not adequately respond to the smoothing procedures alone. That is fairly common for quasi-normals upstream of the first rotor and downstream of the last stator. The same contour smoothing procedures can be very useful in the performance analysis, particularly when the annulus sizing procedure is used. Indeed, these smoothing procedures were used for the two standard repeating stage compressor designs described in Section 10.11 If properly implemented, all of these procedures for refining the design can be employed with very little effort on the part of the designer. If they are not accomplished here, they will almost certainly need to be accomplished manually in the performance analysis. That will involve much more effort, particularly since the blade designs will require modification during the process and the



desired performance parameters will almost certainly be compromised. By incorporating them in the compressor aerodynamic design system, performance goals and blade design are handled automatically. This results in a dramatic reduction in fine-tuning the design using the performance analysis.

11.5 AN AXIAL-FLOW COMPRESSOR DESIGN EXAMPLE The application of this axial-flow compressor aerodynamic design procedure will be illustrated by developing various alternate designs for the second ten-stage compressor designed in Chapter 10 using a standard repeating stage. The procedures in this chapter will be employed to design the end-wall contours and the geometry of the 22 blade rows required. NACA 63-series inlet guide vanes will be used while NACA 65-series blades will be used for all other blade rows. All designs will limit the rotor tip relative Mach numbers to a maximum value of 0.8. All compressor designs will be sized to specified performance parameters with a constant hub-contour radius. The shroud contours will be smoothed and refined as described in the previous section of this chapter. The first example is designated as Compressor A. It is based on stage performance parameters similar to those achieved in Chapter 10 with the standard repeating stage. Figure 11-4 shows the stage mean stream surface performance parameters specified. A simple linear variation with stage number was used,

FIGURE 11-4 Stage Design Data for Compressor A



where data for the first and last stages were obtained from the standard-stage compressor performance analysis shown in Fig. 10-38. Then the work coefficient for the last stage was adjusted by trial and error to produce approximately constant-camber, untwisted stator blades throughout the compressor. After sizing the annulus, the shroud contour was smoothed twice, each time holding the rotor inlet shroud radii constant. Figure 11-5 shows the end-wall contours obtained from that process. Then the flow field was predicted for these end-wall contours using the approximate normal equilibrium model described in Section 7.6 for the meridional through-flow analysis to obtain the final blade row designs. All blades were designed to operate at their design incidence angles, while matching with the predicted flow field. The computerized design system used makes this a rather simple process. Once the initial design specifications had been entered, the entire design process for Compressor A was completed in about five minutes. Then Compressor A was analyzed using the performance prediction methods of Chapter 9. Consistent with the aerodynamic design approach, the approximate normal equilibrium model described in Section 7.6 was used for the performance analysis. The predicted difference between the incidence angles and the design incidence angles on the hub-and-shroud contours is shown in Fig. 11-6. The performance analysis supplies incidence angles rounded off to the nearest 0.1°. Hence, it can be seen that the deviation from the intended incidence angle match was less than 0.15° in all cases. Comparison of the calculated incidence angle

FIGURE 11-5 End-Wall Contours of Compressor A



FIGURE 11-6 Predicted Incidence Angle Match

match with the design specifications is the best way to validate the aerodynamic design system. Even minor differences between the design system and the performance analysis will produce significant differences in the incidence angles that cascade to rather large values in the rear stages of a multistage compressor. The results in Fig. 11-6 are fairly typical of what can be achieved by a properly formulated aerodynamic design system. Figure 11-7 shows a predicted performance map for Compressor A, along with results from Fig. 10-38 for the standard-stage design; The two designs are quite similar. The present design achieves a larger pressure ratio and improved surge margin at design speed, but at the expense of a reduced maximum flow capacity. Careful review of the standard stage compressor performance analysis showed that the stages exhibit slightly positive values of (i – i*) in the front stages, whereas the nearly ideal matching shown in Fig. 11-6 was achieved for Compressor A. Hence the differences in performance are not unexpected nor are they particularly significant. A minor re-matching of the stages in either of these designs could easily result in performance nearly identical to that of the other design. Clearly, the substantial cost advantage offered by the standard repeating stage compressor with its constant camber, untwisted stator blade is achieved with little compromise. The nearly ideal stage matching with completely arbitrary blade geometry used in the design of Compressor A does not result in any significant performance improvement. In this specific example, the most significant benefit



FIGURE 11-7 Performance of Compressor A

obtained from the present aerodynamic design system is that the design of Compressor A was much easier to accomplish than the design of the standard repeating-stage compressor. It should not be concluded that this more general aerodynamic design procedure does not offer the potential for improved performance over a standard stage compressor design. Since the design of Compressor A is based on specifications obtained from a performance analysis of a repeating stage compressor, some similarity to a repeating stage compressor is to be expected. Figure 11-8 shows the basic blade geometry on the hub contour, where the radius is constant. It can be seen that the design does approximate a repeating stage configuration. The geometry for the last stator has been omitted, since it is rather arbitrary, depending directly on the specification of how removal of the swirl from the last rotor is to be split between the last stator and the exit guide vane. Although not obvious from Fig. 11-8, it is simply stated that a standard constantcamber, untwisted blade could easily be substituted for the stator blades designed for Compressor A, but with stagger angles varying through the machine. However, the rotor blade geometry varies too much through the machine to permit substitution of a standard rotor blade. Hence the design of Compressor A was really constrained to approximate the repeating-stage design of Chapter 10. Although the blade geometry was not constrained, this added flexibility was not used to much advantage.



FIGURE 11-8 Hub Contour Blade Geometry

11.6 THE DISTRIBUTION OF STAGE PERFORMANCE PARAMETERS More effective use of the general aerodynamic design procedure is possible by simply applying a distribution of stage performance parameters through the compressor. The variation in stage performance parameters shown in Fig. 11-4 for Compressor A is somewhat misleading, since it really only corrects the stage performance parameters for the variation in mean stream surface radius. A compressor design establishes the annulus area distribution and blade geometry suitable for the design rotation speed. When the compressor is operated at speeds less than the design speed, the work per stage is reduced, such that the increase in gas density is no longer sufficient for the stages to remain matched. The rear stages will operate at flow coefficients higher than their design values, with a corresponding reduction in work per stage. The front stages will have the highest work per stage and will be the first stages to stall. The rear stages contribute much less work and will limit the compressor’s flow capacity. When the compressor is operated at speeds greater than design speed, the opposite trend will occur. The front stages produce more work and a larger increase in gas density than intended. Downstream stages tend to operate at flow coefficients less than their design values, with a corresponding increase in work coefficient. Generally, the rear stages have the highest work per stage and will be the first stages to stall. The front stages will contribute less work input and will limit the compressor’s flow



capacity. In effect, the annulus area distribution is too small for operation at low speeds and too large for operation at high speeds. It follows that the stages least affected by variations in rotation speed are the mid-stages. In a properly designed compressor, the mid-stages will normally operate closer to their design operating conditions than either the front or rear stages, and are less likely to be responsible for the compressor’s stability or flow capacity limits. Hence it is reasonable to employ more highly loaded mid-stages, with more conservative loading of the front and rear stages. Stage loading is not a well-defined term. Some investigators use it to refer to the stage work coefficient; others use it to refer to the flow diffusion across the blade rows. In the present context, both aspects must be considered. Higher work coefficients will be used for the mid-stages. Somewhat greater flow diffusion may be attempted, since the mid-stages are expected to operate over a narrower flow coefficient range than either the front or rear stages. However, excessive flow diffusion must still be avoided. The evaluation of flow diffusion used here will be the relative velocity ratio across the blade rows, W2 / W1, or the effective velocity ratio, WRE, defined in Eq. (9-16). The two parameters are approximately the same, but the latter has been used as the basis of the blade stall criterion developed in Chapter 9. In Chapter 10, it was shown that the work coefficient can be increased without increased flow diffusion if the flow coefficient is also increased. That will require a departure from using Smith’s (1958) recommended flow coefficient of 0.5 based on the recovery ratio of Eq. (10-36). From Figs. 10-7 through 10-9, it is apparent that large positive values of the recovery ratio are achieved for a fairly wide range of flow coefficients. Also, the risk of lower recovery ratios should be much less at the mid-stages, since they are expected to experience less severe operating conditions than the front and rear stages. The present aerodynamic design system makes it relatively simple to develop a suitable distribution of stage performance parameters. To illustrate the process, Compressor B was designed for the same design mass flow, rotation speed and hub contour as that used for Compressor A. The stage performance parameters used are shown in Fig. 11-9. The design system used for this example permits specification of stage performance data at an arbitrary number of stages, with linear interpolation for the other stages. In this case, specifications were provided for stages 1, 2, 3, 4, 8 and 10. The distribution of φc starts with a value of 0.55 for the first stage and ramps up through the front stages to permit an increase in ψc without unacceptable diffusion levels. Initial distributions of Kc and Rc were assumed. Kc was assigned instead of ψc to provide more direct control over the diffusion levels in the stages. The distributions of Kc and Rc were adjusted by trial and error until acceptable diffusion levels were achieved through the compressor. Kc was adjusted to achieve acceptable levels of WRE, and Rc was adjusted to approximately balance the diffusion load between the rotor and stator for all stages. Fig. 11-9 shows the key values of WRE obtained on the hub-and-shroud contours for the design mass flow and rotation speed. Stator shroud values are omitted from figure 11-9 since they pose no problem for the constant swirl vortex type. Similarly, the last stator is omitted since its diffusion load can be controlled independent of the stage performance data. Hence the mid-stages feature higher work and more diffusion than either the front or the rear stages. Based on the previous discussion, it might be expected that the φc



FIGURE 11-9 Stage Design Data for Compressor B

distribution should ramp down to lower values in the rear stages. That is generally not advisable due to adverse effects on the end-wall boundary layers. The end-wall boundary layers are relatively thick in the rear stages, such that a reduction in meridional velocity can easily lead to excessive boundary layer blockage. Indeed, the interaction between the end-wall boundary layer analysis and the annulus sizing calculations can become so severe that convergence is virtually impossible. That is the main reason why the present design system constrains the annulus sizing to preclude an increase in passage area in the streamwise direction. Had φc been ramped down through the rear stages, that constraint probably would have caused the lower values to be ignored anyway. To further illustrate the flexibility available for tailoring the stage performance distributions, Compressor C was designed using the design data shown in Fig. 11-10. This design followed the same strategy as that used for Compressor B, but uses higher stage flow coefficients to permit higher work coefficients. Figure 11-10 also shows the key values of WRE obtained. Note that compressor C achieves higher work per stage than Compressor B, but has similar flow diffusion characteristics. Figure 11-11 compares performance predictions for Compressors A, B and C. Polytropic efficiency is shown for this comparison to avoid the thermodynamic effect due to pressure ratio discussed in Chapter 2. Both the pressure ratio and the stable operating flow range for Compressor B are generally improved relative to Compressor A. There is a slight loss in stable operating range at the design speed, but improved stability at other speeds. There is a slight reduction in efficiency due



FIGURE 11-10 Stage Design Data for Compressor C

FIGURE 11-11 Predicted Performance Maps



to the reduced aspect ratios that result from the higher values of φc and pressure ratio. This requires use of lower values of blade height, h, in the rear stages, resulting in higher predicted loss coefficients as seen from Eq. (6-99). The estimated surge limit for all three compressors is based on stall criterion #3 of Chapter 9. At design speed, this initially occurs in the mid-stages, whereas at other speeds the front stages stall first. The improved stable operating range is particularly noticeable at the lowest speed. As expected, the reduced diffusion in the front stages has permitted the compressor to operate more effectively at low speeds. Note that the improved stable operating range is also present in Compressor C, even though its design pressure ratio is higher than that of Compressor B. Clearly, careful control of the blade row diffusion characteristics is a primary consideration with regard to improved compressor performance. But there are certainly other considerations that may not be reflected by current aerodynamic performance prediction technology. All designers certainly have strong incentive to favor large values of φc and ψc to increase the mass flow per unit frontal area and to reduce the number of stages required. Whether the goal is reduced size and weight or reduced cost, the incentive is always present. Nevertheless, recent design trends appear to favor lower values of φc and ψc based primarily on perceived benefits to performance. Possible explanations were discussed in Section 10.4 of the previous chapter. Other investigators appear to believe that the explanation lies in the end-wall boundary layer behavior. While lower values of φc and ψc may be preferred, the designer has considerable flexibility in regard to stage performance parameters. Rather large values of φc and ψc have been used with success. For example, Fig. 11-12 shows calculated stage performance parameters obtained from the performance analysis of the NACA 10-stage subsonic compressor of Fig. 9-4. This compressor achieved respectable performance, considering the state-of-the-art in axial-flow compressor aerodynamic technology fifty years ago when it was designed. Yet rather large values of φc and ψc were used in most of the stages. For illustration, Compressor D was designed using the highest values of φc and ψc allowed by the rotor tip relative Mach number limit imposed on the previous designs. The same operating conditions and design strategy as those for Compressors B and C were used. Figure 11-13 shows the design parameters used and the flow diffusion characteristics obtained. It was not possible to balance the flow diffusion as well for this case. Blade stall in a constant swirl vortex stage almost always occurs first on the hub contour, usually in the stators. Hence, lower rotor shroud velocity ratios were accepted so that the hub contour flow diffusion characteristics could be maintained in a manner similar to that of Compressors B and C. Indeed, for any compressor, the blade sections at the hub must operate over a much wider incidence angle range than the rotor shroud blade sections. Figure 11-14 compares the predicted performance maps for Compressors B, C and D. Again, polytropic efficiency is shown for this comparison to avoid the thermodynamic effect due to pressure ratio discussed in Chapter 2. It is seen that aspect ratio effects have an adverse effect on efficiency, particularly for Compressor D. A special performance analysis of Compressor D using a larger number of shorter chord length blades in the rear stages produced essentially the same efficiency at design speed as for the other two designs. This indicates that the reduction in design speed efficiency seen in Fig. 11-14 is due to aspect ratio effects rather than being a direct result of



FIGURE 11-12 Actual Stage Performance Data

FIGURE 11-13 Stage Design Data for Compressor D



FIGURE 11-14 Effect of c on Performance

the values of φc and ψc used in the design process. At lower speeds, both efficiency and stability appear to deteriorate as higher values of φc and ψc are used in the design process. Hence the performance analysis of chapter 9 indicates some adverse effects at off-design speeds from designing with higher values of φc and ψc. But it really does not indicate any significant influence of φc and ψc on the performance at design speed. It is now generally accepted that designing with lower values of φc and ψc offers definite performance benefits. It is clear that those benefits are not completely quantified by the performance analysis of Chapter 9. The present design system makes it rather simple to tailor the distributions of stage performance parameters and blade row diffusion characteristics to emphasize aerodynamic features considered important to the design objectives. This is more clearly illustrated by another simple example: Suppose that Compressor B must operate on a load line that lies too close to the surge line at the lower rotation speeds. Compressor E is to be designed with the objective of improving the low-speed surge margin while maintaining the same basic design speed performance as Compressor B. The annulus contour and the number of stages are to be identical to Compressor B. In effect, the design of Compressor E is constrained to be identical to Compressor B except for the blade geometry. Figure 11-15 shows the stage design parameters and flow diffusion characteristics used to achieve this objective. Figure 11-16 illustrates a load line that might require this redesign and compares the predicted performance maps for the two compressors. Comparison with Fig. 11-9 shows that the design of Compressor E differs from that of



FIGURE 11-15 Design Data for Compressor E

FIGURE 11-16 Performance of Compressors B and E



Compressor B by a fairly modest shifting of the stage loading from the front to the rear. It can be seen that a substantial improvement in the low-speed surge margin has been achieved with minimal effect on the design speed performance.

11.7 THE SWIRL VORTEX TYPE Chapter 10 reviews the influence of swirl vortex in the context of a single, repeating stage design. The present aerodynamic design system can be used to evaluate alternate vortex design types in the context of multistage compressors designed with the tailored distributions of stage performance parameters and flow diffusion characteristics. Compressor B is used as the constant swirl vortex design for this comparison. Alternate compressor designs have been generated using the free vortex and constant reaction vortex types. Although the constant reaction vortex type was not very effective for the single stage designs in Chapter 10, the present application is better suited to this type of vortex. The major weakness seen in Chapter 10 is the highly distorted meridional velocity profiles. The higher hub-toshroud radius ratio used in the present case will reduce the influence of that profile distortion. Figures 11-17 and 11-18 summarize the design data used for the alternate designs. The alternate designs are based on the same operating conditions and distribution of φc as Compressor B. The distributions of Rc and ψc are adjusted to produce flow diffusion characteristics similar to those in Fig. 11-9. The free vortex and constant swirl vortex types have very similar flow diffusion

FIGURE 11-17 Free Vortex Design Data



FIGURE 11-18 Constant Reaction Vortex Design Data

FIGURE 11-19 Effect of Vortex Type on Performance



characteristics. Only minor modifications to the data in Fig. 11-9 were needed to obtain the results shown in Fig. 11-17. The flow diffusion characteristics of the constant reaction vortex type are quite different. There the rotor velocity ratios on the shroud had to be balanced against the stator hub values for most of the stages. The stator hub flow diffusion was expected limit stability, which was subsequently confirmed by the performance analysis. Hence emphasis was placed on achieving similar stator hub velocity ratios as those used for the other two vortex types, without excessive diffusion at other locations. Fig. 11-19 compares the predicted performance maps for the three compressor designs. Note that the flow range from the design point to surge at the design speed is essentially identical for all three compressors. The performance predictions indicate that the stator hub flow diffusion characteristics have much more influence on stability than the swirl vortex type. It is also seen that the compressor efficiency level is rather insensitive to the swirl vortex type used. The constant reaction vortex design low speed characteristics are shifted in flow relative to the other two designs, but the efficiency levels are comparable for all designs. The free vortex design produced a slightly lower pressure ratio than the constant swirl vortex design, but otherwise yields performance very similar to Compressor B. The constant reaction vortex type yields a significantly lower pressure ratio than Compressor B. Possibly this could be improved somewhat with further refinement, but it seems clear that the constant reaction vortex design does not offer any advantage over the other two designs. The constant reaction vortex type is much harder to work with and has definite limitations with regard to acceptable hub-to-shroud radius ratios. Hence there seems to be little reason to consider using it for axial-flow compressor design. The blade geometry resulting from the various swirl vortex types is also a significant consideration. Figures 11-20 and 11-21 compare camber and stagger angles for the rotor and stator blades for the first stage of the three compressor designs. The first stage is selected since it has the longest blades, but comparisons of blades in other stages would be quite similar. The rotor blades for the three designs are so similar that there is no clear preference. The stator blade geometry is very dependent on the swirl vortex type. The constant swirl vortex produces a stator that can be well approximated by a constant-camber, untwisted blade. This will be the easiest to manufacture and costs the least. The free vortex stator can be well approximated by a constant-camber blade with a linear twist. This would be the second choice with regard to cost and ease of manufacturing. The constant reaction vortex requires a more complicate camber angle distribution, but a linear twist might be used to approximate the stagger angle distribution. The characteristic increase in camber angle near the shroud is apparent. This would be much more pronounced if a lower hub-to-shroud radius ratio were used, as can be seen in Fig. 10-32. Thus, the complexity of the stator geometry is another reason to avoid the constant reaction vortex type. The present aerodynamic design system makes it relatively simple to investigate the influence of alternate swirl vortex types. Here, three quite different vortex types have been compared for a sample problem quite typical of an industrial axial-flow compressor. Based on this comparison, it is reasonable to conclude that swirl vortex type is not a major factor with regard to the aerodynamic performance of axialflow compressors of this type. If lower hub-to-shroud radius ratios are used, there may be some influence on performance associated with the different maximum



FIGURE 11-20 Comparison of Rotor Blade Geometry

FIGURE 11-21 Comparison of Stator Blade Geometry



inlet relative Mach number levels or extremes in flow diffusion levels produced by the alternate swirl vortex types. But, in general, suitability to the design strategy and objectives, along with the cost and ease of manufacturing of the blades, are more likely to be the basis for selecting the swirl vortex type.

11.8 RISKS AND BENEFITS The benefits from the present axial-flow compressor aerodynamic design procedure should be relatively clear from the examples presented in this chapter. The input specifications required are rather modest considering the complex design problem involved. The input specifications are relatively easy to supply, and provide the designer with substantial control over the aerodynamic features considered most critical to the design objectives. The tedious process of sizing the annulus, selecting blades and fitting them into the available space is left to the computer. The design process is guided by continual feedback from a performance analysis for the design mass flow rate and rotation speed. When a viable candidate design is identified, the design system can create an input file for the aerodynamic performance analysis to permit an evaluation of its off-design performance characteristics. To obtain full benefit from the design system, on-demand monitor screen displays of tabular and graphical data are essential. End-wall contour geometry, blade geometry, relative and absolute flow data, blade row performance data and end-wall boundary layer analysis data are essential to evaluate the designs. Graphical display of the equivalent velocity ratios, WRE, is particularly useful when attempting to tailor the flow diffusion distributions similar to the examples presented in this chapter. It has been noted that the capability to specify stage performance for a few key stages and to interpolate for intermediate stages can simplify the input specifications. The same is true for specifying the fixed endwall contour data. Specifying axial spacing is often simpler than specifying axial coordinates. Provision to enter radii at some stations and to interpolate for intermediate stations is also a useful simplification. It is important to use a representative blade tip clearance, but it is usually sufficient to use a constant value for all blade rows. Monitor screen displays of the actual blade profiles at any radius are also useful, and the design system should have the capability to export blade profile coordinates for the drafting and manufacturing of the blades. Figure 11-22 is an example of these data obtained from the design of Compressor B. It is important to recognize that the substantial benefits of this design system also introduce a level of risk that requires designers to exercise judgment. Chapters 6 through 9 present numerous procedures used in the process of predicting the performance of axial-flow compressors. All of those methods involve a significant level of approximation. A performance analysis is normally qualified against experimental data to fine-tune it until it offers sufficient prediction accuracy for the intended applications. In general, none of the cases considered in such a qualification study will be evaluated by the performance analysis as completely optimized. But the present design system provides the unique situation of producing an axial-flow compressor design that is precisely optimized to the models used in the performance analysis. Since those models are far from exact,



FIGURE 11-22 Typical Blade Profiles

the present design approach will inevitably expose the weaknesses in those models. The most obvious risk is that the predicted design point efficiency is likely to be overestimated. In the examples presented in this chapter, every blade row is operating precisely at its minimum loss condition on all stream surfaces. It is highly unlikely that the empirical models of Chapter 6, the through-flow analysis of Chapter 7 and the boundary layer analysis of chapter 8 are all sufficiently accurate to actually produce that situation throughout a multistage compressor. This design system should provide a very good design, but probably not as good as will be indicated by the performance analysis. Indeed, experienced turbomachinery aerodynamic designers are quite aware of the risk involved with designing directly with the performance analysis that will be used to evaluate the design. The axial-flow compressor is somewhat unique in that it is rather easy to convert a performance analysis to a design system of the type described in this chapter. The substantial benefits provided certainly justify taking advantage of this opportunity, but definitely require considerable judgement by the designer to maintain realistic expectations for the resulting design.


Chapter 12

QUASI-THREE-DIMENSIONAL BLADE PASSAGE FLOW FIELD ANALYSIS

The aerodynamic design and analysis of multistage axial-flow compressors is usually based on empirical cascade performance models as described in Chapters 6 through 11. Although that approach is sufficient for most applications, designers occasionally require the support offered by more fundamental internal flow analyses. When a specific application requires operating a blade row at conditions more extreme than justified by experience, supplemental analysis can greatly increase the designer’s confidence or perhaps prevent a poor decision. In the case of industrial axial-flow compressors, this usually relates to the blade row inlet Mach number level. The industrial compressor designer usually has a number of discrete geometrical scales or frame sizes available. If a specific application slightly exceeds the upper limit of flow capacity for a frame size, the designer must either accept somewhat higher Mach number levels or choose a larger frame size with a substantial increase in cost. Alternatively, transonic blade sections such as double-circular-arc blades might be considered. Usually that also involves increased cost and may be unattractive with respect to structural integrity. Even when the application is well within the normal flow capacity limits of a frame size, an unusual working fluid or a required rotation speed can result in Mach number levels beyond the designer’s experience. In those cases, designers need a more fundamental internal flow analysis to better evaluate probable impact of the higher Mach number levels. Chapter 5 presented several useful techniques for the analysis of the internal flow on stream surfaces within blade passages. Those blade-to-blade flow analyses can provide the desired detailed evaluation, but all of them require that the stream surface geometry and stream sheet thickness be specified in some fashion. An approximate analysis can be conducted using the stream sheet geometry before and after the blade row from the meridional through-flow analysis of Chapter 7, while assuming a linear variation between those stations. That is certainly the fastest and simplest approach and often may be considered sufficient. Alternatively, a more accurate analysis can be conducted using the quasithree-dimensional flow analysis technique mentioned in Chapters 3 and 5. That is a very efficient analysis technique that provides a reasonable approximation to the three-dimensional flow field through a blade row. Originally suggested by Wu



(1952), this technique relies on solving for the two-dimensional flow on the blade-to-blade stream surfaces and on the hub-to-shroud stream surface, with interaction between them until they are consistent with each other. This chapter describes a quasi-three-dimensional flow analysis suitable for that purpose. This analysis was originally developed for centrifugal compressors (Aungier, 2000), where it plays an essential role in the aerodynamic design of impellers. It was subsequently extended to treat arbitrary three-dimensional blade geometry for additional flexibility in centrifugal impeller design. That extension made it possible to use the analysis directly for axial-flow compressor blade rows. The only special extension added for axial-flow compressors is provision for automatic generation of detailed blade geometry for all of the standard blade profiles discussed in Chapter 4. This permits complete blade specification in terms of the radial variation of the basic blade section parameters (e.g., θ, γ, c and tb/c). The same capability is incorporated into the various blade-to-blade flow analyses of Chapter 5 for a single blade section. Hence it is relatively simple to provide the capability to automatically set up input files for either a simple bladeto-blade analysis on a single stream surface or a quasi-three-dimensional flow analysis for a blade passage directly from a completed meridional through-flow analysis. In the author’s design and analysis system, both the performance analysis of Chapter 9 and the design system of Chapter 11 contain that provision. When a more detailed analysis is needed, it can be obtained in a few minutes with very little additional effort.

NOMENCLATURE B* b C D E f g H H1 H2 h I KB M m ˙ m n P q r s T

= = = = = = = = = = = = = = = = = = = = = =

stream surface repositioning damping factor stream sheet thickness absolute velocity stream surface curvature damping factor entrainment function blade force 2πrcosβ′ /Z total enthalpy meridional shape factor = δ 1*/θ11 boundary layer tangential shape factor = δ 2*/θ22 static enthalpy rothalpy blade blockage factor Mach number meridional coordinate mass flow rate normal coordinate pressure inlet dynamic head radius entropy temperature


Quasi-Three-Dimensional Blade Passage Flow Field Analysis • 289

tb W y Z z β ∆A ∆ν δ δc δ 1* δ 2* ε η θ θ11 θ22 κ κm λ ν ρ φ ω

= = = = = = = = = = = = = = = = = = = = = = = =

blade thickness velocity relative to the blade row distance along quasi-normal number of blades axial coordinate flow angle stream sheet area increment in ν due to blade guidance boundary layer thickness blade clearance boundary layer meridional displacement thickness boundary layer tangential displacement thickness angle between quasi-normal and true normal dimensionless tangential coordinate tangential coordinate and streamwise momentum thickness meridional momentum thickness tangential momentum thickness camberline angle with meridional direction stream surface curvature quasi-normal angle (Fig. 12-3) blade force defect thickness fluid density stream surface angle with the axial direction rotation speed, radians/sec.

Subscripts h m s θ

= = = =

parameter at the hub contour meridional component parameter at the shroud contour tangential component

Superscripts ′ = parameter in rotating coordinates = mass-averaged value



12.1 QUASI-THREE-DIMENSIONAL FLOW Figure 12-1 shows a meridional plane view of a blade row to be analyzed. The annular passage is divided into a series of stream sheets, separated by stream surfaces. All stream sheets are defined to contain equal mass flow. For simplicity, it is assumed that all stream surfaces are axisymmetric, i.e., stream sheet twisting will be ignored. The meridional coordinate, m, is measured along a stream sheet’s mean surface, and the stream sheet thickness is designated as b. If the stream sur-



FIGURE 12-1 Hub-to-Shroud Flow Geometry

faces and the distribution of b are known, a complete two-dimensional blade-toblade flow analysis can be conducted using one of the methods described in Chapter 5. Figure 12-2 shows a typical blade-to-blade surface view to illustrate the solution domain for one of the blade-to-blade flow analyses to be conducted. Since blade-to-blade flow analysis has been covered extensively in Chapter 5, it is only necessary to identify the specific method to be used. The best choice for quasithree-dimensional flow analysis is the linearized potential flow analysis of Section 5.4, including the extension to transonic flow. That method offers exceptional computation speed and reliability as well as excellent prediction accuracy for compressor blade rows. To be sure, the two-dimensional blade-to-blade flow analyses offer somewhat better accuracy. But it is still possible to take advantage of those more general analyses when needed. The linearized method is quite adequate to provide support to the hub-to-shroud flow analysis, such that accurate stream surface locations and stream sheet thicknesses are obtained. Once the quasi-three-dimensional flow analysis is complete, all input data needed for the two-dimensional blade-toblade flow analyses are available. It is quite simple to make provision to generate input files for those more exact methods for any or all stream surfaces—that is a



FIGURE 12-2 Blade-to-Blade Flow Geometry

more efficient approach than incorporation of the more exact methods directly into the quasi-three-dimensional flow analysis. Indeed, it is not uncommon for a complete quasi-three-dimensional flow analysis to require significantly less computer time than a blade-to-blade flow analysis for the same problem on a single stream sheet by the two-dimensional blade-to-blade flow analyses. The hub-to-shroud flow analysis usually predicts the average meridional through flow in the passage. Alternatively, the meridional flow field on a mean hub-to-shroud stream surface is sometimes predicted. The prediction of the average through flow will be used in this chapter, since it is more consistent with the assumption of axisymmetric blade-to-blade stream surfaces. The hub-to-shroud flow analysis is similar to the meridional through-flow analysis of Chapter 7, except that the tangential momentum equation is no longer satisfied by conservation of angular momentum. Indeed, the distribution of angular momentum through the blade row must be obtained from the blade-to-blade flow analyses for all stream surfaces.

12.2 HUB-TO-SHROUD FLOW GOVERNING EQUATIONS The hub-to-shroud flow analysis is conducted using the quasi-normal coordinate system introduced in Chapter 7 for the meridional through-flow analysis. The basic nomenclature and construction of that coordinate system developed in Chapter 7 are summarized in the following equations and in Fig. 12-3.



FIGURE 12-3 Quasi-Normal Geometry

tan λ =

∆z zh − zs = ∆r rs − rh

∂r ∂m ε =φ−λ

sin φ =

1 ∂ ∂ ∂  =  − sin ε  ∂m  ∂n cos ε  ∂y ∂φ κm = − ∂m

(12-1) (12-2) (12-3) (12-4) (12-5)

The relevant equations for adiabatic inviscid flow are developed in general form in Chapter 3 in a natural coordinate system. For the present application, Eqs. (3-21), (3-25) and (3-30) apply after they are simplified to their time-steady form. It will be convenient to satisfy conservation of mass in integral form rather than using Eq. (3-21). Conservation of mass along a quasi-normal can be expressed in the form ys

˙ = 2π ∫ K B rρ Wm cos ε dy m

(12-6)

0



KB is the blade blockage factor, which corrects the stream sheet area available for through-flow to account for blade metal blockage. If the angle formed by a tangent to the blade camberline with respect to the meridional direction is denoted by κ, KB is given by K B = 1 − tbZ / (2π r cos κ )

(12-7)

Consistent with the blade-to-blade flow analysis to be used, rothalpy and entropy are assumed to be constant on stream surfaces, but may vary in the direction normal to the stream surfaces. Using Eqs. (3-25) and (3-30), these assumptions can be expressed by

κ mWm2 +

∂I ∂s = =0 ∂θ ∂θ

(12-8)

∂I ∂s = =0 ∂m ∂m

(12-9)

Wθ ∂( rWθ + ω r 2 ) ∂Wm ∂I ∂s + Wm = −T ∂n ∂n r ∂n ∂n

(12-10)

Equation (12-10) can be solved for Wm if Wθ, s and I are known on all stream surfaces. I and s can be supplied through assigned upstream boundary conditions, just as was done for the meridional through-flow analysis in Chapter 7. Equations (12-8) and (12-9) provide values at all other points in the flow field. The distribution of Wθ on the stream surfaces must be obtained from the blade-to-blade flow analysis. Combining Eqs. (3-29) and (12-8), ∂( rWθ + ω r 2 ) ∂Wm = ∂m ∂θ

(12-11)

Chapter 5 develops the blade-to-blade flow analysis for axisymmetric stream surfaces. Indeed, Eq. (12-11) is the principal momentum equation used in the blade-to-blade flow analysis. Clearly, a hub-to-shroud flow analysis restricted to axisymmetric stream surfaces can match only a single distribution of Wθ in the meridional plane. The most logical choice is to use a mass-averaged distribution of Wθ obtained from the blade-to-blade flow analyses on the various stream surfaces. In effect, the hub-to-shroud flow analysis will generate an axisymmetric flow field that is consistent with the mass-averaged angular momentum distribution produced by the blade-to-blade flow analyses. Equation (12-11) is basically ignored by this approximate hub-to-shroud analysis for all points within a blade passage. This equation can be satisfied for points upstream and downstream of the blade passage, where the conservation of the mass-averaged angular momentum is required. In this case, Eq. (12-11) can be written as ∂( rWθ + ω r 2 ) ∂rCθ = =0 ∂m ∂m

(12-12)



Since the linearized blade-to-blade flow analysis of Chapter 5 applies only to points within the blade passage, Eq. (12-12) is required to treat points outside the blade passage. Combining Eqs. (3-1), (12-4), (12-9) and (12-10) yields the normal momentum equation in the quasi-normal coordinate system. sin ε ∂Wm  ∂Wm  Wθ + κ m cos ε −  Wm + ∂y Wm ∂m  rWm  1  ∂I ∂s  =  −T  ∂y  Wm  ∂y

 ∂rCθ ∂rCθ  − sin ε   ∂m   ∂y

(12-13)

For points outside the blade passage where Eq. (12-12) can be applied, it is easily seen that Eq. (12-13) is identical to Eq. (7-12), which is used for the meridional through-flow analysis. For points within the blade passage it is more convenient to express the angular momentum in terms of the relative flow angle, noting that Wθ = Wm tanβ′. This yields  sin ε ∂Wm tan β ′  ∂r tan β ′ + − sin ε κ m cos ε −  2 r  ∂y Wm cos β ′ ∂m  1 1 ∂Wm + + 2ω tan β ′ (cos λ − sin φ sin ε ) = 2 y W ∂ cos β ′ m

∂r tan β ′     Wm ∂m    ∂I ∂s   −T  y y ∂ ∂ 

(12-14)

For points outside the blade passage where Eq. (12-12) applies, Eq. (12-14) should reduce to the form used in the meridional through-flow analysis of chapter 7, i.e., to Eq. (7-14). Introducing Wθ = Wm tanβ′ into Eq. (12-12), it is easily shown that Wm tan β ′ ∂r tan β ′ ∂Wm = − tan 2 β ′ − 2ω sin φ tan β ′ r ∂m ∂m

(12-15)

Substituting Eq. (12-15) into (12-14) does yield Eq. (7-14) as required. While the two forms are mathematically equivalent, some care is recommended when formulating a numerical analysis. Equations (12-13) and (12-14) are necessary when the angular momentum distribution is supplied externally by the blade-to-blade flow analyses. But outside of the blade passage where Eq. (12-12) is applicable, use of Eqs. (7-12) and (7-14) is likely to result in a more accurate numerical solution.

12.3 NUMERICAL INTEGRATION OF THE GOVERNING EQUATIONS The mass and momentum conservation equations can be numerically integrated for a specified stream surface geometry and flow angle distribution through the blade passage. The numerical method used is basically identical to the method used in chapter 7. The momentum equation is written in the form



∂Wm f ( y) = f1( y)Wm + f2 ( y) + 3 ∂y Wm

(12-16)

where f1, f2 and f3 are known functions of y. For quasi-normals within the blade passage,  tan β ′  ∂r tan β ′ ∂r tan β ′   sin ε ∂Wm f1( y) = cos2 β ′ −κ m cos ε − − sin ε   + ∂m   Wm ∂m r ∂ y  

(12-17)

f2 ( y) = −2ω cos β ′ sin β ′ [cos λ − sin φ sin ε ]

(12-18)

 ∂I ∂s  f3 ( y) = cos2 β ′  − T  y y ∂ ∂ 

(12-19)

For quasi-normals upstream and downstream of the blade passage, f1( y) = −κ m cos ε +

sin ε ∂Wm Wm ∂m

(12-20)

f2 ( y) = 0

(12-21)

∂I ∂s Wθ ∂( rCθ ) f3 ( y) = −T − ∂y ∂y ∂y r

(12-22)

At the upstream boundary, it is usually more convenient to supply boundary conditions in the stationary frame of reference, in terms of distributions of H, s and either Cθ or β. The first case is easily modeled by substituting Cθ for Wθ and H for I in Eq. (12-22). When the distribution of β is known, Eqs. (7-19) through (7-21), in the stationary frame of reference, yield  tan β ∂( r tan β ) sin ε ∂Wm  f1( y) = cos2 β −κ m cos ε − +  r ∂y Wm ∂m  

(12-23)

f2 ( y) = 0

(12-24)

 ∂H ∂s  f3 ( y ) = cos2 β  −T  ∂ y ∂ y 

(12-25)

For specified stream surface geometry and flow angle distributions through the blade passage, f1, f2 and f3 will be evaluated at all grid points before integrating the momentum equation. This process is straightforward with the exception of the meridional gradient of Wm in f1, which depends on the solution. Usual practice is to use values of Wm from the previous iteration to evaluate this gradient. That approach can result in numerical stability problems in cases where Wm is changing rapidly between iterations. Since Wm appears in the denominator, it can also produce singular results when Wm approaches zero. Aungier (2000) suggests a better approach based on simple conservation of mass in all stream



sheets. If the local through-flow area of a stream sheet is designated by ∆A, conservation of mass in the stream sheet requires ˙ = ρ Wm ∆A ∆m

(12-26)

Then the meridional gradient in f1 can be replaced by 1 ∂Wm 1 ∂ρ ∆A =− ρ ∆A ∂m Wm ∂m

(12-27)

This expresses the gradient in terms of the stream sheet geometry and gas density. The stream sheet geometry is constant during the process of integrating the mass and momentum conservation equations. It is still necessary to rely on values of gas density from the previous iteration, but density normally does not vary greatly between successive iterations. Since the mass flow rate in all stream sheets is constant, the risk of singular results is also eliminated. Aungier (2000) uses the same approach to remove the singularity in Eq. (12-16) when Wm approaches zero by defining a new function, f4, as

ρ ∆A ˙ ∆m

(12-28)

∂Wm = f1( y)Wm + f4 ( y) ∂y

(12-29)

f4 ( y) = f2 ( y) + f3 ( y) Then Eq. (12-16) can be written as

The solution of this linear differential equation has already been given in Chapter 7 as y

Wm ( y) = Wm (0)F( y) + F( y)∫

0

f4 ( y) dy F ( y)

(12-30)

where y  F( y) = exp  ∫ f1( y)dy   0 

(12-31)

The meridional velocity on the hub contour, Wm(0), is the constant of integration. It is determined from conservation of mass through Eq. (12-6). Equations (12-29) and (12-6) are solved in an iterative numerical scheme, successively improving the estimate of Wm(0) until mass is conserved and the momentum equation is satisfied. This requires calculation of thermodynamic properties such as ρ, using an appropriate equation of state from Chapter 2. At any point, the relative total enthalpy is given by Eq. (3-13), i.e.,



H ′ = I + 12 ( rω )2

(12-32)

The local static enthalpy is given by h = H ′ − 12 W 2

(12-33)

Then static thermodynamic conditions are computed from relative total thermodynamic conditions for the change in enthalpy, (h – H′), while holding entropy constant. The entropy and rothalpy on each stream surface is known from the assigned upstream boundary conditions. Similarly, the angular momentum, rCθ, upstream of the blade on each stream surface is known from the assigned upstream boundary conditions. The angular momentum downstream of the blade on each stream surface is obtained from the solution at the blade trailing edge quasi-normal.

12.4 REPOSITIONING STREAM SURFACES After integrating the mass and momentum conservation equations, the new flow field data is generally not consistent with the resident stream sheet geometry. The correct stream sheet geometry can be computed by integrating Eq. (12-6) across the passage at each quasi-normal and interpolating for the new stream sheet positions such that all stream sheets contain equal mass flow rates. In practice, fairly sophisticated numerical damping procedures are required to avoid numerical instability problems. This writer uses the numerical damping procedure suggested by Novak (1973). For quasi-normals outside of the blade passages Novak recommends 2 (1 − Mm )( ∆y)2 1 = 1+ F B* ( ∆m)2

(12-34)

F is the fraction of the distance between the new and old stream sheet positions that will actually be used in repositioning the stream sheets, ∆y is the hub-toshroud quasi-normal length, ∆m is the minimum meridional spacing between quasi-normals, Mm is the meridional Mach number and B* is an empirical constant. For quasi-normals inside the blade passage, Novak suggests (1 − M′ 2 )(cos β ′∆y)2 1 = 1+ F B* ( ∆m)2

(12-35)

This damping procedure has been found to be quite effective as long as Mm and M′ are limited to be no greater than 0.9. B* values used are typically in the range of 8 to 16, but the numerical procedure should adjust B* automatically based on whether the stream surface position errors are increasing or decreasing on successive iterations.



12.5 THE HUB-TO-SHROUD FLOW ANALYSIS To start the solution, an initial guess must be supplied for all flow field data used in the hub-to-shroud solution. Since no blade-to-blade flow analysis results are available at this point, the initial guess must include an approximate blade-toblade flow definition. This writer uses the following procedure to initialize the flow field data: • Stream surfaces are positioned such that all stream sheets contain the equal through-flow area on each quasi-normal. • Entropy and rothalpy are computed for all stream surfaces from the upstream boundary conditions. • The meridional velocity is computed from local conservation of mass assuming the local gas density is equal to the relative total density. • Cθ upstream of the blade is obtained from conservation of angular momentum, Eq. (12-12), using the upstream boundary conditions. This includes the blade leading edge quasi-normal. • The blade is assumed to provide perfect guidance to the flow over the last 85% of the blade passage length based on the blade camberline angle, κ. • Cθ downstream of the blade is obtained from conservation of angular momentum using the value at the trailing edge quasi-normal. • An “inlet slip” condition is applied over the first 15% of the blade passage length based on the relative flow angle computed at leading edge quasi-normal and the blade camberline angle, κ, i.e., tan β ′ = tan β LE ′ + (tan κ − tan β LE ′ )[(m − mLE ) / (mTE − mLE ) / 0.15]2

(12-36)

This initialization procedure is very conservative and is virtually certain to successfully start the analysis. But Eq. (12-36) is a rather crude approximation for the blade-to-blade flow that often results in difficult and slow convergence of the hub-to-shroud flow analysis. Once the first blade-to-blade flow analysis results are obtained, convergence of the hub-to-shroud analysis consistently improves dramatically. A much more efficient overall analysis is obtained by simply not requiring convergence on the first attempt at the hub-to-shroud flow analysis. This writer simply limits the number of iterations on the hub-to-shroud flow governing equations on this first attempt to, say 10-12, while ignoring the convergence requirement. This is sufficient to obtain a reasonable stream sheet pattern to run the first blade-to-blade flow analysis. The general procedure used for the hub-to-shroud flow analysis is a direct application of the well-known streamline curvature technique. The basic conservation equations are solved at each quasi-normal using the procedures described in Section 12.3. Then the stream surfaces are repositioned as described in Section 12.4. The process is repeated until the stream surface positions and flow field data have converged within an acceptable tolerance. This procedure is adequate for most problems, but Aungier (2000) suggests two simple extensions that can greatly improve the reliability of the analysis to the point that a converged



solution is almost always obtained: First, the hub-to-shroud flow analysis should check for choked flow, and limit the overall mass flow rate to the choke value when it occurs. In the nomenclature of Section 12.3, the flow on any quasi-normal is locally beyond the choke limit when ˙ ∂m ≤0 ∂Wm (0)

(12-37)

where m˙ is the calculated mass flow rate obtained from Eq. (12-6). When this condition is encountered, the solution procedure should determine the mass flow rate for which this gradient term is zero, which is the choke value. The choked flow condition of Eq. (12-37) is not always an indication of true choked flow. Often it is caused by numerical errors during the early iterations. The solution should proceed using the largest acceptable mass flow rate, but not exceeding the specified mass flow rate. In the case of a temporary choke due to numerical error, the mass flow rate will gradually increase to the specified value as the solution converges. This simple procedure avoids the most common cause of solution divergence. Another common cause of solution divergence is numerical errors in the streamline curvature terms. The numerical damping of the stream surface repositioning given in Section 12.4 avoids most of these problems. But that damping procedure is indirect with respect to the curvature terms. This writer also imposes a direct damping procedure that can be expressed as

κ m → (κ mi + Dκ mi−1 ) / (1 + D)

(12-38)

where the subscripts i and i-1 refer to the iteration number and D is a damping factor. Typically, D = 1 is used, but D is increased if successive iterations show a significant increase in convergence errors.

12.6 COUPLING THE TWO BASIC FLOW ANALYSES The quasi-three-dimensional flow analysis is obtained by coupling the hub-toshroud flow analysis described in this chapter with the linearized blade-to-blade flow analysis described in Chapter 5. The hub-to-shroud flow analysis defines the stream sheet geometry and corresponding blade geometry for the blade-to-blade flow analysis. The stream sheet thickness is also required, which is computed from the hub-to-shroud flow data and the specified stream sheet mass flow by simple conservation of mass. ˙ / (2π rK B ρ Wm ) b = ∆m

(12-39)

Similarly, when the blade-to-blade flow analysis is completed, the results are mass-averaged across the blade passage to provide the relative flow angle distribution to be used on the next hub-to-shroud flow analysis. Recalling that the



dimensionless tangential coordinate, η, used in the blade-to-blade flow analysis varies from 0 to 1, the data required are given by 1

Wm =

∫ ρ bWmdη 2

0 1

(12-40)

∫ ρ bWmdη 0

1

Wθ =

∫ ρ bWmWθ dη 0

1

(12-41)

∫ ρ bWmdη 0

tan β ′ = Wθ / Wm

(12-42)

In addition to requiring convergence of the two component analyses, a convergence criterion is needed for the overall quasi-three-dimensional flow analysis. This writer requires that all relative flow angles predicted by Eq. (12-42) on successive blade-to-blade flow analyses converge within an acceptable tolerance. Figure 12-4 shows a basic flow chart for the quasi-three-dimensional flow analysis. The combination of the present hub-to-shroud analysis with the linearized blade-to-blade flow analysis of Chapter 5 provides a very reliable and effi-

FIGURE 12-4 Flow Chart of the Analysis



cient quasi-three-dimensional flow analysis for a wide range of axial-flow compressor blade passage problems of interest. The major limitation is with regard to Mach number levels, since the blade-to-blade flow analysis cannot account for shock waves. That analysis does have the transonic extension described in Chapter 5, so some supersonic flow is acceptable. The other major restriction follows from the basic assumption of inviscid flow. This is a reasonable assumption as long as the incidence angles are reasonable and flow diffusion is not excessive. Extreme incidence angles or excessive diffusion will result in substantial flow separation, which cannot be modeled by an inviscid flow analysis. To obtain meaningful results, those cases require a more fundamental viscous computational fluid dynamics code. This analysis requires a very large number of equation-of-state calculations. The pseudo-perfect gas equation of state model described in Chapter 2 can greatly reduce the computational time relative to real gas models, or even ideal gas models with temperature-dependent specific heats. This model is almost always adequate for blade passage flow analyses in compressors. This follows from the fact that some care is normally taken to avoid any risk of operating compressors with liquid present in the flow. Hence the operating conditions are normally safely in the super-heat zone, where gas properties do not show significant variation over the limited range of conditions needed for a flow field analysis. In contrast, turbine applications may involve two-phase flow, or inlet conditions very close to the vapor saturation line. In either case, the pseudo-perfect gas model is usually not adequate for those problems. Quasi-three-dimensional flow analysis plays a central role in the aerodynamic design of blades for centrifugal compressors, where each blade row is typically a unique design for the specific application (Aungier, 2000). As discussed at the beginning of this chapter, its role for axial-flow compressors is more likely to involve an evaluation of a specific blade row operating under conditions more extreme than justified by experience. To be effective in that role, the analysis must be easily applied as well as fast and reliable. The author’s analysis includes the methods described in Chapter 4 to very precisely define the blade geometry for standard axial-flow compressor blade camberlines and profiles to minimize the geometry data required. This, in turn, makes it quite simple to provide the performance analysis of Chapter 9 and the aerodynamic design program of chapter 11 with the capability to export a complete input file for the present analysis. Hence a more detailed evaluation of any blade row analyzed in one of those programs can be accomplished with very little effort. The performance analysis conducted on the NACA 10-stage compressor in Chapter 9 will be used to illustrate a typical application of this analysis. With reference to Fig. 9-4, the performance analysis for a mass flow rate of 55 at 100% speed predicts that the first stator operates at a supercritical Mach number along the hub contour. Although the performance analysis does not indicate any significant adverse results, the designer might want to evaluate this situation closely. The empirical models for estimating the critical Mach number are certainly quite approximate, and the NACA 65-series blades used are not well suited to operating with significant supersonic flow. To evaluate this situation, a quasi-three-dimensional flow analysis input file for this stator was exported from the performance analysis. After running the quasi-three-dimensional flow analysis, an input file



FIGURE 12-5 A Blade Loading Evaluation

for the two-dimensional blade-to-blade flow analysis of Chapter 5, Section 5.3, for the hub surface was exported and processed for a more accurate analysis on this critical stream surface. Figure 12-5 presents the results from these two analyses. It is seen that the stator does operate at a supercritical Mach number. As noted in Chapter 5, blade-loading data close to the leading edge predicted by the linearized blade-to-blade flow analysis should generally be ignored. In that context, the internal flow analyses indicate that the flow is only slightly supersonic and should not pose any significant risk of a performance penalty. The entire process of completing two internal flow analyses to evaluate this situation was accomplished in less than two minutes. That is an important advantage offered by inclusion of efficient inviscid internal flow analysis techniques in a well-formulated aerodynamic design and analysis system. Had a fundamental viscous CFD code been used for this purpose, the evaluation might easily have taken several days. This analysis really encourages designers to evaluate areas of concern, which might otherwise be ignored due to the prohibitive effort required for a more detailed evaluation.

12.7 BOUNDARY LAYER ANALYSIS Boundary layer analysis provides a useful extension to an inviscid internal flow analysis. It can provide an approximate evaluation of viscous effects with no significant increase in computation time. Chapter 5 describes a two-dimensional



blade surface boundary layer analysis and loss coefficient model that can be incorporated into the quasi-three-dimensional flow field analysis directly. Those models are strictly applicable to just two-dimensional cascade flows. They ignore the secondary flows that are always present in a real cascade. Nevertheless, they do provide a reasonable qualitative evaluation of viscous effects that can provide useful guidance to the designer. Unlike the case of blade surface boundary layers, there is little merit in considering two-dimensional models for the highly threedimensional end-wall boundary layers. Nor is there much merit today in considering use of inviscid flow analysis coupled with three-dimensional boundary layer analysis models. That approach would involve computation times and complexity comparable to a viscous CFD code, but would offer less generality and accuracy. With the many excellent commercially available viscous CFD codes available today, there is no reason to consider more approximate methods unless they offer substantial advantages in terms of computation speed and reliability. Aungier (2000) uses an end-wall boundary layer analysis that is an early version of the analysis described in Chapter 8. That end-wall boundary layer analysis has since been upgraded to include some of the newer developments described in Chapter 8. This type of axisymmetric, three-dimensional boundary layer analysis is about the only method that can provide meaningful results and offer the computation speed and reliability required. This approximation is known to yield excellent results in vaneless passages [Aungier, 1988(b) and Davis, 1976]. Within the blade passages, the boundary layers are certainly not axisymmetric. There the model seeks to predict the gap-averaged boundary layer behavior (Horlock, 1970) as discussed in Chapter 3. The description of the end-wall boundary layer analysis provided in Chapter 8 is directly applicable here with two very specific exceptions: In Chapter 8, all meridional computing stations lie outside of the blade passages, so when blade rows are encountered, the meridional integration step is always across the blade row. In the present application, there will be many meridional computing stations within the blade passage. The process of numerical integration remains the same, but here the boundary layer equations are integrated over many spatial steps within the blade passage instead of a single spatial step across it. That, in turn, requires revision of the blade force defect model for it to be applicable within the blade passage. Aungier (2000) suggests the following blade force defect thickness model:

ν = Kθ11

(12-43)

Based on experimental data from Koch and Smith (1976) and Hunter and Cumpsty (1982), values of K in the range of 0.7 to 1 were recommended. Since those sets of experimental data are from axial-flow compressor tests, it is not too surprising that this simple model yields reasonable results for axial-flow compressor applications. However, since Aungier (2000) was published, a number of centrifugal compressor applications have been encountered where the results did not appear realistic. The weakness in the model has been most evident during analysis of the boundary layer within a blade passage where both the blades and the wall are rotating. As the angle φ becomes large, the centrifugal forces in the boundary layer also become large, tending to reduce θ11 and, therefore, ν. Under some conditions,



this can produce nearly uniform meridional velocity profiles and highly distorted tangential velocity profiles. It is evident that such highly skewed boundary layer profiles are not reasonable within the blade passages, where blades will tend to guide the flow toward a collateral boundary layer condition. The analysis of Aungier (2000) does not recognize that extreme skewing of the boundary layer flow profiles will be constrained by the flow guidance imposed by the blades. These observed weaknesses in the method described in Aungier (2000) have recently been significantly improved by adapting the blade force defect model of Chapter 8 to the present application. Figure 12-6 shows the same experimental data as Fig. 8-5 and the simplified empirical correlation used in the present application. Since the present analysis is intended for single blade passage analyses, the larger values of displacement thickness observed in the rear stages of multistage compressors will not be encountered. With little likelihood of approaching the maximum postulated in Fig. 8-5, the simpler linear empirical correlation is a reasonable approximation, which eliminates the staggered spacing term in the correlation. Hence the basic blade force defect thickness of Eq. (8-44) is replaced by

ν0 = 0.4δ1* + δ c / 2

(12-44)

It can be seen that Eq. (12-44) is similar to Eq. (12-43), but is based on the displacement thickness rather than the momentum thickness and includes the influence of the blade clearance. Following the approach used in Chapter 8, the

FIGURE 12-6 Tangential Force Defect Thickness



blade force defect thickness is modified by a term ∆ν to account for the influence of the blade guidance

ν = ν0 + ∆ν

(12-45)

This correction term is computed exactly as was done in Chapter 8, using Eqs. (8-53) and (8-57), but with an alternate definition for the parameter, F, that is more suitable to integration within the blade passage. The form used for F is based on the following assumptions: • The blades guide the flow toward collateral flow primarily close to the blade surfaces. It is assumed that the gap-averaged deviation from collateral flow imposed by integrating the boundary layer equations, using only ν0, will be reduced by about 50% due to the blade guidance. • The influence of blade guidance and clearance gap on the blade force defect is negligible when the blade clearance gap is greater than the boundary layer thickness. • The influence of blade guidance and clearance gap on the blade force defect is negligible at the blade row leading edge. This influence is assumed to be proportional to the ratio of the meridional distance to the blade spacing. The second assumption simply recognizes the special case where the boundary layer and blades are not really interacting. Although this situation is probably very rare in a compressor, it can easily occur in the present analysis if the entrance boundary layer thickness is specified small enough. Experience has shown that this will lead to erratic and highly questionable results. The third assumption recognizes that the influence of the blades will not extend across the entire blade passage spacing until the flow penetrates well into the guided passage. This leads to the following approximation for F: F = [1 − δ c / δ ]m / [( g + m] / 2 ; δ c ≤ δ

(12-46)

F = 0; δ c > δ

(12-47)

Equation (12-46) includes the influence of the ratio m/g, where m is the meridional distance from the blade leading edge and g is analogous to the local staggered spacing used in Chapter 8, i.e., g = 2π r cos β ′ / Z

(12-48)

F is used directly in Eq. (8-57) and is also imposed as a correction applied to Eq. (12-44).

ν0 = 0.4δ1* + δ cF

(12-49)

Since there are now several spatial integration steps within the blade passage, the blade force defect and the blade force direction given by Eq. (8-49) both vary



through the blade passage, but the process of numerical integration is the same as that used in Chapter 8. However, the calculation of ∆ν requires more care since it relates to a local blade force correction rather than an overall blade row correction. Locally, the ideal blade force may be extremely small, or even zero. This can produce unrealistic values of ∆ν. To avoid such problems, a constraint is imposed on ∆ν in the form ∆ν ≤ [0.05 + 0.2 ( H1 − 1)]δ

(12-50)

This constrains the blade force correction to be less than about 75% of the meridional displacement thickness, except at values of H1 very close to 1. The displacement thickness will vanish when H1 → 1, so special care is required to avoid suppressing the blade guidance effect entirely in that case. Figures 12-7 and 12-8 show typical boundary layer parameters predicted by this end-wall boundary layer analysis when applied to the axial-flow compressor stator problem used as an example earlier in this chapter. This blade row operates at near optimum incidence angles at the operating conditions considered in this sample case. Hence, the blade row pressure coefficient is relatively mild for this case. Based on the experimental data shown in Fig. 8-2, the blade force defect thickness, averaged over the blade row, is expected to be a fairly large fraction of the meridional displacement thickness, which is at least consistent with

FIGURE 12-7 Hub Contour Boundary Layer



FIGURE 12-8 Shroud Contour Boundary Layer

the present predictions. That certainly does not provide a real validation of the assumptions, nor is there any data available in the literature suitable for validating them. The assumptions are considered reasonable, and they produce results consistent with overall blade row experimental data. Also, they produce no obvious deficiencies such as those seen from the simpler model of Aungier (2000). That is about all that can be said to justify their use.


Chapter 13

OTHER COMPONENTS AND VARIATIONS

The preceding chapters have considered the basic aerodynamic design and analysis functions relative to typical axial-flow compressor configurations. Blade row, stage and compressor design and analysis have been considered in the context of the compressor flow path from the first blade row to the last, including operation at off-design speeds and mass flow rates. This chapter addresses other design and analysis functions that are often required by the specific application. One obvious consideration for industrial axial-flow compressors is dictated by the need to supply the compressor discharge flow to the process through a discharge pipe, typically exiting in the radial direction. This requires the design and analysis of an exhaust diffuser and flow collection system, which can have significant influence on the overall performance of the compressor. Another important consideration is the use of adjustable inlet guide vanes and stators as a means of improving the surge margin. Normally this provides a substantially better surge margin at mass flows that are significantly less than the design flow than can be achieved with simple variable speed operation. Indeed, variable geometry is an essential requirement for many industrial compressor applications. The influence of surface roughness on performance is discussed to provide a basis for its evaluation when required. Finally, the axial-centrifugal compressor is briefly discussed. This configuration includes a centrifugal compressor stage following a series of typical axial-flow compressor stages. Under appropriate operating conditions, this can offer significant cost and performance benefits.

NOMENCLATURE A Am B BBL BSEP b C cp cf D

= = = = = = = = = =

passage area maximum, stall-free passage area fractional area blockage boundary layer fractional area blockage minimum fractional area blockage due to flow separation passage width absolute velocity static pressure recovery coefficient skin friction coefficient 2tanθC



limiting value of D for low diffusion losses characteristic diameter for Reynolds number definition hydraulic diameter diffusion efficiency parameter peak-to-valley surface roughness root-mean-square surface roughness normalized stagger angle adjustment total enthalpy static enthalpy passage curvature loss term passage diffusion loss term adjustable blade row number diffuser meridional length and scroll/collector flow path length meridional coordinate mass flow rate total number of adjustable blade rows exponent in power-law stagger adjustment distribution pressure ratio of stagger angle adjustments on successive stationary rows Reynolds number radius scroll/collector sizing parameter local blade speed = ωr stagger angle boundary layer thickness hub-to-shroud radius ratio polar angle diffuser divergence angle stream surface curvature gas viscosity gas density wall shear stress flow coefficient = Cm /U; also, slope angle of mean stream surface with the axial direction φ0 = characteristic inlet flow coefficient ω = rotation speed, radians/sec. – = total pressure loss coefficient ω

Dm d dH E e erms F H h IC ID i L m ˙ m N n P R Re r SP U γ δ ε θ θC κm µ ρ τw φ

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Subscripts 0 1 2 3 4 l ex

= = = = = = =

parameter at a stage inlet parameter at the diffuser inlet parameter at the diffuser exit parameter at the scroll/collector full-collection plane parameter at the exit cone discharge flange laminar condition exit cone parameter


Other Components and Variations • 311

m r s sf t tip θ

= = = = = = =

meridional component fully rough wall condition smooth wall value skin friction parameter total thermodynamic condition or turbulent condition blade tip parameter tangential component

13.1 ADJUSTABLE BLADE ROWS There are two important considerations when using adjustable stationary blade rows: The number of adjustable blade rows to be used must be selected and the distribution of the stagger angle adjustments among those rows must be established. These decisions can be conveniently made using a performance analysis such as that described in Chapter 9. It is relatively easy to include the capability to impose adjustments to the base stagger angles for specific blade rows. It is most convenient to specify those adjustments as a fraction of a reference stagger angle adjustment, e.g., the adjustment to be made to the first adjustable blade row. If that is done, performance predictions for a series of different adjustable blade row settings can be accomplished by simply changing the value of the reference stagger angle adjustment. The most obvious design strategy would be to select one specific setting of the adjustable blades, such as the most extreme adjustment to be used, and optimize the distribution of the adjustments among all adjustable rows. In practice, that is a rather ineffective approach. Adjustable blade rows can produce rather extreme incidence angles on both rotors and stators to substantially increase the uncertainty of the performance analysis and the reliability of any estimate of the surge limit based on that analysis. If an adjustment strategy is established to optimize the most extreme adjustment, prediction of an overall map of adjustments is likely to be totally lacking in credibility. The most common result is to find the estimated surge margin deteriorating at adjustments even slightly different from the optimized adjustment. The real problem lies with the performance predictions for the optimized adjustment. Off-design performance analysis at extreme incidence angles involves too much uncertainty for the optimized adjustments to really be believable. The very precise stage matching chosen to enhance surge margin is very unlikely to be achieved in the actual compressor. Hence the designer who attempts such an optimization strategy is really exceeding the capabilities of off-design performance analysis technology. In effect, that is an excellent way to expose all of the weaknesses of the performance analysis, which is hardly a good basis for design. Even if the performance analysis could accurately model this situation, a surge line that achieves enhanced surge margin very local to one adjustment setting would not be very useful. A better approach is to establish general stagger-angle adjustment distributions for this purpose and evaluate the results with the performance analysis. These general distributions can offer considerable flexibility and yet be very unlikely to produce a misleading “optimum stage matching” distribution. Three general distributions found to be effective for this purpose will be presented and



demonstrated here. If there are N adjustable blades, let Fi represent the adjustment to the ith adjustable blade row in the form ∆γ i = ∆γ ref Fi

(13-1)

where ∆γref is the reference adjustment, which will be assumed to be the adjustment to the first adjustable row, such that F1 = 1. One useful distribution is a simple linear variation such that the adjustment to the stator following the last adjustable stator would be zero, i.e., Fi = ( N + 1 − i) / N

(13-2)

A simple variation on the linear distribution will be referred to as the power-law distribution, given by Fi = [( N + 1 − i) / N]n

(13-3)

Another useful distribution is the fractional distribution, which simply adjusts each blade row as a specified ratio, R, of the adjustment on the preceding adjustable blade, i.e., Fi = R (i −1)

(13-4)

Figure 13-1 illustrates these three stagger angle distributions for a case where N = 7. The linear and fractional distributions are adequate for most applications, but the power-law distribution provides some additional flexibility if needed. When these models are incorporated into the performance analysis, investigation of alternatives for adjustable geometry is rather simple. It is only necessary to specify the adjustment distribution type, the number of adjustable blade rows and the stagger angle adjustment on the first row. Simply varying the last parameter permits analysis of alternate settings of the adjustable rows. The ten-stage axial-flow compressor design of Fig.10-38 will be employed to illustrate the use of adjustable stationary blades. This is a repeating stage design with inlet guide vanes. Figure 13-2 compares the variable speed performance map from Fig. 10-38 with a predicted performance map for a seven-row linear stagger angle distribution (inlet guide vane and six stators). It can be seen that variable geometry offers a significant improvement in surge margin compared to variable speed operation at flow rates less than the design flow. Figure 13-3 illustrates the influence of the number of adjustable blade rows when the linear adjustment distribution is used for this compressor. Increasing N has the expected effect of greater flow capacity control. It can be noted that the best surge margin appears to be associated with N = 7. Generally, there is an optimum value of N to enhance the surge margin, typically around one-half of the total number of stationary rows available for adjustment, excluding exit guide vanes, which are ineffective for this purpose. In this case, there are eleven rows available (inlet guide vane and ten stators), so an optimum N around 6 might be



FIGURE 13-1 Alternate Adjustment Strategies

FIGURE 13-2 Variable Geometry and Variable Speed



FIGURE 13-3 Influence of N with Linear

expected. This really follows from the mechanism by which variable geometry improves surge margin. Basically, the adjustable blades close down the front stages, which would normally stall at the lower flows, to shift the load to the rear stages. Stated differently, adjustable blades reduce the load on the front stages to permit operation at lower volume flow rates where the rear stages will operate closer to their design conditions. Hence variable geometry in the rear stages is seldom effective for this purpose. Figure 13-4 illustrates the fractional adjustment distribution for a series of values of R. It can be seen that increasing R results in increased capacity control, while yielding essentially identical surge lines for all values considered. But there is a slight adverse effect on surge margin and on efficiency with increasing R, suggesting that larger values would not be good for this compressor. As a general rule, R ≈ 0.8 is usually a good choice for the fractional adjustment distribution. Figure 13-5 illustrates the influence of N with the fractional distribution with R = 0.8. Increasing N from 5 to 7 increases both capacity control and surge margin. But only two values of N are shown, since the surge estimate for the higher value is based on stall in one of the adjustable blade rows. Hence, adding more adjustable blade rows with the fractional adjustment distribution on this compressor cannot further improve the surge margin. Comparing Figs. 13-3 and 13-5 shows that the fractional adjustment distribution can achieve a surge margin improvement comparable to the linear style for this compressor. It can be concluded that the use of adjustable stationary blade rows is a very powerful method for achieving increased surge margin at volume flow rates less



FIGURE 13-4 Influence of R with Fractional

FIGURE 13-5 Influence of N with Fractional



than the design value. By including the provision to impose these adjustments within the aerodynamic performance analysis, it is relatively simple to arrive at an appropriate adjustment distribution style and the appropriate number of adjustable blade rows to be used. This also makes it rather easy to generate a predicted performance map to evaluate a range of adjustable blade row settings. The alternative of accomplishing these adjustments manually on each blade row can make this a very tedious process, with little likelihood of arriving at an optimum choice.

13.2 THE EXHAUST DIFFUSER The industrial axial-flow compressor normally supplies the compressed gas to some process through an exit pipe flange. Often, the final destination will involve redirecting the flow from the axial flow leaving the compressor to a pipe oriented normal to the compressor axis. Hence the overall performance of the compressor is strongly influenced by the performance of the exhaust-end components. Generally, the flow is first diffused in an exhaust diffuser to recover some of the kinetic energy in the form of static pressure. This may be an annular, axial exhaust diffuser such as that shown in figure 13-6. Alternatively, curved diffusers may be used to redirect the flow in the radial direction for collection by a scroll or collector, such as illustrated in figure 13-7. Consequently, a very wide variety of exhaust diffusers may be employed, requiring a very general performance analysis technique. One aerodynamic performance analysis method that can handle these various configurations is the vaneless passage analysis of Aungier (1993, 2000). That analysis has been extensively qualified for centrifugal compressor performance

FIGURE 13-6 Axial Exhaust Diffuser Geometry



FIGURE 13-7 Curved Exhaust Diffuser

analysis, where it has been used to treat the swirling flows in vaneless diffusers and 180° crossover bends. With one minor modification, it is directly applicable to the axial-flow compressor exhaust diffuser problem. The area requiring modification is the modeling of the end-wall boundary layer growth and the associated viscous blockage effects. For axisymmetric flow in centrifugal compressors, the rather ideal form of the tangential velocity profile in the boundary layer makes that profile an obvious choice to estimate boundary layer growth. But that choice is not appropriate for an axial-flow compressor exhaust diffuser, which is likely to have little or no tangential velocity component. Similarly, viscous blockage effects are less critical in the centrifugal compressor. Cases where substantial blockage may be induced by excessive rates of diffusion generally occur only in the crossover bend, where performance is primarily described by total thermodynamic conditions rather than static. If the total pressure loss is predicted with sufficient accuracy, the overall performance predictions are relatively insensitive to the viscous blockage prediction. Again, the axial-flow compressor exhaust diffuser often involves quite different requirements. A simple dump collector may be used to collect the flow following an axial exhaust diffuser, such that little or



none of the kinetic energy is recovered. Thus, static conditions can be extremely important, requiring a more careful consideration of viscous blockage effects. Fortunately, generalization of the centrifugal compressor vaneless passage performance analysis proved to be relatively straightforward, since most of the fundamental models were already included for the purpose of total pressure loss prediction. The modified analysis used for axial-flow compressor exhaust diffusers is presented in this section. The analysis is a mean-streamline or one-dimensional flow analysis with wall friction and empirical corrections for diffusion and curvature effects. Figure 13-6 illustrates the mean streamline in a typical axial exhaust diffuser configuration and the nomenclature used in the analysis. The governing equations for one-dimensional flow with skin friction are developed in Aungier (1993, 2000) as ˙ 2π rbρ Cm (1 − B) = m d( rCθ ) = − rCCθ cf dm 1 dP Cθ2 sin φ dCm CCm cf dI D = − Cm − − − IC ρ dm r dm b dm bCm

H = h + 12 C 2

(13-5) (13-6) (13-7) (13-8)

The blockage, B, specifies the fraction of the passage area unavailable for the inviscid through flow, due to viscous effects. The skin friction coefficient, cf, supplies a correction for the effect of the wall shear stress, τw. cf =

τw 1 2

ρ C2

(13-9)

The terms ID and IC are introduced in Aungier (1993, 2000) to account for losses due to diffusion and passage curvature, respectively. In other respects, this set of equations follows directly from the governing equations for inviscid flow presented in Chapter 3, when simplified to time-steady, axisymmetric, one-dimensional flow in a stationary coordinate system. From classical two-dimensional diffuser technology (e.g., Reneau et al., 1966) it is known that the low loss regime is closely related to the well-known diffuser divergence angle, 2θC. This parameter can be generalized to annular diffusers by 2θ C = 2 tan −1[b1( A2 / A1 − 1) / (2L)]

(13-10)

The nomenclature shown in Fig. 13-6, where L is the length of the mean streamline and A = 2πrb is the passage area, is used. Aungier (1993, 2000) introduces a divergence parameter, D, defined as D = 2 tan θ C = b1( A2 / A1 − 1) / L

(13-11)



The experimental results in Fig. 8(b) of Reneau et al. (1967) show that diffusion losses are low when D < Dm, where Dm = 0.4( b1 / L)0.35

(13-12)

Aungier (1993, 2000) uses an analogy with the classical diffuser parameters, D and Dm, but defined as local diffusion parameters, i.e., D=−

b dC C dm

Dm = 0.4 cos β ( b1 / L)0.35

(13-13) (13-14)

Based on detailed comparisons with predicted and measured losses in a substantial number of centrifugal compressor vaneless diffusers, an empirical model was developed for a diffusion efficiency parameter, E, in the form E = 1; D ≤ 0

(13-15)

E = 1 − 0.2( D / Dm )2 ; 0 < D < Dm

(13-16)

E = 0.8 Dm / D ; D ≥ Dm

(13-17)

Then the streamwise diffusion term in Eq. (13-7) is given by 1 dC dI D = −2( Pt − P )(1 − E) ρ C dm dm

(13-18)

It was necessary to check for excessive diffusion in the meridional direction as well as the streamwise direction, again using an analogy with classical diffuser technology. The maximum stall-free local passage area, Am, is estimated from Am = A1[1 + 0.1925m / b1]

(13-19)

which corresponds to a local divergence angle limit of 2θC ≈ 11°. The highly swirling flows in centrifugal compressors required a slightly more conservative limit of 9°, but that is considered unnecessary for the exhaust diffuser application. A minimum value of ID is estimated when A > Am I D ≥ 0.65( Pt − P )[1 − Am / A] / ρ

(13-20)

This value is imposed as a lower limit on ID obtained by integrating Eq. (13-18). The passage curvature, κm, is used to estimate IC as IC = κ m ( Pt − P )Cm / (13ρ C )

(13-21)



κm = −

dφ dm

(13-22)

Aungier (1993, 2000) employs simple one-seventh power law velocity profiles as the boundary layer approximation. This continues to be used, since a one-dimensional flow analysis really does not provide an adequate basis for modeling boundary layer shape factors in any reasonable fashion. As noted, it is not possible to rely on the wall friction effects on the tangential momentum balance as a basis for estimating boundary layer growth in an exhaust diffuser. Instead, the meridional velocity profile is used with a simple flat-plat boundary layer growth model, including adjustment for variations in radius and the boundary layer edge meridional momentum. Turbulent boundary layer growth along a flat plate can be estimated from (Pai, 1957) dδ ≈ 5cf dm

(13-23)

This is generalized to the annular passage with two end-wall boundary layers in the form 2 d[rbρ Cm (2δ / b)] = 10cf ρ rCCm dm

(13-24)

The radius and the boundary layer edge meridional momentum corrections in this equation are similar to momentum thickness corrections of Eq. (3-43). This follows from the approximation that the boundary layer shape factor is constant, which requires that the ratio of the boundary layer thickness to the momentum thickness be constant. The boundary layer thicknesses estimated from Eq. (13-24) must be limited by the fully developed viscous flow profile condition, i.e., 2δ ≤ b. The fractional area blockage due to the two end-wall boundary layers will be designated as BBL. The boundary layer blockage for one-seventh power law profiles can be shown to be BBL = (2δ / b) / 8

(13-25)

For the exhaust diffuser, it must also be recognized that viscous blockage may increase substantially if the rate of diffusion becomes excessive, i.e., if the maximum stall-free local passage area, Am, is exceeded. If that occurs, it is assumed that further diffusion of Cm is suppressed by a minimum blockage, BSEP, where BSEP = 0 ; A ≤ Am

(13-26)

BSEP = 1 − Am / A ; A > Am

(13-27)

BSEP can be imposed as a lower limit on the blockage estimated from Eq. (13-25), but it has been found that a smooth transition between these two values is obtained from the following empirical equation:


Other Components and Variations • 321 2 B = 1 − (1 − BSEP )(1 − BBL + BSEP )

(13-28)

Equations (13-24) through (13-28) contain all of the modifications to the original analysis of Aungier (1993, 2000) developed specifically for the exhaust diffuser application. Finally, solution of the governing equations requires a model for the skin friction coefficient. Following Aungier (2000), a generalized pipe friction factor model is employed for this purpose. While that models fully developed pipe or channel flow, it is applicable to boundary layers by replacing the pipe diameter (or channel width) with 2δ. For this problem, the relevant Reynolds number is Re =

2ρ Cδ µ

(13-29)

The other parameter required is the peak-to-valley surface roughness height, e. Usually surface roughness is measured as a root-mean-square value, erms. The two values can be approximately related by assuming a sine-wave variation for the roughness to yield e = erms / 0.3535

(13-30)

If Re is less than 2000, the laminar skin friction coefficient applies. cf = cfl = 16 / Re

(13-31)

When Re > 2000 and the wall is smooth (e = 0), the well-known log-law profile yields 1 4cfs

 2.51 = −2 log10   Re 4c fs 

   

(13-32)

In the limit when the flow is turbulent and the wall is fully rough, 1 4cfr

  e = −2 log10   δ . ( ) 3 71 2  

(13-33)

The experimental results of Nikuradse (1930), show that surface roughness becomes significant when Ree = (Re− 2000)e / (2δ ) > 60

(13-34)

Hence, a general turbulent skin friction coefficient is given by cft = cfs ; Ree < 60

(13-35)



cft = cfs + ( cfr − cfs )(1 − 60 / Ree ); Ree ≥ 60

(13-36)

If Re ≥ 4000, cf is defined by Eq. (13-35) or (13-36). For 2000 < Re < 4000, the flow is in the transition zone and the skin friction coefficient is approximated by cf = cfl + ( cft − cfl )(Re / 2000 − 1)

(13-37)

These empirical equations yield values in very close agreement with the experimental results of Nikuradse (1930). The performance analysis of the exhaust diffuser involves integrating Eqs. (13-5) through (13-8) along the diffuser length, supported by the supplemental relations provided in this section. Input specifications required include the passage geometry, mass flow rate and inlet values of the total thermodynamic conditions, tangential velocity and normalized boundary layer thickness, 2δ / b. To show at least a qualitative evaluation of the method, the experimental data for two-dimensional diffusers of Reneau et al (1967) will be used. In general, twodimensional and annular diffusers perform in a reasonably similar manner as long as appropriate dimensionless parameters are used in any comparison. In particular, the diffuser divergence angle definition of Eq. (13-10) should be used. The highest inlet blockage level results in Fig. 4(d) of Reneau et al. (1967) are the most representative of exhaust diffusers. These show an optimum static pressure recovery coefficient, cp, around L / b1 ≈ 0.8 and A2 / A1 ≈ 2.4, as well as a strong variation of cp with both diffuser divergence angle and area ratio. The static pressure recovery coefficient is defined as cp = ( P2 − P1) / ( Pt1 − P1)

(13-38)

The exhaust diffusers for the analysis were specified with a constant radius hub wall and a conical shroud wall. The inlet boundary layer thickness was defined as 2δ / b = 0.4 to match the experimental inlet blockage of B1 = 0.05. The inlet tangential velocity was set to zero, and the other inlet data were selected to produce the Mach number and Reynolds number levels indicate in the reference. Figures 13-8 and 13-9 show comparisons of results that pass close to the optimum performance condition and illustrate the influence of variations in both 2θC and A2/A1. Clearly the performance analysis shows reasonable agreement with the experimental results. Although this is a rather qualitative evaluation, it seems evident that the analysis adapted from centrifugal compressor technology should be applicable to the evaluation of exhaust diffusers for axial-flow compressors. From its use in centrifugal compressors, it is also known to be applicable to more general diffuser configurations, such as the curved diffuser illustrated in Fig. 13-7.

13.3 THE SCROLL OR COLLECTOR For industrial axial-flow compressors, the flow from the diffuser may be collected in a scroll or a collector and discharged into an exit pipe. Figure 13-7 presents the side-view schematic of a curved diffuser discharging into a scroll or a collector.



FIGURE 13-8 Influence of Area Ratio on cp

FIGURE 13-9 Influence of Divergence Angle on cp



Figure 13-10 shows the front view of a scroll, where the area increases in the circumferential direction. By contrast, a collector has a circumferentially uniform passage area. Figure 13-11 is a schematic of an axial exhaust diffuser with a collector similar to the arrangement used on the NASA 10-stage compressor (Budinger and Thomson, 1952). The performance of scrolls and collectors can be evaluated using models developed for centrifugal compressors (Aungier, 2000). All flow conditions are known at the entrance station, 2, from the exhaust diffuser exit conditions. The next key location is designated as station 3, which is the cross-section where the flow has been completely collected, as shown in Fig. 13-10. The flow is then discharged through an exit cone to the exit flange, station 4. The velocity at stations 3 and 4 are computed from conservation of mass. For simplicity, the gas density will be assumed to be constant in the scroll or collector. While that is not a necessary assumption, Mach number levels are almost always sufficiently low to justify its use in estimating the loss coefficient. Hence C3 = Cm2 A2 / A3

(13-39)

FIGURE 13-10 Side View of a Scroll



FIGURE 13-11 An Axial Diffuser and Collector

C4 = Cm2 A2 / A4

(13-40)

The overall total pressure loss coefficient for the scroll or collector is defined as

ω = ( Pt2 − Pt4 ) / ( Pt2 − P2 )

(13-41)

This loss coefficient is computed as the sum of four component loss coefficients. First, it is assumed that the meridional velocity head entering the scroll or collector will be lost. The meridional component of the velocity will develop into a secondary flow pattern in the scroll or collector, as illustrated in Fig. 13-11, and eventually dissipate as a loss. Hence the meridional loss coefficient is given by

ω m = (Cm2 / C2 )2

(13-42)

The tangential component of velocity can be smoothly recovered in a properly sized scroll. The ideal size is determined by requiring conservation of angular momentum between stations 2 and 3, i.e., r3C3 = r2Cθ 2

(13-43)



A sizing parameter, SP, is defined as SP = r2Cθ 2 / ( r3C3 )

(13-44)

SP = 1 is the ideal sizing condition. SP > 1 indicates an oversized scroll that is attempting to overdiffuse the flow relative to the ideal condition. SP < 1 is an undersized scroll that is attempting to accelerate the flow relative to the ideal condition. The tangential loss coefficient is given by

ωθ =

1 r3C32 [SP 2 − 1]; SP ≥ 1 2 r2C22

ωθ =

r3C32 r2C22

[SP − 1]2 ; SP < 1

(13-45)

(13-46)

Equations (13-45) and (13-46) are the same as those used in Aungier (2000), but Eq. (13-44) has been used to remove a singularity at SP = 0 in the original reference. Unlike in the centrifugal compressor, Cθ2 = SP = 0 is a condition likely to be encountered in an axial-flow compressor. In the case of a collector, the equations for the tangential loss coefficient apply only to the full collection station. When a constant passage area collector first starts to collect the flow, the passage area is greatly oversized, such that the collector attempts to diffuse the tangential velocity essentially to zero, which implies the local SP becomes infinite. A corresponding tangential loss coefficient can be computed by combining Eqs. (13-44) and 13-45) and taking the limit as SP approaches infinity. The collector tangential loss coefficient is set to an average of the two extreme values, i.e., (ωθ )coll =

ωθ r2Cθ22 + 2 2r3C22

(13-47)

A wall skin friction loss coefficient is given by

ω sf = 4 cf (C3 / C2 )2 L / dH

(13-48)

L is the average path length of the flow in the scroll and dH is the mean hydraulic diameter of the passage. L = π ( r2 + r3 ) / 2

(13-49)

dH = 2A3 / π

(13-50)

The conventional definition of hydraulic diameter is four times the passage area divided by the wetted perimeter of the passage. That is used in Eq. (13-50),



assuming that the cross-section of the passage is circular and the mean passage area is one-half of A3. For a constant passage area collector, the relations used are L = π r3

(13-51)

dH = 4 A3 / π

(13-52)

The skin friction coefficient is calculated in the same fashion as that for the exhaust diffuser, but with the Reynolds number and surface roughness referenced to dH rather than 2δ. Finally, an exit cone loss coefficient is given by

ω ex = [(C3 − C4 ) / C2 ]2 ; C3 > C4

(13-53)

ω ex = 0 ; C3 ≤ C4

(13-54)

The overall total pressure loss coefficient and exit total pressure for the scroll or collector are given by

ω sc = ω m + ωθ + ω sf + ω ex

(13-55)

Pt4 = Pt2 − ω sc ( Pt2 − P2 )

(13-56)

Conservation of energy and an appropriate equation of state from Chapter 2 yields H4 = H2 = H(Tt2, Pt2), from which Tt4 = T(Pt4, H2). Then a simple mass balance on A4 yields the discharge velocity and static conditions. The scroll or collector static pressure recovery coefficient is given by cpsc = ( P4 − P2 ) / ( Pt2 − P2 )

(13-57)

An overall exhaust system loss coefficient and static pressure recovery coefficient for the diffuser and the scroll or collector combined can also be calculated, i.e.,

ω = ( Pt1 − Pt4 ) / ( Pt1 − P1)

(13-58)

cp = ( P4 − P1) / ( Pt1 − P1)

(13-59)

One useful application of the performance analysis results is to define a discharge loss coefficient for use by the axial-flow compressor performance analysis of Chapter 9. That analysis basically predicts the compressor’s performance up to the diffuser entrance, i.e., the station 1 data in the notation in this chapter. A discharge pressure can be computed as Pdis = Pt1 − ω dis ( Pt1 − P1)

(13-60)



The appropriate definition of the discharge loss coefficient depends on whether a total-to-total or a total-to-static evaluation is appropriate, i.e., whether or not the kinetic energy at the discharge is valuable to the application served by the compressor. In the case of total-to-total evaluation, Eq. (13-58) provides the appropriate definition. In the case of total-to-static evaluation, the appropriate discharge loss coefficient definition is

ω dis = 1 − cp

(13-61)

where cp is given by Eq. (13-59).

13.4 REYNOLDS NUMBER AND SURFACE ROUGHNESS EFFECTS The exhaust diffuser performance analysis provides a good basis to discuss methods to include the effects of surface roughness on axial-flow compressor performance. This is often an important consideration for industrial centrifugal compressors, which accounts for its detailed treatment in the exhaust system analyses adapted from centrifugal compressor technology. Surface finish effects are much less likely to be important in typical axial-flow compressor applications. Hence, Reynolds number correction models from the literature seldom considered it [e.g., the models in Chapter 6 and in Wassell (1968)]. But it can become significant for applications involving very high Reynolds numbers or very poor surface finishes. Fortunately, there is a fairly simple method to externally impose a correction for surface roughness on any Reynolds number correction model. This can be easily accomplished by limiting the Reynolds number used in the correction to values for which the surfaces involved are hydraulically smooth. A reasonable approximation for this purpose is to limit the Reynolds number used in the correction by Red ≤ 60d / e

(13-62)

where d is the characteristic length dimension used to define the Reynolds number. Figure 13-12 applies this simple method to correct the skin friction coefficient from Eq. (13-32) and compares it to results from Eqs. (13-30) through (13-37). Clearly, this simple correction is in rather good agreement with the more precise empirical model derived from the pipe friction factor charts of Nikuradse (1930).

13.5 THE AXIAL-CENTRIFUGAL COMPRESSOR The axial-centrifugal compressor consists of a series of axial-flow compressor stages followed by a centrifugal compressor stage. Under appropriate circumstances, this can offer definite advantages. Centrifugal compressor design and analysis is covered in depth in Aungier (2000). The discussion here is limited to



FIGURE 13-12 Simple Surface Roughness Correction

the concept of combining the two compressor types. The relative merits and characteristics of the axial-flow compressor and the centrifugal compressor were discussed briefly in Chapter 1. To review, the following are key differences between these two types of compressors: • The axial-flow compressor can achieve significantly higher mass flow rates per unit frontal area. • The centrifugal compressor can achieve significantly larger pressure ratios per stage. • The centrifugal compressor is generally the more rugged and lower-cost compressor type. • The axial-flow compressor generally offers higher efficiency, although the centrifugal compressor has become rather competitive in recent years. • The centrifugal compressor offers stable operation over a much wider mass flow range. • The axial-flow compressor offers a much steeper pressure-mass flow characteristic. • The centrifugal compressor is much better suited to matching with a scroll or collector at the discharge. These differences need definite qualification when considering a combination of the two types in a single compressor. Axial-flow and centrifugal compressor



stages achieve their optimum performance at quite different dimensionless performance parameters. This difference is typically expressed in terms of specific speed or the characteristic stage inlet flow coefficient, φ0. The latter should not be confused with the definition of flow coefficient, φ, used earlier in this book. Here, the characteristic stage inlet flow coefficient is defined as 2 ˙ / (π rtip φ0 = m Utipρt0 )

(13-63)

Figure 13-13 shows stage efficiency levels that should be achievable with optimized centrifugal compressor stages as a function of φ0. It is adapted from generalized performance charts in Aungier (2000) to emphasize simple radial discharge centrifugal stages. Operation at higher values of φ0 is certainly possible, but is likely to require a mixed-flow design to achieve adequate efficiency levels. A mixed-flow design is a compromise between axial-flow and radial-flow styles such that the flow exits the impeller with velocity components in both the axial and radial directions. At sufficiently high values of φ0, an axial-flow stage will be required. A meaningful extension of the chart in Fig. 13-13 to cover mixed flow and axial-flow stages is not available. Balje (1981) presents a specific speed chart intended to include the three compressor stage types. But his use of total-to-static efficiency severely limits the value of his chart as a guide for multi-stage compressor design. In Chapter 10, it was noted that many investigators now favor relatively low stage flow coefficients, φ, around 0.5. The flow

FIGURE 13-13 Centrifugal Stage Performance



coefficients φ0 and φ can be approximately related using the stage hub-to-shroud radius ratio, ε.

φ0 ≈ 12 (1 − ε 2 )(1 + ε )φ

(13-64)

ε = rh / rs

(13-65)

Typical values of ε for front stages in a compressor are in the 0.4 to 0.6 range. From Eq. (13-64) this suggests that φ0 in the range of about 0.25 to 0.3 are about the lowest values to be expected for optimized axial-flow stages. The stage efficiency is expected to be essentially unaffected by variation of ε from 0.4 to 0.6, yet it results in substantial variation in φ0. Indeed, φ0 is rather ambiguous as a guide to achievable efficiency levels for axial-flow stages. In the rear stages of an axialflow compressor, ε may become large enough to produce values of φ0 in the range normal for centrifugal stages. But in that case, the performance of an axial-flow stage is likely to be far from optimum due to aspect ratio effects associated with the very short blades. It is clear that axial-flow stages generally operate at values of φ0 that are considerably higher than centrifugal stages. In addition, a centrifugal stage substituted for an axial-flow stage will normally operate at a lower value of φ0 than the axial-flow stage it replaced. This follows from the fact that rtip and Utip will typically be greater for the centrifugal stage. Consequently, it may be beneficial to replace some of the rear axial-flow stages in an axial-flow compressor with a centrifugal stage. The higher stage pressure ratio offered by the centrifugal stage may permit replacement of several axial-flow stages to substantially reduce cost. If aspect ratio effects are significant in the replaced axial-flow stages, performance may also be improved. It is rather easy to evaluate the potential benefits from this approach. Simply compute φ0 in the rear stages and estimate rtip of a replacement centrifugal stage to compute φ0 for the centrifugal stage. If the upstream axial-flow stages have reduced φ0 to a value suitable for a centrifugal stage, the replacement may be appropriate. The expected centrifugal stage efficiency can be estimated from Fig. 13-13, or by using the more refined charts or the performance analysis in Aungier (2000). Some industrial axial-flow compressor designers believe that a final centrifugal stage is beneficial, even without obtaining a cost or performance benefit directly from the substitution. This may involve using a rather modest pressureratio centrifugal stage primarily to reduce the exhaust system losses. It is clear that a scroll or collector following typical axial-flow stages cannot be wellmatched, since the sizing parameter of Eq. (13-44) will always be close to zero. Hence the scroll or collector will always be grossly undersized. There is little that can be done in terms of scroll or collector design except to minimize the harm done to the performance by the tangential loss and exit cone loss. Hence the designer’s ability to minimize the exhaust system loss is primarily limited to the exhaust diffuser design. It can be argued that a centrifugal stage can efficiently turn the flow to the radial direction to permit the design of a radial diffuser with a well-matched scroll or collector. No doubt there is some merit to that approach, although it may be difficult to quantify. If the axial length available is limited, e.g., to favor rotor dynamics, the benefit may be more obvious. For example, if



space limitations require reducing the length of the axial diffuser of Fig. 13-11 or replacing it with the curved diffuser of Fig. 13-7, the alternative of a final centrifugal stage may be easier to justify.

EXERCISES 13.1 Flow exits an exhaust diffuser with Cθ = 0. What will be the minimum possible value of the loss coefficient for a scroll or collector following the diffuser? What options do you have to minimize this loss coefficient? Of the diffuser types shown in Figs. 13-7 and 13-11, which one yields the lowest loss in the scroll/collector? 13.2 A curved diffuser similar to that of Fig. 13-7 is replaced by a very low pressure-ratio centrifugal stage with a vaneless diffuser. The diffuser has an exit passage area and radius identical to the original diffuser and the scroll sizing parameter is 1. This substitution reduces the scroll total pressure loss coefficient from 1.1 to 0.4. What can you conclude about the overall exhaust system losses for the two configurations? Which contributions to the overall loss are likely to be reduced by this substitution? What steps are needed to justify this substitution? 13.3 Assuming the diffusers of Figs. 13-7 and 13-11 have the same inlet and discharge areas and Cθ = 0, which diffuser is likely to have the lower loss? Under what circumstances would you choose the higher loss configuration? Would your answers be different if Cθ > 0? 13.4 A general Reynolds number correction model is applied to correct low Reynolds number test performance to high Reynolds number operating conditions. Surface roughness is found to be significant, so the approximate correction of Section 13.4 is imposed. How confident can you be in the results? What areas of uncertainty need to be considered? 13.5 Polishing blade and end-wall surfaces to improve surface finish can add significant cost to the manufacturing process. How can you use the skin friction coefficient model of Eqs. (13-30) through (13-37) to determine whether the added cost is justified? How would you adapt this procedure when the characteristic length is not a passage width or diameter (e.g., blade chord Reynolds number)?


ANSWERS TO THE EXERCISES

1.1 Flow parameter: ˙ / ( ρt0 kRTt0 ) ∝ mRT ˙ t0 / ( Pt0 kRTt0 ) ∝ m ˙ RTt0 / k / Pt0 Q / at0 ∝ m Speed parameter:

N / at0 = N / kRTt0

1.2 If β1 = 0, then Cθ1 = 0 and

ψ = Cθ 2 / U R = 1 − Cθ 2 / (2U ) = 1 − ψ / 2 So if R = 0.5, then ψ = 1 is the only acceptable value. From Eq. (124), the rotor discharge relative flow angle must be zero. Since the absolute flow angles into and out of the stage are zero, no inlet or exit guide vanes will be required. 1.3 DB = 1.2DA and AB = 1.44AA. Therefore, NB = NA / 1.2 QB = 1.44QA For equivalence with the original compressor at 3,600 rpm, the scaled compressor must operate at 3,000 rpm and will supply 44% more flow capacity than the original. 1.4 For a 20% increase in flow capacity, a 20% increase in A0 is needed, which requires a scale factor of √1.2 . Hence the speed must be reduced by a factor of √1.2 . 2.1 Equations (2-19), (2-27), (2-28) and (2-53) combine to yield a12 = kRT1 = cp ( k − 1)T1



2cpT1 = 2a12 / ( k − 1) Tt1 / T1 = 1 + ( k − 1)Cz21 / (2a12 ) = 1 +

k −1 M2 1 2

Hence, from Eqs. (2-28) and (2-55), ( a1 / at1)2 = 1 / (1 + k − 1 M12 ) 2 ( ρ1 / ρt1)k −1 = 1 / (1 +

k −1 M2 ) 1 2

2.2 From Eq. (1-17), H2 = H1 + UCθ 2 For a thermally and calorically perfect gas, Tt is a function of only H. Hence Eq. (2-52) yields Tt2id = Tt2 = Tt1 + ( H2 − H1) / cp In the absence of losses, the flow is isentropic and Eq. (2-55) yields k

Pt2id = Pt1(Tt2id / Tt1) k −1 Equation (2-68), using the inlet velocity pressure as relevant base kinetic energy term, requires Pt2 = Pt2id − ω ( Pt1 − P1) 2.3 For complete Mach number equivalence, the discharge Mach number and flow angle must be equivalent, i.e., Cθ2/a2 and Cz2/a2 must be equivalent. As shown in Exercise 2.1, when Mach number equivalence exists, this requirement can be restated to require equivalence on Cθ2/at2 and Cz2/at2. Mach number equivalence at the inlet requires equivalence on U/at1. From Exercise 2.2, ∆H = H2 − H1 = UCθ 2 ∆H / at21 = (U / at1)(Cθ 2 / at2 )at2 / at1 ( at2 / at1)2 = Tt2 / Tt1 = 1 + ( kR / cp )( ∆H / a12 ) 2 is required to produce equivalence on Hence equivalence on ∆H/at1 Cθ2/at2, Tt2/Tt1 and at2/at1. Neglecting losses, Eq. (2-55) requires


Answers To the Exercises • 335 k

Pt2 / Pt1 = (Tt2 / Tt1) k −1 1

ρt2 / ρt1 = (Tt2 / Tt1) k −1 so equivalence on Pt2/Pt1 and ρt2/ρt1 is also achieved. Hence ˙ / ( ρt2at2 A2 ) = [m ˙ / ( ρt1at1A1)][ρt1at1A1 / ( ρt2at2 A2 )] m N / at2 = N / at1[at1 / at2 ] Since the last terms on the right-hand side of the previous two equations satisfy equivalence, Mach number equivalence at the inlet of the blade row will produce Mach number equivalence at the discharge. 2.4 Denote the base case with the original working fluid by subscript B, and the new case by subscript N. Let Tt2B and ρt2B be the rotor exit conditions for the base case. For mass flow and speed equivalence at the rotor inlet, ˙ / ( ρt kRT ) = constant m N / kRT = constant Since the inlet conditions and R are identical for the two fluids, the mass flow and speed for the new case must be ˙N =m ˙ B 1.38 / 1.4 m NN = NB 1.38 / 1.4 For speed equivalence at the rotor exit, NN = NB (1.38Tt2N ) / (1.4Tt2 B ) Hence Tt2N = Tt2B is required to satisfy the equivalent speed condition at both the rotor inlet and exit. Mass flow equivalence at the rotor exit with Tt2N = Tt2B requires ˙N =m ˙ B 1.38 / 1.4 ( ρt2N / ρt2 B ) m Thus, to achieve mass flow equivalence at both locations, ρt2N = ρt2B is required. But Eq. (2-55) with Tt2N = Tt2B requires ( ρt2 B / ρt1)0.4 = Tt2 B / Tt1 = ( ρt2N / ρt1)0.38 ρt2N / ρt1 = ( ρt2 B / ρt1)0.4 / 0.38



Hence the two conditions for Mach number equivalence cannot be satisfied at both the rotor inlet and exit. Since Mach number equivalence at both stations cannot be achieved for the same speed and mass flow, complete Mach number equivalence through the compressor is not achieved in this case. 2.5 Using Eq. (2-42), the temperature at which liquid will begin to form can be estimated from  P 3 Tc log10   = 1− 7(1 + ω ) T  Pc  For P = 200 kPa, T > 247.6° K is required to avoid liquid phase formation. 2.6 Use Eq. (2-56) to compute the temperature ratio, TR, from pressure ratio, PR, and ηad. Use Eq. (2-57) to compute the polytropic efficiency. Hence PR = 3.0: TR = 1.4338, ηp = 87.11% PR = 5.0: TR = 1.6868, ηp = 87.95% 2.7 Use Eq. (2-56) to compute the stage temperature ratio, TR, from the stage pressure ratio and stage ηad. This yields a stage temperature ratio of 1.032477. For the three-stage compressor, the pressure ratio is (1.1)3 = 1.331 and the temperature ratio is (1.032477)3 = 1.10063. Then Eq. (2-56) yields the overall adiabatic efficiency of 84.59%. If the efficiencies are all polytropic, the stage temperature ratio, TR, can be computed using Eq. (2-57). Then Eq. (2-57) can be used to compute the overall compressor efficiency as

ηp =

k −1 ln(1.13 ) / ln(T 3 ) R k

=

k −1 ln(1.1) / ln(T ) R k

So the overall compressor efficiency is identical to the individual stage efficiency. 2.8 From Eqs. (2-63) and (2-68), the discharge static and total pressures are Pd = 200 + 0.6(30) = 218 Ptd = 230 − 0.1(30) = 227 For a thermally perfect gas with no work or heat transfer, Ttd = Tti = 300. Equation (2-55) yields the inlet and discharge static temperatures. Ti = 300(200 / 230)(0.4 /1.4) = 288.26


Answers To the Exercises • 337

Td = 300(218 / 227)(0.4 /1.4) = 296.55 For adiabatic reversible (isentropic) flow, the discharge temperature would be Td,rev = 300(218 / 230)(0.4 /1.4) = 295.44 Hence the diffuser efficiency is given by Eqs. (2-52) and (2-62)

ηdiff =

cp (295.44 − 288.26) cp (296.55 − 288.26)

= 86.6%

3.1 From Eq. (3-29) for axisymmetric, time-steady flow, ∂( rWθ + ω r 2 ) ∂rCθ = =0 ∂m ∂m Similarly, Eq. (3-25) requires ∂I =0 ∂m Inserting these results into Eq. (3-28) yields ∂s =0 ∂m 3.2 Direct substitution of Wθ = Wm tanβ′ into Eq. (3-30) yields ∂Wm Wm2 tan β ′ ∂r tan β ′ + r ∂n cos2 β ′ ∂n ∂I ∂s 2 +κ mWm + 2ω Wm tan β ′ cos φ = −T ∂n ∂n Wm

For the stationary coordinate system, ω = 0, W → C, β′ → β and I → H. Hence 2 ∂Cm Cm ∂H ∂s tan β ∂r tan β 2 + + κ mCm = −T r ∂n ∂n ∂n cos β ∂n

Cm 2

3.3 From the definition of the dot product and Eq. (3-61), r r V = V ⋅ V = Vm2 + Vn2 + Vθ2



r

1 ∇V 2 2

=V

∂V r ∂V r V ∂V r em + V en + eθ r ∂θ ∂m ∂n

3.4 For the stated conditions, Eqs. (3-21) and (3-27) combine to yield ∂brρ Wm =0 ∂m Equation (3-22) simplifies to

Wm

∂Wm sin φ 1 ∂P − [Wθ + ω r]2 = − r ∂m ρ ∂m

Equation (3-23) and the definition of the angle φ yield ∂( rWθ + ω r 2 ) =0 ∂m and Eq. (3-25) simplifies to ∂I =0 ∂m 3.5 From the uniform flow assumption, the total mass flow is b

δ

0

0

∫ rρ udy = r ρeue (b − 2δ ) + ∫ rρ udy +

b

∫

rρ udy

b −δ

where –r is the average radius. Using the standard boundary layer approximation that r is constant for the last two terms and Eq. (3-35), yields b

˙ / (2π ) = ∫ rρ udy = r ρ eue ( b − 2δ * ) m 0

Similarly, the momentum flux is b

δ

0

0

2 2 2 ∫ rρ u dy = r ρeue (b − 2δ ) + ∫ rρ u dy +

b

∫

rρ u2dy

b −δ

and Eq. (3-38) yields


Answers To the Exercises • 339 b

∫ rρ u dy = r ρeue (b − 2δ 2

2

*

− 2θ )

0

3.6 Conservation of mass for incompressible flow before and after mixing requires rbρ eumix = r ρ eue ( b − 2δ * ) umix = ue (1 − 2δ * / b) Conservation of momentum with constant static pressure requires 2 rb( Pe + ρ eumix ) = rbPe + r ρ eue2b[1 − 2(δ * + θ ) / b]

Introducing the incompressible relation for total pressure, 2 Pt,mix + 12 ρ eumix = Pt,e + ρ eue2 [ 12 − 2(δ * + θ ) / b]

Introducing umix from the mass balance equation Pt,mix + 12 ρ eue2 (1 − 2δ * / b)2 = Pt,e + ρ eue2 [ 12 − 2(δ * + θ ) / b] Pt,e − Pt,mix = ρ eue2 [ 12 (1 − 2δ * / b)2 − Pt,e − Pt,mix =

1 2

1 2

+ 2(δ * + θ ) / b]

ρ eue2 [(2δ * / b)2 + 4θ / b]

5.1 From Eqs. (5-1), (5-2), (5-3) and (5-9) ∂ ∂ξ ∂ ∂η ∂ 1 ∂ tan β ∂S ∂ = + = − ∂m ∂m ∂ξ ∂m ∂η cos β ∂ξ S ∂m ∂η ∂ ∂ξ ∂ ∂η ∂ r ∂ = + = ∂θ ∂θ ∂ξ ∂θ ∂η S ∂η Substitution of these derivatives into the steady form of Eq. (3-21) using Eq. (3-27) yields 1 ∂brρ Wm tan β ∂brρ Wm 1 ∂brρ Wθ =0 − + cos β ∂ξ S ∂η S ∂η Converting to finite-difference form, using the difference approximation form given in Eqs. (5-35) and (5-36) and multiplying through by 4∆ξ∆η, yields



2∆η [( brρ Wm )ξ + ∆ξ ,η − ( brρ Wm )ξ − ∆ξ ,η ] cos β 2∆ξ rb tan β [( ρ Wm )ξ ,η + ∆η − ( ρ Wm )ξ ,η − ∆η ] − S 2∆ξ rb [( ρ Wθ )ξ ,η + ∆η − ( ρ Wθ )ξ ,η − ∆η ] = 0 + S where the overbar designates values at point (ξ, η). Noting that 2∆m – = 2∆ξcosβ , 2∆η S [( brρ Wm )ξ + ∆ξ ,η − ( brρ Wm )ξ − ∆ξ ,η ] r −2b ∆m tan β [( ρ Wm )ξ ,η + ∆η − ( ρ Wm )ξ ,η − ∆η ] +2b ∆m[( ρ Wθ )ξ ,η + ∆η − ( ρ Wθ )ξ ,η − ∆η ] = 0 Checking the result for the control volume in Fig. 5-4, it can be seen that the second term in this difference equation does not precisely – balance mass. The term tan β should be evaluated at each boundary, instead of using a mean value. Thus there will be an inherent error in the numerical approximation to the continuity equation. 5.2 The required Taylor series are

ψ (m + ∆m) = ψ (m) + ψ ′(m)∆m + 12 ψ ′′(m)( ∆m)2 + 16 ψ ′′′(m)( ∆m)3 + ⋅ ⋅ ⋅ ψ (m − ∆m) = ψ (m) − ψ ′(m)∆m + 12 ψ ′′(m)( ∆m)2 − 16 ψ ′′′(m)( ∆m)3 + ⋅ ⋅ ⋅ Subtract the second equation from the first and divide by 2∆m to obtain

ψ ′( m ) =

ψ (m + ∆m) − ψ (m − ∆m) + O[( ∆m)2 ] 2∆m

Hence the difference approximation is of second-order accuracy in ∆m. Add the two Taylor series and divide by (∆m)2 to obtain the second derivative difference approximation, which is also of secondorder accuracy in ∆m. 5.3 The required Taylor series are

ψ (m + 2∆m) = ψ (m) + 2ψ ′(m)∆m + 2ψ ′′(m)( ∆m)2 + 43 ψ ′′′(m)( ∆m)3 + ⋅ ⋅ ⋅ ψ (m + ∆m) = ψ (m) + ψ ′(m)∆m + 12 ψ ′′(m)( ∆m)2 + 16 ψ ′′′(m)( ∆m)3 + ⋅ ⋅ ⋅



Multiply the second equation by 4, subtract the first equation from it and divide by 2∆m to obtain

ψ ′( m ) =

4ψ (m + ∆m) − 3ψ (m) − ψ (m + 2∆m) + O[( ∆m)2 ] 2∆m

which is also of second order accuracy in ∆m. 5.4 The required Taylor series is u(m + ∆m) = u(m) + u′(m)∆m + 12 u′′(m)( ∆m)2 + 16 u′′′(m)( ∆m)3 + ⋅ ⋅ ⋅ Solve for the first derivative to obtain u′(m) =

u(m + ∆m) − u(m) + O( ∆m) ∆m

Hence this difference approximation is of first-order accuracy in ∆m. 5.5 Under the conditions stated, the flow upstream of the blade at near steady-state conditions will be approximately axisymmetric and time steady. From Eq. (5-72), the quantity SbρWm will be approximately constant, so its second derivative in Eq. (5-102) will be approximately zero. Similarly, the quantities I and rCθ = rWθ +ωr2 will be approximately constant, as seen from Eqs. (3-25) and (3-29), so their second derivatives in Eqs. (5-103) and (5-104) will be approximately zero. Had the quantity Sbρ been moved outside of the second derivative in Eq. (5-102), the stabilizing term would no longer be approximately zero unless Sb is constant. In contrast, if Sbρ were moved inside of the second derivatives in Eqs. (5-103) and (5-104), those stabilizing would cease to be near zero unless Sb is constant. The derivative terms in the stabilizing terms are chosen as the quantities most likely to be nearly constant at near steady-state conditions. Although these terms should normally be small, any numerical stability problem will cause them to become large, basically introducing as much numerical damping as needed for a stable solution. 5.6 (a) In general, estimate θ from the upstream value using Eq. (5-106) with δ* and τw set to zero for the purpose of this initial guess. If the previous station is the leading edge, where θ = 0, that procedure won’t work. In that case, a reasonably safe initial guess could be obtained by setting θ from an assumed value of Reθ that is well below transition, e.g., 50. (b) The steps in the iteration to solution at each station might be as follows:



• • • • • • • •

Compute b0 from Eq. (5-121) Compute K from Eq. (5-122) Compute Λ from Eq. (5-124) Compute δ′ from Eq. (5-115) Compute δ* from Eqs. (5-116) through (5-120) Compute τw from Eqs. (5-109) and (5-123) Recompute θ from Eq. (5-106) Repeat above steps until converged

5.7 (a) Estimate θ from the upstream value using Eq. (5-106) with δ* and τw set to zero for the purpose of this initial guess. Estimate (δ – δ*) from Eq. (5-127) with E set to zero for the purpose of this initial guess. (b) The steps in the iteration to solution at each station might be as follows: • • • • • • •

Compute H1 from Eq. (5-128) Compute Hk from Eq. (5-132) Compute E from Eq. (5-133) Compute H and δ* from Eqs. (5-131) and (5-129) Compute cf from Eqs. (5-134) and (5-135) Recompute θ and (δ – δ*) from Eqs. (5-106) and (5-127) Repeat above steps until converged

6.1 For Eq. (6-8), i* = α* + γ – κ1 = α* – θ / 2 = f(σ, θ). For Eq. (6-12), i* = f(β1*,σ,θ). But since β1* = κ1 + i*, the true independent variables can be expressed as i* = f(κ1,σ,θ) 6.2 For Eq. (6-8), β1* = α* + γ = f(σ,θ,γ). For Eq. (6-12), β1* = i* + κ1 = f(σ,θ,κ1) = f(σ,θ,γ). 6.3 Both Fig. 6-2 and 6-14 are from the same reference, and use the same definition of α*. Hence if the positive and negative stall incidence angles are to be properly computed, Eq. (6-8) must be solved for α*. 7.1 Convert the data at base (constant radius) blade angle data to κ1 and κ2. Interpolate for c and tb / c at the mean radius, κ1 at the inlet radius and κ2 at the discharge radius. Estimate the stream surface angle from tan φ = ( r2 − r1) / ( z2 − z1) The effective geometry data in the stream surface are c → c / cos φ s = π ( r2 + r1) / Z tb / c → cos φ tb / c tan κ1 → cos φ tan κ1 tan κ 2 → cos φ tan κ 2



7.2 Calculation of the meridional gradient of Wm with simple finite differences will usually be based on data at points on opposite sides of blade rows. This may cause the computed gradient to be meaningless, since Wm may be strongly influenced by discontinuous changes in the swirl velocity, etc., imposed by the blades. 7.3 A choke condition in an axial-flow compressor will almost always be caused by choke in a blade row, with an associated abrupt increase in loss. The approach to the Mach number limit in Eq. (7-29) is normally just an indication that blade row choke has occurred. In the unlikely event of a true annulus choke, this should still be reasonably accurate, since hub-to-shroud gradients in Wm are seldom extreme in an axial-flow compressor. If the through-flow analysis were to be applied within the blade passage, the more rigorous choke criterion might be necessary. 7.4 The approximation should be quite accurate on interior stream surfaces, assuming a reasonable number of stream surfaces are used. However, it is somewhat of an extrapolation for the end-wall surfaces. Even there, it will offer acceptable accuracy unless the meridional gradient of Wm becomes excessive near the end-walls. By the definition, a positive mass flow rate passes between adjacent stream surfaces, so the singularity will occur only on end-walls, unless it is caused by numerical errors in the early iterations of the solution process. If an end-wall boundary layer solution is included, a singularity in the inviscid through-flow analysis should be suppressed by the abrupt increase in end-wall blockage prediction when the endwall velocity becomes small. 8.1 From Eq. (8-11), the mass flow rate in a boundary layer is given by δ

∫ ρVmdy =ρeVme (δ − δ1 ) *

0

Hence a mass balance combining the two incoming flows yields ˙ leak 2πr( ρ eVme )in (δ − δ1* )+ = 2π ( rρ eVme )in (δ − δ1* )in + m which results in Eq. (8-58). 8.2 From Eqs. (8-11) and (8-12), the meridional momentum flux in a boundary layer is given by δ

∫ ρVmdy =ρeVme (δ − δ1 − θ11) 2

2

*

0

Hence the leakage flow contributes no meridional momentum, so


344 • AXIAL-FLOW COMPRESSORS 2 2 2πr( ρ eVme )in (δ − δ1* − θ11)+ = 2π ( rρ eVme )in (δ − δ1* − θ11)in

Combining with the result in the previous exercise yields Eq. (8-60). 8.3 From Eqs. (8-12) and (8-13), the tangential momentum flux in a boundary layer is given by δ

∫ ρVmVθ dy =ρeVmeVθe (δ − δ1 − θ12 ) *

0

Hence to balance tangential momentum of the two incoming flows, 2πr( ρ eVmeVθe )in (δ − δ1* − θ12 )+ ˙ )leak = 2π ( rρ eVmeVθe )in (δ − δ1* − θ12 )in + (mU Combining with the result of Exercise 8.1 yields Eq. (8-62) for the case of leakage flow entering the boundary layer. 8.4 The process is the same as for the previous three exercises, except for the tangential momentum and direction of the leakage flow. 8.5 Substituting the power-law profiles into the designated equations yields n δ  y  δ δ1* = ∫ 1 −    dy = δ − δ   n +1  0  2n n δ  y  y  n θ11 = ∫   −    dy = δ δ  ( n + 1)(2n + 1) δ    0 

which combine to yield Eq. (8-20). n n+ m  δ  y  y m  dy = δ θ12 = ∫   −   δ  ( n + 1)( n + m + 1) δ    0 

which yields Eq. (8-21). 9.1 Trapezoidal-rule integration of the uncorrected data between stream surfaces 1 and 3 yields ˙ I = 12 [( ∆pt′ )1 + 2( ∆pt′ )2 + ( ∆pt′ )3 ]∆m



Integration of the corrected data yields ˙ Ic = 12 [( ∆pt′ )1,c + 2( ∆pt′ )2,c + ( ∆pt′ )3,c ]∆m Introducing Eq. (9-10), ˙ = 2( ∆pt′ )2,c ∆m ˙ Ic = 12 [2( ∆pt′ )2,c − ( ∆pt′ )3,c + 2( ∆pt′ )2,c + ( ∆pt′ )3,c ]∆m From Eq. (9-9), ˙ =I Ic = 12 [( ∆pt′ )1 + 2( ∆pt′ )2 + ( ∆pt′ )3 ]∆m 9.2 For a circular-arc camberline, Eq. (4-7) gives the arc radius of curvature, RC, as Rc = c/[2 sin(θ/2)] Hence the camberline length, L, is given by L = RCθ = cθ / [2 sin(θ / 2)] Dividing by the staggered spacing, s cosγ, yields Eq. (9-15). 10.1 For a constant-work, repeating stage, C3 = C1 and U2ψ = constant. Hence, Eqs. (10-6) and (10-7) yield h2 − h1 = Uc2ψ c − 12 (C22 − C12 ) h3 − h1 = Uc2ψ c Hence Eqs. (10-3) and (10-5) yield

R = 1−

Cz22 + Cθ22 − Cz21 − Cθ21 2Uc2ψ c

Differentiating with respect to r, ∂R 1 = ∂r Uc2ψ c

 ∂Cz1 ∂C ∂C ∂C  + Cθ1 θ1 − Cz2 z2 − Cθ 2 θ 2  Cz1 ∂r ∂r ∂r ∂r  

From Eq. (10-24),



Cz

∂Cz ∂C C2 + Cθ θ = − θ ∂r ∂r r

Hence

rc

∂R 1 rc  Cθ22 Cθ21  = −   ∂r ψ c r  Uc2 Uc2 

Substituting Eqs. (10-20) and (10-21) to eliminate the velocity terms yields Eq. (10-45). 10.2 For a constant-work stage, Eq. (10-24) can be written as 2 r 2C ∂[rCθ 2 / ( rcUc )] rc ∂(Cz2 / Uc ) = − c θ2 rUc ∂r ∂r 2

Equations (10-20) and (10-21), with n = -1 and m = 1, yield Cθ 2 / Uc = (1 − Rc )( r / rc ) + 12 ψ c ( rc / r ) rc rc

∂[rCθ 2 / ( rcUc )] = 2(1 − Rc )( r / rc ) ∂r

∂(Cz2 / Uc )2 = 4(1 − Rc )[(1 − Rc )( r / rc ) + 12 ψ c ( rc / r )] ∂r

Integration of the last equation yields Eq. (10-48). 10.3 In the answer to Exercise 10.1, it is shown that

rc

∂R 1 rc  Cθ22 Cθ21  = −   ∂r ψ c r  Uc2 Uc2 

Hence constant reaction requires 1 rc  Cθ22 Cθ21  − =0  ψ c r  Uc2 Uc2  Hence Cθ22 − Cθ21 Uc2

=

(Cθ 2 + Cθ1)(Cθ 2 − Cθ1) Uc2

=0



For a constant-work stage, H2 = H1, so Eq. (10-2) can be used to yield (Cθ 2 + Cθ1)ψ c =0 Uc Since reaction is undefined for the trivial case of ψc = 0, the required condition is Cθ 2 = − Cθ1 This condition also must be satisfied at the reference radius. Equations (10-9) and (10-10) are valid at the reference radius, so 2(1 − Rc) = 0 The only case where this is true is that of R = 1. 13.1 From Eq. (13-42) it is clear that the loss coefficient cannot be less than one. About the only design option available is to minimize the tangential loss, and possibly the exit cone loss, through the choice of r3 and A3. If the diffusers achieve the same discharge flow conditions, the diffuser type used has no effect on the scroll/collector loss. 13.2 From the information given, no firm conclusions can be reached about overall exhaust loss. The reduced scroll loss coefficient is due at least in part to the increase in kinetic energy supplied by the centrifugal impeller. The absolute exhaust system loss could actually be higher for the centrifugal stage configuration. The substitution is almost certain to reduce the curvature loss of the original diffuser and the tangential loss in the scroll. To justify the substitution, a performance analysis of the centrifugal stage and its exhaust system must be compared to the exhaust system analysis without the substitution, including any benefits from the additional pressure rise supplied by the centrifugal stage. 13.3 The axial diffuser will have the lower loss since there will be no curvature loss contribution. The main reason a curved diffuser might be chosen is to reduce the overall axial length of the compressor. If the flow exiting the compressor has significant Cθ, the curved diffuser could be more effective. The higher discharge radius will yield greater diffusion of Cθ through conservation of angular momentum. 13.4 The results will basically be as dependable as the correction model itself. While not all losses in a compressor are skin-friction-related, that simply means not all losses are Reynolds-number-related. If the correction model used accounts for that, the imposed roughness cor-



rection should be valid. If anything, it might be slightly conservative if the Reynolds number lies in a zone of transition from smooth to rough skin friction. The greatest uncertainty lies in the ability to assign a surface roughness that is consistent with the characteristic Reynolds number used in the correction model. 13.5 Calculate the value of surface roughness, e, that results in Ree = 60. It is unnecessary to polish the surfaces to achieve a surface roughness less than that value. The constant, 2,000, in the definition of Ree is rather insignificant and can be omitted in Eq. (13-34) for more general applications. Hence, the Reynolds number based on e is the significant parameter.


REFERENCES

Aungier, R. H., 1968, “A Time-Dependent Numerical Method for Calculating the Flow About Blunt Bodies,” Technical Report AFWL-TR-68-52, Air Force Weapons Laboratory, Kirtland AFB, NM. Aungier, R. H., 1970, “A Computational Method for Exact, Direct and Unified Solutions for Axisymmetric Flow Over Blunt Bodies of Arbitrary Shape (Program BLUNT),” Technical Report AFWL-TR-70-16, Air Force Weapons Laboratory, Kirtland AFB, NM. Aungier, R. H., 1971(a), “A Computational Method for Unified Solutions to the Inviscid Flow Field About Blunt Bodies,” The Entry Plasma Sheath and Its Effects on Space Vehicle Electromagnetic Systems, (Proceedings of the Fourth Plasma Sheath Symposium), NASA SP-252, NASA, Washington, DC, pp. 241–260. Aungier, R. H., 1971(b), “A Computational Method for Two-Dimensional, Axisymmetric and Three-Dimensional Blunt Body Flows (Program ATTACK),” Technical Report AFWL-TR-70-124, Air Force Weapons Laboratory, Kirtland AFB, NM. Aungier, R. H., 1988(a), “A Systematic Procedure for the Aerodynamic Design of Vaned Diffusers,” Flows In Non-Rotating Turbomachinery Components, FEDVol. 69, ASME, New York, NY, pp. 27–34. Aungier, R. H., 1988(b), “A Performance Analysis for the Vaneless Components of Centrifugal Compressors,” Flows in Non-Rotating Turbomachinery Components, FED-Vol. 69, ASME, New York, NY, pp. 35–43. Aungier, R. H., 1993, “Aerodynamic Design and Analysis of Vaneless Diffusers and Return Channels,” Paper No. 93-GT-101, ASME, New York, NY. Aungier, R. H., 1994, “A Fast, Accurate Real Gas Equation of State for Fluid Dynamic Analysis Applications,” Contributed Papers In Fluids Engineering 1994, FED-Vol. 182, ASME, New York, NY, pp 1–6. Aungier, R. H., 1995, “A Fast, Accurate Real Gas Equation of State for Fluid Dynamic Analysis Applications,” Trans., Journal of Fluids Engineering, ASME, June, pp. 277–281. Aungier, R. H., 1998, “Thermodynamic State Relations,” The Handbook of Fluid Dynamics (R. W. Johnson, editor), CRC Press LLC, Boca Raton, FL, pp. 4-29–4-34. Aungier, R. H., 2000, Centrifugal Compressors: A Strategy for Aerodynamic Design and Analysis, ASME Press, New York, NY. Balje, O. E., 1981, Turbomachines—A Guide to Design, Selection and Theory, Wiley & Sons, New York, NY.



Balsa, T. F. and Mellor, G. L., 1975, “The Simulation of Axial Compressor Performance Using an Annulus Wall Boundary Layer Theory,” Trans. Journal of Engineering for Power, ASME, July, pp. 305–317. Barnes, F. J., 1973, Ph.D. thesis, Department of Chemical Engineering, University of California, Berkeley, CA. Budinger, R. E. and Thomson, A. R., 1952, “Investigation of A 10-Stage Subsonic Axial-Flow Compressor; II—Preliminary Analysis of Over-All Performance,” NACA Research Memorandum RM E52C04, NACA, Washington, DC. Courant, R., Friedricks, K. O. and Lewy, H., 1928, “Uber die Partiellen Differenzengleichungen der Mathematischen Physik,” Math. Ann., Vol. 100, p. 32. Cumpsty, N. A., 1989, Compressor Aerodynamics, Longman Scientific and Technical, Essex, United Kingdom. Davis, W. R., 1976, “Three-Dimensional Boundary-layer Computation on the Stationary End-Walls of Centrifugal Turbomachinery,” Trans. Journal of Fluids Engineering, Sept., ASME, pp. 431–442. Dean, D. E. and Stiel, L. I., 1965, AIChE Journal, Vol. 11, p. 526. de Haller, P., 1953, “Das Verhalten von Tragflugelgittern in Axialverdichtern und in Windkanal,” Brennstoff-Warme-Kraft 5, Heft 10. Denton, J. D., 1982, “An Improved Time-Marching Method for Turbomachinery Flow Calculation,” ASME Paper No., 82-GT-239, ASME, New York, NY De Ruyck, J., Hirsch, C. and Kool, P., 1979, “An Axial Compressor End-Wall Boundary Layer Calculation Method,” Trans. Journal of Engineering for Power, April, ASME, Vol. 101, pp. 233–249. De Ruyck, J. and Hirsch, C., 1980, “Investigations of an Axial Compressor EndWall Boundary Layer Prediction Method,” ASME Paper No. 80-GT-53, ASME, New York, NY. Dunavant, J. C., Emery, J. C., Walch, H. C. and Westphal, W. R., 1955, “HighSpeed Cascade Tests of the NACA 65-(12A10)-10 and NACA 65-(12A2I8b)-10 Compressor Blade Sections,” NACA Research Memorandum RM-L55I08, NACA, Washington, DC. Dunavant, J. C., 1957, “Cascade Investigation of a Related Series of 6-Percent Thick Guide-Vane Profiles,” NACA TN 3959. Dunker, R., Rechter, H., Starken, H. and Weyer, H., 1984, “Redesign and Performance Analysis of a Transonic Axial Compressor Stator with Subsonic Controlled Diffusion Airfoils,” Trans. Journal of Engineering for Gas Turbines and Power, ASME, Vol. 106, April, pp. 279–287. Egli, A., 1935, “The Leakage of Steam Through Labyrinth Glands,” Trans. ASME, Vol. 57, pp. 115–122. Emery, J. C., Herrig, L. J., Erwin, J. R. and Felix, A. R., 1958, “Systematic TwoDimensional Cascade Tests of NACA 65-Series Compressor Blades at Low Speeds,” NACA Report 1368, NACA, Washington, DC. Geye, R. P., Budinger, R. E. and Voit, C. H., 1953, “Investigation of a High-Pressure-Ratio Eight-Stage Axial-Flow Research Compressor With Two Transonic Inlet Stages; II—Preliminary Analysis of Overall Performance,” NACA Research Memorandum RM E53J06, NACA, Washington, DC. Gopalakrishnan, S. and Bozzola, R., 1973, “Numerical Representation of Inlet and Exit Boundary Conditions in Transient Cascade Flow,” ASME Paper No. 73-GT-55, ASME, New York, NY.


References • 351

Green, J. E., 1968, “The Prediction of Turbulent Boundary Layer Development In Compressible Flow,” Journal Of Fluid Mechanics, Vol. 31, p. 753. Gruschwitz, E., 1950, “Calcul Approche de la Couche Limite Laminaire en Ecoulement compressible Sur Une Paroi Non-conductrice de la Chaleur,” Office National d’Etudes et de Recherche Aeronautiques (ONERA), Paris, Publication No. 47. Head, M. R., 1958, “Entrainment In the Turbulent Boundary Layer,” R&M 3152, Aeronautical Research Council, London, United Kingdom. Head, M. R., 1968, “Cambridge Work on Entrainment,” Proceedings of Computation of Turbulent Boundary Layers, Thermosciences Division, Stanford University, CA. pp. 188–194. Herrig, L. J., Emery, J. C., and Erwin, J. R., 1957, “Systematic Two-Dimensional Cascade Tests of NACA 65-Series Compressor Blades at Low Speeds,” NACA TN 3916, NACA, Washington, DC. Hirsch, C., 1974, “End-Wall Boundary Layers in Axial Compressors,” Trans. Journal of Engineering for Power, ASME, Vol. 96, Oct., pp. 413–426. Hirsch, C., 1976, “Flow Prediction in Axial Flow Compressors Including EndWall Boundary layers,” ASME Paper No. 76-GT-72, ASME, New York, NY. Hobbs, D. E. and Weingold, H. D., 1984, “Development of Controlled Diffusion Airfoils for Multistage Compressor Application,” Trans. Journal of Engineering for Gas Turbines and Power, ASME, Vol. 106, April, pp. 271–278. Horlock, J. H., 1958, Axial Flow Compressors: Fluid Mechanics and Thermodynamics, Butterworths Scientific Publications, London, United Kingdom. Horlock, J. H., 1970, “Boundary Layer Problems In Axial Turbomachines,” Flow Research on Blading (L. S., Dring, editor), Elsevier Publishing, Amsterdam, pp. 322–371. Howell, A. R., 1942, “The Present Basis of Axial Flow Compressor Design; Part I—Cascade Theory and Performance,” R&M 2095, British Aeronautical Research Council, London, United Kingdom. Howell, A. R., 1945, “Design of Axial Compressors,” Proceedings of the Institution of Mechanical Engineers, Vol. 153, London, United Kingdom. Howell, A. R., 1947, “Fluid Dynamics of Axial Compressors,” (Lectures on the Development of the British Gas Turbine Jet Unit), War Emergency Proc. No. 12, Institution of Mechanical Engineers, London, United Kingdom [American Edition published by the American Society of Mechanical Engineers], pp. 441–452. Hunter, I. H. and Cumpsty, N. A., 1982, “Casing Wall Boundary-Layer Development Through an Isolated Compressor Rotor,” Trans. Journal of Engineering for Power, ASME, Vol. 104, Oct., pp. 805-818. Huntington, R. A., 1985, “Evaluation of Polytropic Calculation Methods for Turbomachinery Performance,” ASME Paper no. 85-GT-13, ASME, New York, NY. Jansen, W., 1967, “The Application of End-Wall Boundary Layer Effects in the Performance Analysis of Axial Compressors,” ASME Paper No. 67-WA/GT-11, ASME, New York, NY. Jansen, W. and Moffatt, W. C., 1967, “The Off-Design Analysis of Axial-Flow Compressors,” Trans. ASME, Journal of Engineering for Power, Vol. 89, Oct., pp. 453–462.



Johnsen, I. A., 1952, “Investigation of a 10-Stage Subsonic Axial-Flow Compressor; I—Aerodynamic Design,” NACA Research Memorandum RM E52B18 NACA, Washington, DC. Johnsen, I. A. and Bullock, R. O., editors, 1965, “Aerodynamic Design Of Axial Flow Compressors,” NASA SP-36, NASA, Washington, DC. Katsanis, T., 1964, “Use of Arbitrary Quasi-Orthogonals for Calculating Flow Distribution in the Meridional Plane of a Turbomachine,” NASA TN D-2546, NASA, Washington, DC. Katsanis, T., 1968, “Computer Program for Calculating Velocities and Streamlines on a Blade-to-Blade Stream Surface of a Turbomachine,” NASA TN D4525, NASA, Washington, DC. Katsanis, T., 1969, “Fortran Program for Calculating Transonic Velocities on a Blade-to-Blade Stream Surface of a Turbomachine,” NASA TN D-5427, NASA, Washington, DC. Koch, C. C., 1981, “Stalling Pressure Rise Capability of Axial Flow Compressor Stages,” Trans. Journal of Engineering for Power, ASME, Vol. 103, pp. 411–424. Koch, C. C. and Smith, L. H. Jr., 1976, “Loss Sources and Magnitudes In AxialFlow Compressors, Trans. ASME, Journal of Engineering for Power, Vol. 98, pp. 411–424. Kovach, K. and Sandercock, D. M., 1954, “Experimental Investigation of FiveStage Axial-Flow Research Compressor With Transonic Rotors In All stages; II—Compressor Over-All Performance,” NACA Research Memorandum RME54G01, NACA, Washington, DC. Kovach, K. and Sandercock, D. M., 1961, “Aerodynamic Design and Performance of Five-Stage Transonic Axial-Flow Compressor,” Trans. Journal of Engineering for Power, ASME, Vol. 83, July, pp. 304-321. Lax, P. D., 1954, “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation,” Commun. Pure and Appl. Math., Vol. 7, pp. 159-193. Lax, P. D., and Wendroff, B., 1964, “Differencing Schemes for Hyperbolic Equations With High Order of Accuracy,” Commun. Pure and Appl. Math., Vol. 17, pp. 381–398. Lieblein, S., Schwenk, F. C. and Broderick, R. L., 1953, “Diffusion Factor for Estimating Losses and Limiting Blade Loadings in Axial-flow Compressor Blade Elements,” NACA RM E53D01, NACA, Washington, DC. Lieblein, S. and Roudebush, W. H., 1956, “Theoretical Loss Relations for LowSpeed Two-Dimensional-Cascade Flow,” NACA TN 3662, NACA, Washington, DC. Lieblein, S., 1959, “Loss and Stall Analysis of Compressor Cascades,” Trans., Journal of Basic Engineering, ASME, Vol. 81, Sept., pp 387–400. Lieblein, S., 1960, “Incidence and Deviation-Angle Correlations for Compressor Cascades,” Trans. Journal of Basic Engineering, ASME, Vol. 82, Sept., pp 575–587. Ludwieg, H. and Tillmann, W., 1950, “Investigations of the Wall-Shearing Stress in Turbulent Boundary Layers,” NACA TM 1285, NACA, Washington, DC. Mallen, M. and Saville, G., 1977, “Polytropic Processes in the Performance Prediction of Centrifugal Compressors,” Paper No. C183/77, Institution of Mechanical Engineers, London, United Kingdom, pp. 89–96. Mellor, G. L., and Wood, G. M., 1971, “An Axial Compressor End-Wall Boundary Layer Theory,” Trans. Journal of Basic Engineering, ASME, pp.300–316.


References • 353

Miller, G. R., Lewis, G. W. Jr. and Hartmann, M. J., 1961, “Shock Losses in Transonic Compressor Blade Rows,” Trans. Journal of Engineering for Power, ASME, Vol. 83, July, pp. 235–242. Moretti, G. and Abbett, M., 1966, “A Time-Dependent Computational Method for Blunt Body Flows,” AIAA Journal, Vol. 4, pp. 2136–2141. Nelson, L. C. and Obert, E. F., 1954, “Generalized pvT Properties of Gases,” Trans. ASME, Vol. 76, pp. 1057–1066. Nikuradse, J., 1930, “Laws of Resistance and Velocity Distributions for Turbulent Flow of Water in Smooth and Rough Pipes,” Proceedings, 3rd International Congress for Applied Mechanics, Stockholm, Sweden, pp. 239–248. Novak, R. A., 1967, “Streamline Curvature Computing Procedures for Fluid-Flow Problems,” Trans. Journal of Engineering for Power, ASME, Vol. 89, Oct., pp. 478–490. Novak, R. A., 1973, “Axisymmetric Computing Systems for Axial Flow Turbomachinery,” Lecture 25, ASME Turbomachinery Institute Fluid Dynamics of Turbomachinery, Iowa State University, Ames, Iowa. Pai, S., 1957, Viscous Flow Theory, II-Turbulent Flow, Van Nostrand, Princeton, NJ. Pitzer, K. S., Lippmann, D. Z., Curl, R. F., Huggins, C. M. and Peterson, D. E., 1955, “The Volumetric and Thermodynamic Properties of Fluids. II Compressibility Factor, Vapor Pressure and Entropy of Vaporization,” American Chemical Society, Vol. 77, pp. 3427–3440. Pohlhausen, K., 1921, “Zur Naherungsweisen Integration der Differential-Gleichung der Laminare Reibungsschicht,” ZAMM, Vol. 1, p. 235. Pollard, D. and Gostelow, J. P., 1967, “Some Experiments At Low Speed on Compressor Cascades,” Trans. Journal Of Engineering for Power, ASME, Vol. 89, pp. 427–436. Redlich, O. and Kwong, J., 1949, “On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions,” Chemical Review, Vol. 44, pp. 233–244. Reneau, L., Johnston, J. and Kline, S., 1967, “Performance and Design of Straight Two-Dimensional Diffusers”, Trans. Journal of Basic Engineering, ASME, Vol. 89, pp. 141–150. Ried, R. C. and Sherwood, T. K., 1966, The Properties Of Gases And Liquids, McGraw-Hill, New York. Ried, R. C., Prausnitz, J. M., and Sherwood, T. K., 1977, The Properties of Gases and Liquids, McGraw-Hill, New York, NY. Ried, R. C., Prausnitz, J. M. and Poling, B. E., 1987, The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, NY. Rotta, J. C., 1966, “Recent Developments in Calculation Methods for Turbulent Boundary Layers With Pressure Gradients and Heat Transfer,” Trans. Journal of Applied Mechanics, ASME, Vol. 88, p. 429. Sandercock, D. M., Kovach, K. and Lieblein, S., 1954, “Experimental Investigation of a Five-Stage Axial-Flow Research Compressor With Transonic Rotors In All stages; I – Compressor Design,” NACA Research Memorandum RME54F24, NACA, Washington, DC. Schlichting, H., 1968, Boundary-Layer Theory, 6th Ed., McGraw-Hill, New York, NY. Schlichting, H., 1979, Boundary-Layer Theory, McGraw-Hill, New York, NY.



Schultz, J. M., 1962, “The Polytropic Analysis of Centrifugal Compressors,” Trans. Journal Of Engineering for Power, ASME, Vol. 84, Jan., pp. 69–82. Schumann, L. F., 1985, “A Three-Dimensional Axisymmetric Calculation Procedure for Turbulent Flows in a Radial Vaneless Diffuser,” ASME Paper No. 85GT-133, ASME, New York, NY. Senoo, Y., Kinoshita, Y. and Ishida, M., 1977, “Axisymmetric Flow in Vaneless Diffusers of Centrifugal Blowers,” Trans. Journal of Fluids Engineering, ASME, Vol. 99, March, pp. 104–114. Sheppard, D. G., 1956, Principles of Turbomachinery, Macmillan, New York, NY. Smith, D. J. L. and Frost, D. H., 1969, “Calculation of the Flow Past Turbomachine Blades,” Proc. Inst. Mech. Eng., Vol. 184, Paper 27, London, United Kingdom. Smith, L. H.. Jr., 1958, “Recovery Ratio—A Measure of the Loss Recovery Potential of Compressor Stages,” Trans. ASME, Vol. 80, pp. 517–524. Smith, L. H.. Jr., 1970, “Casing Boundary Layers in Multi-Stage Axial-Flow Compressors,” Flow Research on Blading, (L.S. Dring, editor), Elsevier Publishing, Amsterdam, pp. 275–304. Soave, G., 1972, “Equilibrium Constants From a Modified Redlich-Kwong Equation of State,” Chemical Eng. Science, Vol. 27, 1197–1203. Sovran, G. and Klomp, E. D., 1967, “Experimentally Determined Optimum Geometries for Rectilinear Diffusers With Rectangular, Conical or Annular Cross-Section,” Fluid Mechanics of Internal Flows, (G. Sovran, editor), Elsevier Publishing, Amsterdam, pp. 270–319. Stratford, B. S., 1967, “The Use of Boundary Layer Technique to Calculate the Blockage From the Annulus Boundary Layer in a Compressor,” ASME Paper No. 67-WA/GT-7, ASME, New York, NY. Summer, W. J. and Shanebrook, J. R., 1971, “Entrainment Theory for Compressible Turbulent Boundary Layers on Adiabatic Walls,” AIAA Journal, Vol. 9, p. 330–332. Swan, W. C., 1961, “A Practical method of Predicting Transonic-Compressor Performance,” Trans. Journal of Engineering for Power, Vol. 83, July, pp. 322-330. Vavra, M. H., 1960, Aero-Thermodynamics And Flow In Turbomachines, Wiley, New York. Voit, C. H., 1953, “Investigation of a High-Pressure-Ratio Eight-Stage Axial-Flow Research Compressor With Two Transonic Inlet Stages; I – Aerodynamic Design,” NACA Research Memorandum RM E53I24, NACA, Washington, DC. Von Neumann, J. and Richtmyer, R. D., 1950, “A Method for the Numerical Calculation of Hydrodynamic Shocks,” Journal of Applied Physics, Vol. 21, pp. 232–237. Walsh, J. L., Ahlberg, J. H., and Nilson, E. N., 1962, “Best Approximation Properties of the Spline Fit,” Journal of Mathematics and Mechanics, Vol. 11, pp. 225–234. Wassell, A. B., 1968, “Reynolds Number Effects in Axial Compressors,” Trans. Journal of Engineering for Power, ASME, April, Vol. 90, pp. 149–156. Wilson, G. M., 1966, “Calculation of Enthalpy Data From a Modified RedlichKwong Equation of State,” Adv. Cryogenic Eng., Vol. 11, p. 392. Wu, C. H., 1952, “A General Theory of Three-Dimensional Flow In Subsonic And Supersonic Turbomachines of Axial-, Radial- and Mixed-Flow Types,” NACA TN 2604, NACA, Washington, D.C. Yaws, C. L., 1999, Chemical Properties Handbook, McGraw-Hill, New York, NY.


ABOUT THE AUTHOR

Mr. Aungier is the manager of Advanced Technology for Elliott Turbomachinery Company Inc., Ebara Group in Jeannette, Pennsylvania. He has been active in fluid mechanics research and development for more than 36 years, 32 of those in turbomachinery aerodynamics, specializing in centrifugal compressors, axialflow compressors and radial-inflow turbines. He has numerous publications in this field, primarily through the American Society of Mechanical Engineers. He is a graduate of Cornell University, where he received a masters degree in Aerospace Engineering and a bachelors degree in Engineering Physics. Mr. Aungier started his career in 1966 as an officer in the U.S. Air Force, conducting research in hypersonic re-entry vehicle aerodynamics at the Air Force Weapons Laboratory in Albuquerque, New Mexico. He is the author of numerous Air Force and NASA publications, some of which are the basis for one of the analysis techniques described in this book. In 1970, Mr. Aungier joined the Research Division of Carrier Corporation in Syracuse, New York, where he spent 11 years managing and conducting applied research on the fluid dynamics of turbomachinery and air handling equipment. Most of his individual research was focused on the interests of Elliott Company (then a division of Carrier), including development of aerodynamic performance analysis techniques for axial-flow compressors, centrifugal compressors and radial-inflow turbines. In 1981, Mr. Aungier transferred to Elliott Company as manager of Compressor Development, where his interests expanded to include the development of systematic and efficient techniques for aerodynamic design of turbomachinery. His responsibilities were extended to include turbine aerodynamic development in 1983 and mechanical design and analysis in 1987. He continues to be an active contributor to turbomachinery aerodynamic technology, specializing in comprehensive aerodynamic design and analysis systems. In 2000, ASME Press published his first book, describing his centrifugal compressor aerodynamic design and analysis system. The present book provides a similar treatment of his axial-flow compressor aerodynamic design and analysis system.



INDEX

Accentric factor, 26 Adiabatic efficiency; see efficiency, adiabatic Adiabatic process; see isentropic process Adjustable blade rows, 311–316 Angle of attack Defined, 62, 64, 119–120 Design, 121,124 Stall, 135 Angular momentum, conservation of, 157–158, 162, 293, 325 Annulus sizing, 169–171, 263, 266–268; also see end-wall contours Axial-centrifugal compressor, 328–332 Blade; also see blade camberline; blade profile and blade row Angles, 60–62, 64–65, 70–71, 76, 119–120 Construction, 62–71 Controlled diffusion airfoil, 59, 71–73, 117 Geometry on a stream surface, 202 Leading edge, 89–90 Loading, 129, 222–224, 273 Blade camberline Circular-arc, 65–66 NACA A4K6, 70–71 NACA 65–series, 62–65 Parabolic-arc, 66–68 Blade profile C-series, 68–69 Double-circular-arc, 69–70

NACA 63–series, 70–71 NACA 65–series, 62–65 Blade row; also see blade and cascade Design, 221–222, 264–266 Diffusion limit, 130, 132, 204–206 Force defect; see boundary layer, blade force defect Shrouded, 6, 147–149, 191–193 Throat, 73–75, 201 Types, 3, 201 Blade-to-blade flow; see flow, bladeto-blade Blockage, aerodynamic, 41, 145, 157, 195, 318, 320–321 Blockage, blade, 293 Boundary layer Axisymmetric, 54 Axisymmetric-three-dimensional, 54–56, 175–197, 302–307 Blade force defect, 55–56, 179–180, 182, 187–191, 303–306 Clearance gap, 179, 187, 305 Displacement thickness, 52, 56, 108, 178, 182 Energy thickness, 109 Enthalpy thickness, 109 Entrainment, 54, 56, 181, 187 Laminar, 108–110 Momentum-integral equation, 54, 56, 107, 181–182 Momentum thickness, 53, 56, 108–109, 182 Profiles, 108, 183–184 Profile loss coefficient, 112 Separation, 110–111, 184



Boundary layer (continued) Shape factors, 108, 110–111, 183–184 Shroud seal leakage effects, 191–193 Transition, 110–111, 186, 194 Turbulent, 110–111, 175–197, 302–307 Two-dimensional, 51–54, 107–113 Velocity thickness, 109 Camber angle; see blade, angles Camberline; see blade camberline Cascade; see also blade row Empirical performance models, 121–151 Geometry, 60–62, 75–76, 119–120 Centrifugal compressor; see compressor types CFL stability criterion, 103 Characteristics, 98–101 Choked flow, 161, 299 Collector, 322–327 Compressibility factor, 26 Compressible flow analysis; see flow, compressible Compressor design; see design, detailed, multistage compressor Compressor performance analysis; see performance analysis, compressor Compressor types, 1, 328–332 Computational fluid dynamics (CFD) Euler (inviscid flow) codes; see flow, inviscid Viscous CFD codes, 42, 50, 114, 303 Computerized design system, useful features Aerodynamic performance analysis, 200–202, 311, 327–328 Blade geometry database, 73 Cascade performance models, 149 Equation-of-state package, 37–38 Internal flow analysis, 288, 290–291, 301 Meridional through-flow, 167–171 Multi-stage compressor, 261–268 Stage, 257

Conservation of mass, 157, 160, 292, 296, 299; also see continuity equation and boundary layer, entrainment Continuity equation, 47, 83–84, 97, 159 Contours; see end-wall contours Controlled diffusion airfoil; see blade, controlled diffusion airfoil Critical Mach number, 138 Critical point, 23–24 Curvature effects, 47, 157, 292, 319–320 Departure functions, 30–31, 37 Design, detailed Adjustable blade rows; see adjustable blade rows Multistage compressor, 251–257, 259–285 Stage, 215–257 Deviation angle Defined, 62, 120 Design, 125–128 Off-design, 142–144 D-factor, see diffusion factors Diffuser Divergence angle, 318–319 Divergence parameter, 318–319 Exhaust, 316–322 Two-dimensional, 204 Diffusion Factors, 129–132, 206 Dimensionless parameters, 10–11, 13–14, 217, 262 Displacement thickness; see boundary layer, displacement thickness Divergence angle; see diffuser, divergence angle Drag coefficient, 133 Efficiency Adiabatic, 20–21, 32 Compressor, 11, 20–22, 32 Diffuser, 33 Nozzle, 35 Polytropic, 21–22, 32 Elliptic equations, 49, 81


Index • 359

End-wall boundary layer; see boundary layer, axisymmetricthree-dimensional End-wall contours Designing, 169–171, 252, 263, 266 Smoothing, 266–267 Energy equation, 19, 47, 97, 158, 293 Energy thickness; see boundary layer, energy thickness Enthalpy, 19, 25, 31, 45–46 Enthalpy thickness; see boundary layer, enthalpy thickness Entrainment; see boundary layer, entrainment Entropy, 19, 25, 31 Equation of state Aungier’s modified Redlich-Kwong, 26–29 Caloric, 22, 24–25 Calorically perfect gas, 25, 31–32 Comparison of, 28 Perfect gas, 23–26, 31–32 Pseudo-perfect gas, 32–33 Real gas, 26, 30–31 Redlich-Kwong, 26–29 Thermal, 22, 26 Thermally perfect gas, 23–26, 31–32 Equivalent diffusion factor, see diffusion factors Equivalent performance; see similitude Euler turbine equation, 13, 44–45 Finite-difference approximations; see numerical approximations Flow Blade-to-blade, 41, 49, 77–107, 290–291 Compressible, 316–322; see also flow, inviscid Hub-to-shroud, 41, 49, 153–172, 291–299; also see normal equilibrium Inviscid, 48–50, 77–107, 113–114, 153–169, 288–302 Irrotational, 49, 81, 83–84 One-dimensional, 41, 318–322 Quasi-three-dimensional, 41, 50, 287–302 Transonic, 89

Flow coefficient, 14, 217, 219, 262, 330–331 Flow work, 19 Fluid turning, 64, 120, 141 Gas constant, 10, 23 Gas mixtures, 29 Gas property data, 24, 29 Gas viscosity; see viscosity Head Adiabatic, 20 Defined, 11 Polytropic, 22 Helmholtz energy, 30 Hub-to-shroud flow; see flow, hub-toshroud Hydraulic diameter, 326–327 Hyperbolic equations, 49, 81 Incidence angle Choking, 136–137 Defined, 62, 120 Design, 122–124 Minimum loss, 138 Stall, 134–137 Internal energy, 18, 24–25 Inviscid flow analysis; see flow, inviscid Irrotational flow; see flow, irrotational Isentropic efficiency, see efficiency, adiabatic Isentropic process, 20, 46 Kutta condition, 87, 94–95 Labyrinth seal, 148–149 Lift coefficient, 63, 65–66, 70–71, 120, 133 Loss coefficient Blade tip clearance, 146–147 Collector, 327 Defined, 36 Design, 128, 130, 132–134 Discharge, 327–328 End-wall, 133, 150 Minimum, 138



Loss coefficient (continued) Off-design, 138, 144–145, 150 Profile, 112, 128, 130, 132 Reynolds number effect, 150–151 Scroll, 327 Shock wave, 138–141 Shroud seal leakage, 147–149 Smoothing, 203 Supercritical Mach number, 138

Polytropic process, 21–22 Potential flow; see flow, irrotational Power, 18–19, 44 Pressure recovery coefficient, 34, 322, 327

Mach number, 9, 160; also, see critical Mach number Mass conservation; see conservation of mass Matrix methods, 88–89 Meridional coordinate; see stream surface Meridional through-flow; see flow, hub-to-shroud Momentum equations, 46–47, 97, 157, 293 Momentum-integral equation; see boundary layer, momentumintegral equation Momentum thickness; see boundary layer, Momentum thickness

Ratio of specific heats, 10, 26, 31 Reaction, 14, 217–219, 262 Recovery ratio, 226–229 Relative conditions, 3–6, 43–46 Repeating stage, 215, 219, 251–257 Reversible process, 11, 19 Reynolds number, 10, 110, 150–151, 321, 328 Rotating coordinate system, 43, 55 Rotating stall, see stall Rothalpy, 45

Natural coordinates, 44, 57, 156 Normal equilibrium Approximate, 49, 168, 207–211 Simple, 48–49, 168, 219, 221 Full, 48–49, 165–167, 211–213 Numerical approximations, 87–88, 94, 104, 171–172 Numerical stability, see stability, numerical Passage curvature; see curvature effects Performance analysis Compressor, 199–214 Diffuser, 316–322 Volute and collector, 322–327 Performance characteristics, 7–13 Periodicity condition, 86 Pitch, 62 Polytropic efficiency; see efficiency, polytropic

Quasi-normal, 155–157, 201, 291–292 Quasi-three-dimensional flow; see flow, quasi-three-dimensional

Saturation line, 23, 30 Scroll, 322–327 Seal leakage, 147–149, 191–193 Shear stress, 51–54, 182; see also skin friction coefficient Similitude, 7–11 Sizing parameter, 326 Skin friction coefficient Calculation of, 109, 111, 186, 321–322 Defined, 108, 318 Loss calculation from, 150, 326 Solidity, 62, 120 Sound speed, 10, 26, 31 Specific heat, 24, 31; also see ratio of specific heats Stability, aerodynamic; see stall and surge Stability, numerical, 102–105, 165, 297, 299 Stage design; see design, stage Stage loading, 273 Stage matching, 11–13, 259, 270 Stagger angle; see blade, angles Staggered spacing, 179, 206 Stall, 204–207, 222–224


Index • 361

Stall incidence angles; see incidence angle, stall Stokes’ theorem, 84, 129; also, see also flow, irrotational Stream function, 85, 93 Streamline curvature method; see normal equilibrium, full Stream surface curvature; see curvature effects Stream surface, 43, 79–80, 155, 289 Surface roughness, 321–322, 328 Surge, 11–12, 204–207 Swirl vortex types; see vortex types Thermodynamic properties Real gas, 30–31 Rotating-to-stationary coordinate conversion, 45–46 Thermally and calorically perfect gas, 25, 31 Thermally perfect gas, 24 Total-to-static conversion, 25, 31, 45–46 Thermodynamics, 17–40 First law of, 18 Second law of, 19 Throat; see blade row, throat Time-marching method, 49, 81, 96–107 Total thermodynamic conditions, 19, 45–46

Torque, 44 Transition; see boundary layer, transition Vapor saturation conditions; see saturation line Vector operators, 57 Velocity diagrams, 5–8 Velocity thickness; see boundary layer, velocity thickness Viscosity, 37 Viscous flow analysis; see boundary layer and computational fluid dynamics (CFD), viscous CFD codes Volume ratio effects, 9, 12 Volute; see scroll Vortex types, Assigned flow angle, 245 Comparison of, 247–251, 280–284 Constant reaction vortex, 235–241, 280 Constant swirl vortex, 242–245, 280 Defining equation, 220, 262–263 Exponential vortex, 242 Free vortex, 230–235, 280 Work input, 13, 225 Work coefficient, 13, 217, 219, 262


23217.Axial-Flow Compressors a Strategy for Aerodynamic Design and Analysis

Recommend Documents