Logan's turbomachinery : flowpath design and performance fundamentals [Third edition.] 9781138198203, 113819820X

Table of contents :
Cover
Half title
Series Page
Title Page
Copyright Page
Dedication
Contents
Author
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
1.
Turbomachinery History, Classifications, and Applications
1.1.
Introduction
1.2.
History
1.3.
Classifications of Turbomachines
1.3.1.
General Classifications
1.3.2.
Typical Examples
1.3.3.
Gas Turbines
1.4.
Applications and Technology Development
1.5.
Concluding Remarks
References
Bibliography
2.
Basic Concepts and Relations of Aerothermodynamics
2.1.
Introduction
2.2.
Incompressible versus Compressible Flow
2.2.1.
Total Temperature and Pressure
2.2.2.
Mass Flow Rate
2.2.3.
Compressible Mass Flow Functions
2.3.
Energy Equation
2.4.
Linear Momentum Equation
2.5.
Angular Momentum Equation
2.5.1.
Euler’s Turbomachinery Equation
2.6.
Velocity Diagram
2.7.
Applications
2.7.1.
Axial-Flow Impulse Turbine
2.7.2.
Axial-Flow Compressor
2.7.3.
Centrifugal Pump
2.7.4 Hydraulic Turbine
2.8.
Discussion on Further Applications
2.9.
Concluding Remarks
Worked Examples
Problems
Reference
Bibliography
Nomenclature
3.
Dimensionless Quantities
3.1.
Introduction
3.2.
Turbomachine Variables
3.3.
Similitude
Worked Examples
Problems
References
Nomenclature
4.
Centrifugal Pumps and Fans
4.1.
Introduction
4.2.
Impeller Flow
4.3.
Efficiency
4.4.
Performance Characteristics
4.5.
Design of Pumps
4.6.
Fans
Worked Examples
Problems
References
Bibliography
Nomenclature
5.
Centrifugal Compressors
5.1.
Introduction
5.2.
Impeller Design
5.3.
Diffuser Design
5.4.
Performance
Worked Examples
Problems
References
Bibliography
Nomenclature
6.
Axial-Flow Pumps, Fans, and Compressors
6.1.
Introduction
6.2.
Stage Pressure Rise
6.3.
Losses
6.4.
Pump Design
6.5.
Fan Design
6.6.
Compressor Design
6.7.
Compressor Performance
Worked Examples
Problems
References
Bibliography
Nomenclature
7.
Radial-Flow Gas Turbines
7.1.
Introduction
7.2.
Basic Theory
7.3.
Design
Worked Example
Problems
References
Bibliography
Nomenclature
8.
Axial-Flow Gas Turbines
8.1.
Introduction
8.2.
Basic Theory
8.3.
Design
Worked Examples
Problems
References
Bibliography
Nomenclature
9.
Steam Turbines
9.1.
Introduction
9.2.
Impulse Turbines
9.3.
Reaction Turbines
9.4.
Design
Problems
References
Bibliography
10.
Hydraulic Turbines
10.1.
Introduction
10.2.
Pelton Wheel
10.3.
Francis Turbine
10.4.
Kaplan Turbine
10.5 Cavitation
Problems
References
Bibliography
11.
Wind Turbines
11.1.
Introduction
11.2.
Actuator Theory
11.3.
Horizontal-Axis Machines
11.4.
Vertical-Axis Machines
Problems
References
Bibliography
12.
Gas-Turbine Exhaust Diffusers
12.1.
Introduction
12.2 Roles of an Exhaust Diffuser
12.3.
Diffusion Process
12.3.1.
Diffusion in a Constant-Area Duct
12.3.2.
Pressure Recovery in a Dump Diffuser
12.4.
Performance Evaluation
12.4.1.
Diffuser Isentropic Efficiency
12.4.2.
Pressure Rise Coefficient
12.4.3.
Axial Stream Thrust Coefficient
12.5.
Six Simple Design Rules
12.6.
Concluding Remarks
References
Bibliography
Nomenclature
13.
Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design
13.1.
Introduction
13.2.
CFD Methodology
13.3.
The Common Form of Governing Conservation Equations
13.3.1.
The Common Equation Form
13.4.
Turbulence Modeling
13.4.1.
Reynolds Equations: The Closure Problem
13.4.2. High-Reynolds-Number Two-Equation k-ε Model
13.5.
Boundary Conditions
13.5.1.
Inlet and Outlet Boundary Conditions
13.5.2.
Wall Boundary Conditions: The Wall-Function Treatment
13.5.3.
Alternative Near-Wall Treatments
13.5.4.
Choice of a Turbulence Model
13.6.
Physics-Based Post-processing of CFD Results
13.6.1.
Large Control Volume Analysis of CFD Results
13.6.2.
Entropy Map Generation
13.7.
Turbomachinery Aerodynamic Design Process
13.7.1.
3-D Flow Field
13.7.2.
Preliminary Design
13.7.3.
Detailed Design
13.7.4.
Role of 3-D CFD
13.8.
Concluding Remarks
References
Bibliography
Nomenclature
Appendix A: Tables of Conversion Factors, Pump Efficiency, and Compressor Specific Speed
Table A.1: Conversion Factors
Table A.2:
Pump Efficiency as a Function of Specific Speed and Capacity
Table A.3:
Compressor Specific Diameter as a Function of Specific Speed and Efficiency
Appendix B: Derivation of Equation
for Slip Coefficient
Appendix C: Formulation of Equation for Hydraulic Loss in Centrifugal Pumps with Backward-Curved Blades
Reference
Appendix D: Viscous Effects on Pump Performance
Table D.1:
Viscosities of Liquids at 70°F
Table D.2:
Correction Factors for Oil with Kinematic Viscosity of 176 Centistokes
Appendix E: Comparison of Formulas for Compressor Slip Coefficient, μs = Vu2′/Vu2
References
Appendix F: Molecular Weight of Selected Gases
Table F.1:
Molecular Weight of Selected Gases
Appendix G: Pressure and Temperature Changes in Isentropic Free and Vortex Vortices
G.1.
Introduction
G.2.
Free and Forced Vortices
G.3.
Changes in Static Temperature and Pressure
G.3.1.
A Simple Approach
G.4.
Changes in Total Temperature and Pressure
Reference
Appendix H: Dimensionless Velocity Diagrams for Axial-Flow Compressors and Turbines
H.1.
Introduction
H.2.
Performance Parameters
H.2.1.
Flow Coefficient
H.2.2.
Loading Coefficient
H.2.3.
Stage Reaction
H.3.
Dimensionless Velocity Diagrams
H.3.1.
Derivations of Equations to Compute Velocities and Angles of Dimensionless Velocity Diagram
H.3.2.
Using φ, ψ, and R to Quickly Draw a Dimensionless Velocity Diagram
Appendix I: Throughflow Design with Simple Radial Equilibrium Equation
I.1.
Introduction
I.2.
Radial Equilibrium Equation
I.3.
Vortex Energy Equation
I.4.
Free-Vortex Design
I.4.1.
Degree of Reaction in an Axial-Flow Turbomachine
I.5.
General Vortex Design
I.6.
Throughflow Design Project
I.6.1.
Axial-Flow Compressor Layout and Nomenclature
I.6.2.
Governing Equations
I.6.3.
Case 1: Single-Stage Compressor with Potential Flow (Free-Vortex Design)
I.6.4.
Case 2: Single-Stage Compressor with Specified Flow Angles and Efficiency
Table I.1:
Radial Distributions of Exit Absolute Flow Angles and Stage Efficiency
References
Nomenclature
Appendix J: Review of Necessary Mathematics
J.1.
Suffix Notation and Tensor Algebra
J.1.1.
Summation Convention
J.1.2.
Free and Dummy Indices
J.1.3.
Two Special Symbols
J.2.
Gradient, Divergence, Curl, and Laplacian
J.2.1.
Gradient
J.2.2.
Divergence
J.2.3.
Curl
J.2.4.
Laplacian
J.3.
Dyad in Total Derivative
J.4.
Total Derivative
J.5.
Vector Identities
Bibliography
Index

Citation preview

Logan’s Turbomachinery

Mechanical Engineering Series Editor: L. L. Faulkner

Battelle Memorial Institute and The Ohio State University

A Series of Textbooks and Reference Books

PUBLISHED TITLES High-Vacuum Technology: A Practical Guide, Second Edition Marsbed H. Hablanian Shaft Alignment Handbook, Third Edition John Piotrowski Applied Combustion, Second Edition Eugene L. Keating Introduction to the Design and Behavior of Bolted Joints, Fourth Edition: Non-Gasketed Joints John H. Bickford Design and Optimization of Thermal Systems, Second Edition Yogesh Jaluria Mechanical Tolerance StacKup and Analysis, Second Edition Bryan R. Fischer Vehicle Dynamics, Stability, and Control, Second Edition Dean Karnopp Pump Characteristics and Applications, Third Edition Michael Volk Principles of Composite Material Mechanics, Fourth Edition Ronald F. Gibson Handbook of Hydraulic Fluid Technology, Second Edition George E. Totten and Victor J. De Negri Mechanical Vibration: Analysis, Uncertainties, and Control, Fourth Edition Haym Benaroya, Mark Nagurka, and Seon Han Blake’s Design of Mechanical Joints, Second Edition Harold Josephs and Ronald L. Huston Logan’s Turbomachinery: Flowpath Design and Performance Fundamentals, Third Edition Bijay K. Sultanian For more information about this series, please visit: www.crcpress.com

Logan’s Turbomachinery Flowpath Design and Performance Fundamentals

Third Edition

Bijay K. Sultanian

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-19820-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com eResource material is available for this title at https://www.crcpress.com/9781138198203.

9781138198203_FM.indd 4

27/12/18 10:53 AM

To my dearest friend Kailash Tibrewal whose mantra of “joy in giving” continues to inspire me; my wife, Bimla Sultanian; our daughter, Rachna Sultanian, MD; our son-in-law, Shahin Gharib, MD; our son, Dheeraj (Raj) Sultanian, JD, MBA; our daughter-in-law, Heather Benzmiller Sultanian, JD; and our grandchildren: Aarti Sultanian, Soraya Zahra Gharib, and Shayan Ali Gharib for the privilege of their unconditional love and for immensely enriching my life.

Contents Author������������������������������������������������������������������������������������������������������������������������������������������� xiii Preface to the Third Edition��������������������������������������������������������������������������������������������������������xv Preface to the Second Edition���������������������������������������������������������������������������������������������������� xix Preface to the First Edition��������������������������������������������������������������������������������������������������������� xxi

1. Turbomachinery History, Classifications, and Applications����������������������������������������1 1.1 Introduction������������������������������������������������������������������������������������������������������������������� 1 1.2 History����������������������������������������������������������������������������������������������������������������������������1 1.3 Classifications of Turbomachines������������������������������������������������������������������������������� 3 1.3.1 General Classifications������������������������������������������������������������������������������������ 3 1.3.2 Typical Examples��������������������������������������������������������������������������������������������� 4 1.3.3 Gas Turbines�����������������������������������������������������������������������������������������������������6 1.4 Applications and Technology Development������������������������������������������������������������� 8 1.5 Concluding Remarks����������������������������������������������������������������������������������������������������9 References�������������������������������������������������������������������������������������������������������������������������������10 Bibliography���������������������������������������������������������������������������������������������������������������������������10 2. Basic Concepts and Relations of Aerothermodynamics���������������������������������������������� 13 2.1 Introduction����������������������������������������������������������������������������������������������������������������� 13 2.2 Incompressible versus Compressible Flow������������������������������������������������������������� 14 2.2.1 Total Temperature and Pressure������������������������������������������������������������������ 14 2.2.2 Mass Flow Rate���������������������������������������������������������������������������������������������� 15 2.2.3 Compressible Mass Flow Functions������������������������������������������������������������ 16 2.3 Energy Equation���������������������������������������������������������������������������������������������������������� 18 2.4 Linear Momentum Equation������������������������������������������������������������������������������������� 19 2.5 Angular Momentum Equation��������������������������������������������������������������������������������� 20 2.5.1 Euler’s Turbomachinery Equation���������������������������������������������������������������22 2.6 Velocity Diagram�������������������������������������������������������������������������������������������������������� 25 2.7 Applications����������������������������������������������������������������������������������������������������������������� 27 2.7.1 Axial-Flow Impulse Turbine������������������������������������������������������������������������ 28 2.7.2 Axial-Flow Compressor�������������������������������������������������������������������������������� 29 2.7.3 Centrifugal Pump������������������������������������������������������������������������������������������ 30 2.7.4 Hydraulic Turbine������������������������������������������������������������������������������������������ 32 2.8 Discussion on Further Applications������������������������������������������������������������������������� 32 2.9 Concluding Remarks�������������������������������������������������������������������������������������������������� 33 Worked Examples������������������������������������������������������������������������������������������������������������������34 Problems����������������������������������������������������������������������������������������������������������������������������������44 Reference���������������������������������������������������������������������������������������������������������������������������������49 Bibliography���������������������������������������������������������������������������������������������������������������������������49 Nomenclature������������������������������������������������������������������������������������������������������������������������� 49

vii

viii

Contents

3. Dimensionless Quantities�������������������������������������������������������������������������������������������������� 53 3.1 Introduction����������������������������������������������������������������������������������������������������������������� 53 3.2 Turbomachine Variables�������������������������������������������������������������������������������������������� 53 3.3 Similitude��������������������������������������������������������������������������������������������������������������������� 55 Worked Examples������������������������������������������������������������������������������������������������������������������58 Problems����������������������������������������������������������������������������������������������������������������������������������60 References�������������������������������������������������������������������������������������������������������������������������������63 Nomenclature������������������������������������������������������������������������������������������������������������������������� 63 4. Centrifugal Pumps and Fans���������������������������������������������������������������������������������������������65 4.1 Introduction�����������������������������������������������������������������������������������������������������������������65 4.2 Impeller Flow���������������������������������������������������������������������������������������������������������������65 4.3 Efficiency���������������������������������������������������������������������������������������������������������������������� 68 4.4 Performance Characteristics�������������������������������������������������������������������������������������� 70 4.5 Design of Pumps��������������������������������������������������������������������������������������������������������� 74 4.6 Fans�������������������������������������������������������������������������������������������������������������������������������77 Worked Examples������������������������������������������������������������������������������������������������������������������78 Problems����������������������������������������������������������������������������������������������������������������������������������84 References�������������������������������������������������������������������������������������������������������������������������������88 Bibliography���������������������������������������������������������������������������������������������������������������������������88 Nomenclature������������������������������������������������������������������������������������������������������������������������� 89 5. Centrifugal Compressors���������������������������������������������������������������������������������������������������� 91 5.1 Introduction����������������������������������������������������������������������������������������������������������������� 91 5.2 Impeller Design����������������������������������������������������������������������������������������������������������� 93 5.3 Diffuser Design����������������������������������������������������������������������������������������������������������� 96 5.4 Performance����������������������������������������������������������������������������������������������������������������� 99 Worked Examples����������������������������������������������������������������������������������������������������������������102 Problems��������������������������������������������������������������������������������������������������������������������������������108 References����������������������������������������������������������������������������������������������������������������������������� 112 Bibliography������������������������������������������������������������������������������������������������������������������������� 112 Nomenclature����������������������������������������������������������������������������������������������������������������������� 112 6. Axial-Flow Pumps, Fans, and Compressors����������������������������������������������������������������� 115 6.1 Introduction��������������������������������������������������������������������������������������������������������������� 115 6.2 Stage Pressure Rise��������������������������������������������������������������������������������������������������� 118 6.3 Losses�������������������������������������������������������������������������������������������������������������������������� 120 6.4 Pump Design������������������������������������������������������������������������������������������������������������� 121 6.5 Fan Design����������������������������������������������������������������������������������������������������������������� 125 6.6 Compressor Design�������������������������������������������������������������������������������������������������� 127 6.7 Compressor Performance���������������������������������������������������������������������������������������� 131 Worked Examples����������������������������������������������������������������������������������������������������������������133 Problems��������������������������������������������������������������������������������������������������������������������������������138 References�����������������������������������������������������������������������������������������������������������������������������142 Bibliography�������������������������������������������������������������������������������������������������������������������������143 Nomenclature����������������������������������������������������������������������������������������������������������������������� 143

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7. Radial-Flow Gas Turbines������������������������������������������������������������������������������������������������ 147 7.1 Introduction��������������������������������������������������������������������������������������������������������������� 147 7.2 Basic Theory��������������������������������������������������������������������������������������������������������������� 148 7.3 Design������������������������������������������������������������������������������������������������������������������������� 157 Worked Example������������������������������������������������������������������������������������������������������������������159 Problems��������������������������������������������������������������������������������������������������������������������������������160 References�����������������������������������������������������������������������������������������������������������������������������162 Bibliography�������������������������������������������������������������������������������������������������������������������������163 Nomenclature����������������������������������������������������������������������������������������������������������������������� 163 8. Axial-Flow Gas Turbines�������������������������������������������������������������������������������������������������� 167 8.1 Introduction��������������������������������������������������������������������������������������������������������������� 167 8.2 Basic Theory��������������������������������������������������������������������������������������������������������������� 170 8.3 Design������������������������������������������������������������������������������������������������������������������������� 175 Worked Examples����������������������������������������������������������������������������������������������������������������180 Problems��������������������������������������������������������������������������������������������������������������������������������183 References�����������������������������������������������������������������������������������������������������������������������������186 Bibliography������������������������������������������������������������������������������������������������������������������������� 186 Nomenclature����������������������������������������������������������������������������������������������������������������������� 186 9. Steam Turbines������������������������������������������������������������������������������������������������������������������� 189 9.1 Introduction��������������������������������������������������������������������������������������������������������������� 189 9.2 Impulse Turbines������������������������������������������������������������������������������������������������������ 190 9.3 Reaction Turbines������������������������������������������������������������������������������������������������������ 193 9.4 Design������������������������������������������������������������������������������������������������������������������������� 195 Problems��������������������������������������������������������������������������������������������������������������������������������195 References�����������������������������������������������������������������������������������������������������������������������������196 Bibliography�������������������������������������������������������������������������������������������������������������������������196 10. Hydraulic Turbines������������������������������������������������������������������������������������������������������������ 197 10.1 Introduction��������������������������������������������������������������������������������������������������������������� 197 10.2 Pelton Wheel�������������������������������������������������������������������������������������������������������������� 197 10.3 Francis Turbine���������������������������������������������������������������������������������������������������������� 199 10.4 Kaplan Turbine���������������������������������������������������������������������������������������������������������� 201 10.5 Cavitation������������������������������������������������������������������������������������������������������������������� 202 Problems��������������������������������������������������������������������������������������������������������������������������������203 References�����������������������������������������������������������������������������������������������������������������������������203 Bibliography�������������������������������������������������������������������������������������������������������������������������203 11. Wind Turbines��������������������������������������������������������������������������������������������������������������������� 205 11.1 Introduction��������������������������������������������������������������������������������������������������������������� 205 11.2 Actuator Theory�������������������������������������������������������������������������������������������������������� 205 11.3 Horizontal-Axis Machines��������������������������������������������������������������������������������������� 209 11.4 Vertical-Axis Machines�������������������������������������������������������������������������������������������� 212 Problems��������������������������������������������������������������������������������������������������������������������������������214 References�����������������������������������������������������������������������������������������������������������������������������214 Bibliography������������������������������������������������������������������������������������������������������������������������� 214

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12. Gas-Turbine Exhaust Diffusers��������������������������������������������������������������������������������������� 215 12.1 Introduction��������������������������������������������������������������������������������������������������������������� 215 12.2 Roles of an Exhaust Diffuser����������������������������������������������������������������������������������� 215 12.3 Diffusion Process������������������������������������������������������������������������������������������������������ 217 12.3.1 Diffusion in a Constant-Area Duct����������������������������������������������������������� 218 12.3.2 Pressure Recovery in a Dump Diffuser���������������������������������������������������� 221 12.4 Performance Evaluation�������������������������������������������������������������������������������������������223 12.4.1 Diffuser Isentropic Efficiency��������������������������������������������������������������������223 12.4.2 Pressure Rise Coefficient���������������������������������������������������������������������������� 224 12.4.3 Axial Stream Thrust Coefficient���������������������������������������������������������������� 228 12.5 Six Simple Design Rules������������������������������������������������������������������������������������������� 229 12.6 Concluding Remarks������������������������������������������������������������������������������������������������ 231 References�����������������������������������������������������������������������������������������������������������������������������231 Bibliography�������������������������������������������������������������������������������������������������������������������������232 Nomenclature����������������������������������������������������������������������������������������������������������������������� 232 13. Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design����������������������������������������������������������������������������������������������������������������� 233 13.1 Introduction��������������������������������������������������������������������������������������������������������������� 233 13.2 CFD Methodology����������������������������������������������������������������������������������������������������234 13.3 The Common Form of Governing Conservation Equations������������������������������ 236 13.3.1 The Common Equation Form�������������������������������������������������������������������� 236 13.4 Turbulence Modeling����������������������������������������������������������������������������������������������� 237 13.4.1 Reynolds Equations: The Closure Problem���������������������������������������������� 237 13.4.2 High-Reynolds-Number Two-Equation k-ε Model��������������������������������� 239 13.5 Boundary Conditions����������������������������������������������������������������������������������������������� 240 13.5.1 Inlet and Outlet Boundary Conditions����������������������������������������������������� 240 13.5.2 Wall Boundary Conditions: The Wall-Function Treatment������������������� 241 13.5.3 Alternative Near-Wall Treatments������������������������������������������������������������ 243 13.5.4 Choice of a Turbulence Model������������������������������������������������������������������� 244 13.6 Physics-Based Post-Processing of CFD Results���������������������������������������������������� 244 13.6.1 Large Control Volume Analysis of CFD Results������������������������������������� 245 13.6.2 Entropy Map Generation���������������������������������������������������������������������������� 246 13.7 Turbomachinery Aerodynamic Design Process��������������������������������������������������� 246 13.7.1 3-D Flow Field���������������������������������������������������������������������������������������������� 246 13.7.2 Preliminary Design������������������������������������������������������������������������������������� 247 13.7.3 Detailed Design�������������������������������������������������������������������������������������������� 248 13.7.4 Role of 3-D CFD������������������������������������������������������������������������������������������� 251 13.8 Concluding Remarks������������������������������������������������������������������������������������������������ 251 References�����������������������������������������������������������������������������������������������������������������������������251 Bibliography�������������������������������������������������������������������������������������������������������������������������252 Nomenclature����������������������������������������������������������������������������������������������������������������������� 253 Appendix A: Tables of Conversion Factors, Pump Efficiency, and Compressor Specific Speed��������������������������������������������������������������������������������������� 257 Table A.1 Conversion Factors���������������������������������������������������������������������������������������������257 Table A.2 Pump Efficiency as a Function of Specific Speed and Capacity������������������258 Table A.3 Compressor Specific Diameter as a Function of Specific Speed and Efficiency������������������������������������������������������������������������������������������������������������ 258

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Appendix B: Derivation of Equation for Slip Coefficient������������������������������������������������� Appendix C: Formulation of Equation for Hydraulic Loss in Centrifugal Pumps with Backward-Curved Blades�������������������������������������������������������������������������� 261 Reference�������������������������������������������������������������������������������������������������������������������������������262 Appendix D: Viscous Effects on Pump Performance�������������������������������������������������������� 263 Table D.1 Viscosities of Liquids at 70°F���������������������������������������������������������������������������� 263 Table D.2 Correction Factors for Oil with Kinematic Viscosity of 176 Centistokes����������������������������������������������������������������������������������������������������������263 Appendix E: Comparison of Formulas for Compressor Slip Coefficient, μs = Vu2′/Vu2�������������������������������������������������������������������������������������������������������������������������� 265 References�����������������������������������������������������������������������������������������������������������������������������265 Appendix F: Molecular Weight of Selected Gases�������������������������������������������������������������� 267 Table F.1 Molecular Weight of Selected Gases���������������������������������������������������������������� 267 Appendix G: Pressure and Temperature Changes in Isentropic Free and Vortex Vortices������������������������������������������������������������������������������������������������������������ 269 G.1 Introduction������������������������������������������������������������������������������������������������������������������� 269 G.2 Free and Forced Vortices���������������������������������������������������������������������������������������������� 269 G.3 Changes in Static Temperature and Pressure����������������������������������������������������������� 270 G.3.1 A Simple Approach����������������������������������������������������������������������������������������� 272 G.4 Changes in Total Temperature and Pressure������������������������������������������������������������ 273 Reference�������������������������������������������������������������������������������������������������������������������������������274 Appendix H: Dimensionless Velocity Diagrams for Axial-Flow Compressors and Turbines������������������������������������������������������������������������������������������������������������������������ 275 H.1 Introduction������������������������������������������������������������������������������������������������������������������� 275 H.2 Performance Parameters���������������������������������������������������������������������������������������������275 H.2.1 Flow Coefficient����������������������������������������������������������������������������������������������275 H.2.2 Loading Coefficient����������������������������������������������������������������������������������������275 H.2.3 Stage Reaction�������������������������������������������������������������������������������������������������276 H.3 Dimensionless Velocity Diagrams������������������������������������������������������������������������������ 278 H.3.1 Derivations of Equations to Compute Velocities and Angles of Dimensionless Velocity Diagram���������������������������������������������������������� 280 H.3.2 Using φ, ψ, and R to Quickly Draw a Dimensionless Velocity Diagram���������������������������������������������������������������������������������������283 Appendix I: Throughflow Design with Simple Radial Equilibrium Equation������������ 287 I.1 Introduction���������������������������������������������������������������������������������������������������������������������287 I.2 Radial Equilibrium Equation����������������������������������������������������������������������������������������287 I.3 Vortex Energy Equation������������������������������������������������������������������������������������������������� 290 I.4 Free-Vortex Design��������������������������������������������������������������������������������������������������������� 291 I.4.1 Degree of Reaction in an Axial-Flow Turbomachine����������������������������������� 292 I.5 General Vortex Design��������������������������������������������������������������������������������������������������� 292 I.6 Throughflow Design Project����������������������������������������������������������������������������������������� 293

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I.6.1 Axial-Flow Compressor Layout and Nomenclature�����������������������������������293 I.6.2 Governing Equations���������������������������������������������������������������������������������������295 I.6.3 Case 1: Single-Stage Compressor with Potential Flow (Free-Vortex Design)������������������������������������������������������������������������������������297 I.6.4 Case 2: Single-Stage Compressor with Specified Flow Angles and Efficiency�������������������������������������������������������������������������������������������������������298 Table I.1 Radial Distributions of Exit Absolute Flow Angles and Stage Efficiency��������������������������������������������������������������������������������������������������299 References�����������������������������������������������������������������������������������������������������������������������������300 Nomenclature�����������������������������������������������������������������������������������������������������������������������300 Appendix J: Review of Necessary Mathematics������������������������������������������������������������������ 303 J.1 Suffix Notation and Tensor Algebra����������������������������������������������������������������������������303 J.1.1 Summation Convention�����������������������������������������������������������������������������������303 J.1.2 Free and Dummy Indices��������������������������������������������������������������������������������303 J.1.3 Two Special Symbols����������������������������������������������������������������������������������������304 J.2 Gradient, Divergence, Curl, and Laplacian����������������������������������������������������������������304 J.2.1 Gradient�������������������������������������������������������������������������������������������������������������304 J.2.2 Divergence��������������������������������������������������������������������������������������������������������� 305 J.2.3 Curl���������������������������������������������������������������������������������������������������������������������305 J.2.4 Laplacian������������������������������������������������������������������������������������������������������������306 J.3 Dyad in Total Derivative�����������������������������������������������������������������������������������������������306 J.4 Total Derivative���������������������������������������������������������������������������������������������������������������306 J.5 Vector Identities��������������������������������������������������������������������������������������������������������������307 Bibliography�������������������������������������������������������������������������������������������������������������������������307 Index���������������������������������������������������������������������������������������������������������������������������������������������309

Author

Bijay (BJ) K. Sultanian, PhD, PE, MBA, ASME Life Fellow is a recognized international authority in gas-turbine heat transfer, secondary air systems, and computational fluid dynamics (CFD). Dr. Sultanian is founder and managing member of Takaniki Communications, LLC (www.takaniki.com), a provider of high-impact, web-based and live technical training programs for corporate engineering teams. Dr. Sultanian is also an adjunct professor at the University of Central Florida, where he has taught graduate-level courses in turbomachinery and fluid mechanics since 2006. He has been an active member of ASME IGTI’s Heat Transfer Committee since 1994 and received the ASME IGTI Outstanding Service Award at ASME Turbo Expo 2018, Lillestrom, Norway. He is the author of two graduate-level textbooks: Fluid Mechanics: An Intermediate Approach, which was published in 2015, and Gas Turbines: Internal Flow Systems Modeling (Cambridge Aerospace Series), published in 2018. During his three decades in the gas-turbine industry, Dr. Sultanian has worked in and led technical teams at a number of organizations, including Allison Gas Turbines (now Rolls-Royce), GE Aircraft Engines (now GE Aviation), GE Power Generation (now GE Water & Power), and Siemens Energy (now Siemens Power & Gas). He has developed several physics-based improvements to legacy heat transfer and fluid systems design methods, including new tools to analyze critical high-temperature gas-turbine components with and without rotation. He particularly enjoys training large engineering teams at prominent firms around the globe on cutting-edge technical concepts and engineering and project management best practices. During his initial 10-year professional career, Dr. Sultanian made several landmark contributions toward the design and development of India’s first liquid rocket engine for a surface-to-air missile (Prithvi). He also developed the first numerical heat transfer model of steel ingots for optimal operations of soaking pits in India’s steel plants. Dr. Sultanian is a Life Fellow of the American Society of Mechanical Engineers, a registered Professional Engineer (PE) in the State of Ohio, a GE-certified Six Sigma Green Belt, and an Emeritus Member of Sigma Xi, The Scientific Research Society. Dr. Sultanian received his BTech and MS in Mechanical Engineering from the Indian Institute of Technology, Kanpur, and the Indian Institute of Technology, Madras, respectively. He received his PhD in Mechanical Engineering from the Arizona State University, Tempe, and his MBA from the Lally School of Management and Technology at Rensselaer Polytechnic Institute.

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Preface to the Third Edition Professor Earl Logan, Jr., who was a professor in the Department of Mechanical and Aerospace Engineering at the Arizona State University, Tempe, during the period I was pursuing my PhD degree there (1981–1984), published the first edition of this book in 1981. This unbelievable little book, presenting the essentials of various turbomachinery designs in a little over 100 pages, became one of the best textbooks for instructing senior undergraduate students. Published in 1993, the second edition of the book maintained its concise and easy-to-read style as its hallmark. While the basic concepts remain ageless, turbomachinery design has considerably advanced over the last two decades in concert with various technological advances in this area. Most airfoils in modern turbomachines feature 3-D geometries, aided in large part by the application of computational fluid dynamics (CFD) as an indispensable design tool, while utilizing impressive advances in materials and manufacturing technologies, including additive manufacturing (3-D printing). The classic textbook on turbomachinery by Professor Logan must therefore embody this reality in order to continue to serve engineering students in universities and practising engineers in a diverse turbomachinery industry. Hopefully, the revisions implemented in this edition go a long way in achieving this objective without changing the core characteristics of simplicity and conciseness of this time-tested textbook. During my tenure as the manager of Compressor and Turbine Aero Design, CFD, and Methods Design Engineering at GE Power Generation, Schenectady, in mid-1990s and while teaching a graduate course on turbomachinery (EML5402—Turbomachinery) at the University of Central Florida, Orlando, as an adjunct faculty over the last decade, I realized that a good understanding of the basic concepts of aerothermodynamics has been continuously deteriorating among engineering students and designers. Overdependence on various design tools, including their integration and automation to keep pace with the shrinking design cycle time, has been further adding insult to injury for practicing design engineers. Our best hope to reverse this trend is to reinforce undergraduate instruction with a renewed emphasis on the total understanding of the related flow and heat transfer physics of turbomachinery design. Engineering students with an improved understanding of the fundamental concepts and conservation laws when transitioning from universities to turbomachinery companies will have the mindset that all design methods, tools, and practices should be physics based and data driven; otherwise, the resulting design will not be physically realizable. I am gratitude to Mr. Jonathan Plant, acquiring executive editor at Taylor & Francis, who believed in my passion to author my first dream book (Fluid Mechanics: An Intermediate Approach), which was published in 2015, and later gave me the honor to author this book, which is the third edition of Professor Logan’s classical textbook on turbomachinery. I wish to thank Mr. Edward Curtis, Ms Revathi Vishwanathan and her highly capable team, and all the staff at Taylor & Francis for their exemplary support and professional communications during the entire book-production process. In this generally updated and substantially expanded edition, Chapter 2 is completely rewritten to provide a more comprehensive but easy-to-understand review and reinforcement of some of the key concepts of fluid mechanics (both incompressible and compressible flows), a stronger foundation in control volume analyses of various conservation laws, xv

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Preface to the Third Edition

and an introduction to the concept of rothalpy, which also provides an easy way to convert total temperature and total pressure between stator and rotor reference frames. This edition also includes two new chapters (12 and 13) and four new appendices (G, H, I, and J). Chapter 12 presents gas-turbine exhaust diffusers, which are used in gas turbines for power generation in both simple- and combined-cycle operations. This chapter includes six simple rules for a world-class aerodynamic design of these diffusers, which are considered the low-hanging fruits for improving the overall cycle efficiency, which is approaching 65% for a combined gas-turbine and steam-turbine cycle operation. Chapter 13 provides an introduction to computational fluid dynamics and the vital role this technology continues to play in developing more efficient 3-D flowpath designs of all types of turbomachines. The presentation on how 1-D meanline analysis (the major focus of the entire book) in the preliminary design phase works in concert with axisymmetric throughflow analysis, 2-D blade-to-blade analysis, and 3-D CFD analysis in the detailed design phase for a typical state-of-the-art aerodynamic design of turbomachines is very instructive. The chapter also includes a physics-based method to post-process CFD results, including entropy map generation from these results. This entropy map for the turbomachinery flowpath provides a valuable insight into regions of excess loss for further design improvement. Appendix G presents the must-know concepts of free and forced vortices, which are ubiquitous in all rotating machines. For isentropic free and forced vortices, this appendix presents a simple approach to computing changes in static temperature and static pressure between any two points in the vortex. In fact, the approach is so simple that one need not remember any complicated equations and may obtain the results from “back of the envelop” calculations. While teaching the graduate-level course EML5402—Turbomachinery at the University of Central Florida, I developed a quick graphical technique to draw dimensionless velocity diagrams for axial-flow compressors and turbines, directly using their design parameters: loading coefficient, flow coefficient, and stage reaction. To my knowledge, this technique is not currently published in any turbomachinery textbook; it is included here in Appendix H. My course syllabus also required students to complete both cases of a project on the throughflow design of a turbofan engine compressor using a simple radial equilibrium equation. Case 1 assumed irrotational free-vortex flow, and case 2 specified radially varying absolute flow angles and efficiencies. Essential details on this project are given in Appendix I. By completing this course project, my students were found to gain valuable insight into the 3-D nature of a turbomachinery flow. Appendix J is intended for students to review and reinforce their math skills. This expanded edition is suitable for turbomachinery courses taught at both undergraduate and graduate levels. For each level, I suggest the following syllabus for a three-credit course in a 16-week semester: Undergraduate-Level Course in Turbomachinery: Weeks 1–4: Chapters 1, 2, and 3 Week 5: Chapter 12 Weeks 6–7: Chapter 4 Weeks 8–9: Chapter 5 Weeks 10–11: Chapter 6 and Appendix H Weeks 12–13: Chapter 7

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Weeks 14–15: Chapter 8 and Appendix H Week 16: Course Project (Appendix I, Case 1) Graduate-Level Course in Turbomachinery: Weeks 1–2: Chapters 1, 2, and 3 Week 3: Chapter 12 Weeks 4–5: Chapter 4 Weeks 6–7: Chapter 5 Weeks 8–9: Chapter 6 and Appendix H Weeks 10–11: Chapter 7 Weeks 12–13: Chapter 8 and Appendix H Week 14: Chapter 13 Weeks 15–16: Course Project (Appendix I, both Case 1 and Case 2) However, course instructors are free to fine-tune the suggested syllabi and reinforce them with their notes and/or additional reference material to meet their specific instructional needs. Dr. Bijay K. Sultanian Founder and Managing Member, Takaniki Communications, LLC Adjunct Professor, University of Central Florida

Preface to the Second Edition Use of the first edition during the past 12 years has led to the development of supplementary material for classroom instruction. This material has been integrated into the present edition. The new material comprises equations, graphs, symbol lists, and illustrative examples that clarify the theory and demonstrate the use of basic relations in performance calculations and design. Additionally, a large number of problems have been added in Chapters 2 through 8. Most of these problems were developed as numerical or analytical exercises; however, a few were generated for design projects. In the latter case, the designs correspond to existing hardware for which performance data and dimensions are available. Some material used in the first edition was relocated in the second edition, and some was expanded to provide a complete but concise picture of current knowledge useful in preliminary design. The axial-flow pump material from Chapter 6 has been relocated to Chapter 4. The axial-flow fan and compressor material has been combined in Chapter 6. Chapter 7 treats radial-flow gas turbines only and is an expansion of material formerly in the single gas-turbine chapter. Chapter 8 was enlarged and handles axial-flow gas turbines exclusively. The reorganization has evolved from attempts to find packages of material suitable for classroom instruction that are optimal in both size and content. Although the book was spawned from classroom instruction of fourth-year engineering students, it can be used by practicing engineers outside the classroom and by engineering technology students in the classroom environment. It is assumed that the student or practicing engineer has studied basic fluid mechanics and thermodynamics. With this academic background, one should be able to undertake and complete a study of this volume. There is sufficient material in the first eight chapters for a one-semester course. It is recommended that the first four weeks be devoted to Chapters 1 through 3, three weeks to Chapter 4, and two weeks each to Chapters 5, 6, 7, and 8. Experience with the material presented in this edition has indicated that, after 15 weeks of instruction in a three-semesterhour course, students are able to perform a very satisfactory preliminary design of any of several types of turbomachines. Earl Logan, Jr.

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Preface to the First Edition This book is intended as a text for undergraduate students of engineering. The student should have had introductory courses in fluid mechanics and thermodynamics prior to taking a course for which this book is the assigned text. The book should also find usefulness in the reference libraries of graduate engineers who may not have studied the subject while in school. The plan of the book is first to present the basic principles of turbomachine theory and then to apply these principles to specific devices. The centrifugal pump is studied first, since it is both a common and a simple device. Then more complex machines are considered, concluding with the basic types of hydraulic turbines. Each chapter seeks to address the questions of how the principles may be applied in design and how they may be used to predict the performance of the turbomachine under consideration. A conscious effort to minimize theoretical detail has been made, with the objective of maintaining a clear and unified exposition of the basics. References are cited wherein the student may find more information, if so desired. The contribution of the problems on wind turbines by Professor Robert H. Kirchhoff of the University of Massachusetts is gratefully acknowledged. The author is indebted to many teachers and students who have, over the years, shed light on the subjects discussed. Many sources have been drawn from in preparing the text, and these sources have been cited at appropriate places. Appreciation is expressed to the Arizona State University and, in particular, to Drs. George C. Beakley, Warren Rice, and Darryl E. Metzger, who during their tenure as Chairmen of Mechanical Engineering have permitted me to teach the subject matter contained herein. Earl Logan, Jr.

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1 Turbomachinery History, Classifications, and Applications

1.1 Introduction Turbomachines, collectively called turbomachinery, constitute a large class of machines and, as energy conversion devices, are an integral part of our civilized world. They include such devices as pumps, turbines, fans, and compressors. Each turbomachine involves mechanical energy transfer, in the form of shaft work, between a continuously flowing fluid and a bladed rotor. If the energy transfer is from the rotor to the fluid, the device is called a pump (liquid flow), fan (low pressure rise), or compressor (high pressure rise); if it is from the fluid to the rotor, the machine is called a turbine (gas, steam, or hydraulic). The fluid flow in a turbomachine could be incompressible (liquid or gas flow at Mach numbers less than 0.3) or compressible (gas flow at high subsonic and transonic Mach numbers). The purpose of the aforementioned energy conversion process is either to pressurize the fluid or to produce power. Useful work done by the fluid on the turbine rotor appears outside the casing as work done in turning—for example, turning the rotor of a generator. A pump, however, receives energy from an external electric motor and imparts this energy to the fluid by the dynamic action of its rotating impeller. The temperature and pressure of the fluid are increased by a pumping-type turbomachine, and the same properties are reduced in its passage through a work-producing turbomachine. A water pump might be used to raise the pressure of water, causing it to flow up into a reservoir through a pipe against the resistance of frictional and gravitational forces. The pressure at the bottom of a reservoir could be used to produce a flow through a hydraulic turbine, which would then produce the rotor torque against the resistance to turning offered by the connected electric generator.

1.2 History According to the historical account by Logan (2003), the knowledge of turbomachines has evolved slowly over centuries without the benefit of sudden and dramatic breakthroughs. Turbomachines such as windmills and waterwheels are centuries old. Waterwheels, which dip their buckets (blades) into moving water, were employed in ancient Egypt, China, and Assyria (see Daugherty, 1920). They appeared in Greece in the second century B.C. and in the Roman Empire during the first century B.C. A 7 ft diameter waterwheel at Monte Cassino was used by the Romans to grind corn at the rate of 150 kg of corn per hour, and 1

2

Logan’s Turbomachinery

waterwheels at Arles ground 320 kg of corn per hour (see Gimpel, 1976). The Doomsday Book, based on a survey ordered by William the Conqueror, indicates that there were 5,624 water mills in England in 1086. Besides the grinding of grain, waterwheels were used to drive water pumps and to operate machinery. Agricola (1950) shows by illustrations how in the 16th century waterwheels were used to pump water from mines and to crush metallic ores. In Marly, France, in 1685, Louis XIV had 221 piston pumps installed for supplying 3200 m3 of Seine River water per day to the fountains of the Versailles palace. The pumps were driven by 14 waterwheels, each 12 m in diameter, which were turned by the currents of the Seine (see Klemm, 1959). The undershot waterwheel, which had an efficiency of only 30%, was used up until the end of the 18th century. It was replaced in the 19th century by the overshot waterwheel, which had an efficiency of 70 to 90%. By 1850, hydraulic turbines began to replace waterwheels (see Daugherty, 1920). The first hydroelectric power plant was built in Germany in 1891 and utilized waterwheels and direct-current power generation. However, the waterwheels were soon replaced with hydraulic turbines and alternating-current electric power (see Thirring, 1958). Although the use of wind power in sailing vessels appeared in antiquity, the widespread use of wind power for grinding grain and pumping water was delayed until the 7th century in Persia, the 12th century in England, and the 15th century in Holland (see Johnson, 1945). In the 17th century, Gottfried Leibniz proposed using windmills and waterwheels together to pump water from mines in the Harz Mountains of Germany (see Klemm, 1959). Dutch settlers brought Dutch mills to America in the 18th century. This led to the development of a multiblade wind turbine that was used to pump water for livestock. Wind turbines were used in Denmark in 1890 to generate electric power. Early in the 20th century, American farms began to use wind turbines to drive electricity generators for charging storage batteries. These wind-electric plants were supplanted later by electricity generated by centrally located steam-electric power plants, particularly after the Rural Electric Administration Act of 1936 (see Johnson, 1945). In the second century B.C., Hero of Alexandria invented rotors driven by steam (see Klemm, 1959) and by gas (see Sawyer, 1945), but these machines produced insignificant amounts of power. During the 18th and 19th centuries, the reciprocating steam engine was developed and became the predominant prime mover for the manufacturing and transportation industries. In 1883, Gustaf de Laval constructed the first steam turbines, which achieved speeds of 26,000 rpm (see Stodola, 1927). In 1984, a steam turbine that ran at 17,000 rpm and comprised 15 wheels on the same shaft was designed and built by Charlie Parsons. The gas turbine was conceived by John Barber in 1791, and the first gas turbine was built and tested in 1900 by Franz Stolze (see Sawyer, 1945). Sanford Moss built a gas turbine in 1902 at Cornell University. At Brown Boveri in 1903, Réné Armengaud and Charles Lemale combined an axial-flow turbine and a centrifugal compressor to produce a thermal efficiency of 3% (see Sawyer, 1945). In 1905, Hans Holzwarth designed a gas turbine that utilized constant-volume combustion. This turbine was manufactured by Brown Boveri and Thyssen until the 1930s. In 1911, the turbocharger was built and installed in diesel engines by Sulzer Brothers. In 1918, the turbocharger was utilized to increase the power of military aircraft engines (see Sawyer, 1945). In 1930, Frank Whittle was granted a patent for a turbojet engine, followed by Hans von Ohain’s patent of the first operational turbojet engine in 1936. The first flight using a turbojet engine took place in 1939. That same year Brown and Boveri installed the first combustion gas turbine in Switzerland. A similar turbine was used in Swiss locomotives in 1942 (see Seippel, 1953). The aircraft gas turbine engine (turbojet) was developed by Junkers in Germany around 1940.

Turbomachinery History, Classifications, and Applications

3

The groundbreaking developments of the 1940s gave birth to the era of gas turbines. Since that time, tremendous advances have been made in the technology of gas turbines and their widespread use in aircraft propulsion, electric power plants, mechanical drives for industrial applications, oil and gas pipelines, marine propulsion and power generation, and other applications. Garvin (1998) traces the technical history of GE gas engines for aircraft propulsion, as do Connors (2010) for Pratt & Whitney engines and Rolls-Royce (2005) for Rolls-Royce engines. It is not an exaggeration to say that modern gas turbines have become synonymous with turbomachinery. In the early 1940s, who would have thought that, in 2018, we would have a turbofan engine (GE9XTM) with a 10:1 bypass ratio and a 60:1 overall pressure ratio for Boeing 777X airplanes. According to Vandervort, Wetzel, and Leach (2017), in April 2016, under the auspices of Guinness World Records, a 9HA.01 GTCC set a world record for the combined-cycle efficiency of 62.22% while producing more than 605 MW of electricity. In June 2016, GE and Électricité de France (Electricity of France) officially inaugurated the first 9HA combined-cycle power plant in Bouchain, France, and achieved a combinedcycle efficiency of over 62%.

1.3  Classifications of Turbomachines 1.3.1  General Classifications As shown in Figure 1.1, there are three ways to classify turbomachines. By function, all turbomachines fall into two categories. Those in the first, including the pump, fan, and compressor, absorb power to raise fluid pressure and temperature. Those in the second, including the turbine (gas, steam, and hydraulic), produce work while reducing fluid pressure and temperature. Another way to classify turbomachines is on the basis of the type of fluid they use (liquid or gas) and the flow regime they handle (incompressible or compressible). While the liquid flow can be treated as incompressible with constant density, the flow of a gas, which is a compressible fluid, can be approximated as incompressible at

FIGURE 1.1 General classifications of turbomachines.

4

Logan’s Turbomachinery

Mach numbers less than 0.3. At higher Mach numbers, we must consider the compressibility effects with the possibility of choking (Mach number equals 1) and normal shock formation in the supersonic regime. Yet another way to classify turbomachines is according to the direction of the primary flow of the working fluid in the axial-radial plane, which is also known as the meridional plane. Accordingly, all turbomachines are axial-flow types, radial-flow types, or mixed-flow types. 1.3.2  Typical Examples A typical turbomachine rotor, a centrifugal pump impeller, is shown schematically in Figure 1.2a. Liquid enters the eye E of the impeller axially, then turns to the radial direction, and finally emerges at the discharge D with both radial and tangential velocity components. The blades B impart a curvilinear motion to the fluid particles, thus setting up a radial centrifugal force, which is responsible for the outward flow of fluid against the resistance of wall friction and pressure forces. The blades mounted on the rotor impart energy to the fluid by virtue of pressure forces on their surfaces, which are undergoing a displacement as rotation takes place. Energy from an electric motor is supplied at a constant rate through the shaft S, which is assumed to be turning at a constant angular velocity. If the direction of the fluid flow shown in Figure 1.2a is reversed, the rotor becomes part of a turbine, and power is delivered through the shaft S to an electric generator or other load. Typically, hydraulic turbines have such a configuration (see Figure 1.2d) and are used to generate large amounts of electric power by admitting high-pressure water stored in dams to the periphery of such a rotor. A pressure drop occurs between the turbine’s inlet and outlet; the water exits axially and is discharged at atmospheric pressure. If the fluid flowing through the impeller of Figure 1.2a were a gas, then the device would be a centrifugal compressor, blower, or fan, depending on the magnitude of the pressure rise occurring from inlet to outlet. For the case with a radially inward flow, the machine would be called a radial-flow gas turbine or turboexpander. A different type of turbomachine is shown in Figure 1.2b. Here the flow direction is generally axial—i.e., parallel to the axis of rotation. Depending on the direction of energy flow and the kind of fluid present, the machine shown in this figure could be an axial-flow compressor or blower or, with a different blade shape, an axial-flow gas or steam turbine. In all of the machines mentioned thus far, the working fluid undergoes a change in pressure in flowing from inlet to outlet. Generally, the pressure changes in a diffuser, nozzle, and rotor. However, there is a class of turbines in which the pressure does not change in the rotor. These are called impulse, or zero-reaction, turbines, as distinguished from the so-called reaction turbines, which allow a pressure decrease in both the nozzle and the rotor. A hydraulic turbine with zero reaction is shown in Figure 1.2c, and a reaction-type turbine is shown in Figure 1.2d. Centrifugal, or radial-flow, turbomachines are depicted in Figures 1.2e through 1.2g, and axial-flow turbomachines are depicted in Figures 1.2h through 1.2j. A mixed-flow pump is shown in Figure 1.2k. This class of machine is partway between the centrifugal types and the axial-flow types. Figure 1.2l schematically shows a basic gas turbine, which consists of a compressor, a combustor, and a turbine. The role of the compressor is to increase the pressure and temperature of the ambient air. This high-pressure air enters the combustor, where the fuel chemical energy is released into the air flow by combustion. Hot gases at high

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Turbomachinery History, Classifications, and Applications

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FIGURE 1.2 (a) Pump impeller, (b) axial-flow blower, (c) Pelton wheel, (d) Francis turbine, (e) centrifugal pump, (f) centrifugal compressor, (g) centrifugal blower, (h) Kaplan turbine, (i) steam turbine, (j) axial-flow compressor, (k) mixedflow pump, and (l) basic gas turbine.

6

Logan’s Turbomachinery

pressure and high temperature are then expanded through the turbine to produce work, part or all of which is used to drive the compressor. Using this basic gas turbine, as discussed in Section 1.3.3, a variety of propulsion and shaft-power gas turbines are configured. Turbomachinery sizes vary from a few centimeters to several meters in diameter. Fluid states vary widely as well. Steam at near-critical conditions may enter one turbine, while cool river water enters another. Room air may enter one compressor, while cold refrigerant is drawn into a second. The materials encountered in the machines are selected to suit the temperatures, pressures, and chemical natures of the fluids involved, and the manufacturing methods include welding, casting, machining, and additive manufacturing (3-D printing). 1.3.3  Gas Turbines The core engine configuration shown in Figure 1.2l is modified for various applications in air-breathing propulsion and shaft-power generation. Because of their high power-toweight ratio, gas turbine engines are exclusively being used for all kinds of aircraft propulsion, both commercial and military. 1.3.3.1  Propulsion Gas Turbines Turbojet engine In this engine, the turbine produces enough shaft-power to drive the compressor. As shown in Figure 1.3a, high-pressure hot gases exiting the turbine are expanded in a nozzle to create a high-velocity jet, producing thrust for aircraft propulsion. Turboprop engine A turboprop engine is schematically shown in Figure 1.3b, where the high-pressure (HP) compressor is driven by the HP turbine. The low-pressure (LP) turbine drives the extended propeller at a lower rpm. Turboprops usually refer to gas turbine engines that provide shaft power to a propeller for fixed-wing aircraft propulsion. Gas turbine engines that provide power for rotary-wing aircraft, such as a helicopter, are referred to as turboshafts. Turbofan engine In this engine, shown in Figure 1.3c, the power output is split between propulsive thrust and shaft power to turn the fan (enclosed) in front of the engine. Most large commercial aircraft use turbofan engines operating at a high bypass ratio. 1.3.3.2  Shaft-Power Gas Turbines Marine and industrial gas turbines Due to their lighter weight, smaller footprint, and ability to use many different types of fuel, the aero-derivative engines, typically derived from a turbofan engine by eliminating the fan, are widely used on ships (for both propulsion and electric power generation), offshore oil platforms, and land vehicles.

Turbomachinery History, Classifications, and Applications

7

FIGURE 1.3 Propulsion gas turbines: (a) turbojet, (b) turboprop or turboshaft, and (c) turbofan.

Power generation gas turbines: simple cycle Gas turbine power plants use the shaft power of a turbine to turn a generator to produce electricity, as shown in Figure 1.4a. In simple-cycle operation, these power plants offer the significant advantages of a quick start and flexible loading to meet grid electricity demand. They generally operate at a fixed rotational speed, either 3000 rpm for 50 Hz (Europe) AC electricity or 3600 rpm for 60 Hz AC electricity (North America). Since these machines are designed for the same turbine tip speed, the gas turbines operating at 3000 rpm are larger in size than the ones rotating at 3600 rpm. The thermal efficiency of a state-of-the-art gas turbine used for power generation is in the mid-40% range.

8

Logan’s Turbomachinery

FIGURE 1.4 Shaft-power gas turbines: (a) simple cycle and (b) combined-cycle operation with steam turbine.

Power generation gas turbines: combined cycle To further boost the thermal efficiency of electric power generation, gas turbines are used in combination with steam turbines. As shown in Figure 1.4b, exhaust gases from the gas turbine are used to convert water into steam in the heat recovery steam generator (HRSG) unit. This steam is then used in the steam turbine to generate additional power. The thermal efficiency of a state-of-the-art combined gas turbine/steam turbine power plant is around 62%.

1.4  Applications and Technology Development The importance of turbomachines to our way of life cannot be overemphasized. The steam power plant, which remains a dominant means of electrical power generation in the world, can be used to illustrate this basic fact. The steam power plant consists of a prime mover driving a large electric generator. A steam turbine is usually used as the prime mover. Steam for the turbine is supplied from a boiler at high pressure and temperature. Water for

Turbomachinery History, Classifications, and Applications

9

making the steam is forced into the boiler by means of a multistage centrifugal pump. Fuel for creating the heat in the boiler is supplied by a pump, compressor, or blower, depending on the nature of the fuel. Air for combustion of the fuel enters the boiler through a large centrifugal fan. After the steam has been generated in the boiler and has expanded in the turbine, it is exhausted into a condenser where it is condensed and collected as condensate. Pumps are used to remove the condensate from the condenser and deliver it to feedwater heaters, from which it is drawn into the boiler feed pumps to repeat the cycle. The condensation process requires that large amounts of cooling water be forced through the tubes of the condenser by large centrifugal pumps. In many cases, the cooling water is itself cooled in cooling towers, which are effective because large volumes of outside air are forced through the towers by axial-flow fans. Gas turbines, due to their high power-to-weight ratio, have become the unchallenged leader in aircraft propulsion. Compared to steam turbines, gas turbines feature rapidstart capabilities, and their heavier and more durable designs for industrial applications ensure over a million hours of operation without a major overhaul. Aero-derivative gas turbines are the preferred source of propulsion and power generation on boats and ships. In the modern economy, applications of small and large gas turbines include the oil and gas industry, offshore platforms, utility peak-load power generation, emergency power, military tank propulsion, and distributed electricity generation (schools, hospitals, etc.). To achieve higher thermal efficiency of power generation, the modern trend is to use gas turbines in a combined-cycle operation with steam turbines (cogeneration), minimizing heat loss of gas turbine exhaust. The continued success of turbomachines, particularly gas turbines, since the 1940s can be attributed to advances made in materials technology, aerodynamics, cooling technology, and manufacturing methods, including additive manufacturing (3-D printing). In April 1956, the American Society of Mechanical Engineers (ASME) launched an international annual meeting, the ASME International Gas Turbine Conference and Exhibit, to present and discuss all aspects of turbomachinery technology. In 1988, this conference was renamed ASME TURBO EXPO. This conference continues to be held annually in a 5-day format with over 3000 attendees, presentations of over 1000 technical papers, workshops, tutorials, and topical panel sessions. In the review process, a number of papers from this conference are accepted for publication in either the ASME Journal of Engineering for Gas Turbines and Power or the ASME Journal of Turbomachinery. The conference also features a world-class exhibit showcasing the latest products and technology from leading original equipment manufacturers of turbomachinery and from ancillary industries.

1.5  Concluding Remarks Thus, we see that many turbomachines are required to operate the simplest form of modern steam-electric generating station and that gas turbines are used both for power generation in simple and combined cycles and for aircraft propulsion. No doubt the modern world and the entire economy depend on turbomachines in these and many other applications. Our consideration herein of the subject of turbomachines includes a wide variety of forms and shapes, made of a variety of materials using a number of techniques. This book does not attempt to deal with all the problems encountered by the designer or user of turbomachines but only with the most general aspects of the total problem. The present

10

Logan’s Turbomachinery

treatment is concerned with the specification of principal dimensions and forms of those turbomachines encountered frequently in industry. Before we consider the specifics of pumps, compressors, and turbines, we must deal with the fundamentals underlying their design and performance. In Chapters 2 and 3, we will develop these underlying principles of all turbomachines by starting our discussion with the first principles of thermofluids: i.e., conservation of mass, linear momentum, angular momentum, and energy.

References Agricola, G. 1950. De Re Metallica (trans. H.C. Hoover and L.H. Hoover). New York: Dover. Connors, J. 2010. The Engines of Pratt & Whitney: A Technical History (ed. N. Allen). Reston, VA: AIAA. Daugherty, R.L. 1920. Hydraulic Turbines. New York: McGraw-Hill. Garvin, R.V. 1998. Starting Something Big: The Commercial Emergence of GE Aircraft Engines. Reston, VA: AIAA. Gimpel, J. 1976. The Medieval Machine. New York: Penguin. Johnson, G.L. 1945. Wind Energy Systems. New York: Prentice-Hall. Klemm, F. 1959. A History of Western Technology. New York: Scribner. Logan, E., Jr. 2003. Introduction. In Handbook of Turbomachinery, 2nd edition (ed. E. Logan, Jr., and R.P. Roy). Boca Raton, FL: Taylor & Francis. Rolls-Royce. 2005. The Jet Engine. Hoboken, NJ: Wiley. Sawyer, R.T. 1945. The Modern Gas Turbine. New York: Prentice-Hall. Seippel, C. 1953. Gas turbines in our century. Transactions of the ASME 75: 121–122. Smith, G.G. 1944. Gas Turbines and Jet Propulsion for Aircraft. New York: Atmosphere. Stodola, A. 1927. Gas Turbines (vol. 1). New York: McGraw-Hill. Thirring, H. 1958. Energy for Man: Windmills to Nuclear Power. Bloomington: Indiana University Press. Vandervort, C., T. Wetzel, and D. Leach. 2017. Engineering and validating a world record gas turbine. Mechanical Engineering 12(139): 48–50.

Bibliography Balje, O.E. 1981. Turbomachines: A Guide to Design, Selection and Theory. New York: Wiley. Bashta, T.M., S.S. Rudnev, and B.B. Nekrasov. 1982. Hydraulic and Hydraulic Machines. (in Russian). Moscow: M. Mashinostronic. Baskharone, E.A. 2006. Principles of Turbomachinery in Air-Breathing Engines. New York: Cambridge University Press. Bathie, W.W. 1995. Fundamentals of Gas Turbines, 2nd edition. New York: Wiley. Bidard, R., and J. Bonnin. 1979. Energetique et Turbomachines. Paris, France: Eyrolles. Bölcs, A., and P. Suter. 1986. Transsonische Turbomaschinen. Karlsruhe, Germany: G. Braun. Brennen, C.E. 2011. Hydrodynamics of Pumps. New York: Cambridge University Press. Csanady, G.T. 1964. Theory of Turbomachinery. New York: McGraw-Hill. Cumpsty, N.A. 2004. Compressor Aerodynamics, 2nd edition. Malabar, FL: Krieger. Cumpsty, N., and A. Heyes. Jet Propulsion: A Simple Guide to the Aerodynamics and Thermodynamic Design and Performance of Jet Engines, 3rd edition. New York: Cambridge University Press. Dixon, S.L., and C. Hall. 2013. Fluid Mechanics and Thermodynamics of Turbomachinery, 7th edition. Waltham, MA: Elsevier.

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Eck, B. 1973. Fans. Elmsford, NY: Pergamon Press. Eckert, B., and E. Schnell. 2013. Axial- und Radialkompressoren: Anwendung/Theorie/Berechnung, 2nd edition. New York: Springer. Flack, R.D. 2010. Fundamentals of Jet Propulsion with Applications. New York: Cambridge University Press. Gostelow, J.P. 1984. Cascade Aerodynamics. Elmsford, NY: Pergamon Press. Harman, R.T.C. 1981. Gas Turbine Engineering: Applications, Cycles and Characteristics. New York: Wiley. Hawthorne, W.R. 1964. Aerodynamics of Turbines and Compressors. Princeton, NJ: Princeton University Press. Holschevnikov, K.V., O.N. Emin, and V.T. Mitrohin. 1986. Theory and Design of Blade Machine (in Russian). Moscow: M. Mashinostronic. Horlock, J.H. 1973. Axial Flow Turbines: Fluid Mechanics and Thermodynamics. Malabar, FL: Krieger. Horlock, J.H. 1982. Axial Flow Compressors: Fluid Mechanics and Thermodynamics. Malabar, FL: Krieger. Japikse, D., and N.C. Baines. 1997. Introduction to Turbomachinery. White River Junction, VT: Concepts ETI, Inc. Lakshminarayana, B. 1995. Fluid Dynamics and Heat Transfer of Turbomachinery. New York: Wiley. Li, G., N.X. Chen, and G. Qiang. 1980. Aerothermodynamics of Axial Turbomachinery for Marine Gas Turbines (in Chinese). Beijing: Defence Industry Press. Oates, G. 1997. Aerothermodynamics of Gas Turbine and Rocket Propulsion, 3rd edition. Reston, VA: AIAA. Rangwala, A.S. 2005. Turbo-machinery Dynamic: Design and Operation. New York: McGraw-Hill. Rangwala, A.S. 2013. Theory and Practice in Gas Turbines, 2nd edition. London: New Academic Science Limited. El-Sayed, A.F. 2017. Aircraft Propulsion and Gas Turbine Engines, 2nd edition. Boca Raton, FL: Taylor & Francis. Soares, C. 2014. Gas Turbines: A Handbook of Air, Land and Sea Applications, 2nd edition. New York: Elsevier. Turton, R.K. 1995. Principles of Turbomachinery, 2nd edition. New York: Chapman & Hall. Vavra, M.H. 1960. Aerothermodynamics and Flow in Turbomachinery. New York: Wiley. Wallis, R.A. 1983. Axial Flow Fans and Ducts. New York: Wiley. Walsh, P.P., and P. Fletcher. 2004. Gas Turbine Performance, 2nd edition. Malden, MA: Blackwell. Whitefield, A., and N.C. Baines. 1990. Design of Radial Turbomachines. New York: Longman Group. Wilson, D.G., and T. Korakianitis. 2014. The Design of High-Efficiency Turbomachinery and Gas Turbines, 2nd edition. Cambridge, MA: MIT Press. Wislicenus, G.F. 1965. Fluid Mechanics of Turbomachinery. New York: Dover.

2 Basic Concepts and Relations of Aerothermodynamics

2.1 Introduction All turbomachinery flows are governed by the laws of conservation of mass, momentum, energy, and so forth. A good understanding and an accurate application of these laws are critical to the physics-based design of each turbomachine. Such an understanding cannot be relegated to any design tool (software), which should be viewed simply as a computational aid (the modern version of a slide rule) to bring speed into the design process. This chapter is primarily focused on the review and reinforcement of the key conservation laws, related concepts, and mathematical relations, which are helpful for the aerothermodynamic design of turbomachinery primary flowpaths. They also enable accurate performance evaluation of these machines against design specifications. An important aspect of turbomachinery flowpath design is to understand the sources and mechanisms of various losses and how to minimize them for a nearly isentropic design with maximum efficiency. Rotation is a common feature of all turbomachines. The work transfer under a linear motion results from the product of the force and the displacement in the direction of the force. Similarly, the work transfer under rotation results from the product of torque and angular displacement. Accordingly, the angular momentum equation, which is simply the moment of the linear momentum equation about the axis of rotation, plays an important role in the analysis of turbomachines. Using the angular momentum equation in the inertial reference frame and the steady-flow energy equation, Euler’s turbomachinery equation is derived in this chapter. This equation leads to a useful quantity called rothalpy, which remains constant between any two points in the isentropic flow through a blade passage. In this chapter, we also introduce a general velocity diagram along with its sign conventions. These velocity diagrams constructed at blade inlet and outlet are helpful in computing design performance parameters such as the flow coefficient, loading coefficient, and reaction. They are widely used in the meanline analysis and design of each stage. For axial-flow compressors and turbines, we present in Appendix H a quick method to draw an inlet-outlet composite velocity diagram directly from the known three design performance parameters mentioned here.

13

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Logan’s Turbomachinery

2.2  Incompressible versus Compressible Flow Turbomachines deal with both incompressible and compressible flows. In hydraulic turbomachines, the liquid flow with constant density is considered incompressible. In fans and blowers, the flow of air, which is a compressible fluid, may be approximated as an incompressible flow at M ≤ 0.3. In compressors and gas turbines, the gas flow at M > 0.3 must be treated compressible. Another fundamental difference between an incompressible flow and a compressible flow pertains to the coupling between their internal and external energies. In an incompressible flow, these energies are not coupled. The extended Bernoulli equation, which is a mechanical energy equation, becomes the steady-flow energy equation for an incompressible flow. In a compressible flow, both internal and external energies are coupled with interchangeability, also called compressibility, as measured by the flow Mach number M = V/C = V/ γ RT . The square of the Mach number yields M 2 = V 2/( γ RT) , which is a measure of the ratio of external flow energy to its internal energy. Thus, the Mach number is a measure of how the total energy of a compressible flow is partitioned between its internal and external energies. For a compressible flow with constant total temperature (total enthalpy), as the flow Mach number increases, more of the flow internal energy appears as its external energy, resulting in a decrease in flow static temperature and an increase in flow velocity. A well-known example of a compressible flow is the isentropic flow in the convergent-divergent nozzle of a rocket engine. 2.2.1  Total Temperature and Pressure Assuming a calorically perfect fluid with constant specific heats, we can compute total, or stagnation, temperature of both incompressible and compressible flows as T0 = T +

V2 2c p

(2.1)

where the second term on the right-hand side is called the dynamic temperature. Note that Equation 2.1 does not involve density and the stagnation process needs to be only adiabatic, not isentropic (both adiabatic and reversible). For a compressible flow, we can express Equation 2.1 in terms of Mach number as



T0 γ −1 2 = 1+ M T 2

(2.2)

For an incompressible flow, total (stagnation) pressure is calculated as



p0 = p +

1 2 ρV 2

(2.3)

where the second term on the right-hand side is called the incompressible dynamic pressure, which is the difference between the total pressure and the static pressure. Note that the total pressure given by Equation 2.3 assumes an isentropic (loss-free) stagnation process in which the entire dynamic pressure is converted into pressure.

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Basic Concepts and Relations of Aerothermodynamics

Let us express Equation 2.3 as p 1  p0 = ρ  + V 2  ρ 2 

(2.4) where p/ρ is the specific (per unit mass) flow work (see Sultanian, 2015) and V 2/2 is the specific kinetic energy. Thus, the right-hand side of Equation 2.4 represents the total mechanical energy per unit volume of an incompressible flow. Since the density increases under isentropic stagnation of a compressible flow, we cannot use Equation 2.3 to compute its total pressure. In this case, we use the isentropic relation between the total-to-static pressure ratio and total-to-static temperature ratio along with Equation 2.2 to express γ



γ

p0  T0  γ − 1  γ − 1 2  γ −1 M  =  = 1+   p  T 2

(2.5)

For an ideal gas flow at a low Mach number (M ≤ 0.3) with a constant density given by the equation of state p/ρ = RT , we can express Equation 2.3 as p0 γ M2 = 1+ p 2

(2.6) For example, for γ = 1.4 and M = 0.3, we obtain p0 /p = 1.06443 from Equation 2.5 and p0 /p = 1.06300 from Equation 2.6, an error of 0.134%. Note that the quantity ρV 2/2 represents the dynamic pressure only for an incompressible flow, but it may be interpreted as twice the inertia force per unit area for both incompressible and compressible flows, as used in the definition of impulse pressure given by pi = p + ρV 2 (see Sultanian, 2015).

2.2.2  Mass Flow Rate We compute mass flow rate by the equation  = ρVφ A m (2.7) where ρ is the uniform fluid density and Vφ is the uniform velocity component normal to the flow area A. Note that the other velocity components, if present, do not contribute to mass flow rate; they do, however, contribute to their respective momentum flow rates through the section. Accordingly, Vφ may be interpreted as the mass velocity, as discussed in Section 2.4. When the properties are not uniform over the section, we obtain the mass flow rate by integration over the flow area as

 = m

∫ ρV dA A

φ

(2.8)

For a flow to occur at any section, we must have total pressure higher than the static pressure—i.e., a nonzero dynamic pressure. Combining Equations 2.3 and 2.7, we can compute the ideal mass flow rate of an incompressible flow through a section of area A as  = C V A 2ρ(p0 − p) m where the velocity coefficient C V = Vφ /V .

(2.9)

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2.2.3  Compressible Mass Flow Functions In a compressible flow, the mass flow functions provide a convenient means of computing the mass flow rate at a section without explicitly using the gas density, which does not remain constant in such a flow. This approach is widely used by practicing engineers in turbomachinery industries. Using the static-pressure mass flow function, we can write mass flow rate at a section as  = m

C V AFˆ f p C V AFf p = RT0 T0

(2.10)

where



γ − 1 2 Fˆ f = M γ  1 + M    2

(2.11)

γ γ − 1 2 M  1+  R 2

(2.12)

and



Ff = M

For a calorically perfect gas, both Fˆ f and Ff are functions of the Mach number only, the former being dimensionless and the latter having the dimensions of 1/ R . Equation 2.10 (see Problem 2.1 for its derivation) shows that, for a given Mach number, the mass flow rate at a section is proportional to the static pressure and inversely proportional to the square root of total temperature. Both the static pressure and the total temperature are uniform local values at that section. Using the total-pressure mass flow function, we can write mass flow rate as  = m

C V AFˆ f0 p0 C V AFf0 p0 = RT0 T0

(2.13)

where Fˆ f0 = M

γ

(2.14)

γ +1

 1 + γ − 1 M2  γ −1   2

and Ff0 = M

γ

(2.15)

γ +1

γ − 1 2  γ −1 R  1 + M    2



For a calorically perfect gas, both Fˆ f0 and Ff0 are functions of the Mach number only, the former being dimensionless and the latter having the dimensions of 1/ R .

Basic Concepts and Relations of Aerothermodynamics

17

FIGURE 2.1 Variations of mass flow functions Fˆ f0 and Fˆ f with Mach number for γ = 1.4 .

Equation 2.13 (see Problem 2.2 for its derivation) shows that, for a given Mach number, the mass flow rate at a section is proportional to the total pressure and inversely proportional to the square root of total temperature. Both the total pressure and the total temperature are uniform local values at that section. Figure 2.1 shows that Fˆ f0 increases as the Mach number increases for M < 1 (subsonic flow) and decreases as the Mach number increases for M > 1 (supersonic flow) and that it has a maximum value of 0.6847 at M = 1 (sonic flow) corresponding to the choked flow condition at the section. For ambient air at 20°C and 1.013 bar, the maximum possible mass flow rate computed by Equation 2.13 through an area of 1.0 m 2 is 239.2 kg/s. Figure 2.1 further shows that each value of Mach number yields a single value of mass flow function Fˆ f0 but that each value of mass flow function Fˆ f0 corresponds to two values (except at M = 1) of Mach number—one subsonic and the other supersonic. While the calculation of Fˆ f0 by Equation 2.14 for a given value of Mach number is direct, we need to use an iterative method (e.g., “Goal Seek” in MS Excel) to compute subsonic and supersonic Mach numbers for a given value of Fˆ f0 . Figure 2.1 shows that, unlike Fˆ f0 , Fˆ f increases monotonically with Mach number and does not exhibit a maximum value at M = 1 . For a given value of Fˆ f , we can use the direct method presented here to calculate the Mach number. 2.2.3.1  Finding the Mach Number for a Given Static-Pressure Mass Flow Function Squaring both sides of Equation 2.11, we obtain γ − 1 2 Fˆ f2 = M 2 γ  1 + M  = γ M 2 + 0.5 γ ( γ − 1)M 4   2

0.5 γ ( γ − 1)M 4 + γ M 2 − Fˆ f2 = 0

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Logan’s Turbomachinery

which is a quadratic equation in M 2 with its positive root given by M2 =

M=

−γ + γ 2 + 2 γ ( γ − 1)Fˆ f2 γ ( γ − 1) −γ + γ 2 + 2 γ ( γ − 1)Fˆ f2 γ ( γ − 1)

(2.16)



2.2.3.2  Using Isentropic Flow Tables to Compute Mass Flow Functions Not all isentropic flow tables list mass flow functions, but they all list quantities like pressure ratio (p0 /p) and area ratio (A/A * ) against Mach number. We know that, under isentropic flow condition, the total pressure and total temperature remain constant in a variable-area duct. In a duct flow, we can equate the mass flow rate at a give section with that at the throat with M = 1 (imaginary if outside the duct), giving  = m Fˆ f0 =

AFˆ f0 p0 A * Fˆ f*0 p0 = RT0 RT0 Fˆ f*0 0.6847 = A    A  *   *  A A

(2.17)

and  p0  Fˆ f = Fˆ f0    p



(2.18)

Knowing the Mach number at a section, we can look up p0 /p and A/A * in isentropic flow tables and use Equation 2.17 to compute Fˆ f0 and then Equation 2.18 to compute Fˆ f .

2.3  Energy Equation The steady-flow energy equation embodies the conservation of energy principle based on the first law of thermodynamics. The usual forms of specific energy (energy per unit mass), which must be accounted for in a turbomachine, are potential energy zg, internal energy u, flow work p/ρ, kinetic energy V 2/2 , heat transfer q, and work transfer w. A word statement of the energy equation is the following: Energy at section 1 + Heat transfer = Energy at section 2 + Work done between 1 and 2 which in the equation form can be written as



z1g + u1 +

(See, e.g., Sultanian, 2015.)

p1 V12 p2 V22 + + q = z2 g + u2 + + +w ρ1 2 ρ1 2

(2.19)

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Basic Concepts and Relations of Aerothermodynamics

Frequently, the internal energy is combined with the flow work to form the enthalpy (h = u + p/ρ), giving

z1g + h 1 +

V12 V2 + q = z2 g + h 2 + 2 + w 2 2

(2.20)

Usually, in turbomachinery applications, the potential energy and the heat transfer terms are neglected. Denoting the specific work w by E as the specific energy transfer term, we write

h1 +

V12 V2 = h2 + 2 + E 2 2

(2.21)

We can combine the specific static enthalpy h and the specific kinetic energy (specific dynamic enthalpy) V 2/2 to form the total specific enthalpy h 0. Thus, Equation 2.21 becomes h 01 = h 02 + E



(2.22)

Compressors and pumps increase h 0 , so that h 02 > h 01, which renders the energy transfer E in Equation 2.22 negative. Turbines, however, decrease h 0, with h 02 < h 01, making E positive. The work per unit mass calculated from Equation 2.22, when divided by the gravitational acceleration g, becomes head H, which is the preferred form of work in pump or hydraulic turbine applications.

2.4  Linear Momentum Equation A linear momentum equation is essentially an expression of the force and linear momentum balance on a control volume. Since momentum is a vector quantity, we use a linear momentum equation in each coordinate direction. For an inertial (nonaccelerating) control volume with steady flow, we can express the linear momentum equation as the following word statement: (Sum of surface forces—e.g., pressure force and shear force) + (Sum of body forces—e.g., gravity and centrifugal force under rotation) = (Total linear momentum flow rate leaving the control volume) − (Total linear momentum flow rate entering the control volume) In equation form, we can write the above statement for x-momentum as FSx + FBx =

∑ M − ∑ M x

Noutlet

N inlet

(2.23)

x



While there is little confusion in the evaluation of both the surface force and the body force on the left-hand side of Equation 2.23, the linear momentum flow rate terms on the right-hand side require careful evaluation at control volume inlets and outlets. Because the linear momentum flow rate is the product of mass flow rate and velocity, we can prevent any error in its evaluation by identifying mass velocity and momentum velocity at each inlet and outlet. In this approach, the mass flow rate associated with each inlet and outlet becomes a positive

20

Logan’s Turbomachinery

FIGURE 2.2 A control volume with multiple inlets and outlets.

scalar product of mass velocity, area, and fluid density. The sign (positive or negative) of the linear momentum flow rate is then entirely determined by the momentum velocity relative to the positive direction of the chosen coordinate axis. Using this approach, the x-momentum flow rate at each inlet and outlet of the control volume shown in Figure 2.2 is summarized in Table 2.1. This table shows that, at both outlet 1 and inlet 3, the x-momentum flow rate is negative. The mass velocity is different from the x-momentum velocity at outlet 5.

2.5  Angular Momentum Equation With its axial coordinate direction aligned with the axis of rotation, a cylindrical coordinate system is a natural choice for all turbomachinery flows. As shown in Figure 2.3, the tangential direction in this coordinate system is perpendicular to the meridional plane formed by the axial and radial directions. The figure further shows that Va , Vr , and Vu are, respectively, the axial, radial, and tangential velocity components of the absolute velocity V at any point in the flow; similarly, Wa , Wr , and Wu are, respectively, the axial, radial, and tangential velocity components of the corresponding relative velocity W at that point. The angular momentum vector at point A in the flow about the turbomachinery axis of rotation is the cross product of the radial vector r and the absolute velocity vector V. Because the radial velocity Vr is collinear with the radial vector, its contribution to angular momentum will be zero. The contribution of the axial velocity Va to angular momentum will be in the tangential direction and not along the axis of rotation. Thus, only the tangential TABLE 2.1 x-momentum Flow Rates at Inlets and Outlets of the Control Volume of Figure 2.2 Inlet/Outlet

Mass Velocity

x-momentum Velocity

x-momentum Flow Rate

1 (outlet)

V1

− V1

− V1 (ρA 1 V1 )

2 (inlet)

V2

V2

V2 (ρA 2 V2 )

3 (inlet)

V3

− V3

− V3 (ρA 3 V3 )

4 (outlet)

V4

V4

V4 (ρA 4 V4 )

5 (outlet)

V5

V5 cos α

V5 cos α(ρA 5 V5 )

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Basic Concepts and Relations of Aerothermodynamics

FIGURE 2.3 Cylindrical coordinate system used in turbomachinery flows.

velocity Vu will contribute to the angular momentum vector along the rotation axis, giving a =m  rVu . the angular momentum flow rate H The angular momentum equation is not a new conservation law but simply a moment of the linear momentum equation. In word form, the angular momentum equation reads (Total torque from surface forces—e.g., pressure force and shear force) + (Total torque from body forces—e.g., gravity and centrifugal force under rotation) = (Total angular momentum flow rate leaving the control volume) − (Total angular momentum flow rate entering the control volume) In equation form, we can write the above statement as Γ S a + Γ Ba =

∑ H − ∑ H a

Noutlet

N inlet

(2.24)

a



Similar to the approach presented in the previous section for the evaluation of linear momentum at an inlet or outlet of a control volume by using the concept of mass velocity and momentum velocity, we can evaluate each angular momentum flow rate by taking the product of mass velocity and specific angular momentum, which is considered positive if it is in the counterclockwise direction and negative if in the clockwise direction around

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Logan’s Turbomachinery

TABLE 2.2  z-angular Momentum Flow Rates at Inlets and Outlets of the Control Volume of Figure 2.2 Mass Velocity

Specific z-component Angular Momentum

z-component Angular Momentum Flow Rate

1 (outlet)

V1

a1 V1

a1 V1 (ρA 1 V1 )

2 (inlet)

V2

a2 V2

a2 V2 (ρA 2 V2 )

3 (inlet) 4 (outlet)

V3 V4

−a3 V3 −a 4 V4

−a3 V3 (ρA 3 V3 ) −a 4 V4 (ρA 4 V4 )

5 (outlet)

V5

b5 V5 sin α − a5 V5 cos α

(b5 V5 sin α − a5 V5 cos α)(ρA 5 V5 )

Inlet/Outlet

the axis for which we are writing the angular momentum equation. Table 2.2 summarizes the flow rate of angular momentum around the z axis for each inlet and outlet of the control volume shown in Figure 2.2, demonstrating the application of the approach proposed here. 2.5.1  Euler’s Turbomachinery Equation Euler’s turbomachinery equation is widely used in all types of turbomachinery. This equation, founded in the angular momentum equation discussed in the previous section, determines power transfer between the fluid and the rotor blades. In pumps, fans, and compressors, the power transfer occurs to the fluid from the rotor, increasing its outflow angular momentum over its inflow value; in turbines, the power transfer occurs to the rotor from the fluid, resulting in the reduction of its outflow rate of angular momentum over its inflow rate. Power transfer to or from the fluid is simply the product of the torque and the rotor angular speed in radians per second. Let’s consider a steady adiabatic flow in a rotating passage formed between adjacent blades of the rotor, as shown in Figure 2.4. Each of the velocity vectors, V1 at inlet 1 and

FIGURE 2.4 Flow through an axial-radial turbomachinery passage between adjacent blades.

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Basic Concepts and Relations of Aerothermodynamics

(a)

  

(b)

FIGURE 2.5 (a) Velocity diagram at rotor inlet and (b) velocity diagram at rotor outlet.

V2 at outlet 2, has components in axial, radial, and tangential directions. The meridional velocity, which is the resultant of the axial and the radial velocities, is the mass velocity at sections 1 and 2. Let Vu1 be the tangential velocity at inlet 1 and Vu2 the tangential velocity at outlet 2. As shown in Figure 2.3, the vector addition of the relative velocity W to the blade velocity U yields the absolute fluid velocity V. The relation can be expressed by V = W + U. The graphical representation of the addition of U1 to W1 at the rotor inlet (Figure 2.4) is shown in Figure 2.5a. The corresponding velocity diagram at the rotor outlet is shown in Figure 2.5b.  Using Equation 2.24, the angular momentum equation for a constant mass flow rate m through the blade passage yields

 ( r2 Vu2 − r1 Vu1 ) Γ=m

(2.25)

The aerodynamic power transfer, which is the rate of work transfer due to the aerodynamic torque acting on the fluid control volume, is given by

 ( r2 Vu2 − r1 Vu1 ) ω = m  ( U 2 Vu2 − U1 Vu1 ) P = Γω = m

(2.26)

where U1 and U 2 are rotor tangential velocities at inlet 1 and outlet 2, respectively. Using the steady-flow energy equation in terms of total enthalpy at sections 1 and 2, we can also write  ( h 02 − h 01 ) (2.27) P=m Combining Equations 2.26 and 2.27 yields

− E = ( h 02 − h 01 ) = (U 2 Vu2 − U1 Vu1 )

(2.28)

which is known as Euler’s turbomachinery equation. This equation simply states that, under adiabatic conditions, the change in specific total enthalpy of the fluid flow between any two sections of a rotor equals the difference in the products of the rotor tangential velocity and the flow tangential velocity at these sections. Equation 2.28 further reveals that, for turbines, where work is done by the fluid, we have h 02 < h 01 and U 2 Vu2 < U1 Vu1 ; for compressors, fans, and pumps, where the work is done on the fluid, we have h 02 > h 01 and U 2 Vu2 > U1 Vu1. For axial-flow machines, where r1 ≈ r2 and U1 ≈ U 2 , Equation 2.28 reveals that the change in total enthalpy is entirely due to the change in flow tangential velocity—i.e., ∆h 0 = U∆Vu —requiring blades with camber (bow). For radial-flow machines (centrifugal compressors), however, the change in total enthalpy results largely from the change in rotor tangential velocity due the change in radius—i.e., ∆h 0 = ∆UV u.

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Logan’s Turbomachinery

2.5.1.1 Rothalpy The concept of rothalpy is grounded in Euler’s turbomachinery equation presented in the previous section. Rearranging Equation 2.28, we obtain h 01 − U1 Vu1 = h 02 − U 2 Vu2



(2.29)

which reveals that the quantity (h 0 − UVu ) at any point in a rotor flow remains constant under adiabatic conditions (no heat transfer). This quantity is called rothalpy, expressed as I = h 0 − UVu = h +



V2 − UVu 2

(2.30)

where both h 0 and Vu are in the inertial (stationary) reference frame. Let us now convert Equation 2.30 into the rotor reference frame. With reference to Figure 2.3, we can write

V 2 = Va2 + Vr2 + Vu2

(2.31)



W 2 = Wa2 + Wr2 + Wu2

(2.32)



Vu = Wu + U

(2.33)

where U is the local tangential velocity of the rotor. Substituting for Vu from Equation 2.31 into Equation 2.32 and noting that Wa = Va and Wr = Vr , we obtain V 2 = Wa2 + Wr2 + Wu2 + 2Wu U + U 2



(2.34)

Using Equations 2.32 and 2.34, we rewrite Equation 2.30 as I=h+



Wx2 + Wr2 + Wu2 + 2Wu U + U 2 − U(Wu + U) 2

=h+

Wx2 + Wr2 + Wu2 U 2 − 2 2

=h+

W2 U2 − 2 2

Thus,



I=h+

W2 U2 U2 − = h 0R − 2 2 2

(2.35)

where h 0R is the specific total enthalpy in the rotor reference frame. For a calorically perfect gas with constant c p , we can also write Equation 2.35 as



I = c p T0R −

U2 2

where T0R is the fluid total temperature in the rotor reference frame.

(2.36)

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Basic Concepts and Relations of Aerothermodynamics

2.5.1.2  An Alternate Form of Euler’s Turbomachinery Equation For the adiabatic flow in a rotor, the rothalpy remains constant (i.e., I1 = I 2 ). Using Equation 2.35, we can write h1 +

W12 U12 W2 U2 − = h2 + 2 − 2 2 2 2 2

(2.37)

 W2 W2   U2 U2  h2 − h1 =  1 − 2  +  2 − 1  2   2 2   2



which expresses the change in static enthalpy in a rotor in terms of changes in flow relative velocity and rotor tangential velocity. From the definition of total enthalpy, we can write its change between locations 1 and 2 as  V2 V2  h 02 − h 01 = (h 2 − h 1 ) +  2 − 1  2   2



(2.38)

Substituting for (h 2 − h 1 ) from Equation 2.37 into Equation 2.38 yields the following alternate form of the Euler’s turbomachinery equation earlier derived as Equation 2.28.



 W2 W2   U2 U2   V 2 V 2  h 02 − h 01 =  1 − 2  +  2 − 1  +  2 − 1  2   2 2   2 2   2

(2.39)

The terms on the right-hand side of Equation 2.39 have the following physical interpretations:  W12 W22   2 − 2  ≡ Static enthalpy change due to change in relative kinetic energy in the blade passages  U 22 U12   2 − 2  ≡ Static enthalpy change due to rotation of the rotor  V22 V12   2 − 2  ≡ Dynamic enthalpy change due to change in flow absolute velocity

2.6  Velocity Diagram Figure 2.3 depicts the axial, radial, and tangential components of the absolute flow velocity V and the relative flow velocity W. The plane formed by the axial and radial directions is called the meridional plane, and the component of V or W in this plane is called the meridional velocity Vm or Wm, respectively. The absolute angle α is between V and Vm, and the relative angle β is between W and Wm. For an axial-flow machine, we have Vr = Wr = 0, giving Vm = Va = Wa. Similarly, for a radial-flow machine, we have Va = Wa = 0, giving Vm = Vr = Wr. Various relations of the absolute velocity and its components and the relative velocity and its components are summarized in Table 2.3. Each of the velocity diagrams in Figure 2.6 represents the fact that the absolute flow velocity V is a vector sum of the relative flow velocity W and the rotor tangential

26

Logan’s Turbomachinery

TABLE 2.3  Relations of Absolute and Relative Flow Velocities Absolute Velocity

Relative Velocity

V =V +V +V =V +V

W = Wa2 + Wr2 + Wu2 = Wm2 + Wu2

V =V +V

Wm2 = Wa2 + Wr2 Wm = W cos β

2

2 a

2 r

2 m

2 u

2 a

2 m

2 u

2

2 r

Vm = V cos α Vu = V sin α

Wu = W sin β

velocity U. Note that, in the velocity triangle shown in Figure 2.6b, V and W are joined at their tails, while in Figure 2.6c, they are joined at their tips. For the analysis of turbomachinery designs using velocity diagrams, it is important to adopt a sign convention and use it consistently. According to the sign convention used here, Vu and Wu are positive if they are in the same direction as the rotor (blade) tangential velocity U; otherwise, they are negative. In addition, the absolute flow angle α and the relative flow angle β are positive if they produce tangential velocity components that are positive; otherwise, these angles are negative. This sign convention is depicted in Figure 2.7. For the velocity diagram in Figure 2.7a, all quantities are positive. In Figure 2.7b, however, both β and Wu are negative; the rest are positive. For the velocity diagram shown in Figure 2.7b, simply adding U and Wu without using the present sign convention will result in an incorrect value of Vu . The velocity diagrams at inlet and outlet for a compressor blade are shown in Figure 2.8, and those for a turbine blade are shown in Figure 2.9. For triangle ABC in Figure 2.7a, we can write

W 2 = V 2 + U 2 − 2UV cos γ

(2.40)



V cos γ = Vu

(2.41)

By eliminating V cos γ between Equations 2.40 and 2.41, we obtain

(2.42)

W 2 = V 2 + U 2 − 2UVu

Now, applying Equation 2.42 at two locations in a rotor flow and subtracting them yields



 W2 W2   U2 U2   V 2 V 2  U 2 Vu2 − U1 Vu1 = h 02 − h 01 =  1 − 2  +  2 − 1  +  2 − 1  2   2 2   2 2   2

(2.43)

which is identical to Equation 2.39.

FIGURE 2.6 Velocity diagrams relating absolute flow velocity V, relative flow velocity W, and rotor tangential velocity U: (a) V = W + U, (b) V and W are joined at their tails, and (c) V and W are joined at their tips.

Basic Concepts and Relations of Aerothermodynamics

27

FIGURE 2.7 Velocity diagrams showing the present sign convention.

FIGURE 2.8 Velocity diagrams at the inlet and outlet of a compressor blade.

FIGURE 2.9 Velocity diagrams at the inlet and outlet of a turbine blade.

2.7 Applications Let us apply various relations derived in the foregoing to a number of common turbomachines: namely, an axial-flow impulse turbine, an axial-flow compressor, a centrifugal pump, and a hydraulic turbine.

28

Logan’s Turbomachinery

FIGURE 2.10 Velocity diagram for an impulse turbine.

2.7.1  Axial-Flow Impulse Turbine The flow in an impulse turbine is generally in the axial direction, and the blade velocity is the same at the entrance and exit of the rotor. Figure 2.10 shows a typical blade cross section and the corresponding velocity diagram. Steam or hot gas leaves a nozzle with a velocity V1 at a nozzle angle α, measured from the tangential direction, and enters the region between the blades with relative velocity W1. Ideally, no pressure drop occurs in the blade passage, and the relative velocity W2 is equal in magnitude to W1. This is what is meant by the term impulse turbine, also called a zero-reaction turbine. The absolute velocity V2 at the blade-passage exit is much reduced and is typically less than half of V1. With U1 = U 2 for an axial-flow turbine, the specific energy transferred from the fluid to the rotor is easily found from Equation 2.39 as



E=

1 2 1 V1 − V22 + W22 − W12 2 2

(

)

(

)

(2.44)

For an impulse turbine with W1 = W2, Equation 2.44 further reduces to



E=

1 2 V1 − V22 2

(

)

(2.45)

which shows that maximizing energy transfer means minimizing V2 or requiring V2 to be in the axial direction only—i.e., α 2 = 0. The result of Equation 2.45 is also obtainable from Equation 2.21 if h 1 = h 2. Equal enthalpy implies no change of temperature and pressure in the flow, which agrees with the original assumption of zero reaction—i.e., no pressure drop in the rotor. If the degree of reaction of a turbine stage, denoted by R, is defined as the ratio of the enthalpy drop in the rotor to the enthalpy drop in the stator plus that in the rotor—namely, R=



h1 − h2 he − h2

(2.46)

where h e is the enthalpy at the nozzle (stator) inlet—then we can write



he +

Ve2 V2 = h2 + 2 + E 2 2

(2.47)

29

Basic Concepts and Relations of Aerothermodynamics

as the energy balance for the entire stage, and R becomes R=

V22/2 − V12/2 + E V22/2 − Ve2/2 + E

(2.48)



Substituting Equation 2.44 into Equation 2.48 yields



R=

W22 − W12 V12 − Ve2 + W22 − W12

(2.49)

which is generally applicable to axial-flow machines. Quite commonly, in the analysis of multistage machines, it is assumed that the fluid velocity V2 leaving the rotor is the same as that from the stage immediately upstream—i.e., Ve = V2. The degree of reaction would then be expressed as



R=

W22 − W12 V − V22 + W22 − W12

(2.50)

2 1

which we will utilize for axial-flow machines. We have learned that the blade profile of the impulse turbine is designed to make W1 = W2. Clearly, Equation 2.50 confirms the earlier assumption that R = 0. 2.7.2  Axial-Flow Compressor An axial-flow compressor blade and the corresponding velocity diagram are shown in Figure 2.11a. Compared with the turbine blade deflection, the fluid in this case is deflected only slightly by the blade. Another difference is that the pressure rises in the flow direction both in the stator and in the rotor. Pressure rise is related to enthalpy rise, and the latter is dependent on the deflection of the fluid by the blade. Equating rothalpies given by Equation 2.35 at the inlet and outlet of an axial-flow compressor, we obtain

h2 − h1 =

(a)

1 W12 − W22 2

(

)

  

(2.51)

(b)

FIGURE 2.11  (a) Velocity diagram for an axial-flow compressor and (b) tangential components of relative velocity.

30

Logan’s Turbomachinery

which is interpreted as an enthalpy rise associated with a decrease in relative kinetic energy in the blade passages. The associated pressure ratio is easily obtained from the enthalpy rise through the use of a polytropic exponent n. Using the perfect gas relation ∆h = c p ∆T , we obtain  p2  h 2 − h 1 = c p T1    p1 



(n − 1)/n

 − 1 

(2.52)

from which the pressure ratio, and hence the pressure rise, may be determined. It is observed that the pressure rise depends on the change of relative velocity, which is directly related to the compressor blade shape—i.e., to the angle of deflection of the fluid. Using Equation 2.33, we can write E = U(Vu1 − Vu2 ) = U(Wu1 − Wu2 )





(2.53)

which shows that the energy transfer is also related to the difference in the tangential components of the relative velocity. Figure 2.11b shows that this difference is proportional to the deflection angle θ = β1 − β 2. A typical compressor velocity diagram is constructed by making V1 = W2 and V2 = W1. Referring to Figure 2.11a, it is seen that the triangles would be symmetrical about the common altitude (Va ). Such symmetry, whether in a turbine or a compressor diagram, results in R = 0.5, as determined from Equation 2.50. This condition is also termed a 50% reaction. Physically, this means that 50% of the compression (or enthalpy rise) takes place in the rotor of the compressor and 50% in the stator. This degree of reaction is optimum for minimizing the aerodynamic drag losses of rotor and stator blades in both turbines and compressors. 2.7.3  Centrifugal Pump The centrifugal pump is and has been an extremely important machine to humans, and one would think it to be theoretically complex. However, it is extremely simple to analyze. It was discussed in Chapter 1 and is illustrated in Figure 2.12a. The inner and outer radii r1 and r2, respectively, define the inlet and outlet of the control volume. At the inlet, the flow, assumed to be incompressible, has purely radial velocity V1, which implies that Vu1 = 0. The impeller imparts angular momentum to the fluid, so that the flow exits with both radial and tangential velocity components. Because r2 > r1 and the angular speed is constant, we have U 2 > U1. The energy transfer E, or the head H, is calculated from Equation 2.28 as gH = − E = U 2 Vu2



(2.54)

From Figure 2.12b, we can write (2.55) Vu2 = U 2 − Vm2 tan β 2 in which the meridional component Vm2 equals the volume flow rate Q divided by the flow area 2 πr2 b2 and U 2 = Nr2 . The head is thus expressed as Nr2 {Nr2 − [Q/(2 π r2 b2 )]tan β 2 } g where b2 is the impeller tip width. H=

(2.56)

31

Basic Concepts and Relations of Aerothermodynamics

(a)

  

(b)

FIGURE 2.12 (a) Centrifugal pump and (b) velocity diagram for a centrifugal pump.

An important performance curve, the head-capacity curve, for a centrifugal pump is constructed by plotting H as a function of Q. Equation 2.56 expresses this relationship analytically and provides an ideal head-capacity curve for comparison with actual curves. Since β 2 is usually about 65°, the theoretical relation indicates decreasing head with increasing flow rate, a situation realized in practice. This equation indicates that H goes up as the square of N, which also agrees with experience. It is interesting to note that an actual pump impeller can be measured and the measurements used to predict the expected flow rate. Figure 2.12b shows that such a prediction can be easily made from the knowledge of β1 , N, r1 , and b1, since

Q = 2 π r12 b1 N cot β1

(2.57)

The enthalpy change in a fluid is determined from the thermodynamic equation T ds = dh −



dp ρ

(2.58)

which, upon integration for the isentropic compression of a liquid with constant density, yields



h2 − h1 =

p 2 − p1 ρ

(2.59)

Substituting Equation 2.59 into Equation 2.21 gives H=

p2 − p1 V22 − V12 + ρg 2g

(2.60)

which is useful in calculating the pressure rise across the pump impeller. Of course, the pressure can be raised further in the pump casing by reducing V2 in a passage of increasing cross-sectional area—i.e., a diffuser.

32

Logan’s Turbomachinery

FIGURE 2.13 Hydraulic turbine.

2.7.4  Hydraulic Turbine The radial-flow hydraulic turbine, depicted in Figure 2.13, is the reverse of the centrifugal pump. In this case, water enters the turbine at the larger radius r1 from a stator that controls the flow angle at the rotor inlet. Ideally, the absolute velocity V2 at the outlet is purely radial, so that the energy transfer is simply E = gH = U1 Vu1



(2.61)

Since Vu1 = U1 − Vm1 tan β1 and Vm1 = Q/A 1, we can write the turbine head as H=

Nr1 {Nr1 − [Q/(2 π r1 b1 )]tan β1 } g

where b1 is the runner tip width. The flow rate Q is given, as in Equation 2.57, by

Q = 2 π r22 b2 N cot β 2

(2.62)

2.8  Discussion on Further Applications Similarities of other turbomachines to the four examples discussed in Section 2.7 should be noted. The axial-flow reaction turbines, which include most steam and gas turbines, are like the impulse turbine example given except that an expansion of the fluid also occurs in the rotor. This means that an enthalpy drop occurs there and the degree of reaction R is greater than zero (typically R  = 1 2). It should also be noted that steam and gas turbines used to drive large loads, such as electric generators, include many stages in series, frequently with many rotors mounted on a single shaft. The energy transfer term for each rotor (stage) must be added to obtain the total work done per unit mass of steam or gas flowing. The turbine power is then obtained from the product of the total specific work and fluid mass flow rate. The axial-flow compressor example indicated calculations for a single stage. Compressors usually involve many stages, and the pressure ratio for each must be multiplied to obtain the overall pressure ratio of the machine. In addition, the relations developed for the compressor stage apply to axial-flow blowers, fans, and pumps. The difference is that

33

Basic Concepts and Relations of Aerothermodynamics

the isentropic enthalpy rise is calculated from Equation 2.59 for the approximately incompressible flows usually assumed in these machines. The centrifugal pump is geometrically similar to the centrifugal compressor, centrifugal blower, and centrifugal fan. However, the flow in the compressor must be modeled as compressible, and the pressure ratio should be calculated from an equation like Equation 2.52. Usually, however, total properties p0 and T0 are used to formulate a working equation for the calculation of the total pressure ratio in centrifugal compressor stages. Since Vu1 = 0 at the inlet of the centrifugal compressor, Equation 2.28 simplifies to

h 02 − h 01 = c p (T02 − T01 ) = U 2 Vu2

Using a polytropic process to relate the end states, we have



p02  T02  = p01  T01 

n/(n − 1)

(2.63)

(2.64)

Combining Equations 2.63 and 2.64 results in the following working equation for the total pressure ratio of a centrifugal compressor stage:



p02  V U  =  1 + u2 2  p01  c p T01 

n/(n − 1)

(2.65)

Although this equation expresses the essential form for the calculation of the total pressure ratio for a stage, some additional refinement is required and will be added in Chapter 5. As observed in Figure 2.12a, the fluid enters the eye of the centrifugal impeller axially—i.e., Vu1 = 0—which means that Equation 2.54 is also valid for the case where the pump or compressor blades are extended into the eye of the impeller. However, it should be noted that, in this case, the cylindrical flow area 2 πr1 b1 used in Equation 2.57 must be replaced by a circular or annular flow area. Moreover, the meridional velocity Vm1, which appears in this equation as Nr1 cot β1 must be replaced by the axial velocity V1 entering the eye of the impeller. A similar situation exists at the outlet of the hydraulic turbine rotor depicted in Figure 2.13; namely, the fluid can be made to exit axially, which implies that neither tangential nor radial velocity components are present and the blades can be extended into the exit plane of the rotor. For an assumed axial exit at station 2, Vu2 = 0, as was assumed in the development of Equation 2.61. Likewise, Equation 2.61 can be applied to the radial-inflow gas turbine, which is of the same geometry as the inward-flow hydraulic turbine. Hence, the blades are usually extended into the exhaust plane, where there is no swirl of the exhaust gases and Vu2 = 0. For the gas turbine, the relative gas angle β1 at the turbine inlet is zero, which implies that Vu1 = U1. These conditions define the so-called 90° (radial entry) IFR (inward-flow radial) gas turbine, which is a design commonly employed in industry.

2.9  Concluding Remarks Using numerous worked examples; we have reviewed in this chapter foundational material on the conservation laws of mass, linear momentum, angular momentum, and energy, which are common to the primary flowpath design of all turbomachines. General velocity

34

Logan’s Turbomachinery

diagrams at the blade inlet and outlet introduced in this chapter form the basis for the meanline aerodynamic design and the evaluation of design performance parameters: namely, flow coefficient, loading coefficient, and degree of reaction. The chapter further introduces a useful quantity called rothalpy, which is derived from Euler’s turbomachinery equation, the backbone of all turbomachinery flows. Like the total enthalpy of an adiabatic flow remains constant in a stationary duct (with or without friction), the rothalpy of an adiabatic flow remains constant in a rotating duct, automatically accounting for work transfer to the fluid due to duct rotation. The rothalpy expressed in both stator and rotor reference frames provides a useful means of converting total temperature and total pressure between these two reference frames, noting that the static properties do not change across reference frames (see Problem 2.28). By way of applications of various aerothermodynamic equations developed in this chapter, four typical turbomachinery designs are analyzed: namely, an axial-flow impulse turbine and an axial-flow compressor, both of which involve compressible flows, and a centrifugal pump and a hydraulic turbine, both of which involve incompressible flows. Based on these four typical applications, the chapter includes a helpful discussion on how one can easily analyze other types of turbomachines, which are presented in greater detail in subsequent chapters in the book. Before diving deep into the remaining chapters, it behooves each reader to master the various concepts and relations of aerothermodynamics, including worked examples and chapter-end problems, presented in this chapter.

Worked Examples EXAMPLE 2.1 A constant-area (A = 0.0025 m 2 ) pipe with steady adiabatic air flow is shown in Figure 2.14. The pipe wall has friction. Inlet total pressure p01 and total temperature T01 are 1.5 bar and 500 K, respectively. The flow exits the pipe at the static pressure of p2 = pamb = 1.0 bar . Under these boundary conditions, the measured loss in total pres through the pipe. sure from inlet to exit is 0.15 bar. Calculate the mass flow rate m (For air: γ = 1.4 and R = 287 J/(kg ⋅ K).) SOLUTION: In this problem, for a steady air flow in the pipe, the conservation of mass (continuity equation) requires that each section have the same mass flow rate. Since the flow is adiabatic (no heat transfer), air total temperature remains constant from pipe inlet to exit.

FIGURE 2.14  Adiabatic air flow in a constant-area pipe with friction (Example 2.1).

Basic Concepts and Relations of Aerothermodynamics

35

Note that the wall friction does not increase air total temperature; it only reduces air total pressure. The air flow is assumed one-dimensional only at section 2 (exit) and not anywhere else in the pipe. Total pressure at section 2:  p02 = p01 − ∆p0 = 1.5 − 0.15 = 1.35 bar = 1.35 × 105 Pa Pressure ratio and Mach number at section 2: 



p02 1.35 = = 1.35 p2 1.0

γ −1    2   p02  γ   2   1.35 1.41.4− 1 − 1 = 0.669 M2 =  1 − )   =    (   1.4 − 1    γ − 1   p 2     

Static-pressure mass flow function at section 2:



γ −1 2 1.4 − 1 Fˆ f2 = M 2 γ  1 + M 2  = 0.669 1.4  1 + (0.669)2  = 0.8263     2 2

Mass flow rate at section 2: 2= m

A 2 Fˆ f2 p2 0.0025 × 0.8263 × 1.0 × 105 = = 0.545 kg/s RT02 287 × 500

 = 0.545 kg/s m

EXAMPLE 2.2 In this example, shown in Figure 2.15, a short conical diffuser of area ratio 1.5 with negligible friction and heat transfer is added at the exit of the pipe of Example 2.1, with identical inlet boundary conditions. The exit boundary conditions at section 2 in Example 2.1 now prevail at section 3 in this example. In this example also, the mea sured loss in total pressure from inlet to exit is 0.15 bar. Calculate the mass flow rate m through the pipe and the total-to-static pressure ratio p01 /p2 between sections 1 and 2, and compare these values with the corresponding values in Example 2.1. (For air: γ = 1.4 and R = 287 J/(kg ⋅ K) .)

FIGURE 2.15  Adiabatic air flow in a constant-area pipe with friction, extended with a short conical diffuser (Example 2.2).

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Logan’s Turbomachinery

SOLUTION: Since the boundary conditions at the exit (section 3) in this example are identical to the boundary conditions at the exit (section 2) in Example 2.1, we obtain M 3 = 0.669 and Fˆ f3 = 0.8263. 3= Mass flow rate at section 3:  m

A 3 Fˆ f3 p3 0.0025 × 1.1 × 0.8263 × 1.0 × 105 = = 0.599 kg/s RT03 287 × 500

 = 0.599 kg/s m

Mach number at section 2: 2= m

A 2 Fˆ f02 p02 RT02

 2 RT02 m 0.599 287 × 500 or Fˆ f02 = = = 0.6733 A 2 p02 0.0025 × 1.35 × 105 Fˆ f02 = M 2



γ

γ +1

 1 + γ − 1 M2  γ −1 2   2

= 0.6733

Using an iterative solution method (e.g., “Goal Seek” in MS Excel) yields M = 0.863

Static pressure at section 2: γ

1.4

1.4 − 1 p02  1.4 − 1 γ − 1 2  γ −1  M2  (0.863)2  = 1+ = 1+ = 1.6264     p2 2 2



p2 =

p02 1.35 = = 0.83 bar 1.6264 1.6264

Total-to-static pressure ratio between sections 1 and 2: p01 1.5 = = 1.807 p2 0.83



p01 1.5 = = 1.5. The results here show that adding a p2 1.0 short diffuser section to the pipe flow of Example 2.1 increases the mass flow rate from 0.545 kg/s to 0.599 kg/s and the overall total-to-static pressure ratio between sections 1 and 2 from 1.5 to 1.807. This example reveals two important roles an exhaust diffuser plays in the design and aerothermodynamic performance of all land-based gas turbines for power generation: (1) to increase the overall pressure ratio for the turbine, thereby improving its thermal efficiency, and (2) to increase the mass flow capacity of the machine if not choked upstream. In Example 2.1, we obtained

Basic Concepts and Relations of Aerothermodynamics

37

EXAMPLE 2.3 As shown in Figure 2.16, air flows through two identical convergent nozzles into a large plenum and exits from it sideways through a choked divergent nozzle. The throat area of the divergent nozzle equals twice the throat area of each convergent nozzle. The supply total pressure and total temperature are 8 bar and 436.5 K, respectively, for each convergent nozzle. Find the mass flow rate through the choked divergent nozzle. All walls are adiabatic and frictionless. (For air: γ = 1.4 and R = 287 J/(kg ⋅ K).) SOLUTION: The dynamic pressure of the flow from each convergent nozzle is lost in the plenum. As a result, the inlet total pressure of the choked divergent nozzle equals the static pressure at the exit of each convergent nozzle. Since there is no heat transfer, the air total temperature remains constant throughout the flow. From mass conservation, the mass flow rate exiting the divergent nozzle equals the  dn = 2m  cn). The mass sum of mass flow rates through both convergent nozzles (i.e., m flow rate through each nozzle can be written as follows:  cn = Each convergent nozzle:  m

AFˆ f p RT0

 dn = Choked divergent nozzle:  m

 dn = m

2AFˆ f*0 p RT0 AFˆ f p 2AFˆ f*0 p  cn = 2 = 2m RT0 RT0

Fˆ f = Fˆ f*0

FIGURE 2.16  Adiabatic air flow entering a plenum through two convergent nozzles and exiting through one divergent nozzle (Example 2.3).

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Logan’s Turbomachinery

Since the flow is choked (M = 1) at the throat of the divergent nozzle, we obtain Fˆ f*0 = 0.6847 = Fˆ f



Mach number at each convergent nozzle throat:



 −γ + γ 2 + 2 γ ( γ − 1)Fˆ 2 f M= γ ( γ − 1) 



M = 0.561

1

 2  −1.4 + 1.42 + 2 × 1.4 × (1.4 − 1) × (0.6847)2  = 1.4 × (1.4 − 1)  

1

 2  

Static pressure at each convergent nozzle throat: γ

1.4

1.4 − 1 p0  1.4 − 1 γ − 1 2  γ −1  M  (0.561)2  = 1+ = 1+ = 1.238    p  2 2



 p 8 p = p0   = = 6.46 bar = 6.46 × 105 Pa  p0  1.238

Choked mass flow rate through the divergent nozzle:  dn = m

2AFˆ f*0 p 2 × 0.002 × 0.6847 × 6.46 × 105 = RT0 287 × 436.5

 dn = 4.999 kg/s m

EXAMPLE 2.4 Figure 2.17 shows a premixing chamber of an industrial combustor. Air flows through the central pipe, and fuel flows through six peripheral tubes placed in a hexagon pattern. A uniform mixture of air and fuel exits the premixing chamber into the combustion chamber (not shown here). Using the following data, neglecting wall friction, and assuming the flow to be incompressible, calculate the total pressure loss, which

FIGURE 2.17  Premixing of air and fuel in an industrial combustor (Example 2.4).

Basic Concepts and Relations of Aerothermodynamics

39

primarily results from the complex shear-layer mixing between the high-momentum air flow and the low-momentum fuel flow, in the premixing chamber over sections 1 and 2. 3 A a = 0.0024 m 2 , A f = 0.0004 m 2, A 2 = 0.0216 m 2, ρa = ρf = 1.2 kg/m , Data:  5 Va = 50 m/s, Vf = 20 m/s, and p1 = 2 × 10 Pa

SOLUTION: When a high-momentum flow mixes with a low-momentum flow, additional production of entropy due to complex shear-layer interactions in such a flow field leads to a loss in total pressure, which is also called mixing loss in turbomachinery design. In this example, we can compute the loss in total pressure in the premixing chamber by the application of continuity and momentum equations. Continuity equation  a = ρa A a Va = 1.2 × 0.0024 × 50 = 0.144 kg/s Air mass flow rate:  m  f = ρf A f Vf = 1.2 × 0.0004 × 20 × 6 = 0.0576 kg/s Fuel mass flow rate through six tubes:  m The continuity equation yields 2 =m  a +m f m  2 = ρm A 2 V2 = 1.2 × 0.0216 × V2 = 0.144 + 0.0576 = 0.2016 kg/s m

V2 =

0.2016 = 7.778 m/s 1.2 × 0.0216

Momentum equation  a =m  aVa = 0.144 × 50 = 7.200 N Air momentum flow rate:  M  f =m  f Vf = 0.0576 × 20 = 1.152 N Fuel momentum flow rate through six tubes:  M  1=M  a +M  f == 7.200 + 1.152 = 8.352 N Total momentum flow rate at section 1:  M  2 =m  2 V2 = 0.2016 × 7.778 = 1.568 N Mixture momentum flow rate at section 2:  M Assuming the static pressure p1 prevails over the entire inlet section 1, with A 1 = A 2, the momentum equation yields  2 −M 1 p1 A 1 − p2 A 2 = (p1 − p2 )A 2 = M (p2 − p1 ) =

 1−M  2 8.352 − 1.568 M = = 314 Pa A2 0.0216

p2 = p1 + 314 = 2 × 105 + 314 = 2.00314 × 105 Pa

Dynamic pressures at sections 1 and 2 1 ρa Va2 = 0.5 × 1.2 × (50)2 = 1500 Pa 2 1 Fuel dynamic pressure at section 1:  pdf = ρf Vf2 = 0.5 × 1.2 × (20)2 = 240 Pa 2 Air dynamic pressure at section 1:  pda =

40

Logan’s Turbomachinery

Mass-weighted average dynamic pressure at section 1:  pd1 =



pd1 =

 a p da + m  f pd f m   (ma + m f )

0.144 × 1500 + 0.0576 × 314 = 1140 Pa (0.144 + 0.0576)

Mixture dynamic pressure at section 2:  pd2 =

1 ρm V22 = 0.5 × 1.2 × (7.778)2 = 36.3 Pa 2

Total pressures at sections 1 and 2 Total pressure at section 1:  p01 = p1 + pd1 = 2 × 105 + 1140 = 2.0114 × 105 Pa Total pressure at section 2:  p02 = p2 + pd2 = 2.00314 × 105 + 36.3 = 2.00350 × 105 Pa Loss in total pressure over sections 1 and 2: p01 − p02 = 2.0114 × 105 − 2.00350 × 105 = 790 Pa

EXAMPLE 2.5 Figure 2.18 shows a wooden log of square cross-section in a two-dimensional crossflow with a uniform velocity U in the x-direction. The flow leaves the control volume ABCDE in the x-direction through DE with a linear velocity profile. There is no flow crossing CD. The fluid has constant density ρ. Calculate the total drag force (in the x-direction) acting on both the log and the flat plate (per unit depth) within the control volume. SOLUTION: Continuity equation  AB = 2aρU Mass inflow rate through AB:  m

 AE Mass outflow rate through AE:  m

a

U 1 ρ   y dy = aρU  a 2 1 3  AB − m  DE = 2aρU − aρU = aρU =m 2 2

 DE = Mass outflow rate through DE:  m

∫

0

FIGURE 2.18  Drag force on a square log resting on a flat plate in a two-dimensional, uniform, incompressible cross-flow (Example 2.5).

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Basic Concepts and Relations of Aerothermodynamics

Momentum equation  AB = 2aρU 2 x-momentum inflow rate through AB:  M  DE = x-momentum outflow rate through DE:  M

∫

a

2

U 1 ρ   y 2 dy = aρU 2  a 3 0

 AE = 3 aρU 2 x-momentum outflow rate through AE:  M 2 Net force in the x-direction acting on the control volume:  DE + M  AE − M  AB Fx = M Fx =



1 3 aρU 2 + aρU 2 − 2aρU 2 3 2

Total drag force on the log and the flat plate (per unit depth):  FD = − Fx =

1 aρU 2 6

EXAMPLE 2.6 Figure 2.19 shows an air jet (diameter d j = 10 mm ) emanating from a rotating pipe at a radius of 0.5 m. The air enters the pipe at the origin (r1 = 0) at a total pressure of 5 bar and a total temperature of 300 K. The ambient pressure is 1.0 bar. The pipe is rotating at N = 3600 rpm. Assuming an isentropic flow in the pipe and neglecting frictional torque resisting rotation, calculate (a) the mass flow rate of air through the pipe and (b) the total power needed to rotate the pipe at the given rpm. (For air: γ = 1.4 and R = 287 J/(kg ⋅ K).) SOLUTION: Due to work transfer by pipe rotation, the air total temperature and total pressure will increase from the inlet at 1, which is at origin, to the exit at 2, which is at r2 = 0.5 m . Preliminary calculations: Pipe angular speed:  ω =

2 πN π × 3600 = = 377 rad/s 60 30

FIGURE 2.19  Air flow through a rotating pipe (Example 2.6).

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Logan’s Turbomachinery

πd 2j π(0.010)2 = = 7.854 × 10−5 m 2 4 4 J R( γ − 1) π(0.010)2 Air specific heat at constant pressure:  c p = = = 1004.5 γ 4 kg K

Air jet area:  A j =

Air total temperature at 1 relative to the rotating pipe: T02R = T01R +



r22 ω 2 (0.5 × 377)2 = 300 + = 318 K 2c P 2 × 1004.5

Air total pressure at 2 relative to the rotating pipe: For isentropic work transfer in the rotating pipe, we can write γ

1.4

p02R  T02R  γ − 1  318  1.4− 1 = = = (1.059)3.5 = 1.222  300  p01R  T01R  p02R = 1.222 × p01R = 1.222 × 5 × 105 = 6.11 × 105 pa



Mass flow rate through the rotating pipe: For the given pressure ratio, the air flow is choked at the pipe exit with M = 1 and Fˆ f*01 = 0.6847 .  = m

A j Fˆ f*01 p01R 7.854 × 10−5 × 0.6847 × 6.11 × 105 = = 0.1088 kg/s RT01R 287 × 318

Power needed to rotate the pipe.  Since the air jet comes out radially from the pipe, its tangential velocity relative to the rotating pipe is zero (Wu1 = 0). The tangential velocity of the rotating pipe at 2 is calculated as U 2 = r2 ω = 0.5 × 377 = 188.5 m/s Absolute tangential velocity of the air jet at 1:  Vu2 = U 2 + Wu2 = 188.5 m/s  2 Vu2 = 0.1088 × 0.5 × 188.5 = 10.256 Nm Torque needed to rotate the pipe:  Γ = mr Power needed to rotate the pipe:  P = Γω = 10.256 × 377 = 3866 W

EXAMPLE 2.7 As shown in Figure 2.20, a high-pressure rotary arm is used for impingement air cooling of a cylindrical surface. The total pressure and total temperature of air inside the rotary arm are 4.7 bar and 27 o C , respectively. The static pressure outside the rotary arm is 1 bar, and that at each nozzle exit is 248292 Pa. Note that each nozzle is choked at the exit with a jet velocity of 316.938 m/s and a mass flow rate of 0.0861 kg/s. At the maximum rotational speed N max, the rotary arm needs to overcome a frictional torque of 1.4 Nm. For the given geometric data, calculate N max . Assume that the given total pressure and total temperature are relative to the arm and independent of arm rotation. (For air: γ = 1.4 and R = 287 J/(kg ⋅ K).) Geometric Data:  Jet diameter (d j ) = 10 mm, r1 = 40 cm, r2 = 50 cm, and a = 20 cm

Basic Concepts and Relations of Aerothermodynamics

43

FIGURE 2.20  Impingement air cooling of a cylindrical surface with a rotary arm (Example 2.7).

SOLUTION: The absolute-velocity-based angular momentum efflux at location 1 (in the counterclockwise direction) is given by

  1 (r1ω − 0) ω − Wj ) + mr ma(a

The absolute-velocity-based angular momentum efflux at location 2 (in the counterclockwise direction) is given by

 2 (r2 ω − 0) mr

The torque due to the pressure force at location 1 (in the counterclockwise direction) is given by

A ja(p* − pamb ) where p* = 248292 pa

Since the arm is rotating in the counterclockwise direction, the friction torque must be acting in the clockwise direction. Thus, noting that the pressure force at location 2 does

44

Logan’s Turbomachinery

not contribute to any torque on the control volume, the torque and angular momentum balance on the combined control volume yields   1 (r1ω − 0) + mr  2 (r2 ω − 0) −Γ friction + A ja(p* − pamb ) = ma(a ω − Wj ) + mr



ω=

 maW j − Γ friction + A j a(p* − pamb )  a2 + r12 + r22 m

ω=

0.0861 × 0.2 × 316.938 − 1.4 + 11.647 = 164.848 rad/s 0.0861 (0.2)2 + (0.40)2 + (0.50)2





(

{

)

}

N max = 1574.184 rpm

Problems 2.1 Using the equation of state for a calorically perfect gas and isentropic flow equations, derive Equations 2.11 and 2.12 for the static-pressure mass flow functions that are used in Equation 2.10 to calculate mass flow rate through a section. 2.2 Using the equation of state for a calorically perfect gas and isentropic flow equations, derive Equations 2.14 and 2.15 for the total-pressure mass flow functions that are used in Equation 2.13 to calculate mass flow rate through a section. 2.3 In Figure 2.21, compressed air at a total pressure of 8 bar and a total temperature of 300 K enters the plenum through the inlet pipe at A and leaves the outlet pipe at D to the ambient pressure of 1 bar. Both pipes are identical in length and diameter. Going from inlet to outlet, including a sudden expansion at B and a sudden contraction at C (neglect any vena contracta effect near C), the flow suffers a total pressure loss of 0.5 bar. The entire flow system is adiabatic. Based on your understanding of a compressible flow, choose a section (A, B, C, or D) at which the air flow will choke (M = 1). Give the reason for your choice.

FIGURE 2.21 Impingement air cooling of a cylindrical surface with a rotary arm (Problem 2.3).

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Basic Concepts and Relations of Aerothermodynamics

2.4 Two air streams of different properties enter a long duct of constant flow area of 0.002 m 2. At the duct inlet, each stream occupies equal area. The uniform static pressure at the duct inlet equals 895,614 Pa, and that at the outlet equals 1,000,000 Pa. Both streams are fully mixed at the duct outlet. Total temperature (K) Total pressure (Pa)

Stream 1 1173 1,000,000

Stream 2 1473 1,500,000

Calculate a. the mass flow rate of each stream. b. the mass-weighted average total temperature at the duct inlet. c. the average total pressure at the duct inlet. d. the percentage drop in total pressure from the duct inlet to the duct outlet. e. the total wall shear force acting on the duct flow between the inlet and outlet. 2.5 Under cold ambient conditions, an inlet bleed heat system is used to raise the compressor inlet temperature to prevent ice formation in IGVs (inlet guide vanes). This hot air is bled from an intermediate compressor stage and uniformly mixed into the engine inlet air. Let’s consider such a system, as shown schematically in Figure 2.22, in this design problem.   Ambient pressure and temperature are 1 bar and 20°C, respectively. The air flow Mach number at the IGV inlet is 0.6. To prevent ice formation, the static temperature at this section is required to be 2°C. The air mass flow rate entering the engine inlet system (before the high-pressure bleed heat section) at a low Mach number (M < 0.2) is 275 kg/s. The total temperature and pressure of the compressor bleed air are 269°C and 8 bar, respectively. Neglect any pressure and temperature changes in the bleed air supply system. a. Calculate the compressor bleed air mass flow rate to meet the design objective at the compressor IGV inlet. b. There are 200 nozzles (for bleed air injection) uniformly placed in the inlet duct cross-section to promote uniform mixing of the hot compressor bleed air with the cold inlet ambient air. Find the effective flow area (discharge coefficient = 1) of each nozzle.

FIGURE 2.22 Impingement air cooling of a cylindrical surface with a rotary arm (Problem 2.5).

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Logan’s Turbomachinery

2.6 The total temperature and total pressure at the inlet to a three-stage axial-flow turbine are 1000°C and 10 bar, respectively. The flow conditions at the turbine’s last stage exit, before the flow enters the annular diffuser, correspond to the total velocity Mach number of 0.6, the static pressure of 0.85 bar, and the total temperature of 400°C. The swirl velocity (tangential velocity) at the diffuser inlet equals 20% of the total velocity. The flow exits the diffuser fully axially. The total temperature over the diffuser section drops by 25°C due to heat transfer, and the loss in total pressure due to wall friction and secondary flows equals 5500 N/m2. The design calls for the static pressure at the diffuser exit to be 1.018 bar to allow the exhaust gases to discharge into the ambient air through a downstream ducting. See Figure 2.23. Calculate a. the annular diffuser exit-to-inlet flow area ratio. b. the diffuser actual pressure rise coefficient, and compare it with the theoretical value based on total flow velocities at the diffuser inlet and exit planes. These pressure rise coefficients are defined as follows: V  C p, theoretical = 1 −  2   V1 

2

and

C p, actual =

p 2 − p1 p01 − p1

2.7 Starting with Equation 2.58, show that the entropy change between any two points in a gas flow is given by





 p2  T  s 2 − s1 = c p ln  2  − R ln    T1   p1  which can also be expressed in terms of total temperatures and pressures at these points as  p02  T  s 2 − s1 = c p ln  02  − R ln   T01   p01 



2.8 Construct the velocity diagram for an axial-flow gas turbine having a degree of reaction of 0.25 and minimal leaving kinetic energy (V22 /2).

FIGURE 2.23 Impingement air cooling of a cylindrical surface with a rotary arm (Problem 2.6).

Basic Concepts and Relations of Aerothermodynamics

47

2.9 Derive the relationship between torque and speed for an axial-flow impulse turbine. 2.10 Determine energy transfer E for an axial-flow turbine in terms of blade speed U when the degree of reaction is 0.5 and the leaving kinetic energy is minimal. 2.11 Repeat Problem 2.3 for the impulse turbine with minimal leaving kinetic energy. 2.12 Sketch the head-capacity curves for centrifugal pumps having β 2 between 0° and 90°, equal to 0°, and less than 0°. 2.13 Consider a single-stage air turbine with an air flow rate of 2 kg/s. The gas turbine is a 90° IFR type. The relative velocity of the air entering the rotor is purely radially directed. The exhaust is purely axially directed. The rotor has an O.D. (outside diameter) of 0.3 m, and the tip blade speed is 350 m/s. Find a. the turbine power in kW. b. the shaft speed in rpm. c. the shaft torque in Nm. Hint: The turbine is similar to the hydraulic turbine depicted in Figure 2.13 with Vu1 = U1 and Vu2 = 0 . 2.14 The impeller of a centrifugal pump has an O.D. of 0.30 m. Oil having a specific gravity of 0.81 enters the impeller at a rate of 63.0833 L/s. The relative velocity at the impeller exit has no tangential component. The impeller rotates at 4800 rpm. Find a. the impeller tip speed in m/s. b. the energy transfer in J/kg. c. the power input in kW. 2.15 An axial-flow steam-turbine rotor receives steam at 20° to the tangential direction and exhausts it axially. The axial component of the steam velocity, which is assumed constant, is 0.7 times the blade velocity at the mean radius of the blade and has a magnitude of 450 ft/s. The mass flow rate of steam through the stage is 5.75 lbm/s. For the stage at the mean radius, find a. the relative velocity leaving the rotor in ft/s. b. the relative velocity entering the rotor in ft/s. c. the absolute velocity leaving the nozzles in ft/s. d. the degree of reaction. e. the energy transfer in ft · lbf/slug. f. the power produced in hp. 2.16 An axial-flow gas-turbine stage has a degree of reaction of 0.5. The blade speed is 600 ft/s at the mean radius, and the mean radius is 1.5 ft. Gas enters the stage at a nozzle angle of 27°. The exhaust velocity is directed axially over the entire annulus, which has a throughflow area of 3.927 ft2. The density of the exhaust gas is 0.135 lbm/ft3. The axial velocity component is constant through the rotor. Find a. the mass flow rate of gas in lbm/s. b. the energy transfer in ft · lbf/slug. c. the rotational speed in rpm. d. the power output in hp.

48

Logan’s Turbomachinery

2.17 Consider an axial-flow air-compressor stage. Air enters the stage from the atmosphere at a static temperature of 540°R. The blade speed at the mean radius is 1000 ft/s. The air deflection angle in both the rotor and the stator is 30°. The absolute velocity leaving the rotor makes an angle of 60° with the axial direction. The degree of reaction of the stage is 50%. Assume that the stage is a repeating stage (i.e., V1 = V3), that the axial component is the same at all three stations, and that the compression is isentropic. Find the stage (static) pressure ratio. 2.18 An axial-flow gas-turbine stage is to be designed for 50% reaction and minimum exhaust gas kinetic energy. The rotor is to turn at 3600 rpm with a mean blade velocity of 1000 ft/s. The nozzle blades are set to produce a gas nozzle angle of 24°. Assume that c p = 0.27 Btu/(lbm · °R) and that the stage exhausts to the atmosphere at 14.1 psia and 900°F. Find a. the total temperature drop in the stage. b. the energy transfer in Btu/lbm. c. the power produced in hp for a blade length of 8 in. 2.19 Solve for the stage horsepower of an air turbine with Vu1 = U and Vu2 = 0. The mean blade velocity is 1200 ft/s, and the mass rate of flow of air through the stage is 50 lbm/s. The nozzle angle α is 15°. For an annular throughflow area A 2 of 4.99 ft2, find the density ρ2 of the exhaust gas in lbm/ft3. 2.20 For the axial-flow compressor, show that the degree of reaction is R = − Wum /U, where Wum = (Wu1 + Wu2 )/2. 2.21 Show that the degree of reaction for axial-flow turbines is also given by R = − Wum /U. 2.22 Consider an axial-flow gas turbine with R = 1 2. The exhaust velocity V2 makes an angle of 10° with the axial direction. The decrease of absolute tangential velocity in the rotor is ∆Vu = 1250 ft/s. The blade speed is U = 1100 ft/s. Find a. the energy transfer E. b. the axial component of velocity Va. c. the nozzle exit velocity V. d. the nozzle angle α. 2.23 A centrifugal pump handles 2400 gallons of water per minute (density = 62.4 lbm/ft3). The impeller diameter D 2 is 19 in., the impeller tip width b2 is 1.89 in., the rotational speed is 870 rpm, and the head H is 80 ft. Find a. the impeller tip speed U 2 in ft/s. b. the meridional velocity Wm2 in ft/s. c. the tangential velocity component Vu2 in ft/s. d. the relative fluid angle β 2 in degrees. 2.24 A centrifugal water pump with backward-curved vanes runs at 1500 rpm and delivers 0.6688 ft3/s. The tip width of the vanes is b2 = 0.5 in. The tip diameter is U 2 = 0.5106 ft . The head is H = 28 ft . The density of water is 62.4 lbm/ft3. Find a. the tangential velocity component at the impeller tip Vu2 in ft/s. b. the tip speed U 2 in ft/s. c. the angle β 2 between W2 and U 2 in degrees.

Basic Concepts and Relations of Aerothermodynamics

49

2.25 An axial-flow gas-turbine stage produces 1000 hp with a gas flow rate of 10 lbm/s. The degree of reaction is 50%. The exhaust velocity is V2 = 600 ft/s and is directed axially. Find the blade speed U and the nozzle angle α. 2.26 The nozzle angle (angle between V1 and U) is 25° in an axial-flow gas turbine. The throughflow velocity is Va = 400 ft/s, and the mean blade speed is U = 800 ft/s. The velocity diagram is symmetrical (W1 = V2 ). Find a. the relative fluid angles β1 and β 2. b. the energy transfer E. c. h 1 − h 2. 2.27 Use the throughflow and blade velocity data from Problem 2.26 to construct the velocity diagram for an impulse turbine with zero swirl in the exhaust gas (Vu2 = 0). Find a. the nozzle angle α. b. the energy transfer. c. h 1 − h 2. 2.28 Using the expressions of rothalpy in the rotor and stator reference frames, derive the equations to convert fluid total temperature between these reference frames. Then, using the fact that the local static pressure and temperature are independent of the reference frame, derive equations to convert fluid total pressure between the rotor and stator reference frames.

Reference Sultanian, B.K. 2015. Fluid Mechanics: An Intermediate Approach. Boca Raton, FL: Taylor & Francis.

Bibliography Allen, T., and R.L. Ditsworth. 1972. Fluid Mechanics. New York: McGraw-Hill. Jones, J.B., and G.A. Hawkins. 1986. Engineering Thermodynamics. New York: Wiley. Pritchard, P.J., and J.W. Mitchell. 2015. Fox and McDonald’s Introduction to Fluid Mechanics, 9th edition. New York: Wiley. Reynolds, W.C., and P. Colonna. 2018. Thermodynamics: Fundamentals and Engineering Applications. Cambridge: Cambridge University Press.

Nomenclature A b1 b2

Flow area at control surface Width of vane at r = r1 in pump or turbine rotor Width of vane at r = r2 in pump or turbine rotor

50

cp cv C CV E

Logan’s Turbomachinery

Specific heat of gas at constant pressure Specific heat of gas at constant volume Speed of sound Velocity coefficient = ratio of mass velocity to total velocity Specific energy transfer = w(positive if it is from the fluid and negative if it is to the fluid) Ff Static-pressure mass flow function having dimensions of 1/ R Dimensionless static-pressure mass flow function Fˆ f Ff0 Total-pressure mass flow function having dimensions of 1/ R Fˆ f0 Dimensionless total-pressure mass flow function FSx x-component of the surface force FBx x-component of the body force g Acceleration due to gravity h Specific enthalpy of the fluid h e Specific enthalpy at the nozzle (stator) entry h 0 Specific total (stagnation) enthalpy of the fluid h 0R Specific total enthalpy in the rotor reference frame H Head = − E/g  a Flow of the axial component of angular momentum H I Rothalpy IFR Inward-flow radial  m Mass flow rate M Mach number  Flow rate of x-component of linear momentum Mx n Polytropic exponent (n = γ for isentropic processes) N Rotational speed p Static pressure pi Impulse pressure p0 Total (stagnation) pressure pamb Ambient pressure Power to or from rotor P q Heat transfer per unit mass of flowing fluid r Radial coordinate perpendicular to axis of rotation R Position vector; origin on the axis of rotation R Degree of reaction R Gas constant T Absolute static temperature T0 Absolute total (stagnation) temperature T0R Fluid total (stagnation) temperature in the rotor reference frame u Specific internal energy of fluid U Blade speed V Absolute flow velocity Va Component of V in axial direction Vm Component (meridional) of V perpendicular to control surface Vr Component of V in radial direction Component of V in tangential direction Vu Vφ Component of V normal to the flow area A w Specific work done on or by the fluid W Velocity relative to moving blade

Basic Concepts and Relations of Aerothermodynamics

Wu z α α1 α2 β1 β2 γ Γ ρ θ ω

51

Tangential component of W Altitude above an arbitrary plane in a direction opposite to g Nozzle angle; angle between V1 and U Angle between V1 and Vm1 (Vm1 = Va in axial-flow machines) Angle between V2 and Vm2 (Vm2 = Va in axial-flow machines) Angle between W1 and Vm1 (Vm1 = Va in axial-flow machines; Vm1 = Vr1 in radial-flow machines) Angle between W2 and Vm2 (Vm2 = Va in axial-flow machines; Vm2 = Vr2 in radial-flow machines) Ratio of specific heats = c p /c v Torque Fluid density Angle between W1 and W2 (axial-flow compressor) Angular velocity

3 Dimensionless Quantities

3.1 Introduction In plotting the results of turbomachinery tests and in analyzing performance characteristics, it is useful to use dimensionless groups of variables. Appropriate groups of variables are found by application of a dimensional methodology (dimensional analysis), and it is known from the Buckingham pi theorem that the dimensionless groups so formed have a functional relationship, although the nature of the relationship is frequently unknown except by experimentation. An important benefit of dimensional analysis is that the results of model studies so analyzed and plotted may then be used to predict full-scale performance. This is important for efforts to reduce the cost of the development of turbomachines. It is also useful in the analysis of data from full-sized machines when the aim is to predict the performance of other full-sized machines that are of a different size than those tested or that operate under different conditions.

3.2  Turbomachine Variables The important variables in turbomachine performance are shown in Table 3.1. The Buckingham pi theorem applied to the four variables and two dimensions indicates that two dimensionless groups can be formed. Of the several possible groups that can be formed, the most useful combinations of variables are the flow coefficient ϕ, defined by

ϕ=

Q ND 3

(3.1)

ψ=

gH N 2D 2

(3.2)

and the head coefficient ψ, defined as

where E = gH has been used in the analysis. Three other dimensionless groups are also used extensively by engineers. However, they may be easily derived from ϕ and ψ. Specific speed N s is formed in the following way: 1



Ns =

1

ϕ2 NQ 2 3 3 = ψ 4 (gH) 4

(3.3) 53

54

Logan’s Turbomachinery

TABLE 3.1 Primary Turbomachine Variables Variable

Symbol

Dimensions

Head or energy transfer Volume flow rate Angular speed Rotor diameter

gH or E

L2/T2 L3/T 1/T L

Q N D

As ϕ and ψ are the nondimensional flow rate and head, respectively, so N s is the nondimensional speed. In fact, if Q and H are unity, we observe that N s = N. The other particularly useful dimensionless groups are the specific diameter D s and the power coefficient CP , as defined below: 1

Ds =

ψ 4 D(gH) = 1 1 Q2 ϕ2

CP = ϕψ =

1

4

(3.4)

P

(3.5)

3

ρN D 5



where power P = ρQgH . Recall that four variables are assumed to be of primary importance. If additional variables are added to the list in Table 3.1, then a new dimensionless group can be formed that will contain each—i.e., an additional group for each new variable. For example, if kinematic viscosity ν is added, we have the Reynolds number Re, defined as Re =



ND 2 ν

(3.6)

Another example is the inlet fluid temperature or the inlet specific enthalpy h 1. Since T1 is proportional to the square of the acoustic speed in gases, we would expect the Mach number M to emerge as the appropriate dimensionless group—namely, M=

ND = γRT1

ND h 1 ( γ − 1)

(3.7)

where we have used the relations h 1 = c p T1 and c p /R = γ /( γ − 1). This is an important variable in turbomachines involving the high-speed flow of gases. If the number of variables is increased to seven by adding the inlet pressure p1, the nature of the dimensionless groups is changed but not the number. This is because pressure involves the force dimension, which is not present in the others. Thus, the number of groups remains four. However, since density ρ1 can be introduced as a combination of p1 and T1 (from h 1), it appears in the groups—for example, in combination with Q as the mass flow rate m ˙ . In this case, the mass flow coefficient can be written as



Cm =

 m (p1ρ1 )1 2 D 2

(3.8)

55

Dimensionless Quantities

Other forms of the head coefficient are the ratio of outlet pressure to inlet pressure p1/p2 and the ratio of outlet temperature to inlet temperature T1/T2 . Clearly, these ratios are equivalent, since head is proportional to enthalpy difference, which in turn is proportional to temperature difference in gas-flow machines and pressure difference in incompressible-flow machines. In gas-flow turbomachines, either p1/p2 or T1/T2 could be used, since the two are related through isentropic or polytropic process relations. Efficiency η has many specialized definitions, but generally it is defined as output power divided by input power. It, too, can be included in the list of variables, and since it is already dimensionless, it is also included in the list of dimensionless groups.

3.3 Similitude Flow similarity occurs in turbomachines when geometric, kinematic, and dynamic similarities exist between a model (i.e., a small-sized turbomachine) and its larger prototype. Thus, ratios of dimensions of corresponding parts are the same throughout. Velocity triangles at corresponding points in the flow fields are also similar, as are ratios of forces acting on the fluid elements. Similar velocity triangles, for example, imply equal flow coefficients:



 Q  = Q      ND 3  p  ND 3  m

(3.9)

In contrast, similar force triangles are equivalent to equal head coefficients:



 gH  =  gH     2 2  N D p  N 2D 2  m

(3.10)

The equality of dimensionless groups resulting from similitude has important practical consequences. It allows a most compact presentation of graphical results. One example of this is seen from a consideration of head-capacity curves for centrifugal pumps, which typically appear as shown in Figure 3.1. A separate curve is needed for each shaft speed when plotting the primary variables. On the other hand, if the head coefficient is plotted against the flow coefficient, the curves collapse to a single curve with a single relationship between ψ and ϕ, regardless of speed.

FIGURE 3.1 Head-capacity curves.

56

Logan’s Turbomachinery

The so-called pump laws also follow from the similarity conditions expressed by Equa­tions 3.9 and 3.10. If one wants to know how a given pump will perform at another speed when its performance at one speed is known, we simply cancel D 3 in Equation 3.9 and find

Q1 Q 2 = N1 N 2

(3.11)

which is a pump law; it implies that capacity Q varies directly with speed N. In a similar manner, we see from Equation 3.10 that the head H or pressure rise is governed by

H1 H 2 = N 12 N 22

(3.12)

i.e., head varies as the square of the speed. Power is the product of Q and H, and the third pump law states that

P1 P = 23 3 N1 N 2

(3.13)

Laws for scaling up or down—i.e., varying diameter D while keeping the speed constant— follow in a similar manner after canceling the factors containing N. Thus, we find

Q1 Q 2 = D13 D 32

(3.14)



H1 H 2 = D12 D 22

(3.15)



P1 P = 25 5 D1 D 2

(3.16)

Performance curves are frequently plotted from dimensionless or quasi-dimensionless groups. Compressor maps, for example, are usually presented in graphs of the form shown in Figure 3.2. The abscissa is determined from Equation 3.8 by dropping the diameter, since it is not a variable in the performance of a single machine, and by using the same gas constant. Similarly, the parameter N/ T1 is a variation of the machine Mach number

FIGURE 3.2 Compressor map.

57

Dimensionless Quantities

TABLE 3.2 Specific Speeds Turbomachine Pelton wheel Francis turbine Kaplan turbine Centrifugal pumps Axial-flow pumps Centrifugal compressors Axial-flow turbines Axial-flow compressors

Specific Speed Range 0.03–0.3 0.3–2.0 2.0–5.0 0.2–2.5 2.5–5.5 0.5–2.0 0.4–2.0 1.5–20.0

formed from Equation 3.7 by eliminating specific heat and the rotor diameter, which are both constants for a given map. Dimensionless groups are also useful in design and machine selection. For example, specific speed is commonly used to indicate the type of machine appropriate to a given service. Table 3.2 gives ranges of specific speeds corresponding to efficient operation of the turbomachines listed. Sizes of turbomachines required for a given service are also determinable from specific speed-diameter plots of the type shown in Figure 3.3.

FIGURE 3.3 Cordier diagram. (Source: G.T. Csanady, Theory of Turbomachines. Copyright 1964 by McGraw-Hill, Inc. Used with the permission of McGraw-Hill Book Company.)

58

Logan’s Turbomachinery

This correlation, developed by Csanady (1964), of the optimum specific speeds of various machines as a function of specific diameter is useful in determining an appropriate size for a given set of operating conditions. To enter the diagram, called the Cordier diagram, a specific speed can be selected from Table 3.2. The rotor diameter can be determined from the specific diameter found from the Cordier diagram. A machine so selected or designed would be expected to have high efficiency.

Worked Examples EXAMPLE 3.1 Use dimensional analysis to derive Equation 3.1 from the variables and dimensions in Table 3.1. SOLUTION: There are four variables and two dimensions in Table 3.1. The Buckingham pi theorem states that the number of independent dimensionless groups equals the number of variables minus the number of dimensions. Since two dimensionless groups can be formed, Q and gH are chosen to serve as nuclei for the groups. Denoting the groups as ϕ and ψ, we form the following arrangements of variables: ϕ = QN aD b

and

ψ = gHN cDd



where the exponents of N and D are to be determined. Since φ is dimensionless, we can write the ϕ equation dimensionally as L0 T 0 = L3 T −1T − a Lb



Equating the exponents of L, we have 3+ b = 0



Similarly, equating the exponents of T gives −1 − a = 0



The above two equations yield a = −1 and b = −3 , resulting in ϕ = QN −1D −3

which is identical to Equation 3.1.

59

Dimensionless Quantities

EXAMPLE 3.2 Calculate the dimensionless and the dimensional values of specific speed N s for a centrifugal water pump whose design-point performance is the following:

Q = 2400 gal/min



H = 70 ft



N = 870 rpm

SOLUTION: The first step is to convert each quantity to consistent units. Use the conversion factors in Table A.1. 2400 Q= = 5.3476 cfs 448.8

gH = 32.174 × 70 = 2252.18 ft 2/s 2 N=



870 = 91.106 rad/s 9.5493

Dimensionless specific speed: 1

Ns =

91.106(5.3476) 2 = 0.644 3 (2252.18) 4

Dimensional specific speed: 1

2  s = NQ3 N H4

1



2 rpm(gal/min)  s = 870(2400) = 1761.170 N 3 3 4 ft 4 (70)

1

2

EXAMPLE 3.3 Use the relations for fluid friction losses in pipes to derive a scaling law for the efficiency of hydraulic turbines. The scaling laws already derived are for Q, H, and P and are given by Equations 3.14–3.16. Such laws are based on the principle of geometric, kinematic, and dynamic similarity. Assume this similarity between model and prototype in the present derivation. SOLUTION: Refer to the Moody diagram in a fluid mechanics text (e.g., Pritchard and Mitchell, 2015). The hydraulic loss HL is calculated from a friction factor f, which can be read from the Moody diagram. The hydraulic loss in a rotor flow passage is given by



HL =

fL R W 2 2gD R

(3.17)

60

Logan’s Turbomachinery

Defining efficiency as output over input, we write η=

g ( H − HL ) gH

(3.18)



where gH is the mechanical energy given up by the fluid during its passage through the turbine. The numerator of Equation 3.18 represents the mechanical energy extracted by the turbine rotor. The dimensionless loss is given by 1− η =

H L fL R W 2  N 2D 2  = H 2gD R H  4U 2 

(3.19)

The similarity between model and prototype implies equality of L R /D R , W/U, and gH / N 2D 2 . Thus, we find

(

)

1 − ηp fp = 1 − ηm fm



(3.20)

where subscripts p and m refer to prototype and model, respectively. In turbomachines, the Reynolds number is typically so high that only the relative roughness of the passage walls affects the friction factor. Because of similarity, D/D R is also constant. If roughness height is assumed invariant, then the friction factor is inversely related to machine diameter, and 1 − ηp  D m  = 1 − ηm  D p 



n

(3.21)

where n is an experimentally determined exponent.

Problems 3.1 Derive expressions for specific speed, specific diameter, and power coefficient by combining flow and head coefficients. Show that each is dimensionless. 3.2 Determine the model-to-prototype diameter ratio for a water turbine that will produce 30,000 hp at 100 rpm with a head of 50 ft, while the model will produce 55 hp under a head of 15 ft. Hint: Use the results of Problem 3.19. 3.3 Determine the speed and flow rate of the model turbine in Problem 3.2. Assume that the turbine efficiency η is 0.90. Hint: Use the turbine power relation P = ηρgQH P. 3.4 A centrifugal fan is to be compared with a larger, geometrically similar fan. The smaller fan delivers 500 ft3/min of air at standard conditions with a pressure rise (head) of 2 in. of water. The smaller fan runs at 1800 rpm, while the larger one operates at 1400 rpm and produces the same head. Determine the diameter ratio of the two fans and the flow rate of the larger.

Dimensionless Quantities

61

3.5 Use the Cordier diagram to estimate the rotor diameter of a pump that, while running at 1000 rpm at a head of 30 ft, will deliver 4500 gal/min. Would you recommend a centrifugal or an axial-flow pump for this service? 3.6 Use dimensional analysis to derive Equation 3.2 from the variables and dimensions in Table 3.1. 3.7 Derive Equation 3.6 by forming a new group, Reynolds number Re, in which the kinematic viscosity appears, along with N and D. The Reynolds number so formed is called the Machine Reynolds number. Hint: Start with Re = ν−1N aD b. Note that the dimensions of ν are L2 /T . 3.8 Derive Equation 3.7 by introducing the specific enthalpy h 1, the enthalpy at the inlet of a machine in which the flow is compressible, as an additional variable. Note that the dimensions of specific enthalpy are FL/M—i.e., energy dimensions divided by mass dimensions. Additionally, Newton’s second law of motion can be written dimensionally as F = ML/T 2. Hence, the dimensions of h are L2 /T 2 , and no new dimensions are introduced. Hint: Form the new dimensionless group, Mach number, by writing M = h −1N aD b. 3.9 Show that the quantity h 1 in Equation 3.7 is the same as the square of the acoustic speed at temperature T1 divided by γ − 1, where γ is the ratio of specific heats. 3.10 Show that the quantity ND, which appears in Equations 3.6 and 3.7, is proportional to the blade speed at the tip of a turbomachine rotor. 3.11 Show that the flow coefficient ϕ defined in Equation 3.1 is proportional to Va /U in an axial-flow turbomachine. The latter form is used to define the flow coefficient in Chapters 6 and 8. 3.12 Show that the flow coefficient ϕ defined in Equation 3.1 is proportional to Wm2 /U 2 in centrifugal pumps and compressors. The latter definition is used to define the flow coefficient in Chapters 4 and 5. Hint: Use the fact that b2 is proportional to D 2 in centrifugal machines. 3.13 Show that the mass flow coefficient Cm defined by Equation 3.8 can be derived from the product of ϕ and M given by Equations 3.1 and 3.7, respectively. Hint: Divide 1/2 ϕM by [( γ − 1)/γ ] , and then multiply by ρ1/ρ1 to obtain Equation 3.8. 3.14 Specific speed can be calculated as a dimensional quantity (e.g., as used in Table A.2) or in dimensionless form (as used to construct the Cordier diagram in Figure 3.3). In dimensional form, its units are (rpm)(gal/min)1/2(ft)−3/4. Note that H is used in lieu of gH in calculating the dimensional form of N s. Convert the dimensional specific speed N s = 1000 into its corresponding dimensionless value. 3.15 If the velocity triangles of a centrifugal pump are similar to those of the model pump, show that the flow coefficients are equal for model and prototype. Hint: Use the results of Problem 3.12. 3.16 If the velocity triangles of an axial-flow compressor are similar to those of the model compressor, show that equality of the flow coefficients is a consequence. Hint: Use the results of Problem 3.11. 3.17 If the velocity triangles of an axial-flow compressor are similar to those of the model compressor, show that equality of the head coefficients is a consequence. Hint: First show that gH/(N 2D 2 ) = E /(4D 2 ), and then use Equation 2.53.

62

Logan’s Turbomachinery

3.18 Use the Cordier diagram (Figure 3.3) to determine a value of impeller diameter D 2 that is suitable for a centrifugal pump whose operating conditions are those given in Example 3.2. Hint: Find the specific diameter D s from the Cordier diagram using the previously calculated N s . Use Equation 3.4 to solve for D, which is the same as D 2 for the pump. Note: The actual pump with the same capacity and head has an impeller diameter of 19 in. 3.19 Derive the relation that states that NP 1/2 H 5/4 is the same for model and prototype. Hint: Divide the square root of the power coefficient CP by the head coefficient ψ to the 5 4 power. The dimensionless group formed involves N, P, g, H, N, and ρ. Equate these groups for model and prototype. Assume the same fluid density for model and prototype. 3.20 Derive a scaling law for the efficiency of centrifugal pumps. Use the method of Example 3.3, but note that pump efficiency is defined by





η=

gH g ( H + HL )

where gH represents the mechanical energy rise of the fluid passing through the pump and the denominator represents the mechanical energy transferred from the impeller to the fluid. 3.21 Use values from Table A.2 to estimate the efficiency of the centrifugal pump described in Example 3.2. Determine the hydraulic loss H L from the definition of pump efficiency given in Problem 3.20. 3.22 Show that Q is proportional to D 3 for similar machines operating at the same specific speed N s and the same rotational speed N. Hint: Use Equation 3.3. Note that head and flow coefficients have the same values for model and prototype. 3.23 Estimate the empirical exponent in the scaling law derived in Problem 3.20 for centrifugal pumps with N s = 500 and Q between 30 and 1000 gal/min. Use data from Table A.2, and assume that Q is proportional to D3. Hint: Plot (1 − η)/η as a function of Q on log-log graph paper to determine the slope of the resulting straight line. 3.24 If the pump described in Example 3.2 were to be operated at 1200 rpm, what capacity Q, head H, and power P would result? 3.25 A model of the centrifugal pump analyzed in Example 3.2 is to be constructed for laboratory study. Assuming that the model and prototype have complete similarity, that the capacity of the model is reduced to 150 gal/min, and that the same rotational speed is to be used for model and prototype, find the impeller diameter of the model, the head produced by the model pump, and the model pump efficiency. 3.26 Use the Cordier diagram to test the manufacturer’s claim that a centrifugal pump with a rotor diameter of 6 in. can supply water at the rate of 10 ft3/s and produce a head of 30 ft at a rotational speed of 980 rpm. 3.27 Examine the claim of pump performance stated in Problem 3.26 by calculating the maximum possible head with Equation 2.56. Assume that radial vanes are used, so that β 2 = 0o. Is the 30 ft head possible even under these idealized conditions? 3.28 Use Equation 2.56 to calculate the maximum possible head—i.e., the ideal head—for the pump described in Example 3.2. Assume D 2 = 19 in., b2 = 0.10D 2, and β 2 = 65o.



Dimensionless Quantities

63

References Csanady, G.T. 1964. Theory of Turbomachines. New York: McGraw-Hill. Pritchard, P.J., and J.W. Mitchell. 2015. Fox and McDonald's Introduction to Fluid Mechanics, 9th edition. New York: Wiley.

Nomenclature a b b 2 c Cm CP d D D 2 Ds DR E f F g gH h h 1 H H L k R L LR M M  m n N N s  s N p1 P Q R

Exponent Exponent Width of blade at r = r2 in pump or turbine rotor Exponent Mass flow coefficient Power coefficient Exponent Tip (radial-flow) or mean (axial-flow) rotor diameter Rotor tip diameter Specific diameter Mean hydraulic diameter of rotor flow passage (four times the flow area divided by the wetted perimeter) Specific energy transfer between rotor and fluid Friction factor Force dimension Gravitational acceleration Loss (turbine) or gain (pump) of fluid-specific mechanical energy during passage through the turbomachine Specific enthalpy of fluid Specific enthalpy of fluid at inlet Head = E /g Hydraulic loss in rotor passage Average height of surface roughness Length dimension Length of rotor passage Mass dimension Mach number Mass flow rate of fluid Empirical exponent in the efficiency scaling law Rotor speed in rpm Dimensionless specific speed Dimensional specific speed Fluid pressure at inlet Power to or from rotor Capacity = volumetric flow rate Gas constant

64

Re T T1 U U 2 Va Wm2 W β 2 γ ν ρ ϕ ψ [=]

Logan’s Turbomachinery

Reynolds number Time dimension Absolute temperature at inlet Blade or runner speed Rotor tip speed Axial component of absolute velocity V Meridional component of W at rotor exit Average velocity relative to moving rotor flow passage Angle between W2 and Wm2 Ratio of specific heats Kinematic viscosity Fluid density Flow coefficient Head coefficient Dimensionally equal to

4 Centrifugal Pumps and Fans

4.1 Introduction Known more commonly as impellers, rotors of centrifugal pumps, blowers, and fans are designed to transfer energy to an incompressible fluid flow. Fans and blowers usually consist of a single impeller spinning within an enclosure, known as the casing. Pumps, on the other hand, may be designed to have several impellers mounted on the same shaft, and the fluid discharging from one is conducted to the inlet of the neighboring rotor, thus making the overall pressure rise of the pump the sum of the individual-stage pressure rises. In cross section, the individual impellers are designed to look somewhat like that shown in Figure 1.2a. An end view of the impeller is shown in Figure 4.1. The blades shown are curved backward, making the angles β1 and β 2 with tangents to the circles at radii r1 and r2 , respectively. Ideally, the relative velocity W2 leaves the blade at the outer edge of the impeller at the blade angle β 2 . Figure 4.2 shows velocity diagrams at the inlet and outlet of the blade passages. For the design-point operation, the relative velocity W1 is approximately aligned with the tangent to the blade surface at angle β1. The absolute velocity V1 at the inlet is shown entering with no whirl. Thus, Vu1 = 0 and Vm1 = V1. The ideal or virtual head H i , which is the ideal energy transfer per unit mass for perfect guidance by the blades, is given by Hi =

U 2 Vu2 g

(4.1)

Equation 4.1 was derived earlier as Equation 2.54. The ideal head Hi is higher than that found in practice. Reasons for this disparity and methods for correction will be given in subsequent sections.

4.2  Impeller Flow Figure 4.1 shows an impeller rotating in the clockwise direction. Fluid next to the blade pressure surface is forced to rotate at the blade speed. Motion in a purely circular path at radius r implies a net pressure force directed radially inward, so that the net pressure

65

66

Logan’s Turbomachinery

FIGURE 4.1 Pump impeller.

force Adp on a differential element of cross-sectional area A balances the centrifugal force (ρAdr)ω 2 r ; thus, the radial pressure gradient is given by



dp ρU 2 = dr r

(4.2)

Since the fluid does not follow the impeller as in a solid body rotation but instead tends to remain stationary relative to the ground, a resultant outward flow along the blade with an accompanying adverse pressure gradient occurs. However, the magnitude of the pressure rise across the rotor is less than that indicated by the integration of Equation 4.2—i.e., less than



p 2 − p1 =

(

ρ U 22 − U12 2

)

(4.3)

A better estimate of the pressure rise is obtained from an equation formed from Equations 2.54 and 2.60:



FIGURE 4.2 Velocity diagrams at inlet and outlet.

p2 − p1 = ρU 2 Vu2 −

V22 − V12 ρ 2

(4.4)

67

Centrifugal Pumps and Fans

Applying the law of trigonometry to the diagrams of Figure 4.2 yields the relations

and

U 2 Vu2 =

U 22 + V22 − W22 2

(4.5)

V12 = W12 − U12



(4.6)

Combining Equations 4.4, 4.5, and 4.6 results in the following expression for pressure rise:



p2 − p1 =

(

ρ U 22 − U12 + W12 − W22 2

)

(4.7)

Although the static pressure at the inlet and outlet of the impeller is expected to be uniform across the opening between the blades, pressures on the two sides of a blade are expected to be different. As the fluid moves radially outward, its angular momentum per unit mass rVu is clearly increased. This means that a moment of some force has been applied to the control volume considered. The source of such a force is obviously a pressure difference between any two points on opposite sides of the control volume at the same radial distance from the axis of rotation. The azimuthal force resulting from this pressure difference is the so-called Coriolis force, which equals 2ωW . This force is applied to the impeller at the pressure surface and the suction surface of the blade. Equation 4.7 applied between the inlet and some intermediate radius less than r2 implies that the greater pressure rise on the pressure surface is accompanied by a lower relative velocity W on that surface. Conversely, a higher relative velocity at the suction surface is indicated. Figure 4.1 shows a circulatory flow, which is radially inward on the pressure surface and radially outward on the suction surface. This secondary flow is superposed on the radially outward main flow. The difference in pressure rise on the two sides of the passage between blades implies a separation, or backflow, region near the outer end of the suction surface. The latter indicates a flow deflection away from the suction surface near the exit of the passage. The change in Vu2 associated with this flow deflection is known as slip. The ratio of the actual Vu2 to the ideal Vu2 is usually known as the slip coefficient µ s. Since the slip depends on the circulation, which in turn is dependent on the geometry of the flow passage, a theoretical relationship expressing µ s as a function of the number of blades n B and exit angle β 2 is not surprising. Shepherd (1956) has given such a relation:



µs = 1 −

πU 2 sinβ 2 Vu2 n B

(4.8)

which is derived in Appendix B. For a finite number of blades, the velocity diagram of Figure 4.2 must be modified to reflect the effect of slip; this effect is illustrated in Figure B.2 in Appendix B. The actual tangential component of V2 is denoted by Vu2′ , which replaces the component Vu2 , which corresponds to perfect guidance by the blades. The fluid angle for perfect guidance is β 2 and is the same as the blade angle. With a finite number of blades and the accompanying slip, the actual fluid angle is different from the blade angle and is denoted by β 2 ′ . The energy transfer with a finite number of blades is given by Vu2 ′ U 2 , and the corresponding input head H in is calculated from H in = U 2

Vu2 ′ g

(4.9)

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Logan’s Turbomachinery

4.3 Efficiency Flow in the impeller or casing passages is accompanied by frictional losses that are proportional to the square of the flow velocity relative to the passage walls. All losses result in a conversion of mechanical energy into thermal (internal) energy. Wall friction causes this transfer through direct dissipation by viscous forces and by turbulence generation that culminates in viscous dissipation within the small eddies. Secondary flow losses occur in regions of flow separation, where circulation is maintained by the external flow, and in curved flow passages, where it is maintained by centrifugal effects. The steady-flow energy equation is applied to a control volume that is bounded by the pump casing and the suction and discharge flanges, as is depicted in Figure 4.3. The enthalpy, kinetic energy, and potential energy are changed by the work input gH in , and the balance of these energies is expressed by hs +

Vs2 V2 + z s g + gH in = h d + d + z d g 2 2

(4.10) where the subscripts s and d refer to properties at the suction and discharge flanges of the pump casing. Enthalpy can be written in terms of internal energy and flow work, so that the work input becomes

gH in =

pd − p s Vd2 − Vs2 + (z d − z s )g + + ud − u s 2 ρ

(4.11)

The hydraulic loss gH L is the loss of mechanical energy or the gain of internal energy per unit mass of fluid passing through the pump. Substituting gH L for the last term in Equation 4.11 and transposing it to the left-hand side, we have pd − p s Vd2 − Vs2 + (z d − z s )g + (4.12) 2 ρ where the output head H is defined as the input head less the hydraulic loss. The righthand side of Equation 4.12 represents the increase in the three forms of mechanical energy: kinetic, potential, and flow work. Typically, only the last term need be considered in computing the output head, so that Equation 4.12 becomes gH = g(H in − H L ) =

H=

FIGURE 4.3 Centrifugal pump with piping.

pd − p s ρg

(4.13)

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Centrifugal Pumps and Fans

According to Csanady (1964), the hydraulic loss can be expressed in terms of loss coefficients k d and k r for the diffuser and rotor, respectively, and the corresponding kinetic energies, so that



gH L =

k d V22′ k r W22′ + 2 2

(4.14)

Equation 4.14 is used in Appendix C to determine the value of Vu2 ′ /U 2 that gives the minimum value of hydraulic loss. For typical flow coefficient values 0.05–0.20, the optimum value of Vu2 ′ /U 2 is approximately 0.5. Although Equation 4.14 is useful in analyzing losses, HL is usually obtained from the hydraulic efficiency, which is defined by



ηH =

H H − HL = in H in H in

(4.15)

Karassik et al. (2007) offer the following empirical correlation for ηH: ηH = 1 −

0.8 1 Q4

(4.16)

where Q is capacity in gallons per minute. According to Equation 4.16, the hydraulic losses, represented by 1 − ηH , vary from 30% for pumps of 50 gal/min capacity to 8% for pumps of 10,000 gal/min capacity. Outside the impeller, where no throughflow occurs, the fluid is forced to move tangentially and radially. This circulatory motion of unpumped fluid results in an additional (disk-friction) loss. A different, but equally nonproductive, use of energy occurs because of a reverse flow (leakage) from the high-pressure region near the impeller tip to the lowpressure region near the inlet. The latter effect is the reason for the introduction of the volumetric efficiency ηv, defined as



ηV =

 m  +m L m

(4.17)

 L is the leakage mass flow rate and m  is the mass flow rate actually discharged where m from the pump. Because of the loss of mechanical energy by the several mechanisms mentioned above, the head H—i.e., the net mechanical energy added to the fluid in the pump as determined by measurement—is less than the head computed from Equation 4.9. Usually, the practical performance parameter as determined by test is the overall pump efficiency η, defined by



η=

 mgH P

(4.18)

where P is the power of the motor driving the pump as determined by dynamometer test. The so-called total head H is determined from the steady-flow energy equation after experimentally evaluating the mechanical energy terms at the suction and discharge sides of the pump.

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Logan’s Turbomachinery

TABLE 4.1 Constants for Equation 4.21 Ns 500 1000 2000 3000

C

N

1.0 0.35 0.091 0.033

0.50 0.38 0.24 0.128

The mechanical efficiency ηm accounts for frictional losses occurring between moving mechanical parts, which are typically bearings and seals, as well as for disk friction and is defined by



ηm =

 +m  L )gH in (m P

(4.19)

Substitution of Equations 4.9, 4.15, and 4.18 into Equation 4.19 yields the simple relationship η = ηm ηv ηH



(4.20)

Karassik et al. (2007) provide data on volumetric efficiency, which is correlated by ηv = 1 −



C Qn

(4.21)

where C and n are constants that depend on the dimensional specific speed N s . Some values of these constants are presented in Table 4.1. Equation 4.21 shows that volumetric efficiencies range from 0.99 for large pumps to 0.85 for pumps of low capacity. The mechanical efficiency can be calculated by formulating disk- and bearing-friction forces or from the knowledge of other efficiencies. The overall pump efficiency can be obtained from Table A.2, and the other efficiencies can be calculated from Equations 4.16 and 4.21. Thus, the mechanical efficiency is the only unknown in Equation 4.20. The process is illustrated in Example 4.1.

4.4  Performance Characteristics Characteristic curves for a given pump are determined by test, and they consist primarily of a plot of head H as a function of the volume flow rate Q. Expressing Equation 4.1 in terms of Q, we obtain Hi = U2

U 2 − Q cot β 2 /A 2 g

(4.22)

A typical characteristic curve is shown schematically in Figure 4.4. The theoretical head from Equation 4.22 is also shown in Figure 4.4. The actual curve is displaced downward as

71

Centrifugal Pumps and Fans

FIGURE 4.4 Pump characteristics.

a result of the losses of mechanical energy previously discussed. However, Equation 4.22 provides the engineer with the upper limit of performance that can be achieved, since it does not account for losses. If the speed is increased, Equation 4.22 indicates that the curve will shift upward, and vice versa. Dividing Equation 4.14 by the square of twice the tip speed ND 2 , we obtain gH i D 2 Q cot β 2   = 0.5  0.5 − 2 2  N D2 πbND 32 



(4.23)

(

)

which indicates a functional relationship between head coefficient gH/ N 2 D 22 and flow coefficient Q/ ND 32 , becoming independent of speed. The actual performance curves, when plotted nondimensionally, also show a functional relationship that is independent of speed; i.e., data for different rotor speeds will collapse into a single head-coefficient versus flow-coefficient curve. We can predict the approximate value of head or flow rate resulting from a change of speed if we assume that the operating state—i.e., the values of head and flow coefficients— is the same before and after the change. Referring to Figure 4.4 and considering a change of speed from N1 to N2, the operating state point on the characteristic plot moves from position 1 to position 2. Since we are assuming similar flows,

(



)

H1 H 2 = N 12 N 22

(4.24)

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Logan’s Turbomachinery

and Q1 Q 2 = N1 N 2



(4.25)

Equations 4.24 and 4.25 express the pump (or fan) laws and together yield



H1 H 2 = Q 12 Q 22

(4.26)

which states that H is proportional to Q 2 . This relation is approximately followed by the external system to which the pump is attached, assuming that no changes have been made in it. Thus, the path from 1 to 2 for a simple change of speed is roughly the locus of similar states, and this fact makes the pump (or fan) laws extremely useful. As indicated by Equation 4.18, the overall efficiency varies with flow rate, and it is required for the computation of brake power. Referring to Figure 4.5, which is a typical variation, we see that the efficiency varies with flow rate from zero at no flow to a maximum value ηmax near the highest flow rate. The actual value of ηmax varies from 70 to 90%, depending primarily on the design flow capacity. Machines handling large flows have higher maximum efficiencies, since frictional head loss decreases proportionately with large flow area. On the other hand, machines of high head and low flow (i.e., low specific speed) tend to have lower efficiencies. High head is associated with large-diameter or high-speed impellers, which feature increased disk-friction losses; low flow implies higher proportional head loss associated with smaller flow area. Characteristically, the latter type of machine has a radial-flow design, while the former is classified as a mixed-flow design. Although the choice of a specific speed may be dictated by design requirements, it is worth noting that test results show that centrifugal pumps with specific speeds between 0.7 and 1.0 seem to have the highest maximum efficiencies (e.g., see Church, 1972). The head-capacity (H–Q) curves can be altered at the high-flow end by cavitation, which occurs when the fluid pressure falls below the vapor pressure, featuring formation and collapse of vapor bubbles. Outward flow in the impeller passage, which is accompanied by pressure rise, results in a collapse of the bubble. Acceleration of fluid surrounding the bubble, which is required to fill the void left by the vapor, results in losses and pressure

FIGURE 4.5 Equal-efficiency contours for centrifugal pumps.

73

Centrifugal Pumps and Fans

waves, which cause damage to solid-boundary materials. Since the energy transfer per unit weight is reduced by the presence of vapor, the head-capacity curve falls off at the flow corresponding to the onset of cavitation. To avoid cavitation, the net positive suction head (NPSH), defined as the atmospheric head plus the distance of liquid level above the pump centerline minus the friction head in the suction piping minus the gauge vapor pressure, is maintained above a certain critical value. A critical specific speed Sc defined as 1

Sc =

NQ 2 3 [g(NPSH)c ] 4

(4.27)

is used to determine the lowest safe value of NPSH. For single-suction water pumps, Shepherd (1956) gives S c = 3, and for double-suction pumps, he gives S c = 4. These form useful rules of thumb for the avoidance of cavitation by designers and users of centrifugal pumps. The effect of high fluid viscosity on pump performance can be determined through the use of correction factors for head, capacity, and overall efficiency. The factors are ratios of head, capacity, and efficiency for viscous fluid pumping to the same parameters with water as the pumped fluid; thus, cH =



H vis H

(4.28)

Q vis Q η c E = vis η

cQ =



(4.29) (4.30)

Table 4.2 illustrates the effect of high viscosity on the head, capacity, and efficiency factors for the case of a centrifugal pump having a design capacity of 2400 gal/min at an output head of 70 ft of water. The table shows that equations and graphs developed from water-pump tests can be used to accurately predict the head and capacity of pumps handling fluids with 100 times the viscosity of water. On the other hand, the efficiency decreases dramatically with increased viscosity. This effect can be explained by a significant increase in disk friction, stemming from the fact that disk-friction power is proportional to the one-fifth power of kinematic viscosity. TABLE 4.2 Effect of Kinematic Viscosity on the Performance of a Centrifugal Pump with H = 70 ft and Q = 2400 gal/min Kinematic Viscosity (centistokes)

cE

cQ

cH

5 10 20 32 65 132 220

0.99 0.97 0.92 0.90 0.84 0.78 0.71

1.0 1.0 1.0 1.0 1.0 0.99 0.97

1.0 1.0 0.99 0.98 0.97 0.94 0.93

74

Logan’s Turbomachinery

Selected values of viscosities of liquids are presented in Table D.1. Values of c H , c Q , and c E for an engine oil are given in Table D.2. Analysis of viscous pumping is facilitated through the use of charts prepared by the Hydraulic Research Institute, which are reprinted in the pump handbook by Karassik et al. (2007).

4.5  Design of Pumps Requirements for a pump comprise the specification of head, capacity, and speed. This section deals with the application of principles to the problem of the determination of the basic dimensions of the impeller and casing. The process outlined below would enable the engineer either to carry out a preliminary design to which the detailed mechanical design could be added or to select a suitable pump from commercially available machines. The impeller design can be started by computing the required specific speed and using this value to determine efficiency from available test data plotted in the form of η as a function of N s with Q as the parameter. Brake power calculated from Equation 4.18 is then used to determine shaft torque from



Γ=

P N

(4.31)

The shaft torque can be used to determine the shaft diameter through the use of a formula for stress in a circular bar under torsion. As shown in Figure 4.6, the hub diameter is larger than the shaft diameter, and the shaft may pass through the entire hub. For single-suction pumps, the shaft may end inside the hub and thereby not pierce the eye of the impeller.

FIGURE 4.6 Double-suction centrifugal impeller.

75

Centrifugal Pumps and Fans

The double-suction impeller shown in Figure 4.6 takes half of the flow in each side. The double-suction type is used to maintain low fluid speeds at the impeller eye and to avoid abrupt turning of the fluid when shroud diameters are large. The shroud diameter should not exceed half of the impeller diameter. Each side of the double-suction impeller should be treated as a single-suction impeller when determining overall, hydraulic, or volumetric efficiency; thus, the determination of the specific speed for efficiency is based on Q/2 rather than Q. On the other hand, when power or tip blade width is to be determined or when the Cordier diagram is being used to determine the specific diameter, the full flow rate Q should be used. The impeller tip speed and the impeller diameter should be determined from the head equation: H=

ηH U 22 (Vu2 ′ /U 2 ) g

(4.32)

Using the results from Appendix C, a value of 0.5–0.55 is substituted for Vu2 ′ /U 2 in Equation 4.32. The hydraulic efficiency ηH is computed from Equation 4.16; thus, U 2 is the only unknown in Equation 4.32 and is readily calculated for the specified head. The impeller diameter is obtained using D 2 = 2U 2 /N. Karassik et al. (2007) recommend that the flow coefficient be chosen in the range



Ns Ns < ϕ2 < + 0.019 21, 600 15, 900

(4.33)

where N s is the dimensional specific speed. The value of the flow coefficient selected is used to determine the meridional velocity using the defining relation: Wm2 = ϕ 2 U 2



(4.34)

Using values of constants from Table 4.1, the volumetric efficiency is determined from  Equation 4.21. The impeller flow rate Q + Q L is determined from Equation 4.17 by dividing m by ρ to obtain Q. The impeller flow rate is used to determine the width at the blade tip from b2 =



Q + QL πD 2 Wm2

(4.35)

Karassik et al. (2007) recommend the following equations for the calculation of the shroud diameter: D  k = 1 −  1H   D1S 



2

(4.36)

and D1S

 Q + QL  = 4.54   kN tan β1s 

1

3

(4.37)

where the shroud diameter is given in inches when N is in revolutions per minute and Q is in gallons per minute. Equation 4.37 applies to single-suction impellers. If this equation is to be used for double-suction impellers, then (Q + Q L ) should be replaced by (Q + Q L )/2 .

76

Logan’s Turbomachinery

The hub-tip ratio used in Equation 4.36 is selected by the designer and can have values from zero to more than 0.5. The constant k approaches unity as the hub-tip ratio decreases to zero and can be taken as unity to approximate the shroud diameter. The hub diameter must exceed the shaft diameter if the shaft passes through the eye. In such cases, the ratio may be taken as 0.5. An optimum hub-tip ratio for minimizing the relative velocity in the eye can be obtained by differentiation of W1S with respect to D1S . The inlet blade angle at the shroud is selected by the designer from the range of 10°–25° recommended by Karassik et al. (2007). When all values are substituted into Equation 4.37, the shroud diameter is determined. The hub-tip ratio is then used to determine the design hub diameter. The next phase of the preliminary design involves iteration on the blade angle β 2. The designer selects a blade angle in the range of 17°–25°, as is recommended by Karassik et al. (2007). Using the selected angles along with the calculated diameters, the optimum number of blades is calculated from an equation recommended by Pfleiderer (1949) and Church (1972):



 D + D1S   β + β2  n B = 6.5  2 sin  1S     D 2 + D1S  2 

(4.38)

The optimum number of blades for pumps lies in the range of 5 to 12. The slip coefficient is combined with its basic definition to yield µs =

π sin β 2  U 2  Vu2 ′ /U 2 = 1− n B  Vu2  Vu2 /U 2

(4.39)

which can be solved for Vu2/U2. The assumed blade angle can now be tested by calculating a new blade angle from β 2 = tan −1

Wm2 /U 2 1 − Vu2 /U 2

(4.40)

When the new value of the blade angle agrees with the assumed value, the design is complete, in that the basic impeller dimensions will have been determined. The fluid exits the impeller with tangential and radial components of absolute velocity and is collected and conducted to the discharge of the pump by the volute or scroll portion of the casing (Figure 4.7). The volute is usually in the form of a channel of increasing crosssectional area. It begins at the tongue with no cross-sectional area and ends at the discharge nozzle. At any angle ϕ, measured from the tongue, the flow rate is (ϕ/360)Q . The angular momentum of the exit flow, r2 Vu2′ , is conserved, so the distribution is approximately

(4.41)

rVu = constant

The angle ϕ corresponding to each radial coordinate r3 is determined from the integrated volume flow equation: ϕQ r = wVu2 ′ r2 ln 3 (4.42) 360 r2 If the channel width w is variable, as in a channel of circular cross-section, then the governing relation should be

ϕQ = Vu2 ′ r2 360

∫

r3

r2

W r

dr



(4.43)

Centrifugal Pumps and Fans

77

FIGURE 4.7 Volute of a pump.

The so-called discharge nozzle, which is really a diffuser, joins the volute to the discharge flange of the pump. For water, the nozzle is typically sized to produce a discharge velocity of 25 ft/s. A radial diffuser may be added between the impeller and the volute for high-pressure pumps. This may take the form of a space of constant width without vanes, or it may include vanes forming diverging passages aligned with the absolute velocity vector.

4.6 Fans Fans produce very small pressure heads measured in inches of water pressure differential and, of course, are employed to move air or other gases. A compressor also handles gases but with large enough pressure rises that significant fluid density changes occur; i.e., if density is increased by 5%, then the turbomachine may be called a compressor. A centrifugal fan, as compared with a pump, requires a much smaller increase in impeller blade speed—i.e., a smaller radius ratio R 2 /R 1 —as may be inferred from Equation 4.7. It requires a volute, of course, but no diffuser is needed to enhance pressure rise. The flow passages between impeller blades are quite short, as indicated in Figure 4.8. The analysis and design of the impeller proceed as with the centrifugal pump. The small changes of gas density are ignored, and the incompressible equations are applied as with pumps. Performance curves are qualitatively the same as for pumps except that the units of head are customarily given in inches of water and those of capacity are typically given in cubic feet per minute. Other differences are that both total head and static (pressure) head are usually shown on performance curves and the fan static efficiency, based on Equation 4.18, is calculated using static head (p1 − p2 )/(ρg) in place of total head H. Similarity laws for pumps are applied and are known as fan laws; these are represented by Equations 4.24 and 4.25.

78

Logan’s Turbomachinery

FIGURE 4.8 Centrifugal fan.

Worked Examples EXAMPLE 4.1 Determine the overall, hydraulic, volumetric, and mechanical efficiency for a centrifugal pump having a capacity of 1000 gal/min and a dimensional specific speed of 1600. SOLUTION: Obtain the overall pump efficiency from Table A.2: η = 0.83



Calculate the hydraulic efficiency using Equation 4.16: ηH = 1 −

0.8 0.8 1 = 1− 1 = 0.858 4 Q (1000) 4

Determine the constants for Equation 4.21 by interpolating between values in Table 4.1: C = 0.195; n = 0.296 . Calculate volumetric efficiency from Equation 4.21:

ηv = 1 −

0.195 = 0.975 (1000)0.296

Finally, determine the mechanical efficiency from Equation 4.20:

ηm =

η 0.83 = = 0.99 ηH ηv (0.858)(0.975)

EXAMPLE 4.2 A single-suction centrifugal pump runs at 885 rpm while delivering water at the rate of 10,000 gal/min. Determine the ideal, input, and output heads if the impeller diameter is 38 in. and the blade angle at tip is 21.6°.

79

Centrifugal Pumps and Fans

SOLUTION: Convert the rotational speed from rpm to rad/s: N = 885( π/30) = 92.68 rps

Determine the tip speed:

U 2 = ND 2 /2 = 92.68 × 38/24 = 146.7 ft/s Find Wm2 . Since the impeller tip width is not known, choose a value for ϕ 2 in the range of design values prescribed in Section 4.5. The head is unknown but will be determined; hence, an assumed value of dimensional specific speed can be checked at the conclusion. Assume N s = 1000. Use the upper limit for the flow coefficient:



ϕ 2 = 1000/15,900 + 0.019 = 0.082 and determine the meridional component:



Wm2 = 146.7 × 0.082 = 12 ft/s Calculate Vu2. Refer to Figures 4.2 and B.2 (in Appendix B):



Vu2 = U 2 − Wm2 cot β 2 = 146.7 − 12 × cot 21.6o = 116.4 ft/s Calculate the ideal head using Equation 4.1:



H i = U 2 Vu2 /g = 146.7 × 116.4/32.174 = 531 ft Calculate the slip coefficient. From Section 4.5, we learn that the optimum number of blades for a centrifugal pump is between 5 and 12. A conservative choice would be 6 blades. Let n B = 6. Now apply Equation 4.8:



µs = 1 −

3.14159 × 146.7 × sin21.6° = 0.757 116.4 × 6

Calculate Vu2′ using the definition of slip coefficient:

Vu2 ′ = µ s Vu2 = 0.757 × 116.4 = 88.1 ft/s Calculate the input head using Equation 4.9:



H in =

146.7 × 88.1 = 401.7 ft 32.174

Calculate the hydraulic efficiency using Equation 4.16: ηH = 1 −

0.8 1 = 0.92 (10, 000) 4

Calculate the output head using Equation 4.15:

H = 401.7 × 0.92 = 369.6 ft

80

Logan’s Turbomachinery

Check the assumed specific speed: 1

Ns =

NQ 2 (885)(10, 000) = 3 3 H4 (369.6) 4

1

2

= 1050

The assumed value is acceptable.

EXAMPLE 4.3 The single-suction centrifugal pump whose characteristics are shown in Figure 4.9 is operating at the design point—i.e., Q = 3100 gal/min, H = 100 ft, and N = 1160 rpm. The suction pipe connecting the pump suction to the supply reservoir has a diameter D su of 8 in. and a length L su of 10 ft. The pipe lifts water from a reservoir 10 ft below the centerline of the pump. The free surface of the reservoir is at 14.7 psia. Determine the suction specific speed, and assess the adequacy of the design. SOLUTION: The net positive suction head is given by NPSH =

p vap patm + zR − h f − ρg ρg

We are given that z R = −10 ft. The specific weight of water ρg is 62.4 lbf /ft 3. The vapor pressure of water is obtained from a table of thermodynamic properties (e.g., from Moran and Shapiro, 1988), which gives p vap = 0.5073 psia at 80°F. The head loss h f is determined from hf =

FIGURE 4.9 Head-capacity curve for a pump.

2 fL su Vsu 2gD su

81

Centrifugal Pumps and Fans

The average velocity Vsu in feet per second in the suction pipe is Vsu =



4Q 4(3100/448.8) = = 19.79 ft/s πD 2su π(8/12)2

The kinematic viscosity of water may be taken as 1.0 cs (centistoke), which is easily converted to English units using the conversion factor from Table A.1: ν = 1.0/92903 = 1.0764 × 10−5 ft 2 /s



The Reynolds number of the pipe flow is



Re =

Vsu D su 19.79(8/12) = = 1.226 × 106 1.0764 × 10−5 ν

The friction factor f is taken from the Moody diagram (e.g., from that in Pritchard and Mitchell, 2015) and is f = 0.0112. The head loss can now be calculated as hf =



0.0112 × 10 × (19.79)2 = 1.02 ft 2 × 32.174 × (8/12)

Substituting into the equation for NPSH, we have



NPSH =

14.7 × 144 0.5073 × 144 + (−10) − 1.02 − = 19.56 ft 62.4 62.4

The definition of the suction specific speed S is 1

NQ 2 S= 3 [g(NPSH)] 4



We calculate S stepwise and obtain





N=

1160 = 121.47 rad/s 9.54929

Q=

3100 = 6.9073 ft 3 /s 448.8

g(NPSH) = 32.174 × 19.56 = 629.3 ft 2 /s 2 121.47 × (6.9073) S= 3 (629.3) 4

1

2

= 2.54

Since the suction specific speed S is less than its critical value S c, we can assume that the cavitation will not occur and that the design—i.e., the elevation of the pump above the supply reservoir—is acceptable.

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Logan’s Turbomachinery

EXAMPLE 4.4 Estimate the performance of the centrifugal pump in Example 4.3 for viscous pumping of a fluid with ν = 150 cs at the same speed. SOLUTION: The pump performance with water is given as Q = 3100 gal/min, H = 100 ft, and η = 0.83. The correction factors in Table D.2 are for a viscosity of 176 cs and will give somewhat more conservative predictions of performance with ν = 150 cs; however, we will use them without adjustment. Interpolating in Table D.2, we compute c E = 0.72, c Q = 0.97, and c H = 0.94



Predictions of performance based on these factors are as follows:     ηvis = c E η = 0.72 × 0.83 = 0.60 Q vis = c Q Q = 0.97 × 3100 = 3007 gal/min   H vis = c H H = 0.94 × 100 = 94 ft



EXAMPLE 4.5 The specifications of a double-suction centrifugal water pump are as follows:

Q = 2400 gal/min



H = 70 ft



N = 870 rpm Find D 2 , b2 , D1S , D1H , β 2 , β1S , and n B for the impeller. SOLUTION: Choose Vu2 ′ /U 2 = 0.5 on the basis of the results of Appendix C. Calculate hydraulic efficiency from Equation 4.16: ηH = 1 −



0.8 1 = 0.864 (1200) 4

Solve the head equation (Equation 4.32) for U 2: 32.174 × 70  U 2 =   0.5 × 0.864 



1

2

= 72.2 ft/s

Determine D 2:

D2 =

2 × 72.2 2U 2 = = 1.585 ft = 19.02 in N 870/9.549

83

Centrifugal Pumps and Fans

Round off the calculated diameter. Choose D 2 = 19 in. Recalculate U 2: 870 × 19 = 72.17 ft/s 9.54 × 24

U2 =



Calculate dimensional specific speed. Use Q/2: Ns =

870(1200) 3 (70) 4

1

2

= 1245 rpm(gal/min)1/2 /ft 3/4

Determine the flow coefficient from Equation 4.33. The upper limit is 1245 + 0.019 = 0.0973 15900

ϕ2 =



Choose ϕ 2 = 0.09. Determine the constants from Table 4.1 by interpolation: C = 0.287



n = 0.346

Calculate volumetric efficiency from Equation 4.21: ηv = 1 −



0.287 = 0.975 (1200)0.346

Calculate the impeller flow rate using Equation 4.17 and dividing it by fluid density: Q + QL =



Q 2400 = = 2461 gal/min ηv 0.975

Calculate the meridional velocity at the tip using Equation 4.34: Wm2 = ϕ 2 U 2 = 0.09 × 72.17 = 6.495 ft/s



Calculate the width at the blade tip from Equation 4.35:



b2 =

Q + QL (2461/448.8) = = 0.1697 ft = 2.036 in πD 2 Wm2 π × (19/12) × 6.495

Choose b2 = 2.0 in . Choose the hub-shroud ratio of 0.5. Calculate k from Equation 4.36:

k = 1 − (0.5)2 = 0.75 Choose β1S = 17°. Calculate the shroud diameter using Equation 4.37: 1



Calculate the hub diameter using the hub-shroud ratio.

1

 (Q + Q L )/2  3  3 1230 D1S = 4.5  = 4.5   0.75 × 870 × tan 17°  = 8.25 in  kN tan β1S   

D1H = 0.5 × 8.25 = 4.125 in Begin iteration by trying a value of 20° for β 2.

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Logan’s Turbomachinery

Calculate the number of blades using Equation 4.38: n B = 6.5



(19 + 8.25) 17 + 20  sin   = 5.2  (19 − 8.25) 2 

Choose n B = 5. Calculate Vu2 /U 2 using Equation 4.8: µs =



π sin β 2 Vu2 ′ /U 2 = 1− Vu2 /U 2 n B Vu2 /U 2

Solve the above equation for Vu2 /U 2 , and adjust Vu2 ′ /U 2 from 0.5 to 0.54: πsin 20 Vu2 = 0.54 + = 0.755 5 U2



Check the assumed value of β 2 using Equation 4.40: β 2 = tan −1



0.09 = 20.17° 0.245

The agreement is satisfactory. Choose β 2 = 20°. Calculate µ s using Equation 4.8, giving µ s = 0.715. Check for the head produced: H= H=



ηH µ s U 22 (Vu2 /U 2 ) g

0.864 × 0.715 × (72.17)2 × 0.755 = 75.5 ft 32.174

The result is a little higher than specified, but the design is acceptable.

Problems 4.1 A centrifugal water pump has the characteristic curves shown in Figure 4.9. Using the 1160 rpm curve, plot the characteristics (H versus Q ) for a geometrically similar pump having twice the speed and half the diameter (of the rotor). Show the calculations that were used to obtain the coordinates plotted. 4.2 Calculate the power required to drive the original pump at 1160 rpm at a flow rate of 3100 gal/min. Also determine the specific speed (dimensionless). 4.3 For a pump impeller with a diameter D 2 of 1.326 ft and an axial width b2 of 2 in., determine the velocity diagram at the exit of the rotor for the conditions in Problem 4.2. The impeller has seven blades, each with the blade angle β 2 of 25°. 4.4 Determine the principal dimensions of a centrifugal pump that can deliver 3100 gallons of water per minute at a 100 ft head. The speed is 1160 rpm.

Centrifugal Pumps and Fans

85

4.5 A centrifugal kerosene pump with backward-curved blades runs at 1876 rpm and delivers 0.678 ft3/s. The tip width of the blades is b2 = 0.498 in . The tip diameter is D 2 = 0.505 ft , and the head is H = 28 ft . The density of the kerosene is 50.8 lbm/ft3. a. Draw and label the velocity diagram. b. Solve for U 2 in ft/s. c. Find Vu2′ in ft/s. d. Find Vm2 in ft/s. e. Find β 2 ′ in degrees. 4.6 Using the concept of rothalpy presented in Chapter 2 and assuming an isentropic flow, derive Equation 4.7 for determining the pressure rise in a centrifugal impeller. 4.7 Derive Equation 4.5 using the outlet velocity triangle in Figure 4.2. 4.8 Show that H i = U 22 /g applies to centrifugal pumps that have an infinite number of blades but have neither throughflow (Q = 0) nor friction loss by a. using Equation 4.22. b. using Equations 2.60 and 4.7. Hint: As Q → 0, W1 → – U1 and V2 → U2. Substitute H = Hi in Equation 2.60. 4.9 A centrifugal water pump produces a head of 70 ft while delivering 2400 gal/min at a speed of 870 rpm. The impeller tip diameter is 19 in., and the width at the blade tip is 2 in. The pump is of the double-suction type with the shroud diameter of 9.50 in. at the impeller eye penetrated by a shaft having a diameter of 2.20 in. with a hub diameter of 4.75 in. The impeller has six backward-curved blades. Determine a. the specific speed, assuming 1200 gal/min entering each side of the impeller. b. the overall pump efficiency. c. the hydraulic efficiency. d. the volumetric efficiency. e. the mechanical efficiency. f. the required shaft power for operation at the design speed. g. the required shaft power for operation at 1200 rpm. 4.10 Construct the actual and ideal velocity triangles for the pump described in Problem 4.9. Find a. the axial component of velocity entering the eye of the impeller. b. the flow coefficient at the impeller exit. c. Vu2 ′ /U 2 for the actual diagram. d. the fluid angle at the impeller tip β 2 ′. e. Vu2 /U 2 for the ideal diagram. f. the slip coefficient. g. the blade angle at the impeller tip β 2.

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4.11 A centrifugal water pump is to be designed to create a pressure rise of 30 psi while delivering 1000 gal/min at a speed of 2000 rpm. Using the flow coefficient 0.122, the blade angle of 25° at the tip, and six impeller blades, find a. the expected overall efficiency. b. the expected hydraulic efficiency. c. the expected slip coefficient. d. the appropriate impeller tip diameter. e. the axial width at the impeller blade tip. 4.12 An impeller of a centrifugal water pump used in the irrigation of farmlands has the following dimensions: Blade angle at the impeller tip β 2 = 25° Blade angle at the leading edge β1 = 12° Shroud diameter at the eye = 4.875 in. Blade leading-edge diameter = 4.875 in. Impeller diameter at the tip = 8.625 in. Blade axial width at the tip = 1.375 in. Blade axial width at the leading edge = 1.50 in. Number of cylindrical, backward-curved blades = 6 Assuming a design speed of 1750 rpm, find a. the design capacity in gal/min. b. the design head in ft of water. c. the drive motor power in hp. 4.13 For a centrifugal pump with backward-curved blades, it may be assumed that the loss of input head H L is given by the expression

gH L = K d V22′ /2 + K r W22′ /2

where K d is the loss coefficient for the diffuser, K r is the loss coefficient for the rotor (impeller), and V2′ and W2′ are the actual fluid velocities. Assuming that K r /K d = 1/3 (as suggested by Csanady, 1964) and that the optimum actual velocity diagram corresponds to the minimum value of H L /H in , determine a. the optimal values of Vu2 ′ /U 2 for tip flow coefficients of 0.05, 0.1, and 0.2. b. the corresponding optimal fluid angles at the impeller tip. 4.14 Consider a centrifugal pump designed for operation at a specific speed in the 1000–3000 range. Appropriate midrange values of the flow coefficient and the blade angle at the tip are ϕ 2 = 0.1 and β 2 = 22°. For an impeller having six blades, determine a. Vu2 /U 2 for the ideal tip velocity diagram. b. the corresponding slip coefficient. c. Vu2 ′ /U 2 for the actual velocity diagram. d. how these practical values compare with those resulting from the analysis of Problem 4.13.

Centrifugal Pumps and Fans

87

4.15 A centrifugal pump having nine backward-curved blades and a tip diameter of 6.06 in. operates at 1500 rpm and delivers 0.6684 ft3/s of water. The blades make an angle of 30° with the tangential direction at the tip of the impeller. The axial width of the blade at the impeller tip is 0.5 in. Calculate a. the tip blade speed in ft/s. b. Vm2 in ft/s. c. Vu2 in ft/s. d. the slip coefficient. e. the input head in ft of water. f. the output head in ft of water. 4.16 A single-sided centrifugal gasoline pump with six backward-curved blades runs at 1876 rpm and delivers 0.678 ft3/s. The tip width of the blades is b2 = 0.498 in . The blades are cylindrical (meaning the blades do not extend into the impeller eye), and the shaft and hub do not extend into the impeller eye. The tip diameter D 2 is 0.505 ft , and the shroud diameter is 0.25 ft. The output head H is 28 ft. The density of the gasoline is 46 lbm/ft3, the kinematic viscosity is 0.000006 ft2/s, and the vapor pressure is 10 psia. a. Draw and label the velocity diagram for perfect guidance by the blades; superimpose the diagram for a finite number of blades. b. Find the flow coefficient. c. Calculate dimensional specific speed in units of rpm(gal/min)1/2/ft3/4 (see Example 4.5). What range of values for the flow coefficient are possible for good design? d. Find the tip blade angle in degrees. e. Calculate the dimensionless and dimensional specific diameters. f. Find the input head in ft of gasoline. g. Find the ideal head in ft of gasoline. h. Find the hp required to drive the pump. i. Find the shaft torque at steady speed. j. Find the lowest acceptable suction pressure in psia. 4.17 Determine the height that water at 80°F can be lifted by the centrifugal pump in Problem 4.4 if the pump is connected by means of an 8 in. diameter suction pipe to a supply reservoir whose free surface is maintained at a pressure of 1 atm. 4.18 A three-stage centrifugal pump operating at 38,000 rpm handles 125 lbm/s of liquid hydrogen (vapor pressure = 14.7 psia; density = 4.43 lbm/ft3) at a suction pressure of 100 psia and a discharge pressure of 6300 psia. Assume that the shroud-to-tip diameter ratio is 0.5 and the hub-to-shroud diameter ratio is 0.44. Find a. the impeller diameter. b. the shroud diameter. c. the hub diameter. d. the mean velocity at the impeller eye. e. the suction specific speed. Hint: Assume equal pressure rises for the three stages.

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4.19 Determine the impeller diameter and suction specific speed of a small centrifugal cryogenic pump with the following performance requirements: 3    Q = 0.016 m /s    H = 2230 m

   N = 45,000 rpm NPSH = 20 m Note: The result shows that S > S c ; thus, an axial-flow stage ahead of the eye of the centrifugal pump is needed. This stage is called an inducer and is frequently required for high-speed pumps. 4.20 Oil having a specific gravity of 0.886 at 130°F is pumped by three centrifugal pumps arranged in parallel at the rate of 2,116,000 barrels (42 gal = 1 barrel) per day from a suction pressure of 26 psig to a discharge pressure of 1150 psig. The pump is driven by a gas turbine operating at 3400 rpm. For this service, design one impeller of a two-stage, double-suction centrifugal pump. Assume that the two impellers are identical. 4.21 Performance curves for a double-suction centrifugal pump indicate the following data at the maximum efficiency of 90%: H = 345 ft, Q = 20, 000 gal/min, and N = 885 rpm . Determine the principal dimensions for such a pump. 4.22 Assume that the velocity of water at 80°F entering one side of the pump in Problem 4.21 is Vsu = 19.1 ft/s. Assume that S = Sc and h f = 0. Find the diameter of the suction pipe and the water pressure at the suction flange.



References Church, A.H. 1972. Centrifugal Pumps and Blowers. Huntington, NY: Krieger. Csanady, G.T. 1964. Theory of Turbomachines. New York: McGraw-Hill. Karassik, I.J., et al. 2007. Pump Handbook, 4th edition. New York: McGraw-Hill Education. Moran, M.J., and H.N. Shapiro. 1988. Fundamentals of Engineering Thermodynamics. New York: Wiley. Pfleiderer, C. 1949. Die Kreiselpumpen. Berlin: Springer-Verlag. Pritchard, P.J., and J.W. Mitchell. 2015. Fox and McDonald’s Introduction to Fluid Mechanics, 9th edition. New York: Wiley. Shepherd, D.G. 1956. Principles of Turbomachinery. New York: Macmillan.

Bibliography Bleier, F.P. 1997. Fan Handbook: Selection, Application, and Design. New York: McGraw-Hill Education. Gülich, J.F. 2014. Centrifugal Pumps. New York: Springer.

Centrifugal Pumps and Fans

89

Nomenclature A 2 b2 cE cH cQ C D 2 Ds D1S D1H D su f g hd h s h f H Hi HL H in H vis k k d k r L su  m  L m n n B N N s NPSH P p d ps patm p vap Q Q L Q vis Re S S c u d us

Flow area at impeller exit Width of blade at r = r2 in pump impeller Efficiency correction factor for viscosity Head correction factor for viscosity Capacity correction factor for viscosity Constant in Equation 4.21 Impeller tip diameter Specific diameter Shroud diameter Hub diameter Diameter of suction pipe Friction factor Gravitational acceleration Specific enthalpy of fluid at the pump discharge flange Specific enthalpy of fluid at the pump suction flange Friction head loss in suction pipe Output head Ideal head Hydraulic loss in impeller and diffuser Input head Output head with viscous pumping Function of hub-tip ratio Loss coefficient in pump diffuser Loss coefficient in pump impeller (or rotor) Length of suction pipe Mass flow rate of fluid at pump discharge flange Mass flow rate of fluid leaked around the outside of the impeller from the highto low-pressure regions Constant in Equation 4.21 Number of blades in the impeller Rotor speed Specific speed Net positive suction head Power to impeller shaft Static pressure of fluid at pump discharge flange Static pressure of fluid at pump suction flange Atmospheric pressure Vapor pressure of fluid Capacity (i.e., volumetric flow rate delivered by pump at discharge flange) Volume flow rate of leaked fluid Flow rate with viscous pumping Reynolds number Suction specific speed Critical suction specific speed Specific internal energy of fluid at pump discharge flange Specific internal energy of fluid at pump suction flange

90

U1 U2 Vd Vs V1 V2 V2′ Vm1 Vm2 Vsu Vu1 Vu2 W1 W2 W2′ W1S Wm2 zd zR zs β1 β2 β2′ β1S Γ η ηH ηm ηv ηvis µs ρ ϕ ϕ ϕ2 ω

Logan’s Turbomachinery

Impeller speed at blade leading edge Impeller speed at blade tip Average velocity at pump discharge flange Average velocity at pump suction flange Absolute velocity at leading edge of impeller blade Absolute velocity leaving the impeller with an infinite number of blades Absolute velocity leaving the impeller with a finite number of blades Meridional component of V1 Meridional component of V2 or V2 ′ = Wm2 Average velocity in suction pipe Tangential component of V1 Tangential component of V2 Relative velocity at leading edge of impeller blade Relative velocity leaving the impeller with an infinite number of blades Relative velocity leaving the impeller with a finite number of blades Relative velocity at shroud Meridional component of W2 or W2′ Elevation of the center of the discharge flange above (positive) or below (negative) the centerline of the pump shaft Elevation of the free surface of the supply reservoir above (positive) or below (negative) the centerline of the pump shaft Elevation of the center of the suction flange above (positive) or below (negative) the centerline of the pump shaft Angle between W1 and U1 Angle between W2 and U 2 ; also the blade angle at the blade trailing edge Angle between W2′ and U 2 ; actual fluid angle Angle between W1S and U1S Torque Overall pump efficiency Hydraulic efficiency Mechanical efficiency Volumetric efficiency Overall efficiency with viscous pumping Slip coefficient Fluid density Flow coefficient Volute angle Flow coefficient at impeller exit = Wm2 /U 2 Rotor angular velocity in rad/s; ω = N( π / 30) where N is in rpm

5 Centrifugal Compressors

5.1 Introduction Although centrifugal compressors are slightly less efficient than axial-flow compressors, they are easier to manufacture and are thus sometimes preferred. Additionally, a single stage of a centrifugal compressor can produce a pressure ratio that is five times that of a single stage of an axial-flow compressor. Thus, the centrifugal machine finds application in ground-vehicle power plants, auxiliary power units, and other small units. The parts of a centrifugal compressor are the same as those of a pump: namely, the impeller, the diffuser, and the volute. The basic equations developed in Chapter 4 apply to compressors with the difference that density does increase and we must consider the thermodynamic equation of state of a perfect gas in the detailed calculations. The main difference in carrying out a compressor analysis, as opposed to a pump analysis, is that an enthalpy term appears in place of the flow work or pressure-head term. It is convenient to use both total and static enthalpy, denoted by h 0 and h, respectively. Thus, energy transfer E is given by

E = ηm (h 03 − h 01 )

(5.1)

E = U 2 Vu2 ′

(5.2)

as well as by

where ηm denotes the mechanical efficiency. When Equations 5.1 and 5.2 are used in the same analysis, the units must be handled with care, since the enthalpy difference in Equation 5.1 may carry units such as Btu/lbm, whereas Equation 5.2 would carry units of v2. Suitable conversion factors do not appear in the equations but must be applied in computations with them. Since thermodynamic calculations are involved in compressor analysis and design, the enthalpy-entropy (h-s) diagram, such as that shown in Figure 5.1, becomes useful. The state at the impeller inlet is indicated by point 1 and that at the impeller outlet by point 2. The diffuser process is indicated between points 2 and 3. Since kinetic energies are usually considerable, the corresponding stagnation properties with subscripts 01, 02, and 03 are also indicated in Figure 5.1. The expression for compressor efficiency appears to be somewhat different from that for the pump efficiency, but the definition is essentially based on the same principle. Both definitions employ the ratio of the useful increase of fluid energy divided by the actual energy input to the fluid. For the compression of a gas to

91

92

Logan’s Turbomachinery

FIGURE 5.1 Enthalpy-entropy diagram.

the actual final pressure p03 , the useful energy input is the work of an ideal, or isentropic, compression. This is calculated from



 p03  ( γ − 1)/γ   E i = c p T0l  − 1   p0l  

(5.3)

which evaluates the work of the isentropic process from state 01 to state i in Figure 5.1. The compressor efficiency can be reduced to ηc =



Ti − T0l T03 − T0l

(5.4)

which is the ratio of E i to E. An underlying assumption in the development of Equations 5.3 and 5.4 is that there is no external work associated with the diffuser flow, nor is there any heat transfer; thus, h 02 = h 03 and T02 = T03 . The compressor efficiency, an experimentally determined quantity, is useful in predicting pressure ratios in new designs. Using Equations 5.2, 5.3, and 5.4, we can obtain the overall pressure ratio:



p03  U V η  =  1 + 2 u2 ′ c  p01  c p T0l ηm 

γ /( γ − 1)

(5.5)

Since relative eddies are present between the blades, as is the case with centrifugal pumps, slip exists in the compressor impeller; consequently, the slip coefficient is used to calculate the actual tangential velocity component, which is given by Vu2 ′ = µ s Vu2



(5.6)

For compressors, however, the Stanitz equation:



µs = 1 −

 1 0.63 π  n B  1 − ϕ 2 cot β 2 

(5.7)

Centrifugal Compressors

93

FIGURE 5.2 Velocity diagram at impeller exit.

is used in place of Equation 4.8 to calculate the slip coefficient. Several such equations are available, as is indicated in Appendix E, but the Stanitz equation is an accurate predictor of slip coefficient for the usual range of blade angles encountered in practice—namely, 45° < β < 90°. Thus, the total pressure ratio for a compressor stage can be determined from the knowledge of the ideal velocity triangle shown in Figure 5.2 at the impeller exit, the number of blades, the inlet total temperature, and the stage and mechanical efficiencies. The mechanical efficiency is defined by Equation 4.19 and accounts for frictional losses associated with bearing, seal, and disk friction. It is assumed that the mechanical energy lost through frictional processes reappears as enthalpy in the outflowing gas; hence, the specific shaft work into the compressor is given by E/η m, as is shown in Equation 5.1.

5.2  Impeller Design The impeller is usually designed with a number of unshrouded blades, given by the Pfleiderer equation (Equation 4.38), to receive the axially directed fluid ( V1 = Vm1) and deliver the fluid with a large tangential velocity component Vu2 ′ , which is less than the tip speed U 2 but which has the same sense of direction. The blades are usually curved near the rim of the impeller, so that β 2 < 90°, but they are usually bent near the leading edge to conform to the direction of the relative velocity W1 at the inlet. The angle β1 varies over the leading edge, since V1 remains constant, while U1 (and r) varies. At the shroud diameter D1S of the impeller inlet, the relative velocity W1S and the corresponding relative Mach number M R1S are the highest. This is because the blade speed U1 increases from hub to tip at the inlet plane and the incoming absolute

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Logan’s Turbomachinery

FIGURE 5.3 Velocity diagram at impeller inlet.

velocity V1 is assumed to be uniform over the annulus. Referring to Figure 5.3, it is clear 1

that W1 = (V12 + U12 ) 2 and that the maximum value of W1 occurs at the shroud diameter.   p01 , and T01—it is easily shown For a fixed set of inlet operating conditions—i.e., N,  m, that the relative Mach number has its minimum where β1S is approximately 32° (see Shepherd, 1956). Referring to Figure 5.3, it is seen that a choice of relative Mach number at the shroud of the impeller inlet allows the inlet design to proceed in the following manner. Since the static temperature, and hence the acoustic speed, remains the same in both absolute and relative reference frames, we compute the absolute Mach number M1 from the relative Mach number M R1S as

M1 = M R1S sin 32°

(5.8)

which is then used to find the static temperature T1 from the given absolute temperature T01 using



T1 =

T01 1 + 0.5( γ − 1)M12

(5.9)

The acoustic speed is then calculated from

a1 = ( γ RT1 )1/2

(5.10)

The absolute and relative velocities V1 and W1S at the shroud are calculated, respectively, from

V1 = M1a1

(5.11)

W1S = M R1S a1

(5.12)

U1S = W1S cos 32 o

(5.13)

and Then U1S is calculated from

It is then possible to determine the shroud diameter, since



D1S =

2U1S N

(5.14)

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Centrifugal Compressors

The hub diameter can be found using the mass flow equation at the impeller inlet: D1H

   2 4m =  D1S − πρ1 V1  

1

2

(5.15)

where the density is determined from the equation of state of a perfect gas: ρ1 =



p1 RT1

(5.16)

The static temperature is found from the total temperature using Equation 5.9, and the static pressure is found using p01 [1 + 0.5( γ − 1)M12 ]γ /( γ − 1) Referring to Figure 5.3, the fluid angle at the hub is calculated from p1 =



 V  β1H = tan −1  1   U1H 

(5.17)

(5.18)

where the blade speed at the hub is given by

U1H =

ND1H 2

(5.19)

To determine the impeller diameter, one should follow the procedure used in Example Problem 5.3: First, the dimensional specific speed is calculated, and Table A.3 is consulted to determine the highest possible compressor efficiency and the corresponding dimensional specific diameter. Next, the impeller diameter D 2 is calculated from the specific diameter, the tip speed U 2 is determined from the impeller diameter, and the energy transfer E is calculated from the ideal energy transfer E i and the compressor efficiency ηc . The actual tangential velocity component Vu2 ′ is calculated from the energy transfer, and the ideal tangential velocity component Vu2 is calculated using the slip coefficient from 0.85 to 0.90. Finally, the selection of a flow coefficient in the range of 0.23 to 0.35 permits the calculation of the blade angle and the number of blades. The compressor efficiency ηc, in addition to its use in Equation 5.5, can be employed to estimate the impeller efficiency ηI . The ratio χ of impeller losses to compressor losses: 1 − ηI (5.20) 1 − ηc can be estimated and lies between 0.5 and 0.6. The definition of impeller efficiency: χ=



ηI =

Ti ′ − T01 T02 − T01

(5.21)

can be used to estimate Ti ′ (see Figure 5.1). The latter total temperature corresponds to the total pressure p02, calculated from



p02  Ti ′  = p01  T01 

γ /( γ − 1)

(5.22)

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Logan’s Turbomachinery

TABLE 5.1 Design Parameters for Centrifugal Compressors Parameter Flow coefficient Shroud-tip ratio Absolute gas angle Diffusion ratio

Source

Recommended Range

Ferguson Whitfield Whitfield Whitfield

0.23 < φ2 < 0.35 0.5 < D1S/D2 < 0.7 60° < α2′ < 70° W1S/W2′ < 1.9

The static pressure p2 is then determined from



p02 = [1 + 0.5( γ − 1)M 22 ]γ /( γ − 1) p2

(5.23)

The static pressure p2 from Equation 5.23 and the static temperature T2 determined from T2 = T02 −

V22′ 2c p

(5.24)

are used to determine density ρ2 at the impeller exit. Finally, the axial width b2 of the impeller passage at the periphery may be found from



b2 =

 m 2 πρ2 r2 Vm2

(5.25)

Ranges of design parameters that are considered optimal by Ferguson (1963) and Whitfield (1990) are presented in Table 5.1. The recommended ranges should be used by the designer to check calculated results for acceptability during or after the design process.

5.3  Diffuser Design A vaneless diffuser, or empty space, between the leading edges of diffuser vanes and the impeller tip allows some equalization of velocity and a reduction of the exit Mach number. The vaneless portion, which may have a width as large as 6% of the impeller diameter, also contributes to a rise in static pressure. As with the pump, the angular momentum rVu is conserved, and the fluid path is approximately a logarithmic spiral. Diffuser vanes are set with the diffuser axes tangent to the spiral paths and with an angle of divergence between them not exceeding 12°. The wedge-shaped diffuser vanes are depicted in Figure 5.4. Since the addition of a vaned portion in the diffusion system results in a small-diameter casing, vanes are preferred in instances where size limitations are imposed. On the other hand, a completely vaneless diffuser is more efficient. If vanes are used, then their number should generally be less than the number of impeller vanes to ensure a uniform flow and high diffuser efficiency in the range of flow coefficient Vm2 /U 2 recommended in the previous section.

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Centrifugal Compressors

FIGURE 5.4 Arrangement of diffusers and impeller.

The vaneless diffuser is situated between circles of radii r2 and r3 . At any radial position r, the gas velocity V will have both tangential and radial components. The radial component Vr is the same as the meridional component Vm . The mass flow rate at any r is given by  = 2 πrbρVm m





(5.26)

For constant diffuser width b, the product ρrVm is constant, and the continuity equation becomes ρrVm = ρ2 r2 Vm2



(5.27)

Since the angular momentum is conserved in the vaneless space, we can write

rVu = r2 Vu2 ′

(5.28)

where the primed subscript is used to indicate the actual value of the tangential velocity component at the impeller exit; however, in the vaneless space, the actual velocity is unprimed. Typically, the flow leaving the impeller is supersonic—i.e., M 2 ′ > 1 —and the flow leaving the vaneless diffuser is subsonic—i.e., M 3 < 1. The radial position at which M = 1 is denoted by r*; similarly, all other properties at the plane of sonic flow are denoted with a starred superscript (e.g., ρ*, Vm* , a*,  T*, and α*). The absolute gas angle α is the angle between V and Vr —i.e., between the direction of the absolute velocity and the radial direction. Since the radial velocity component can be written as

Vr = Vm = Vcos α

(5.29)

ρrVcos α = ρ* r* V * cos α*

(5.30)

the continuity equation becomes

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Logan’s Turbomachinery

Similarly, the angular momentum equation is expressed as rVsin α = r* V * sin α *



(5.31)



Dividing Equation 5.31 by Equation 5.30, we obtain tan α tan α * = ρ ρ*



(5.32)

Assuming an isentropic flow in the vaneless region, we find T  ρ = T*  ρ* 



γ −1

(5.33)

and T=



T0 1 + 0.5( γ − 1)M 2

(5.34)

2T0 γ +1

(5.35)

For M = 1, Equation 5.34 becomes



T* =

Substituting Equations 5.34 and 5.35 into Equation 5.33 yields



ρ*  2  γ −1 2 = M   1 +  ρ γ +1 2

1/( γ − 1)

(5.36)

Substituting Equation 5.36 into Equation 5.32 gives



 2  γ − 1 2 tan α = tan α *  M   1 +  γ + 1 2 

−1/( γ − 1)

(5.37)

The angle α * is evaluated by substituting α = α 2 ′ and M = M 2 ′ into Equation 5.37. Equation 5.31 can be rewritten as



r* sin α * V V a T = = = M    r sin α V* a a* T* 

1

2

(5.38)

Substituting Equations 5.34 and 5.35 into Equation 5.38 yields



 2  r* sin α* γ − 1 2 = M M   1 +  r sin α 2 γ +1

−1

2

(5.39)

The radial position r* can be found from Equation 5.39 by substituting r = r2 and M = M 2 ′, which are known from impeller calculations. Finally, Equation 5.37 can be used to determine α 3 from a known M 3 , and Equation 5.39 can be used to calculate r3 for known values of M 3 and α 3.

99

Centrifugal Compressors

A volute is designed by the same methods outlined in Chapter 4. The volute functions to collect the diffuser’s discharge around the 360° periphery and deliver it through a single nozzle to the connecting gas-piping system or to the inlet of the next compressor stage.

5.4 Performance Typical compressor characteristics are shown in Figure 5.5. Qualitatively, their shape is similar to that of the centrifugal pump, but the sharp fall of the constant-speed curves at higher mass flows is due to choking in some component of the machine. At low flows, operation is limited by the phenomenon of surge. Thus, smooth operation occurs on the compressor map at some point between the surge line and the choke line. The phenomenon of choking is associated with the attainment of a Mach number of unity. In the stationary passages of the inlet or diffuser, the Mach number is based on the absolute velocity V. Thus, for a Mach number of unity, the absolute velocity equals the acoustic speed a, calculated from a = ( γ RT)1/2



(5.40)

The temperature at this point is calculated from the total temperature T0 using

T0 = T 1 + 0.5( γ − 1)M 2 



(5.41)

which with M = 1 yields



FIGURE 5.5 Compressor map.

 2  T* = T0  = Tt  γ + 1 

(5.42)

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Logan’s Turbomachinery

This Mach number is found near the cross section of minimum area, or throat (A t), so that we can estimate the choking mass flow rate from



 γ   = A t pt  m  RTt 

1

2

(5.43)

The static pressure Pt at the throat area may be estimated by assuming an isentropic process from the inlet of the stationary component to the throat area. Thus,



 T  pt = pin  t   Tin 

γ /( γ − 1)

(5.44)



The process of estimating choked flow rate in the impeller is the same except that the relative velocity replaces the absolute velocity. When the relative Mach number W/a is set equal to unity in the energy equation of the rotor—namely,



h 01 = h +

W2 U2 − 2 2

(5.45)

we obtain  U 2  2T01 T* =  1 + = Tt 2c p T01  ( γ + 1) 

(5.46) Using the isentropic relation between pressure and temperature and substituting into the continuity relation, the mass flow rate at the throat section of the impeller is given by





 γ   = A t P0l  m  RT01 

1

( γ + 1)

2

 2  U 2   2( γ − 1) 1+    2c p T01    γ + 1 

(5.47)

Thus, it is clear that the mass flow for choking in stationary components, given by Equation 5.43, is independent of impeller speed but that the mass flow for choking in the impeller, given by Equation 5.47, actually increases with the impeller speed. This is indicated schematically in Figure 5.5. As an alternative approach to arriving at Equation 5.47, we can use Equation 2.13, which for M = 1 and C V = 1 becomes  = m

A t Fˆ f*0 P0R RT0R

(5.48)

where P0R and T0R are the relative total pressure and temperature, respectively, at the throat section of area A t and the total-pressure mass flow function, given by Equation 2.15, reduces to Fˆ f*0 =

γ  γ + 1   2 

(5.49)

γ +1 γ −1



101

Centrifugal Compressors

* * Next, our task is to express P0R and T0R in Equation 5.48 in terms of P01 and T01, given at the stage inlet (stationary component). We assume isentropic flow conditions in both the stationary inlet section and the impeller blade-to-blade passage. As a result of this assumption, both P01 and T01 remain constant in the stationary passage; however, P0R and T0R will change in the rotating passage due to rotational work transfer. Unless the fluid enters the impeller at a tangential velocity equal to the impeller velocity, energy exchange will occur between the two. We can account for this energy transfer at the stator-rotor interface by simply equating the rothalpy values expressed in the stator and rotor reference frames. As discussed in Chapter 2, since the rothalpy for an isentropic flow in the rotor remains constant throughout, the rothalpy at the impeller choked-flow section will be equal to that at the stator-rotor interface. Thus, we can write



2   u = h 0R − U h 01 − UV 2

(5.50)

 u are the interface values, while h 0R and U correspond to the choked-flow  and V where U  u = 0, Equation 5.50 reduces to section. With V



h 01 = h 0R −

U2 2

(5.51)

which is identical to Equation 5.45. In terms of total temperatures, we can express Equation 5.51 as





c p T01 = c p T0R −

U2 2

T0R  U2  = 1+ T01  2c p T01 

(5.52)

From isentropic relations, we can writes



P0R  T0R  = P01  T01 

γ

γ −1

 U2  = 1+ 2c p T01  

γ

γ −1

(5.53)

Substituting Equations 5.49, 5.52, and 5.53 into Equation 5.48 and simplifying the resulting expression, we finally obtain Equation 5.47. Referring to Figure 5.5, point A represents a point of normal operation. An increase in flow resistance in the connected external flow system results in a decrease in Vm2 at the impeller exit and a corresponding increase in Vu2 , which results in an increased head or pressure increase. However, the surge phenomenon results when a further increase in external resistance produces a decrease in impeller flow that tends to move the point beyond C, where stall at some point in the impeller leads to a change of direction of W2 and an accompanying decrease in the head (or pressure rise) in the impeller. A temporary flow reversal in the impeller and the ensuing buildup to the original flow condition is known as surging, which continues cyclically until the external resistance is removed. Surging is an unstable and dangerous condition and must be avoided by careful operational planning and system design.

102

Logan’s Turbomachinery

Worked Examples EXAMPLE 5.1 Data from a performance test of a single-stage centrifugal air compressor are the following: Measured mass flow rate of air = 2.2 lbm/s Test speed = 60,000 rpm Total pressure of air drawn into compressor = 14.7 psia Total temperature of air drawn into compressor = 60°F Total pressure at compressor discharge = 61.74 psia Impeller measurements are the following: Impeller tip diameter = 5.92 in. Inlet hub diameter = 1.35 in. Inlet shroud diameter = 3.84 in. Number of blades = 33 Blade angle with respect to tangent to wheel = 90° Calculate the efficiency of this compressor. SOLUTION: Calculate the tip speed:

U 2 = ND 2/2 = 60, 000 × ( π/30) × (5.92/24) = 1550 ft/s Note that Vu2 = U 2 , since β 2 = 90°. Calculate the slip coefficient using Equation 5.7:



µ s = 1 − 0.63 π/n B = 1 − 0.63 π/33 = 0.94 Next, determine the tangential velocity component:



Vu2 ′ = µ s Vu2 = 0.94 × 1550 = 1457 ft/s Use Equation 5.2 to determine the energy transfer:



E = U 2 Vu2 ′ = 1550 × 1457 = 2, 258, 350 ft 2/s 2

103

Centrifugal Compressors

or

E = 2, 258, 350(ft ⋅ lbf )/slug = 70, 192(ft ⋅ lbf )/lbm Since air is a diatomic gas, γ = 1.4. From Table F.1, we find that the molecular weight of air is 28.97. Compute the specific heat of air at constant pressure using the relations for gases: c p − c v = R and γ = c p/c v . The resulting equation is c p = γ R/( γ − 1)

The gas constant is found from

 = 1545/28.97 = 53.33(ft ⋅ lbf )/(lbm ⋅ °R) R = R u/ M Using the above equation, the specific heat is



c p = 1.4 × 53.33 /0.4 = 186.66(ft ⋅ lbf )/(lbm ⋅ °R) Since no mechanical efficiency is given, we assume that ηm = 0.96. Calculate the actual total temperature rise in the compressor using Equation 5.1; thus, T03 − T01 = E/(c p ηm )



T03 − T01 = 70, 192 /(186.66 × 0.96) = 392°R Next, determine the isentropic specific work from Equation 5.3:  p03  ( γ − 1)/γ  E i = c p T01   −1   p01  



 61.74  (0.4/1.4)  E i = 186.66 × 520 ×   −1    14.70  E i = 49, 195(ft ⋅ lbf )/lbm Calculate the total temperature rise for the isentropic compression:



Ti − T01 = E i/c p = 49, 197/186.66 = 264°R Finally, calculate the compressor efficiency:



ηc =

Ti − T01 264 = = 0.673 T03 − T01 392

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Logan’s Turbomachinery

EXAMPLE 5.2 Using Table A.3, determine the compressor efficiency for the compressor described in Example 5.1. SOLUTION: Determine the density at the inlet. Note that the eye velocity V1 must be found iteratively. Assume V1 = 447 ft/s. Calculate T1: V12 2c p

T1 = T01 −



c p = 186.66 × 32.174 = 6006(ft ⋅ lbf )/(slug ⋅°R) (447)2 = 503°R 2 × 6006

T1 = 520 −

 T  p1 = p01  1   T01  ρ1 =

γ /( γ − 1)

503  = 14.7   520 

3.5

= 13.09 psia

p1 13.09 × 144 = = 0.070 lbm /ft 3 RT1 53.33 × 503

Calculate the flow area at the eye: π 2 2 (D1S − D1H ) 4 π[(3.84)2 − (1.35)2 ] A1 = = 0.0705 ft 2 4 × 144 A1 =



Check the mass flow rate to verify the assumed V1:  = ρ1 V1 A 1 = 0.070 × 447 × 0.0705 = 2.2 lbm/s m



Calculate the inlet volume flow rate: Q1 =



 m 2.2 = = 31.4 ft 3/s ρ1 0.070

Calculate the output head H: H=

E i 49, 197 × 32.174 = = 49, 197 ft g 32.174

Calculate the dimensional specific speed: 1

Ns =

NQ 1 2 (H)

3

4

=

60, 000 × (31.4) (49, 197)

3

4

1

2

1

= 102 rpm (gpm) 2 /ft

3

4

105

Centrifugal Compressors

Calculate the dimensional specific diameter: Ds =

D2 H Q

1

1

2

4

=

(5.92 / 12)(49, 197) (31.4)

1

2

1

4

5

= 1.31 ft 4 /(gpm)

1

2

Interpolate in Table A.3 to find efficiency:

ηc = 0.70 for D s = 1.3 and N s = 102 whose units are given in the foregoing. Note: This is slightly higher than the 0.673 calculated previously. If ηm = 0.99 had been assumed in Example 5.1, the agreement would have been perfect.

EXAMPLE 5.3 A single-stage centrifugal compressor draws in 6.9 lbm of air per second at a total pressure of 14.2 psia and a total temperature of 550°R. It discharges the air at a total pressure of 59.64 psia. The compressor runs at 41,700 rpm. Find the basic dimensions of the impeller. SOLUTION: Assume a slightly supersonic relative Mach number at the inlet shroud. Although creating a region of supersonic flow in the inlet is undesirable, it is necessary to handle the large mass flow. Choose M R1S = 1.2 and β1S = 32°. Calculate the inlet density and the volumetric flow rate using Equations 5.8–5.12: M1 = M R1S sin β1S = 1.2 sin 32° = 0.636



T1 =



550 = 509°R 1 + 0.2 × (0.636)2 1



a1 = (1.4 × 1716 × 509)



V1 = 0.636 × 1106 = 703 ft/s  T  ρ1 = ρ01  1   T01 



Q1 =



= 1106 ft/s

1/( γ − 1)

509  ρ1 = .0697   550 



2

2.5

= 0.0574 lbm/ft 3

 m 6.9 = = 120 ft 3/s ρ1 0.0574

Calculate the output head H:



  59.64  (1/3.5) E i = 6006 × 550 ×  − 1 = 1, 674, 291 ft 2/s 2    14.20  H=

E i 1, 674, 291 = = 52, 038 ft g 32.174

106

Logan’s Turbomachinery

Calculate the dimensional specific speed: 1

Ns =

NQ 1 2 H

3

4

41, 700 × (120)

=

(52, 038)

3

1

2

1

= 132.5 rpm (gpm) 2 /ft

4

3

4

Referring to Table A.3 for highest efficiency, D s = 1.18 and 1

D2 =

Ds Q1 2 H

1

4

=

1.18 × (120) (52, 038)

1

1

2

4

= 0.855 ft = 10.2 in

The corresponding tip speed is U 2 = 4366.8 × (10.2/24) = 1856 ft/s



The average impeller temperature Tav is estimated to be 200°F. The recommended maximum tip speed for this metal temperature is U 2max = 2000 − Tav = 2000 − 200 = 1800 ft/s



Choose a design tip diameter of 9.6 in: U 2 = 4366.8 × (9.6/24) = 1747 ft/s



The corresponding D s is 1.103. From Table A.3, we obtain ηc = 0.723

The energy transfer is E=



ηm E i 0.96 × 1, 674, 291 = 2, 223, 125 ft 2/s 2 = ηc 0.723

The actual tangential velocity component is found from Equation 5.2: Vu2 ′ =



2, 223, 125 = 1273 ft/s 1747

Choose a slip coefficient of 0.9. Then Vu2 =



1273 = 1414 ft/s 0.9

and Wu2 = U 2 − Vu2 = 1747 − 1414 = 333 ft/s



Choose the flow coefficient of 0.30. This is the middle of the design range: Wm2 = ϕ 2 U 2 = 0.3 × 1747 = 524 ft/s

The blade angle is



β 2 = tan −1

Wm2 524  = tan −1  = 57.6°  333  Wu2

107

Centrifugal Compressors

Solve the slip equation (Equation 5.7) for the number of blades: 0.9 = 1 −

 0.63 π  1 n B  1 − 0.3 cot 57.6 

which yields n B = 24



Estimate the impeller efficiency using a loss ratio of 0.55. From Equation 5.20, we have ηI = 1 − 0.55 × (1 − 0.723) = 0.848



Equation 5.1 can be used to find T03: T03 = 550 +



2, 223, 125 = 936°R 6006 × 0.96

Note: T02 = T03 = 936°R . Use Equation 5.21 to find Ti′ :

Ti ′ = 550 + 0.848 × (936 − 550) = 877°R The impeller pressure ratio is found from Equation 5.22. p02  887  =  p01  550 



3.5

= 5.12

The density based on total properties is



ρ02 =

p02 14.2 × 5.12 × 144 = = 0.21 lbm/ft 3 RT02 53.33 × 936

The gas temperature at the impeller exit is T2 = T02 −

V22′ 2c p

V22′ = (1273)2 + (524)2 T2 = 778°R The density at impeller tip is  T  ρ2 = ρ02  2   T02 



1/( γ − 1)

778  ρ2 = 0.21 ×   936 

2.5

= 0.132 lbm/ft 3

Determine the tip width using Equation 5.25:

b2 =

6.9 × 12 = 0.48 in π(9.6/12) × 0.132 × 524

108

Logan’s Turbomachinery

Determine the inlet dimensions from Equations 5.13–5.15: U1S = 1.2 × 1106 × cos32° = 1126 ft/s



D1S =



D1H



24 × 1126 = 6.2 in 4366.8

4 × 6.9 × 144  = (6.2)2 − π × 0.0574 × 703  

1

2

= 2.7 in

Note: D1S/D2 = 0.65, which is within the design limits. Calculate the absolute gas angle: 1273   V  = 67.6° < 70° α 2 ′ = tan −1  u2'  = tan −1   524   Wm2 



The absolute gas angle is inside the acceptable range. Calculate the diffusion ratio:

W1S = a1 M R1S = 1106 × 1.2 = 1327 ft/s



Wu2 = U 2 − Vu2 ′ = 1747 − 1273 = 474 ft/s 2 2 W2 ′ = (Wu2 ′ + Wm2 )



1

2 1



W2 ′ = (474)2 + (524)2 



W1S 1327 = = 1.877 < 1.9 W2 707

2

= 707 ft/s

The diffusion ratio is high but in the acceptable range.

Problems 5.1 Air enters a centrifugal compressor at 1 atm, 58°F, and V1 = 328 ft/s. At the impeller exit β 2 ′ = 63.4°,  Vm2 = 394 ft/s, and U 2 = 1640 ft/s. The mass flow rate is 5.5 lbm/s, the mechanical efficiency is 95%, and the compressor efficiency is 80%. Determine the ratio of total pressures at outlet and inlet and the power required to drive the machine. 5.2 Design a single-stage centrifugal compressor to handle 2.2 lbm/s of air at a pressure ratio of 4.2:1. Use 33 radial blades with an appropriate inducer. Assume that ηc = 0.70 . The machine is to operate at 60,000 rpm and to supply air to the combustion chamber of a turbojet engine. The basic design parameters required are the following: a. Hub diameter b. Shroud diameter at the impeller inlet

109

Centrifugal Compressors

c. Shroud diameter at the impeller exit d. Impeller inlet vane angle e. Vane width at the impeller exit f. Velocity triangle at the impeller inlet (shroud diameter) g. Velocity triangle at the impeller exit The compressor is to have no inlet guide vanes and is to draw in ambient air at 14.7 psia and 520°R. Assume ηm = 0.99. 5.3 Derive Equation 5.3 from Equation 5.1. Hint: h 02 = h 03. 5.4 Consider the high-speed flow of a diatomic gas at the inlet of the impeller of a centrifugal compressor. Show that



 M 3R1S cos 2β1S sin β1S C1 N 2 m = 1 2 2 4 2 [1 + (0.2M R1S )sin β1S ] p01 (T01 ) (1 − k 2 ) where C1 is a constant and k is the hub-tip ratio at the impeller inlet.

5.5 Plot the left-hand side of the equation of Problem 5.4 as a function of β1S with M R1S as parameter. Use subsonic values of the relative Mach number (e.g., 0.5, 0.6, 0.7, 0.8, and 0.9). Show that the maximum ordinate for each curve occurs at β1S = 32° . 5.6 Test data resulting from the test of a single-stage centrifugal air compressor are the following:

p01 = 14.5 psia



T01 = 58°F



p03 = 60.9 psia



T03 = 450°F

Assume ηm = 0.96 and calculate a. the power required to drive the compressor. b. the compressor efficiency. c. the energy transfer. 5.7 A single-stage centrifugal compressor operating at a speed of 15,000 rpm compresses air from an inlet total pressure of 14.7 psia to a discharge total pressure of 24.7 psia. The compressor is driven by an 80 hp motor. The mechanical efficiency of the compressor is 0.96, and the inlet total temperature is 528°R. The volumetric flow rate of air handled, measured at inlet conditions, is 1350 ft3/min. Find a. the mass flow rate of air in lbm/s. b. the energy transfer for isentropic compression in Btu/lbm. c. the output head in ft of air. d. the actual energy transfer in Btu/lbm e. the input head in ft of air. f. the compressor efficiency.

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Logan’s Turbomachinery

5.8 The speed of the compressor described in Problem 5.7 is increased until the discharge total pressure is 29.5 psia. Find a. the new output head in ft of air. b. the new speed in rpm. c. the new volumetric flow rate at inlet conditions in ft3/min. 5.9 Air enters a single-stage centrifugal compressor with a total pressure of 1.013 bar and a total temperature of 288 K. The axial velocity at the impeller eye is 100 m/s. The rotor tip speed is 500 m/s, V2′ is 456 m/s, and Vu2′ is 440 m/s. The impeller efficiency is 0.9, and that of the compressor stage is 0.8. The mechanical efficiency is 0.98. Find the static pressure in bar at the impeller outlet. 5.10 A single-stage centrifugal compressor receives air at a stagnation pressure of 1.013 bar and a stagnation temperature of 288 -K. The rotor tip speed is 500 m/s, the velocity at the impeller eye is 100 m/s, and V2′ is 456 m/s. The energy transfer in the impeller is 220 kJ/kg. The slip coefficient is 0.9. The impeller efficiency is 0.9, and the compressor efficiency is 0.8. The mechanical efficiency is 0.98. Find a. the blade angle at the impeller tip. b. T02. c. T2. d. p02 . e. p2. 5.11 Air enters a single-stage centrifugal compressor at 1 atm and 58°F. The entering flow is axial with V1 = 328 ft/s . At the impeller tip, β 2 ′ = 63.4°,  Vm2 = 394 ft/s, and U 2 = 1640 ft/s. The mass rate of flow is 5.5 lbm/s. The mechanical efficiency is 95%, and the compressor efficiency is 80%. Find a. Vu2′. b. p03/p01. c. the brake power. 5.12 A performance test of a single-stage centrifugal air compressor yielded the following data: p03 = 44 psia,  T03 = 770°R,  p2 = 28 psia,  N = 45, 960 rpm,  p01 = 14.5 psia, T01 = 520°R, and the mass flow of air is 1.4 lbm/s. The impeller has 17 blades, and its principal dimensions are D1 = 6.5 in. and b2 = 394 in . Assuming a mechanical efficiency of 0.96, find a. the compressor efficiency. b. the impeller efficiency. c. the fraction of the overall loss occurring in the impeller. 5.13 A single-stage air compressor with 20 radial vanes (β 2 = 90°) has an efficiency of 78%, an impeller efficiency of 89%, and a mechanical efficiency of 96%. The compressor handles 19.8 lbm of air per second and turns at 17,400 rpm. p01 = 15.96 psia, T01 = 531°R,  Vu2 ′ = 1345 ft/s, and Vm2 = 469 ft/s . Find a. D 2. b. E. c. p03.

Centrifugal Compressors

111

d. p02. e. p2. f. b2. 5.14 Test data for a single-stage centrifugal air compressor are the following: D 2 = 18 in., the blade angle at the impeller tip is β 2 = 60°,  N = 18, 000 rpm, the mass rate of flow is 19 lbm/s, T01 = 520°R,  T02 = 778°R,  p01 = 14.1psia, and p03 = 44.9 psia . Assume that the slip coefficient is 0.90 and the mechanical efficiency is 96%. Find a. the compressor efficiency. b. the flow coefficient Wm2/U 2 . c. the shaft power required in hp. 5.15 Design data for a single-stage centrifugal air compressor are the following:  = 35 lbm/s,  T01 = 58°F,  p01 = 14.6 psia, impeller effiD 2 = 19.7 in.,  N = 16, 200 rpm,  m ciency = 90%, mechanical efficiency = 96%, β 2 = 90°,  n B = 20,  b2 = 1.97 in., and the radial gap of vaneless space r3 − r2 = 1.6 in. Find a. p02. b. T02. c. M 2. d. M 3. e. α 3. This is the angle at which the diffuser vanes should be set. 5.16 A single-stage centrifugal compressor is to handle 2.25 lbm of air per second at a total pressure ratio of 4.15. The inlet air has a total pressure of 14.5 psia and a total temperature of 65°F. At the impeller tip, the 16 full blades and 16 splitters have a blade angle of 90° with the tangential direction. The impeller is to turn at 60,000 rpm. Find a. D1S. b. D1H. c. D 2. d. b2. 5.17 The first stage of a centrifugal air compressor is to handle 6.938 lbm of air per second at 14.168 psia and 560°R (total properties). The impeller is to rotate at 41,730 rpm and to produce a pressure ratio of 4.21. Determine the basic dimensions of the impeller, blade angles, and number of blades. 5.18 Design a vaneless diffuser to match the impeller of Problem 5.2. Assume that D 2 = 5.92 in.,  b2 = 0.281in.,  T2 = 700°R,  Vm2 = 463 ft/s,  M 2 ′ = 1.172, and α 2 ′ = 72.3°. Calculate α 3 and r3 at the diffuser exit if M 3 = 0.80. Assume that b is constant in the vaneless space. 5.19 Verify that substituting Equations 5.49, 5.52, and 5.53 into Equation 5.48 yields Equation 5.47.

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Logan’s Turbomachinery

References Ferguson, T.B. 1963. The Centrifugal Compressor Stage. London: Butterworths. Shepherd, D.G. 1956. Principles of Turbomachinery. New York: Macmillan. Whitfield, A. 1990. Preliminary design and performance prediction techniques for centrifugal compressors. Journal of Power and Energy 204(2): 131–144.

Bibliography Aungier, R.H. 2000. Centrifugal Compressors: A Strategy for Aerodynamic Design and Analysis. New York: ASME Press. Boyce, M.P. 2002. Centrifugal Compressors: A Basic Guide. Tulsa, OK: PennWell Corporation. Braembussche, R.V. 2018. Design and Analysis of Centrifugal Compressors (Wiley-ASME Series). New York: Wiley. Japiske, D. 1996. Centrifugal Compressor Design and Performance. White River Junction, VT: Concepts ETI, Inc.

Nomenclature a a1 b b2 c p D 2 D s D1H D1S E E i g h 01 h 02 h 03 H k  m  L m M  M M1

Local acoustic speed in gas Acoustic speed at impeller inlet Width of vaneless diffuser Width of blade at r = r2 in compressor impeller Specific heat at constant pressure Impeller tip diameter Specific diameter Hub diameter at impeller inlet Shroud diameter at impeller inlet Energy transfer from impeller to fluid Ideal energy transfer for isentropic compression from p01 to p03 (= gH) Gravitational acceleration Total enthalpy of gas entering impeller Total enthalpy of gas leaving impeller Total enthalpy of gas leaving diffuser Output head (= E i/g ) Hub-tip ratio at impeller inlet (= D1H/D1S ) Mass flow rate of gas discharged from compressor Mass flow rate of gas leaked from the high- to low-pressure regions outside of impeller Mach number Molecular weight of gas Absolute Mach number at impeller inlet (= V1/a1)

Centrifugal Compressors

MR MR1S nB N Ns p1 pt p01 p02 p03 pin P Q1 r r2 r3 r4 R Ru T T0 Tt T*

T1 Ti T01 T02 T03 Tin U U1 U2 U1H U1S V1 V2 Vm Vr Vu V2′ Vm2 Vu1 Vu2 Vu2 ′ W W1 W2

Relative Mach number (= W/a) Relative Mach number at impeller inlet shroud (= W1S/a1) Number of blades in impeller Rotor speed Specific speed Static pressure at impeller inlet Static pressure at the throat section of the stationary component feeding the impeller Total pressure at stage inlet Total pressure at impeller outlet Total pressure at diffuser outlet Static pressure at the inlet of the stationary component feeding the impeller Power to impeller shaft Volumetric flow rate based on gas density at impeller inlet Radial position measured from axis of rotation Radial position at impeller tip (= D 2/2 ) Radial position at exit from vaneless diffuser Radial position at exit from vaned diffuser ) Gas constant (= R u/M Universal gas constant Gas temperature Total temperature of gas Static temperature at the throat section of the stationary component feeding the impeller Gas temperature where M = 1 (stationary frame) or M R = 1 (moving frame) Static temperature at impeller inlet Total temperature of gas at the end of an isentropic compression from p01 to p03 Total temperature of gas entering impeller Total temperature of gas leaving impeller Total temperature of gas leaving diffuser Static temperature at the inlet of the stationary component feeding the impeller Blade speed at any r Impeller speed at blade leading edge Impeller speed at blade tip U1 at hub diameter U1 at shroud diameter Absolute velocity at blade leading edge Absolute velocity of gas leaving the impeller with an infinite number of blades Meridional component of V at any r (= Vr ) Radial component of V at any r (= Vm) Tangential component of Vat any r Absolute velocity of gas leaving the impeller with a finite number of blades Meridional component of V2 or V2′ (= Wm2) Tangential component of V1 Tangential component of V2 Tangential component of V2 ′ Velocity relative to impeller blade Relative velocity at impeller blade leading edge Relative velocity of gas leaving the impeller with an infinite number of blades

113

114

W2 ′ W1S Wm2 α α* α3 α 2′ β1 β2 β1S β2′ γ ηc ηm µs ρ ρ* ρ1 ρ2 ϕ2

Logan’s Turbomachinery

Relative velocity of gas leaving the impeller with a finite number of blades Relative velocity of gas entering impeller at shroud Meridional component of W2 ,  W2 ′ ,  V2 , or V2 ′ Absolute gas angle = angle between V and Vr Absolute gas angle where M = 1 Absolute gas angle at r = r3 Absolute gas angle at r = r2; tan −1 (Vu2 ′/Wm2 ) Angle between W1 and U1 Angle between W2 and U 2; also the blade angle at its trailing edge Angle between W1S and U1S Angle between W2 ′ and U 2; actual fluid angle Ratio of specific heats Compressor efficiency Mechanical efficiency Slip coefficient Gas density Gas density where M = 1 Gas density at impeller inlet Gas density at impeller exit Flow coefficient at impeller exit (= Wm2/U 2 )

6 Axial-Flow Pumps, Fans, and Compressors

6.1 Introduction Originally a very inefficient machine, the axial-flow compressor was not used to compress air in the gas-turbine power plant. However, the advances made in aerodynamics and computational fluid dynamics (CFD), which accompanied the development of highperformance aircraft, made possible its present use in gas turbines. Now a highly efficient machine, it must be studied and understood thoroughly by engineers. This machine resembles the axial-flow steam or gas turbine in general appearance. Usually multistage, it has rows of blades on a single shaft with blade length varying monotonically as the shaft is traversed. The difference is, of course, that the blades are shorter at the outlet end of the compressor, whereas the turbine receives gas or vapor on short blades and exhausts it from long blades. A close look at the blades shows that the compressor blade deflects the fluid through only a fraction of the angle that the turbine blade does. This point is illustrated by Figure 6.1, which also indicates that the concave side of the compressor blade moves ahead of the convex side; the reverse is true of the turbine blade. Clearly, the fluid receives energy from the compressor blade and gives up energy to the turbine blade. Aerodynamic analysis must be carried out for compressor blades, since flow in the boundary layer encounters an adverse pressure gradient, which may lead to separation, stall, and the consequent surge phenomenon discussed in connection with centrifugal machines. To avoid separation, the pressure rise must be small for each stage, in contrast with the very large pressure drops found in turbine stages. Typically, about one-half of the static enthalpy rise occurs in the rotor and one-half in the stator, indicating 50% reaction. The approach to compressor stage design is the same as that used for axial-flow pumps and fans except that the compressibility (density change) of the gas must be considered in the overall process of multistage machines. Fortunately, an abundance of theory exists, and many blade shapes have been tested in cascade tunnels, so that the designer has a large stock of data to draw on for design considerations. Axial-flow pumps and fans move liquids and gases with nearly constant density. They are like propellers in that the power is supplied to produce axial motion of the fluid, but they are different in that the fluid being moved is enclosed by a casing. As with propellers, the curvature of the blades is small, so they cause little deflection of the relative velocity vector of a fluid particle as it migrates through the moving passages. Generally, the blades have shapes, or profiles, like that of an airfoil: they are thin, streamlined, and cambered (Figure 6.2). The relative velocity W1 approaches the blade at an angle α (the angle of attack) to the chord line. The exiting fluid with relative velocity W2 has been deflected slightly, and the change of momentum results in a lift force L perpendicular to the mean direction of W1 and W2—i.e., perpendicular to a mean relative velocity Wm. 115

116

Logan’s Turbomachinery

FIGURE 6.1 Blade comparison.

The lift force L is primarily responsible for the transfer of energy, and the drag force D, which is directed parallel to Wm, is strongly associated with blade losses. Lift is maximized by setting the blades at a high angle of attack, but stall occurs if the angle is too high. Such characteristics of blades are determined in a wind tunnel using a representative set of blades arranged in series, known as a cascade. This kind of experimentation provides information not only about optimum incidence but also about optimum spacing for maximum lift and minimum drag. In an axial-flow fan, pump, or compressor, the blade rotates such that the pressure surface (concave) leads the suction surface (convex); the opposite is true for an axialflow turbine: i.e., the suction surface leads the pressure surface. In Figure 6.3, the blade motion is to the right. The tangential component FBu of the blade force can be obtained in terms of the angle β m that Wm makes with the axial direction. Referring to Figure 6.4, it is clear that (6.1)

FBu = L cos β m + D sin β m



The rate of energy transfer E is given by E = U ( L cos β m + D sin β m )





(6.2)

and the energy transfer per unit mass is expressed as E = U ( Vu2 − Vu1 )



(6.3)

where the blade velocity U is the same at the inlet and exit planes, since the flow ideally contains no radial components of velocity. Also, as is evident from Figure 6.4, we have

∆Vu = Vu2 − Vu1 = Wu1 − Wu2 = ∆Wu

FIGURE 6.2 Blade profile.

(6.4)

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Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.3 Blade motion.

An alternative expression for E is obtained by multiplying Equation 6.3 by the mass flow  = ρVa S ) : rate per blade per its unit length ( m (6.5) E = U ( Vu2 − Vu1 ) (ρVa S ) where Va is the axial component of absolute velocity and S is the spacing between two adjacent blades in a row. The blade force equation (Equation 6.1) and the blade power equation (Equation 6.2) should also be interpreted as force and power, respectively, per blade per its unit length. Equating Equation 6.2 with Equation 6.5 and nondimensionalizing, we obtain CL = 2

cos 2β1 S ( tan β1 − tan β2 ) cos β m C

(6.6)

where the lift coefficient CL is defined as

CL =

1

L 2 2 ρW1 C

(6.7)

where C is the chord, the length of a line drawn between the leading edge and the trailing edge of the blade (Figure 6.3). Also, in forming Equation 6.6, we have omitted the drag

FIGURE 6.4 Velocity diagrams.

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Logan’s Turbomachinery

term that appears in Equation 6.1 on the grounds that D V1, which means that the kinetic energy of the fluid has been increased by the action of the blades. In a single-stage machine, outlet guide vanes may turn the fluid back to the axial direction. In multistage machines, the vanes redirect the fluid to its original direction, so that the fluid leaves the stage at absolute fluid angle α 3 = α 1 with absolute velocity V3 = V1. Equation 2.51, applied to an axial-flow compressor, fan, or pump, shows that h 2 > h 1, since W1 > W2. For liquids or gases, the enthalpy rise implies a corresponding pressure rise in the rotor, and for gases, a temperature increase is also indicated. Equation 2.21, applied to the stator or diffuser downstream of the blade row of an axial-flow stage, simplifies to

h2 +

V22 V2 = h3 + 3 2 2

(6.8)

and shows that h 3 > h 2 and p3 > p2; for compressors, however, T3 > T2 is also indicated. Thus, a complete stage accelerates the flow in the rotor and decelerates it in the stator, accompanied by a pressure rise in both. Compressors, pumps, and fans have solidities ranging from as low as 0.1 for two-bladed pumps or fans up to 1.5 for some compressors. Axial-flow machines of low solidity are typically machines of high specific speed, as is shown in Table 6.2. Eck (1973) shows that the optimum solidity (the ratio of blade chord length to blade spacing) is proportional to tan β1 − tan β 2. This implies that the machines of low energy transfer, or head, require low solidities and that low head correlates with high specific speed. Increasing the number of blades increases guidance and thereby increases head, but friction losses also are increased. The optimum solidity is determined by test. Let us consider a control volume, surrounding a single moving blade, of width S and of unit height along the blade, as shown in Figure 6.5. Assuming that the axial component of fluid velocity does not change from inlet to outlet and that the mass flow rate remains constant, we can write the force equilibrium equation as

( p2 − p1 ) S − FBa = 0

(6.9)

Expressing the axial component FBa of the blade force in terms of lift and drag, we obtain for the pressure rise across the blade row:

( p2 − p1 )rotor = LS sin βm − DS cos βm

(6.10)

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Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.5 Control volume for a cascade blade.

In nondimensional form, Equation 6.10 becomes

( p2 − p1 )rotor

1

2 2 ρW1

=

C ( CL sin βm − CD cos βm ) S

(6.11)

Thus, Equation 6.11 allows prediction of pressure rise in terms of aerodynamic coefficients and relative velocities. A similar relation can be obtained for the diffuser section of the stage using the same method. An alternate expression for pressure rise across the blade row is obtained by expressing the axial component FBa in terms of the tangential component FBu. Thus,

FBa = FBu tan (β m − δ )

(6.12)



Since it is small, the tangent of δ may be approximated by δ, which represents the ratio of drag to lift, or CD/CL. Using a trigonometric identity and noting from Figure 6.4 that



tan β m =

Wum R = Va ϕ

(6.13)

we substitute Equations 6.12 and 6.13 into Equation 6.9 to obtain



( p2 − p1 )rotor = ρU2 ϕΛB ϕR +− δϕδR

(6.14)

where the component FBu is replaced by using the change in tangential momentum flow—namely,

FBu = ρVa S ( Vu2 –  Vu1 ) = ρU 2 SΛ B ϕ

(6.15)

where Λ B is the blade loading coefficient ∆Vu/U. A similar derivation may be made for the diffuser vanes to obtain the pressure rise in the stationary part of the stage. Addition of the two leads to the equation for the overall stage pressure rise:



 R − ϕδ 1 − R − ϕδ  ∆p = ρU 2 ϕΛ B  +   ϕ + δR ϕ + δ(1 − R) 

(6.16)

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Logan’s Turbomachinery

The above relation is useful in estimating stage efficiency, which is defined as the ratio of pressure rise with the blade drag accounted for to that with frictionless blades. Elimination of drag means that the terms involving δ in Equation 6.16 vanish. Thus, the stage efficiency ηs is given by ηs =

 R − ϕδ 1 − R − ϕδ  ∆p = ϕ +  ∆pideal  ϕ + δR ϕ + δ(1 − R) 

(6.17)

The drag-lift ratio δ used in Equation 6.17 must be modified to account for several additional losses. These are considered in the next section.

6.3 Losses Boundary layers on the surfaces of airfoils, whether moving or stationary, mark regions of high shear stress; the resultant of the viscous forces produced at the airfoil surface is the drag force. In addition to resisting blade movement, viscous forces retard fluid in the stationary passages and result in total pressure losses. The thickness of the boundary layers on the blade surfaces deflects the main flow and thus changes the effective blade shape. Increasing the pressure in the flow direction slows down the fluid in the boundary layer and promotes separation of the boundary layer from the blade surfaces, concomitantly creating regions of reversed flow. In the latter case, the effective blade shape is drastically distorted, and the flow direction is severely modified. In addition to the blades, boundary layers are formed on the inner and outer surfaces of the annular-flow passage and the cylindrical surfaces at the hub and tip radii. Since the flow actually takes place in the rectangular passage bounded by blades on two sides and by walls of the annulus on the other two, it is expected that losses will depend on the ratio of blade spacing to blade height. Empirically, it has been found that the drag produced by these surfaces is correctly reflected by the relation



CD ′ = 0.02

S h

(6.18)

where h represents blade height and CD′ is the increment to be added to the previous drag coefficient to account for annulus losses. The velocity variation due to boundary layers on the blades and walls of the annulus, coupled with the curvature of the blade surfaces, results in an additional loss. Secondary currents are set up in a plane transverse to the flow, as is indicated in Figure 6.6. Dissipation of the energy of these secondary currents takes place in the blade passage and in the wakes behind the trailing edges via vortices spawned by the interaction of neighboring secondary flow cells as they leave the blade. Because the trailing vortices are similar to wing vortices, it is expected that the corresponding drag is proportional to the square of the lift coefficient. The recommended equation for drag coefficient is then

CD ″ =  0.018C2L

(6.19)

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Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.6 Secondary flow in blade passages.

The difference in pressure on the two sides of the moving blades results in a leakage of fluid around the tip—i.e., through the narrow passage formed between the blade tip and the casing. This loss is accounted for by the empirical formula CD ′′′ = 0.29



cT 3 2 CL h

(6.20)

where c T is the tip clearance. To obtain a more realistic value for the stage efficiency using Equation 6.17, we can artificially increase the blade drag force by an amount proportional to the sum of CD′ , CD″ , and CD″′ . We then substitute for δ in Equation 6.17 using the expression



δ=

CD + CD ′ + CD ′′ + CD ′′′ CL

(6.21)

6.4  Pump Design Axial-flow pumps are used for specific speeds above approximately 3, with centrifugal pumps occupying the range below 2 and mixed-flow pumps filling the gap between the two. They are then machines of low head, high capacity, and a single stage. They require several well-finished blades of airfoil section, as shown in Figure 6.2. As a starting point in the design of an axial-flow pump, we can use that part of a Cordier diagram (Figure 3.3) for which N s > 3. The relationship between specific speed N s and specific diameter D s is given approximately by

Ds =

2.95 N 0.485 s

(6.22)

Calculating N s from specified values of N,  Q, and H, we can arrive at D s from Equation 6.22. This value of the specific diameter is used to compute the rotor tip diameter D t in the following manner: Dt =

Ds Q

1

(gH)

1

2 4

(6.23)

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Logan’s Turbomachinery

TABLE 6.1 Constant for Equation 6.24 as a Function of Hub-Tip Ratio K

Dr /Dt 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

8.38 7.65 6.1 5.1 4.49 3.75 3.38 2.94 2.61

Source:  Stepanoff (1957).

By selecting a suitable hub-tip ratio—i.e., blade-root diameter D r divided by blade-tip diameter D t—we are able to compute the hub diameter, which is a synonymous expression for root diameter. Generally, axial-flow pumps have hub-tip ratios in the range 0.3 to 0.7. The graphical display of the relationship among D r/D t , N s , and solidity C/S, based on current practice, is given by Stepanoff (1957) and may be approximated by the relation σ=



C K = 1.447 S NS

(6.24)

where K is obtained from Table 6.1. The solidity, calculated from Equation 6.24, is based on a suitably chosen value of hub-tip ratio and the required specific speed. It should lie in the range 0.4 to 1.1, and if the calculated solidity lies outside that range, a new choice of D r/D t should be made. The optimal number of blades recommended by Stepanoff (1957) is presented in Table 6.2 as a function of specific speed. The annular flow area and the required flow rate can now be used to determine the axial velocity component Va. Thus, we have Va =

4Q π D 2t − D 2r

(

(6.25)

)

The velocity diagram in Figure 6.7 shows the relationship between the mean fluid angle β1 and the velocities. Because the inlet velocity is axial, it can be determined from Equation 6.25. TABLE 6.2 Optimum Number of Blades for Axial-Flow Pumps Ns

nB

2–3.5 3–4.5 4–5.5 5–6.5

5 4 3 2

Source:  Stepanoff (1957).

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Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.7 Velocity diagram at rotor inlet.

The mean blade speed U is determined from the mean diameter Dm. Thus, at the mean diameter, we can write  D 2 + D 2r  Dm =  t  2 



1

2

(6.26)

and

U=

ND m 2

(6.27)

where N is the rotational speed in radians per second. The required fluid angle β1 is therefore given by



β1 = tan −1

U Va

(6.28)

Similarly, the fluid angle β 2 at the rotor exit is determined by reference to Figure 6.8. Assuming the same annular flow area, and hence the same axial velocity Va, we know U and Va as before. With a known head and with no inlet whirl—i.e., Vu1 = 0, we determine the exit whirl velocity Vu2 from Equation 6.3; thus, we may write



Vu2 =

gH ηH U

(6.29)

The exit flow angle β 2 is easily found from the geometric relation



β 2 = tan −1

U − Vu2 Va

FIGURE 6.8 Velocity diagram at rotor outlet.

(6.30)

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Logan’s Turbomachinery

The mean fluid angle β m is determined from Equation 6.13- with Wum =



Wu1 + Wu2 2

(6.31)

where Wu1 and Wu2 are defined by Figures 6.7 and 6.8, respectively. The hydraulic efficiency ηH is equivalent to ηs and can be calculated from Equation 6.17. The evaluation of the drag-lift ratio δ from Equation 6.21 requires the use of Equation 6.6 in the form



CL =

2 cos β m ( tan β1 − tan β 2 ) σ



(6.32)

In the absence of cascade data, the profile drag coefficient can be determined from



CD =

ζ p cos 3 β m σ

(6.33)

where the profile loss coefficient ζ p is extracted from Figure 6.9. The additional drag coefficients are found from Equations 6.18 through 6.20. The stagger angle—i.e., the angle between the chord line of the profile and the axial direction—is determined from the required incidence, or angle of attack α, necessary to produce the lift coefficient CL calculated from Equation 6.6. Generally, we will select an airfoil section for which the cascade data are available. NACA Report No. 460 by Jacobs et al. (1935) is an example of such

FIGURE 6.9 Typical cascade results. (Source: Balje (1981).)

125

Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.10 Cascade data.

a source. Figure 6.10 shows schematically the sort of cascade results available in NACA reports and elsewhere. Cascade results should also be checked to assure that the angle of attack chosen does, in fact, produce the desired fluid deflection. Wilson (1984) recommends the use of NASA cascade data for double-circular-arc hydrofoils as a basis for the design of axial-flow pumps and presents carpet plots for this purpose. The blade may be twisted if the so-called free-vortex method is employed. In this method, the product of Vu2 and D is kept constant. As a result, Vu2 varies with radius, and so does β 2. The angle β1 varies with U, and the blade may be twisted to provide proper guidance at the trailing edge as well as the correct incidence. Free-vortex design results in approximately uniform energy transfer at all radial positions. However, untwisted blades may be used in the interest of economy of production. The fluid leaving the blades encounters a row of vanes (Figure 6.11). These serve to straighten the flow—i.e., to remove the whirl component Vu2—and to increase the pressure. Referring to Figure 6.8, it is seen that fluid enters the vanes at the angle α 2 and leaves axially. The axial component Va may be reduced by flaring the walls of the annulus by several degrees. The angle formed by the camber line tangent at the leading edge and the vector V2 should vary from root to tip. It should be designed to provide a positive incidence over the operating range, down to 50% of the design flow rate.

6.5  Fan Design The design of an axial-flow fan can proceed in a manner similar to that of the axial-flow pump. Specific speed N s can be determined from specified values of rotational speed N in rad/s, volume flow rate Q in cubic feet per second, and head H in feet. The Cordier curve

FIGURE 6.11 Blades and vanes.

126

Logan’s Turbomachinery

FIGURE 6.12 Eck’s correlation for axial-flow fans. (Source: Eck (1973).)

relation (Equation 6.22) may then be used to determine specific diameter D s. Finally, the blade tip diameter D t is found using Equation 6.23. The root diameter D r is then calculated from the hub-tip ratio D r/D t, which is chosen to lie in the usual range of 0.25 to 0.7. Eck (1973) recommends that the values of specific diameters found in the shaded zone between the curves of Figure 6.12 be used in lieu of those obtained from the Cordier relation for the determination of a tip diameter. The velocity diagrams, as depicted in Figures 6.7 and 6.8, are then constructed as described in the previous section using Equations 6.25 through 6.31. Fan efficiency, estimated from Equation 6.17, would be expected to lie in the range 0.8–0.9. The fluid angles β1 and β 2 required by the velocity diagrams can then be used with cascade data such as those depicted schematically in Figure 6.13 to select a suitable solidity

FIGURE 6.13 Velocity diagram for a compressor stage.

127

Axial-Flow Pumps, Fans, and Compressors

σ for the mean diameter. McKenzie (1988) recommends the following optimal solidity for axial-flow fans: σ opt =

1 9 0.567 − C pi

(

)

(6.34)

where C pi represents the ideal static pressure rise coefficient and is defined as

C pi = 1 −

W22 W12

(6.35)

C pi = 1 −

V32 V22

(6.36)

for the rotor and as

for the stator. The actual chord would be selected to provide an aspect ratio h/C of from 1 to 3. The angle of attack α required to produce CL calculated from Equation 6.6 is determined from wind tunnel cascade data for the airfoil shape and solidity used in the design. Figure 6.10 schematically shows data of this type—i.e., the blade coefficient as a function of angle of attack α. The angle of attack is the angle between W1 for a blade or V2 for a vane and the straight line drawn from the leading edge to the trailing edge of the airfoil profile; the latter line is also called the chord line. The stagger angle α s is the angle at which the airfoil (chord line) is set with respect to the axial direction and is given by β1 − α for blades and by α 2 − α for vanes. NACA cascade data have been plotted by Mellor (1956) in a series of charts that enable the determination of β1 and β 2 for blades or α 2 and α 3 for vanes. There is a separate chart for each solidity and blade profile. Each chart consists of lines of constant stagger angle α s and lines of constant angle of attack α. The coordinates of the point of intersection of these lines are the desired fluid angles. The use of the Mellor charts is further explained by Wilson (1984). The blade is twisted to accord with the angles determined from velocity diagrams for the tip and root diameters. Here, we use the free-vortex condition—i.e., Vu2 D = constant—to establish velocity triangles at the extremities of the blade. For an axial-flow compressor stage, Appendix H presents a quick method to draw the composite dimensionless velocity diagram directly from the knowledge of the flow coefficient, loading coefficient, and degree of reaction.

6.6  Compressor Design It is important to design the compressor stage in such a way as to avoid stall, which occurs on compressor blades as it does on airplane wings. As the angle of attack of a wing or blade is increased, the lift force increases until a maximum value is achieved; if further increases in the angle of attack occur, the wing or blade is said to stall—i.e., to lose lift and, at the same time, to lose pressure rise. The phenomenon of stall occurs as a result of a slowing of the fluid in the boundary layer until the flow stops—or even reverses. Thus, the phenomenon is avoided by keeping the lift force, or blade loading, below a certain limiting value.

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Logan’s Turbomachinery

The quantity used as a measure of blade loading, and hence of the tendency to stall, is the diffusion factor, which, when applied to rotor blades, is defined as



Df = 1 −

W2 ∆Wu + W1 2σW1

(6.37)

For vanes, W2 and W1 are replaced by V3 and V2, respectively. Mattingly (1987) recommends the use of designs having D f < 0.55 to assure avoidance of stall in axial-flow compressors. Wilson (1984) recommends the de Haller criterion, which may be stated for blades as W2/W1 < 0.71 and for vanes as V3/V2 < 0.71. Equation 6.3 can be expressed in an alternate form as

E = UVa (tan β1 − tan β 2 )



(6.38)

Since blade speed U increases with radius, Equation 6.38 shows that, for radially independent throughflow velocity Va and specific energy transfer E, we must vary the fluid angles β1 and β 2 and hence the blade angles. This is, as previously discussed, the free-vortex design constant U∆Vu. As before, with the axial-flow pump, we must have twisted blades in order to achieve this equality of energy transfer along the blade. The variation of blade angles then implies that β m will vary and hence that the degree of reaction R will also vary. The latter has been defined by Equation 2.46 and can vary between zero and unity. R is found empirically, and it has been shown theoretically (Shepherd, 1956) that a value of 0.5 is a near optimum for the degree of reaction producing maximum stage efficiency. Consequently, we find this value frequently used for a design value at the mean diameter. Another design approach is to use a value of 0.5 for R at all radial positions. Both bases for design are used as well as others not discussed here. It can be shown that ϕ = 1 2 is also an optimum value of the flow coefficient when the optimum value of R—R = 1 2 —is selected simultaneously. The theory discussed in the previous section relates energy transfer to fluid angles, blade speed, and axial velocity through the velocity diagrams drawn at the hub, mean, and tip radii. The development of a blade design requires the use of wind tunnel results such as those shown in Figure 6.14 (Herrig et al., 1957). Many such results are available to designers, and they are made for very specific blade shapes. Thus, the designer will generally specify a blade shape for which results exist, and these proportions are given in the report of the wind tunnel results. In addition, the tests are carried out for specific values of solidity C/S and stagger angle. For example, Figure 6.14 gives results for the NACA 68 (18): 10 airfoil shape, a solidity of 0.75, and a fixed fluid angle β1 of 60°. For further information on cascade data, the reader is referred to Horlock (1958), Wilson (1984), and Gostelow (1984). A compressor velocity diagram of the type shown in Figure 6.15, but in nondimensional form, can be started using the chosen values of R and ϕ. The mean relative fluid angle β m can be calculated from Equation 6.13. The blade tip speed can be selected on the basis of the strength of the blade material. Cohen et al. (1987) recommend the stress equation:



 D2  s c = 0.5ρB U12  1 − H2  Dt  

(6.39)

where s c is the design centrifugal stress and ρB is the density of the blade material. To solve for the tip speed, a tentative value of hub-tip ratio must be chosen. Wilson (1984) recommends hub-tip ratios of greater than 0.6 for axial-flow compressors.

Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.14 Cascade results for the NACA Profile 68 (18): 10. (Source: Herrig et al. (1957).)

FIGURE 6.15 Profile loss coefficients. (Source: Balje (1981).)

129

130

Logan’s Turbomachinery

Substituting the chosen hub-tip ratio and a design centrifugal stress for the blade material into Equation 6.39, a safe tip speed and tip radius can be determined. Since a hub-tip ratio was also used, both rt and rH can be determined. Tip speeds of 1500 ft/s and less are typical. Finally, the mean blade speed U is determined from U=



rm U t rt

(6.40)

where rm is determined from Equation 6.26. The mean blade speed is used to convert the dimensionless velocities into dimensional ones. The blade loading ∆Wu or ∆Vu is obtained from the energy transfer using Equation 6.3. Since the energy transfer is required, it must be obtained from the required stage total pressure ratio R pn through the use of the definition of the stage efficiency: η ∆T   R pn =  1 + s 0s   T01 



γ /( γ − 1)

(6.41)

and the steady flow energy equation: E = c p ∆T0s



(6.42)



where ∆T0s represents total temperature rise for the stage. A value of the stage efficiency must be assumed at this point and verified later. The above calculations result in the determination of all the velocities and angles in the velocity diagram. At this point, several checks must be made. The diffusion factor should be calculated to assure that D f < 0.55. One should select solidity σ in the range of 0.66 to 2.0 in Equation 6.37. A typical value of solidity is 1.0. The hub-tip ratio must be checked to assure that the mass flow rate is the required value. Equation 6.25 can be used for this purpose along with the  ρ. Finally, the assumed stage efficiency is checked using volumetric flow relation Q = m/ Equation 6.17. For this calculation, the aspect ratio can be selected in the range of 1 < h/C < 3, and the tip clearance ratio can be taken as 0.02. The choice of blade can be made on the basis of the camber angles presented in Table 6.3. The camber angle is the difference between the angles formed by tangents to the camber line at the ends of the blade profile (Figure 6.16). Thus, (6.43)

θ = γ1 − γ2

TABLE 6.3

Optimum Camber Angles in Degrees for Blades and Vanes of Axial-Flow Compressors S/C

Deflection, ε (degrees)

0.5

1.0

1.5

15 20 25 30

– 15 27 35

14 27 38 47

22 33 46 –

Source:  Horlock (1958).

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Axial-Flow Pumps, Fans, and Compressors

FIGURE 6.16 Compressor blade diagram.

Once the blade having the desired camber angle is selected, the deviation, defined by δ ′ = β2 − γ 2





(6.44)

can be calculated from the Carter rule: δ′ =

θ 4σ

1

(6.45) 2



which is recommended by Mattingly (1987). The angle γ 2 is then calculated from Equation 6.44, and γ 1 is determined from Equation 6.43. Finally, the stagger angle is determined from the approximate relation:

αs =

γ1 + γ2 2

(6.46)

The recommended calculations relate to rotor design; for vane design, the same procedure is followed, but V2, V3, α2, and α3 replace W1, W2, β1, and β2, respectively.

6.7  Compressor Performance Prior to construction and testing of the prototype machine, it is desirable to determine estimated performance characteristics by means of calculation. Normally, stage efficiency ηs and multistage compressor efficiency are based on total temperatures. Thus, referring to Figure 6.17, we have



ηs =

Ti − T01 T03 − T01

(6.47)

This is the same definition used in Equation 5.4 for centrifugal compressors. In Figure 6.17, state 01 denotes conditions at the rotor inlet and state 03 those at the stator outlet. It has been shown, however, by Cohen et al. (1987) that the incompressible definition (Equation 6.17) predicts the stage efficiency well because the rise of total temperature in

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Logan’s Turbomachinery

FIGURE 6.17 Enthalpy-entropy diagram.

the stage is sufficiently small. Cascade test results may be used to determine values of lift and drag coefficients for the blade profile. If cascade data are not available, the lift and drag coefficients can be calculated from Equations 6.18, 6.19, 6.20, 6.32, and 6.33. The energy transfer relation (Equation 6.3) is quite accurate for a single stage because the velocity profile Va (r) is nearly flat. The annular walls create a boundary layer that causes peaking of Va near the mean radius. Thus, Equation 6.3 must be multiplied by a factor λ, called the work-done factor, and the resulting equation:

E = λUVa ( tan β1 − tan β 2 )



(6.48)

can be used for each stage with a constant value of λ. The work-done factor may be approximated from



 − ( N st − 1)  λ = 0.85 + 0.15exp    2.73 

(6.49)

where N st is the number of stages in the compressor. The same basic relations, coupled with cascade data, can be used to predict off-design performance. As the flow rate through the compressor is varied from the design value, the angle of incidence also varies, but the rotor and stator fluid exit angles do not deviate appreciably from their design values. Thus, it is possible to construct velocity diagrams for each off-design flow rate and, from the indicated incidence, to determine values of CL and CD from cascade test results. The overall compressor pressure ratio can be determined from the product of individual stage pressure ratios. Similarly, the overall total temperature rise is the sum of stage total temperature rises where the definitions of Equations 6.41 and 6.42 are applied to the compressor as a whole. Thus,



ηc =

R (pγ − 1)/γ − 1 ∆T0 /Tin

(6.50)

where R and ∆T0 in this equation denote the total pressure ratio and the total temperature rise for the whole machine. The polytropic efficiency ηp is sometimes used in multistage

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Axial-Flow Pumps, Fans, and Compressors

compressors to calculate the outlet temperature; thus, ( γ − 1)/( ηp γ ) replaces the polytropic exponent (n − 1)/n, so that (γ − 1)/(ηp γ )

Tout = Tin R p





(6.51)

The compressor map for an axial-flow compressor will have the same appearance as that shown schematically for the centrifugal compressor in Figure 5.5. This plot of R p  with N as a parameter shows operational limits set by the phenomas a function of m ena of stalling at low flow rates and choking at high flow rates. At low speeds, choking occurs in the rear stages and stalling (due to high incidence) in the front stages, whereas the situation is reversed at high rotor speeds. These phenomena can be predicted in advance using indicators such as the critical Mach number M c based on inlet relative velocity (usually Mc ≈ 0.7–0.8) that indicates the first appearance of sonic flow in the blade passages and the stalling incidence angle corresponding to the maximum value of CL obtained in cascade tests. Since temperatures increase in stages after the first, Mach numbers decrease. Thus, the first stage will be the most likely site of shock losses. The first stage may be designed for supersonic inlet velocities near the tips. The leading edge of such blades will be sharp to accommodate attached oblique shocks, as discussed by Kerrebrock (1977). The blades are called transonic in that they accommodate subsonic flow near the hub. Such a stage may be desirable in aircraft compressors where the crosssectional area is minimal.

Worked Examples EXAMPLE 6.1 In an axial-flow compressor stage, the degree of reaction is 0.5, the mean blade speed is 1000 ft/s, the flow coefficient is 0.4, and the blade loading coefficient is 0.355. Lift and drag coefficients are 1.30 and 0.055, respectively, where CD includes all losses. The entering air density is 0.0024 slug/ft3. Determine the energy transfer and the stage static pressure rise. SOLUTION: Calculate the change in the whirl component of velocity:

∆Vu = Λ B U = 0.355 × 1000 = 355 ft/s

Calculate the energy transfer:

E = 355 × 1000 = 355, 000 ft 2/s 2

Calculate the drag-lift ratio:



δ=

CD 0.055 = = 0.0423 CL 1.30

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Logan’s Turbomachinery

Calculate the stage pressure rise:  R − ϕδ 1 − R − ϕδ  p 3 − p1 = ρ U 2 Λ B ϕ  +   ϕ + δR ϕ + δ(1 − R)   0.5 − 0.4 × 0.0423 1 − 0.5 − 0.4 × 0.0423  = 0.0024 × (1000)2 × 0.355 × 0.4 ×  +   0.4 + 0.0423 × 0.5 0.4 + 0.0423 × (1 − 0.5)  = 783 psf = 5.43 psi

EXAMPLE 6.2 An axial-flow, high-performance, single-stage, experimental air compressor (see Paulon et al., 1991) has no inlet guide vanes and runs at 12,000 rpm. The blade speed at the tip is 1322 ft/s, the hub-tip ratio is 0.70, and the solidity is 1.375. Find the mean blade speed. SOLUTION: Calculate the tip radius:



rt =

Ut 1322 = = 1.052 ft N 12, 000 × (π/30)

The circle of mean radius divides the annular area of the compressor into two equal areas. Thus, the mean radius is calculated from the following equation:



 r 2 + rH2  rm =  t  2 

1

2

 1.49 × (1.052)2  =  2  

1

2

= 0.908 ft

where rH = 0.7rt has been used. Finally, the mean blade speed is

U = Nrm = 12, 000 × (π/30) × 0.908 = 1141 ft/s

EXAMPLE 6.3 Using the data from Example 6.2, find the absolute and relative Mach numbers of the air entering the rotor if the relative air angle at entry is 60° and T01 is 540°R. SOLUTION: Noting that V1 is axially directed, the velocity diagram is as shown in Figure 6.7. Thus,

V1 = U cot β1 = 1141cot 60° = 659 ft/s



W1 = U csc β1 = 1141csc 60° = 1317.5 ft/s

Calculate the static temperature using the relation between static and total temperature:



T1 − T01 −

V12 (659)2 = 540 − = 504°R 2c p 12, 012

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Axial-Flow Pumps, Fans, and Compressors

Calculate the acoustic speed in the inlet air: a1 = ( γ RT1 )

1

2

= (1.4 × 1716 × 504)

1

Finally, calculate the Mach numbers: M1 =



M R1 =

2

= 1100 ft/s

V1 659 = = 0.60 a1 1100 W1 1317.5 = = 1.198 a1 1100

EXAMPLE 6.4 The relative air angle at the rotor exit is 35° in the compressor stage considered in Examples 6.2 and 6.3. Find the energy transfer and stage total pressure ratio, assuming the stage efficiency is 0.87. Note: This high efficiency is realizable in modern, highperformance stages. SOLUTION: A previous calculation gave V1 = 659 ft/s (axially directed). From Figure 6.7, it is clear that Wu1 = U = 1141ft/s



Referring to Figure 6.8, it is seen that Wu1 = V1 tan β 2 = 659 × tan 35° = 461ft/s The energy transfer is given by Equation 6.3 and is

E = U∆Vu = 1141 × (1141 − 461) = 775, 880 ft 2/s 2

The stage pressure ratio is obtained from Equations 6.41 and 6.42 as Rp =

p03  ηE  = 1 + s  p01  C p T01 

γ /( γ − 1)

0.87 × 775880  R p = 1 + 6006 × 540  

3.5

= 1.938

Note: The pressure ratio measured by Paulon et al. (1991) for the same design was 1.95.

EXAMPLE 6.5 Estimate the stage efficiency for the single-stage compressor described in Examples 6.2–6.4. The approximate rotor solidity is 1.375, the approximate rotor blade aspect ratio is 0.72, and the assumed tip clearance ratio is c T/h = 0.02 .

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Logan’s Turbomachinery

SOLUTION: Calculate the degree of reaction:

− Wum = R=



− Wul − Wu2 1141 + 461 = = 801ft/s 2 2 − Wum 801 = = 0.702 U 1141

Calculate the flow coefficient: ϕ=



V1 659 = = 0.5776 U 1141

Calculate the mean relative air angle:

tan β m =

tan β1 + tan β 2 tan 60° + tan 35° = 2 2

β m = 50.57°



Solve for CL using Equation 6.32:

CL = 2 cos β m

tan β1 − tan β 2 σ

CL = 2 cos 50.57°

tan 60° − tan 35° = 0.953 1.375

Use Figure 6.15 to determine the profile loss coefficient. First, find the blade loading coefficient over the flow coefficient:



Λ B ∆Vu Wu1 − Wu2 1141 − 461 = = = 1.03 = ϕ Va Va 659

Next, find the chart abscissa:

− cot ( 180° − β m ) = − cot(180° − 50.57°) = 0.82

Enter the chart and find ζ p = 0.09 . Calculate CD for the profile drag from Equation 6.33:



CD =

ζp 0.09 cos 3 β m = cos 3 (50.57°) = 0.0168 σ 1.375

Next, apply Equations 6.18, 6.19, and 6.20 to determine additional contributions to drag:



CD ′ = 0.02

0.02 0.02 S = = = 0.20 h σ h/C 1.375 × 0.72

CD ′′ = 0.018 CL2 = 0.018 × (0.953)2 = 0.01635 3 3 c CD ′′′ = 0.29  T  CL2 = 0.29 × 0.02 × (0.953) 2 = 0.0054  h

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Axial-Flow Pumps, Fans, and Compressors

Next, solve for the total drag over the lift using Equation 6.21: δ=



0.0168 + 0.020 + 0.01635 + 0.0054 = 0.062 0.953

Finally, compute the stage efficiency using Equation 6.17: 0.702 − 0.5776 × 0.062 1 − 0.702 − 0.062 × 0.5776  = 0.873 ηs = 0.5776 ×  + 0.5776 + 0.062 × 0.298   0.5776 + 0.062 × 0.702



Note: It is interesting that the stage efficiency measured by Paulon et al. (1991) for the corresponding actual machine was 0.872.

EXAMPLE 6.6 Find the overall compressor efficiency for a four-stage compressor in which each stage has the same velocity diagram and stage efficiency. The inlet temperature of the compressor is 540°R, and the velocity diagrams are the same as in Examples 6.2 through 6.4. SOLUTION: Since we have a multistage machine, we must apply a work-done factor. Thus, from Equations 6.48 and 6.49 with N st = 4 , λ = 0.85 + 0.15exp  − ( N st − 1)/2.73  = 0.90

and

E = 0.90 × 1141 × (1141 − 461) = 698, 292 ft 2 /s 2 The individual stage pressure ratios are calculated from Equations 6.41 and 6.42 using the above E and the given ηs . Thus,



698, 292 × 0.87   R pn = 1 + 6006 × T0n  

3.5

applies to each of the four stages when n takes on the value of 1, 2, 3, or 4, corresponding to the stage number. The rise of total temperature is the same across each stage and is given by ∆T0s =

E 698, 292 = = 116.27°R cp 6006

The overall temperature rise for four stages is

∆T0 = 4 ∆T0s = 4 × 116.27 = 465.08°R

The inlet total temperature T0n increases by 116.27° in each subsequent stage. Thus, the inlet temperatures for the four stages are the following: T01 = 540°R, T02 = 656.27°R, T03 = 772.54°R, and T03 = 888.81°R.

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Logan’s Turbomachinery

Substituting these values into the pressure-ratio equation yields the following total pressure ratios for the four stages: R p1 = 1.824, R p2 = 1.652, R p3 = 1.538, and R p4 = 1.458. The product of these pressure ratios gives the overall pressure ratio for the compressor: R p = 1.842 × 1.652 × 1.538 × 1.458 = 6.757 The overall pressure ratio and the overall temperature rise are used in Equation 6.50 to determine the overall efficiency of the four-stage compressor. We thus obtain 1

ηc =



(6.757) 3.5 − 1 = 0.843 (465.08/540)

It is noted that the overall efficiency is a little lower than the stage efficiency, which is always the case for multistage compressors and is just the opposite for multistage turbines. For multistage machines with five or more stages and small stage pressure ratios, it is sometimes assumed that the stage efficiency is equal to the polytropic efficiency. In the present example, the stage pressure ratios are unusually high, and the compressor has only four stages; thus, the approximation would not be expected to apply. However, we will calculate the polytropic efficiency to make a comparison with the stage efficiency for the present case. First, we calculate the compressor outlet temperature in the following way: Tout = T01 + ∆T0 = 540 + 465.08 = 1005°R Then the polytropic efficiency is calculated from Equation 6.51 as ηp =

( γ − 1)ln R p ln (6.757) = = 0.8788 γ ln ( Tout /T01 ) 3.5 × ln (1005/540)

Note: Even in the present case, the polytropic and stage efficiencies are nearly equal.

Problems 6.1 An axial-flow fan is to be designed with a tip diameter of 9.5 in. and a hub-tip ratio of 0.5. Assuming that the fan is driven by a 1.5 hp motor and has an overall efficiency of 80%, determine the flow rate and desirable speed. The fan discharges air into the room through an exit area of 78.54 in2. 6.2 Construct velocity diagrams for the rotor inlet and outlet at the mean diameter for the fan considered in Problem 6.1. 6.3 Find the degree of reaction at the hub and tip of an axial-flow compressor stage of free-vortex design with a hub-tip ratio of 1 ⁄3 and a flow coefficient at the hub of 1.0. In and out of the stage, the flow is purely axial. At the hub, the flow is turned 30° in the rotor blades. The flow is modeled as incompressible. 6.4 Given the data of Problem 6.3, show that the degree of reaction increases from hub to tip and can be written as

R = 1−

E 2(Nr)2

Axial-Flow Pumps, Fans, and Compressors

139

6.5 Find the degree of reaction at the tip of an axial-flow compressor stage of free-vortex design with a hub-tip ratio 0.357, a flow coefficient at the hub of 1.25, hub solidity of 1.89, hub degree of reaction of −0.649, and E = Va2. The blade chord is constant from hub to tip. The flow is modeled as incompressible. Assume that V1 = V3. 6.6 Given the data of Problem 6.5, determine the relative and absolute flow angles at the tip and hub. Draw the velocity diagrams for the tip and hub. 6.7 Given the data of Problem 6.5, find the diffusion factor at the tip of the rotor and at the hub of the stator. 6.8 Find the diffusion factor at the hub and tip for both the rotor and the stator of an axial-flow compressor stage of free-vortex design with a hub-tip ratio of 0.385, a flow coefficient at the hub of 1.43, hub solidity of 1.695, hub degree of reaction of −0.599, and E = Va2 . Assume that V1 = V3. The flow is modeled as incompressible. 6.9 Given the data of Problem 6.8, determine the relative and absolute flow angles at the tip and hub. Draw the velocity diagrams for the tip and hub. 6.10 In an axial-flow compressor stage, R = 0.5, U = 1030 ft/s, Va = 400 ft/s, W1 = 800 ft/s, and W2 = 523 ft/s. The lift and drag coefficients are the same in the rotor and stator and are 1.4 and 0.08, respectively. CD includes all losses. Calculate stage pressure rise if the entering gas density is 0.0040 slug/ft3. 6.11 In the first stage of an axial-flow air compressor, mean values are U = 181m/s, Va = 151m/s, and R = 0.5 . Air in the room at 1.01 bar and 287 K is drawn into the compressor at the rate of 19.98 kg/s. Rotational speed is 9000 rpm, and the total temperature leaving the stage is 308 K. The work-done factor for the stage is 0.961. Find a. the relative air angles. b. the mean radius. c. the blade length. 6.12 Use the cascade results of Figure 6.14 to determine the velocity triangles (mean radius) and static pressure rise for a compressor stage having the following features: U = 1000 ft/s, ∆T0s = 54°F, β1 = 60°, ρ = 0.00237 slug/ft 3 , α = 12°, σ = 0.75, and h/C = 2. 6.13 Determine the mean radius, air angles, and blade length for the first stage of a  = 44 lbm/s, compressor having the following data: N = 150 rad/s, ∆T0s = 36°F, m Va = 492 ft/s, U = 590 ft/s, λ = 0.96, R = 0.5 (mean radius), p01 = 1atm, and T01 = 518°R. 6.14 A axial-flow air compressor stage has a solidity of 0.75, an aspect ratio of 2, an inlet relative air angle β1 = 60°, an outlet relative air angle β 2 = 42°, a flow coefficient of 0.4, inlet air density of 0.00237 slug/ft3, and a mean blade speed of 1050 ft/s. Use the velocity diagram at the mean radius to find a. CL. b. R. c. D . d. θ/C. e. ζ p . f. CD′. g. CD′′. h. CD′′′ . i. ηs .

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Logan’s Turbomachinery

6.15 Use the cascade results for the NACA 68 (18):10 airfoil with a solidity of 0.75, an inlet relative gas angle of 60°, and an outlet relative gas angle of 42°. The mean blade speed is 1000 ft/s, T01 = 520°R , the stage total temperature rise is 54°R, and p01 = 14.1psia. Take the gas to be air with a specific heat of c p = 5999 ft ⋅ lbf /(slug ⋅ °R) . Take V1 = V3 and Va as a constant. Take the aspect ratio to be 2. The ratio of tip clearance to blade height is 0.02. Find a. b. c. d. e. f. g. h. i. j. k. l. m. n.

E. Va . V1. V2 . the air density entering the rotor. the flow coefficient at the mean radius. the blade loading coefficient at the mean radius. R at the mean radius. δ = D/L based on the four losses. the pressure rise for the stage. the density at stage exit. the stage efficiency. the stage total pressure ratio from an incompressible model. the stage total pressure ratio from a compressible model.

6.16 Repeat Problem 6.15 without the use of cascade data. 6.17 Air at p01 = 14.7 psia and T01 = 519°R enters a three-stage compressor with a velocity of 350 ft/s. There are no inlet guide vanes, and the axial component Va remains constant through the compressor. In each stage, the rotor turning angle is 25°. The annular flow passages are shaped so that the mean blade radius is everywhere 9 in. The rotor speed is 9000 rpm. The ratio of specific heats is 1.4, and the polytropic efficiency is constant at 0.90. The blade height at the inlet is 2 in. Draw the velocity diagram and calculate a. the work per unit mass for each stage in ft ∙ lbf/slug. b. the mass flow rate of air in lbm/s. c. the power to run the compressor. d. the stage temperature ratios. e. the overall compressor pressure ratio. f. the blade height at the exit from the third stage. 6.18 Air enters an axial-flow compressor stage axially (no inlet guide vanes) at p01 = 14.696 psia and T01 = 60°F with V1 = 490 ft/s. The air is turned 30° at the mean radius by the blades, and the outlet guide vanes turn the air to the axial direction— i.e., V1 = V3 = Va. The tip of the rotor blade has a radius of 12 in., and the blade length is 2 in. Rotational speed is 6000 rpm, and the stage efficiency is 0.90. Find a. the mean radius of blade. b. the mean blade speed. c. the relative air angles.

Axial-Flow Pumps, Fans, and Compressors

141

d. the absolute air angles. e. the energy transfer. f. the mass flow rate of air. g. the power required to drive the stage. 6.19 The first stage of an axial-flow compressor receives air from the atmosphere that is at 14.7 psia and 520°R when at rest. At the mean radius, the inlet guide vanes turn the air 30° from the axial direction; moreover, the moving vanes reduce the relative air angle by 30°, while the mean blade speed is 1000 ft/s. The degree of reaction is 0.5, and the stage efficiency is 0.90. The mean radius is 10 in., and the blade length is 2 in. Neglect the total pressure loss in the inlet guide vanes. Find a. the rotational speed. b. the relative air angles. c. the flow coefficient. d. the energy transfer. e. the mass flow rate. f. the power required to operate the stage. g. the total-to-total pressure ratio for the stage. 6.20 Each stage of a three-stage axial-flow air compressor has the same velocity diagram, the same stage efficiency, the same mean radius, and the same rotational speed as in Problem 6.19. The same first-stage dimensions and inlet air conditions are also applicable. Neglect the total pressure loss in the inlet guide vanes. Determine a. the overall ratio of total pressure. b. the power to drive the compressor. c. the total temperature at the compressor discharge. 6.21 An axial-flow compressor is to be designed to handle 6.6 lbm of air per second, which is drawn from an ambient state of 14.7 psia and 520°R. The overall ratio of total pressure is to be 4, and the total temperature rise per stage is to be 41.5°R, while the rotational speed is 20,000 rpm. Choose a constant mean radius of 4.5 in. Assume a polytropic efficiency of 0.88. For all stages, use the optimum values of the flow coefficient and degree of reaction: ϕ = 0.5 and R = 0.5 , respectively. Determine a. the number of stages required. b. the length of the first-stage rotor blades. c. the length of the last-stage rotor blades. 6.22 Derive Equation 6.6. 6.23 Show that the optimum degree of reaction for an axial-flow compressor stage is ½. Hint: Find the minimum value of the ratio of the rate of loss of mechanical energy, E L = Wm D R + Vm D S , to the rate of energy input given by E = U ( L R cos β m + D R sin β m ). Assume that D R sin β m p1. Thus, even in a constant-area duct, we find that the pressure recovery occurs when a nonuniform velocity profile diffuses into a uniform one. From the foregoing analysis, we also infer that, with equal average velocity, the linear momentum associated with a nonuniform velocity profile is always higher than that with a uniform one; see Sultanian (2015) for a discussion of the momentum correction factor used in the control volume analysis of the linear momentum equation. 12.3.1.3  Loss of Total Pressure In an incompressible flow, the total pressure at any point is computed by the equation



p0 = p +

ρV 2 2

(12.4)

where the dynamic pressure ρV 2/2 is simply the product of the specific kinetic energy and density, which remains constant.

220

Logan’s Turbomachinery

At section 1 of the duct flow shown in Figure 12.5, we have a nonuniform velocity profile, which will yield a nonuniform distribution of the specific kinetic energy over the section. However, because the streamlines are parallel, the static pressure is uniform over the section. So, in order to compute an average total pressure at section 1, we must first compute the section-average specific kinetic energy. Because the kinetic energy is a scalar quantity, we should use mass-weighted averaging for it. Accordingly, Equation 12.4 at section 1 becomes p01 = p1 + ρ



V2 2

(12.5)

where

∫

V 2 (ρVdA )

∫

V 3 (1 + ε)3dA



V2 1 = 2 2



V2 1 = 2 2



V2 V2 V2 = + 2 2 2



V2 V2 V2 = + 2 2 2



V 2 V 2 3V 2 2 V 2 3 = + ε + ε 2 2 2 2

(1)

(1)

ρVA

VA

∫ ( 3ε

2

(1)

∫

(1)

)

+ ε 3 dA

A

3ε 2dA A

V2 + 2

∫

(1)

ε 3dA A (12.6)

where ε 2 is given by Equation 12.3 and ε 3 is given by ε3 =



1 A

∫

(1)

ε 3dA



(12.7)

Substituting Equation 12.6 into Equation 12.5 yields



p01 = p1 +

ρV 2 3ρV 2 2 ρV 2 3 + ε + ε 2 2 2

(12.8)

ρV 2 2

(12.9)

At section 2, Equation 2.4 yields

p02 = p2 +

From Equations 12.8 and 12.9, we can express the loss in total pressure between sections 1 and 2 as

p01 − p02 = p1 − p2 +

3ρV 2 2 ρV 2 3 ε + ε 2 2

221

Gas-Turbine Exhaust Diffusers

which after substitution for p2 from Equation 12.2 yields

p01 − p02 = −ρV 2 ε 2 + p01 − p02 =

3ρV 2 2 ρV 2 3 ε + ε 2 2

(

)

ρV 2 2 3 ε +ε 2

(12.10)

Because ε is a fraction, ε 3 is typically much less than ε 2 , which is a positive definite quantity. Thus, in the duct flow shown in Figure 12.5, the downstream static pressure recovery given by Equation 12.2 is accompanied by a loss in total pressure given by Equation 12.10, even in the absence of wall friction. Note that this loss in total pressure results from the entropy production in the shear layer mixing within the nonuniform velocity profile, which becomes uniform at section 2. In fact, for ε 3 0.3). A good understanding of the flow physics of a gas-turbine exhaust diffuser holds the key to its world-class design and future upgrades. 3-D CFD is a powerful diagnostic tool in diffuser design. It, however, generally underpredicts loss in total pressure and overpredicts recovery in static pressure. Diffuser C p calculation using the mass-weighted averaging of a nonuniform total pressure distribution at the inlet is not physics based. The axial stream thrust coefficient is a physics-based and improved measure of diffuser performance. The higher the axial thrust generated by an exhaust diffuser, the higher its aerodynamic performance. Six simple design rules introduced in this chapter should be used for the preliminary design of a gas-turbine exhaust diffuser. The entropy map generated from a 3-D CFD analysis (see Sultanian, 2015) in an exhaust diffuser clearly delineates the areas of excessive entropy generation. Eliminating these areas will result in improved diffuser performance. Scaled-model testing of an exhaust diffuser (at less than the design Mach number) is the quickest and cheapest way to qualify a world-class diffuser design.

References Lipstein, N.J. 1962. Low velocity sudden expansion pipe flow. ASHRAE Journal 4: 43–47. Sultanian, B.K. 1984. Numerical modeling of turbulent swirling flow downstream of an abrupt pipe expansion. PhD diss., Arizona State University. Sultanian, B.K. 2015. Fluid Mechanics: An Intermediate Approach. Boca Raton, FL: Taylor & Francis. Sultanian, B.K., S. Nagao, and T. Sakamoto. 1999. Experimental and three-dimensional CFD investigation in a gas turbine exhaust system. Journal of Engineering for Gas Turbines and Power 121: 364–374.

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Bibliography Japikse, D., and N.C. Baines. 2000. Turbomachinery Diffuser Design Technology, 2nd edition. White River Junction, VT: Concepts ETI, Inc. Wilson, D.G., and T. Korakianitis. 2014. The Design of High-Efficiency Turbomachinery and Gas Turbines, 2nd edition. Cambridge, MA: MIT Press.

Nomenclature A c p c v CD C p C pi C p max CV D FD h h 0 K  m p p 0 pamb ∆p0_loss s S Tx T T0 V V Vx x y β ε ηD γ ξ ρ ζ STx

Flow area Specific heat of gas at constant pressure Specific heat of gas at constant volume Drag coefficient Static pressure rise coefficient Static pressure rise coefficient of an ideal diffuser Maximum static pressure rise coefficient of a dump diffuser Control volume Diameter Drag force Specific enthalpy of the fluid Specific total (stagnation) enthalpy of the fluid Total pressure loss coefficient Mass flow rate Static pressure Total (stagnation) pressure Ambient pressure Loss in total pressure Specific entropy Component of stream thrust in the x direction Absolute static temperature Absolute total (stagnation) temperature Absolute flow velocity Section-average flow velocity Component of V in axial direction Cartesian coordinate x Cartesian coordinate y Smaller-to-larger diameter ratio in a sudden pipe expansion Velocity profile parameter Diffuser isentropic efficiency Ratio of specific heats = c p /c v Axial distance from the face of the expansion normalized by two step heights in a sudden pipe expansion = x /(D 2 − D1 ) Fluid density Axial stream thrust coefficient

13 Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design

13.1 Introduction Computational fluid dynamics (CFD) is the numerical prediction of the distributions of velocity, pressure, temperature, concentration, and other relevant properties throughout the computational domain of interest in the flow field. Since its infancy in 1960s, CFD technology has advanced significantly, benefiting from concurrent advances in computing power and technology. Once considered nearly impossible, numerical predictions of practical turbulent flows are now routinely performed using CFD, which remains the only method for computing time-dependent 3-D flows. Since the early 1980s, three key factors have accelerated the industrial use of CFD. The first is the emergence of multiple commercial CFD codes, which initially grew from the needs of specific industries such as the turbomachinery, chemical, and automobile industries. These codes eventually became general-purpose for usage in other industries. Second, the continuous reduction in design cost and cycle time drove industries to adopt CFD technology in their design practices to reduce their testing budget. Third, seeing the emerging industrial trend for the CFD technology, experimentalists and CFD technologists in universities started working as a team in their research activities, which further accelerated the growth in each area with significant benefits to industrial design and development. The rapid popularity and acceptance of commercial CFD codes gave rise to two ancillary software industries, one specializing in computer codes for generating both structured and unstructured CFD grids and the other in codes for common post-processing of results obtained from different leading CFD codes. All these computer codes have become so user friendly and solution robust that anyone trained in the mechanics of running these codes will be able to generate 3-D CFD solutions for many industrial design problems in weeks rather than in months, fully supporting today’s shrinking design cycle time. Notwithstanding the fact that all turbomachines feature complex 3-D flows in their primary flowpaths, the main thrust throughout this book has been on their 1-D aerothermodynamic analysis. These analyses are based on averaged flow parameters at the mean radii of the rotor/stator inlet, outlet, and other intermediate stations. They constitute the essential part of the turbomachinery preliminary design and provide approximate geometry and basic dimensions. The resulting velocity diagrams provide good overall understanding of aerodynamic work transfer in the machine. Changes in pressure and temperature throughout the machine at a given mass flow rate yield its key thermodynamic cycle performance parameters. 233

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One-dimensional flow analyses invariably use various empirical correlations to account for the 3-D nature of the flow field and the associated losses. These correlations are not universally applicable, and their use is limited to the range of their parent empirical data. Therefore, a detailed understanding of a turbomachine flow field, which is 3-D, turbulent, and unsteady, is essential for further improvement in the machine’s isentropic efficiency, which is generally around 90%. Toward this end, CFD has emerged as an indispensible tool in modern turbomachinery design, leveraging concurrent advances in computer technology in terms of speed and capacity. Although not an exact science and not free from empirical data, which are introduced through turbulence models, CFD has successfully reduced hitherto required design iterations, model fabrication, and laboratory tests. Today, many commercially available CFD software packages can be economically applied to turbomachine design and analysis. All in all, the state-of-the-art turbomachinery aerodynamic design system represents a unique blend of 1-D, 2-D, and 3-D CFD analysis methods, providing designers with a better physics-based understanding of their final design from computer-generated results. In this chapter, we first present a brief overview of the CFD technology, including the common form of the governing equations in tensor notations (see Appendix J for a review of tensor notations), the closure problem for the statistical modeling of a turbulent flow needing the use of a turbulence model such as the standard high-Reynolds-number, two-equation k-ε model, and boundary conditions. With the widespread use of various commercial CFD codes, two value-added sections discussed in this chapter cover the physics-based post-processing of computed CFD results and the generation of an entropy map from these results. Then we present a typical turbomachinery aerodynamic design system that leverages the CFD technology. Readers will find comprehensive details on turbomachinery preliminary design, detailed design, and CFD technology in the References and Bibliography sections of this chapter.

13.2  CFD Methodology The key objective of an industrial CFD analysis is to develop a detailed qualitative and quantitative understanding of the flow and heat transfer physics of a design toward its performance improvement. As shown in Figure 13.1, the flow and heat transfer physics

FIGURE 13.1 CFD methodology.

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of design is mathematically modeled using the conservation equations of mass, momentum, and energy. For the applicable boundary conditions relevant to the design physics, these equations are numerically solved by the CFD methods. The numerical results are then properly interpreted and used to make necessary changes in design for better performance. Until the mid-1980s, most industries were engaged in the development and maintenance of their own in-house CFD computer codes. Universities were also engaged in developing their own CFD codes for their CFD-related research activities, including more accurate and robust numerical methods and improved turbulence models. With the emergence of several general-purpose commercial codes, most CFD development activities in industries and universities dramatically subsided over the next 30 years. When one uses a leading commercial CFD code, the task of CFD investigation of an industrial design becomes routine, as depicted in Figure 13.2. A CFD engineer essentially focuses on generating a high-quality and high-fidelity grid in the calculation domain, setting up the correct boundary conditions, and selecting the appropriate physical models (e.g., a turbulence model consistent with the design physics). The commercial CFD code used by the engineer takes care of the rest. The post-processing of CFD results for their design applications remains a major nontrivial task. For details on numerical aspects of CFD, including derivations of the discretization equations and their iterative solution methods, interested engineers may want to study Patankar (1980) and Pletcher et al. (2012). Thompson et al. (1998) provide excellent coverage of the CFD grid generation technology. One may find comprehensive details of statistical turbulence models in Leschziner (2016). Durbin and Shih (2005) present a state-of-the-art review of turbulence modeling.

FIGURE 13.2 Using a commercial CFD code in design.

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13.3  The Common Form of Governing Conservation Equations Continuity Equation: ∂ρ ∂ + ρU j = 0 ∂t ∂x j

(



)

(13.1)

Momentum Equations: ∂ ∂ ∂  ∂U i  ρU i ) + ρU j U i = µ + S Ui ( ∂t ∂x j ∂x j  ∂x j 

(



)

(13.2)

Energy Equation:



∂ ∂ ∂  k ∂ h  ρU j h = (ρh) + + Sh ∂t ∂x j ∂x j  c P ∂x j 

(

)

(13.3)

Substituting h = c P T into Equation 13.3 yields



∂ ∂ ∂  k ∂T  S h ρU j T = (ρT) + + ∂t ∂x j ∂x j  c P ∂x j  c P

(

)

(13.4)

13.3.1  The Common Equation Form In tensor notations, the common form of the conservation equations suitable for a common numerical solution method can be expressed as



∂ ∂ ∂  ∂Φ  ρU j Φ = ΓΦ (ρΦ) + + SΦ  ∂t ∂x j ∂x j  ∂x j 

(

)

(13.5)

where Φ = 1, U i , h, and T yield, respectively, Equations 13.1, 13.2, 13.3, and 13.4. In Equation 13.5, each term is interpreted as follows:







∂ (ρΦ) ≡ Transient term ∂t ∂ (ρU j Φ) ≡ Convection term ∂x j ∂  ∂Φ  ΓΦ ≡ Diffusion term ∂x j  ∂x j  S Φ ≡ Source term

Note that Equation 13.5 is valid whether the flow is laminar or turbulent, incompressible or compressible, and steady or unsteady.

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13.4  Turbulence Modeling There are two ways of numerically predicting turbulent flows: one by simulation (using either direct numerical simulation [DNS] or large eddy simulation [LES]) and the other by statistical modeling. The first is too expensive and time consuming for most practical designs, and the second is not universally accurate and reliable. Despite these difficulties, CFD remains the method for microanalysis of a 3-D complex turbulent flow. For statistical modeling of a turbulent flow, we decompose all its randomly varying properties into their statistically average values and their fluctuating parts. One such decomposition for U = U + u(t) is shown in Figure 13.3. In the top velocity plot, the mean velocity U is time independent, and the flow is considered to be stationary in the mean. In the bottom plot, the mean velocity obtained from ensemble averaging varies with time. 13.4.1  Reynolds Equations: The Closure Problem 13.4.1.1  Reynolds Averaging Let us decompose U i (t), p(t), and the general flow property Φ(t), respectively, as U i = U i + u i, p = p + p ′, and Φ = Φ + φ. The time-averaging (also called Reynolds averaging) of these quantities yields their mean values:



FIGURE 13.3 Velocity at a point in a turbulent flow.

Ui =

1 t2 − t1

p=

1 t2 − t1

Φ=

1 t2 − t1

∫

t2

∫

t2

∫

t2

t1

t1

t1

U i dt pdt Φ dt

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13.4.1.2  The Closure Problem We restrict the rest of our discussion of turbulence modeling to Navier-Stokes equations governing statistically stationary incompressible flows. The Reynolds averaging of Equation 13.2 with the source term S Ui replaced by the pressure gradient term and the bars on the time-averaged quantities U i and p dropped yields

(

)

∂p ∂ ∂ ∂  ∂U i  ρU j U i + ρ ui u j = − + µ ∂x j ∂x j ∂x i ∂x j  ∂x j 

(



(

)

)

 ∂p ∂ ∂  ∂U i ρU j U i = − + µ − ρ u jui   ∂x j ∂x i ∂x j  ∂x j 

(



)

(13.6)

where U i and u i are the mean and fluctuating parts, respectively, of the local instantaneous velocity and −ρ u j u i are the Reynolds stresses that result from the time-averaging of the nonlinear convection terms in the Navier-Stokes equations. Clearly, additional equations are required to determine these stresses and thereby have a closed system—the closure problem. This is the task of turbulence modeling. Thus, the turbulence models essentially model the Reynolds stresses in terms of the mean flow quantities. 13.4.1.3  Boussinesq Hypothesis According to Boussinesq hypothesis, which invokes the gradient transport model as in a laminar flow, the Reynolds stresses are related to the mean velocity gradients by the following equation:  ∂U i ∂U j  2 −ρ u j u i = µ t  +  − ρkδ ij  ∂x j ∂x i  3



(13.7)

where µ t ≡ Turbulent viscosity k ≡ Turbulent kinetic energy  1 for i = j δ ij ≡ Kronecker delta =   0 for i ≠ j Note that the turbulent viscosity used in Equation 13.7, unlike a laminar (molecular) viscosity, is the property not of a fluid but of a flow. However, by analogy with the molecular viscosity used in a laminar flow, we write where

µ t ∼ ρυ

(13.8)

υ ≡ Turbulent velocity scale  ≡ Turbulent length scale Thus, according to Equation 13.8, the challenge of using turbulence modeling to solve the second-order closure problem boils down to finding the turbulent velocity and length scales.

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13.4.2  High-Reynolds-Number Two-Equation k-ε Model The conventional high-Reynolds-number two-equation k-ε model embodies the Boussinesq eddy-viscosity hypothesis, which relates the Reynolds stresses to the mean velocity gradients. This model has been in widespread use for more than four decades and is generally now considered the starting point for most complex shear flow predictions in industrial design applications. Even when using a more advanced turbulence model in design, the baseline predictions are often carried out with this model. The turbulent velocity scale, turbulent length scale, and eddy viscosity in this model are computed using the following equations: Turbulent velocity scale: υ = k

1

2

3

2 Turbulent length scale:  = k ε

µ t = Cµ ρ



k2 ε

(13.9)

where ε ≡ Dissipation rate of turbulent kinetic energy (k) Cµ ≡ Proportionality constant = 0.09 Equation 13.9 is based on the Kolmogorov-Prandtl relation (see Launder and Spalding, 1974) and on the fact that, in a region of a high turbulent Reynolds number, the dissipation ε is essentially controlled by large-scale turbulent motions through an energy cascade. The transport equation for k is given as



∂ ∂  µ t ∂k  ρU j k = + ρ ( Pk − ε ) ∂x j ∂x j  σ k ∂x j 

(

)

(13.10)

where σ k (typically assigned a value 1.0) is the turbulent Prandtl number for k and Pk is the production rate of k given by the equation



 ∂U i ∂U j  ∂U i + Pk = µ t    ∂x j ∂x i  ∂x j

(13.11)

Although a transport equation for ε can be derived, the resulting form needs several modeling assumptions. In view of this, the practice has been to use an ε equation patterned along the lines of the k equation, taking the form



∂ ∂  µ t ∂ε  ε (ρU j ε) = + ρ (Cε 1 Pk − Cε 2 ε) ∂x j ∂x j  σ ε ∂x j  k

(13.12)

where σ ε is the turbulent Prandtl number for ε and Cε 1 Cε 2 are additional model constants. Flow predictions are greatly influenced by the choice of constants that belong to a turbulence model. Rather than allowing them to be used arbitrarily to fit data, they are selected with hopes of having universality. For the k-ε turbulence model, the model constants used in most industrial applications are those of Launder and Spalding (1974). These constants

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are Cµ = 0.09, Cε 1 = 1.44, Cε 2 = 1.92, σ k = 1.00, and σ ε = 1.30. Two well-known shortcomings of this choice of model constants are overprediction of the growth of an axisymmetric free jet and underprediction of the reattachment length in a sudden pipe expansion, as discussed by Sultanian at al. (1987). Note that Equations 13.10 and 13.12 are both expressed in the common form given by Equation 13.5 so as to leverage the common methods used for discretization of the partial differential equations and their numerical solution.

13.5  Boundary Conditions All solutions of a given set of governing equations for the primitive variables and the turbulence model variables are subject to the specified boundary conditions at inflow, outflow, and wall boundaries. Often in a design environment, we may not know the detailed boundary conditions needed for a high-fidelity CFD solution. In such cases, it behooves us to perform sensitivity analyses to understand the variations in the computed CFD results due to uncertainties in critical boundary conditions. In the following discussion, we will consider the high-Reynolds-number k-ε turbulence model widely used as the baseline turbulence model in most industrial CFD applications. 13.5.1  Inlet and Outlet Boundary Conditions As a matter of CFD best practice used in industrial design, inlets and outlets are assigned to the CFD calculation domain where the flow field is expected to be parabolic—that is, free from any reverse flow. At times, the calculation domain is modified with artificial extensions to achieve desirable inflow and outflow boundaries. At inlets, we specify uniform or nonuniform profiles of all dependent variables of the mean flow either from available measurements or from other related analyses. For the k-ε turbulence model, assuming isotropic turbulence at the inlet, k in is computed by the equation k in = 1.5(Tu)2 U 2in



(13.13)

where Tu is the average inlet turbulence intensity and U in is the mean inlet velocity. At each inlet, we specify ε in in one of two ways: Method 1: 3



3

ε in = Cµ4 k in2 /  m

(13.14)

ε in = ρCµ k 2in / µ t

(13.15)

Method 2:

In Equation 13.14,  m is the mixing length determined from the inlet dimensions. In Equation 13.15, µ t is the assumed inlet turbulent viscosity—say, a multiple of the fluid dynamic viscosity µ. In both equations, k in is determined from Equation 13.13. If an outlet is placed far enough downstream, a fully developed boundary condition may be specified. Otherwise, the specifications of all dependent variables at the outlet are required.

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13.5.2  Wall Boundary Conditions: The Wall-Function Treatment The near-wall region in a turbulent flow is characterized by steep gradients in mean flow variables and turbulence quantities. Due to the no-slip condition, the fluid in contact with the wall assumes the wall velocity. Both k and ε are zero at the wall. In order to directly incorporate these simple wall boundary conditions, the conservation equations must be integrated up to the wall. This requirement poses two main difficulties: first, the high-Reynoldsnumber k-ε is not valid in the region of a low turbulent Reynolds number that prevails near a wall, and, second, a very fine grid is required near the wall so that the assumption of a linear profile for each quantity between grid points is valid for a proper numerical integration. The use of a wall function overcomes both these difficulties, since it directly links the near-wall equilibrium region (characteristic of all turbulent wall boundary layers where the local production of turbulent kinetic energy balances its dissipation) with the wall. When a turbulent boundary layer separates from the wall, either under an adverse pressure gradient or due to a step change in wall geometry, a stalled region of flow recirculation occurs at the wall. In this region, turbulence energy production near the wall is negligible, and turbulence energy diffusion toward the wall nearly balances its local dissipation. In spite of some regions of local nonequilibrium, perhaps for reasons of simplicity or in the absence of better information, the wall-function approach, as recommended by Launder and Spalding (1974), is widely used in the simulation of most industrial flows. 13.5.2.1  Logarithmic Law of the Wall In a boundary layer, we define shear velocity U* as U* = τ w /ρ , where τ w is the wall shear stress. In terms of this shear velocity, we further define U + and y + as follows: U+ =



U U*

and

y+ =

yU* υ

Note that y + is a Reynolds number based on the shear velocity U* and the distance y from the wall. Plotted in terms of U + and y +, Figure 13.4 shows the overall structure of a flat plate turbulent boundary layer, which is devoid of influence from any stream-wise pressure gradient and streamline curvature. This boundary layer consists of two regions: 30 25

Inner region

20 Outer region

Viscous sublayer

U+ 15

Log-law zone

10 Buffer zone

5 0

1

10

100 y+

FIGURE 13.4 Variation of velocity in a flat plate turbulent boundary layer.

1000

10,000

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the inner region and the outer region, which interfaces with the nearly potential outer flow. For the wall-function treatment, the inner region, which is further divided into three zones, is of interest. As shown in the figure, the innermost zone in direct contact with the wall is the viscous sublayer, which in the old literature on fluid mechanics was also called the laminar sublayer. The log-law zone at its outer edge interfaces with the outer region and at its inner edge connects with the viscous sublayer through the buffer zone. The loglaw zone is considered somewhat universal in nature, the corresponding logarithmic law of the wall being given by

U + = 5.5 log 10 y + + 5.45 = 2.388 ln y + + 5.45 U+ =

1 ln (Ey + ) K

(13.16)

where K = 0.4187 and E = 9.793. 13.5.2.2  Modified Logarithmic Law for the Velocity Parallel to the Wall While solving for the velocity component parallel to the wall using a wall-function approach, one essentially applies the shear stress (or momentum flux) boundary condition for the near-wall control volume. Since the wall-function treatment in a turbulent flow CFD is targeted for all boundary layers, including those under nonzero stream-wise pressure gradients and with streamline curvature, the logarithmic law of the wall given by Equation 13.16 is modified using the turbulence structure parameter, which in a turbulent boundary layer in local equilibrium is given by (based on measurements) 1 − uv = Cµ2 = 0.3 k



(13.17)

The shear velocity can now be expressed as 1







U* = τ w /ρw = − uv = Cµ4 k

1

2

For the near-wall node shown in Figure 13.5a, we can write U+ = U+ =

( UP − U w ) U

*

=

1 ln Ey + K

( U P − U w ) U*ρ τw

(

=

)

1 ln Ey + K

(

)

FIGURE 13.5 Wall boundary conditions: (a) velocity parallel to wall, (b) k P and ε P, and (c) temperature.

(13.18)

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In the aforementioned equation, substituting for U* from Equation 13.18 and solving for τ w yield

( U P − U w ) K ρ Cµ k 1

τw =

(

ln Ey +



4

1

2

(13.19)

)



The method to obtain k P in Equation 13.19 is discussed in the following. 13.5.2.3  Specifications of kP and εP From an equilibrium boundary layer consideration, ε P is fixed at the near-wall node P shown in Figure 13.5b using the following relation: 3

εP =

3

Cµ4 k 2 U*3 = Ky P Ky P

(13.20) To obtain k P, the k equation is solved assuming k = 0 at the wall, and the dissipation source term is approximated as

∫



yP

0

(

3

ε dy =

Cµ k P2 ln Ey P+

)

K

(13.21)

13.5.2.4 Temperature In the wall-function approach for the energy equation, using the temperatures at the wall and point P (see Figure 13.5c), the wall heat flux is calculated by the equation 1

1

qw =

(Tw − TP )ρc P Cµ4 k P2 Prt U +P + Pf

(

)

(13.22)

where Prt ≡ Turbulent Prandtl number ≈ 0.9 Pf ≡ Jaytilleke’s P-function given by



 Pr  3 4   Pr    − Pf = 9.24  1  1 + 0.28exp  − 0.007    Prt   Prt     

(13.23)

13.5.3  Alternative Near-Wall Treatments The wall-function treatment discussed above has its limitations. The main drawback of the approach is that it is strictly valid for wall boundary layers that satisfy the local equilibrium conditions; that is, the production rate of turbulent kinetic energy is equal to its dissipation rate. Complex geometries associated with industrial flow and heat transfer situations do not generally meet the restrictions of a wall-function treatment, whose

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continued use is primarily driven by a reduced model size as the first near-wall is placed in the y + range from 40 to 400 (see the log-law zone in Figure 13.4). Note that, while using wall functions, any attempt to resolve the near-wall region with lower y + values for the node next to the wall will be counterproductive. To overcome the limitations of wall functions, alternative near-wall treatments—for example, two- and three-layer models and low-Reynolds-number turbulence models— have been developed and used. In the latter, the governing equations for both mean flow variables and turbulence model variables are integrated right up to the wall with the first layer of near-wall nodes placed at a y + of around unity. Luo et al. (2012) recently compared the performance of several commonly used turbulence models using both wall functions and low Reynolds modeling at the wall. It is important to note that, in addition to packing many more grid points near a wall to properly resolve the boundary layer, the turbulence models themselves are modified to model the low-Reynolds-number region and the prevailing nonisotropic turbulence field near the wall. The standard k-ε turbulence model, discussed in Section 13.4.2, is not suitable as such to model the low-Reynolds-number wall region with a wall-integration grid. Turbulent heat transfer predictions even using a low-Reynolds-number turbulence model with wall integration remain a challenge due to inadequate modeling of the variable turbulent Prandtl number in the boundary layer. 13.5.4  Choice of a Turbulence Model The accuracy of a turbulent flow CFD for a given set of boundary conditions depends on three key factors: the quality of the grid, the accuracy of the numerical scheme, and the turbulence model used. Although most CFD engineers in an industry rightly focus on generating a high-quality grid in the computation domain and spend a good deal of time in obtaining a fully converged solution with a high-order numerical scheme, their choice of turbulence model is limited to the options available in the commercial CFD code used and the design cycle time available to conclude the analysis. If the CFD predictions fail to validate with the available test data, calculations are repeated with another turbulence model available in the CFD code. A lot of time and computing resources can be saved if the CFD analysts short-list the available turbulence models based on the flow and heat transfer physics of their design. Sultanian (2015) presents a few illustrative examples of choosing a turbulence model.

13.6  Physics-Based Post-processing of CFD Results CFD, being a microanalysis involving many small control volumes, generates a detailed description of velocity, pressure, temperature, and other flow and heat transfer properties in each control volume. The last step, and perhaps the most important and nontrivial step, in a 3-D CFD analysis is to post-process the computed results for design applications. A post-processing of CFD results is carried out with two initial objectives in mind. First, we use the results to get a clear qualitative understanding of the key features of the computed flow field by generating, for example, a plot of streamlines. Second, we obtain various integral quantities such as section-average values of the static pressure, total pressure,

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static temperature, total temperature, shear force on the bounding walls, and drag force on an internal design feature. For those who want to use the ultimate power of CFD analysis in a design, generating an entropy map from the computed results will clearly identify local areas of high entropy production. Improving these areas in design will certainly improve the component aerodynamic performance efficiency, which is difficult to achieve using other means. 13.6.1  Large Control Volume Analysis of CFD Results Figure 13.6 shows a large parallelepiped control volume drawn around a solid airfoil around which a detailed 3-D CFD analysis has been performed. Let us assume that the main design objective of the CFD analysis is to evaluate the drag force acting on the airfoil in the x direction. A direct method to determine the drag force is to integrate the x component of the forces from both pressure and shear stress distributions on the surface of the airfoil. We can also use an indirect method of first post-processing the CFD results to determine the x-momentum flux and surface force on each of the six faces of the large control volume and then performing the x-momentum control volume analysis, discussed in Chapter 2, to find the drag force on the airfoil. Except for some small numerical error introduced in the post-processing of the CFD results, both methods should essentially yield the same result for the drag force. The second method, however, can also yield sectionaverage values of the 3-D distribution of various quantities available from the CFD results. These section-average values are extremely useful in design applications to determine overall performance of a turbomachinery device. Chapter 12 presents a step-by-by method to post-process 3-D CFD results in an exhaust diffuser to obtain section-average values of static and total pressures at the inlet and outlet in order to compute the key design performance parameter—namely, the pressure recovery coefficient C p—for a gas-turbine exhaust diffuser. The methodology used for computing section-average total pressure avoids the difficulty of determining whether one should use area-weighted averaging or mass-weighted averaging.

FIGURE 13.6 A large control volume representation of 3-D CFD results.

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13.6.2  Entropy Map Generation In the foregoing, we discussed post-processing 3-D CFD results to obtain section-average values in order to determine overall equipment performance and efficiency. The knowledge of local areas of significant irreversibility present in the design, as obtained from the detailed CFD results, could help designers redesign these areas, leading to improved overall design. The conventional approach to assessing the performance of a flow system is to find the loss of total pressure in the system. This approach serves well when dealing with incompressible flows, since the total pressure in such flows represents mechanical energy per unit volume and its loss appears as an increase in fluid internal energy, loosely called thermal energy, which we generally do not track in these flows. For a compressible flow system, the loss in total pressure is used to denote performance loss under adiabatic conditions. Entropy, being a scalar quantity and grounded in the second law of thermodynamics, serves as a more useful quantity for tracking irreversibility in a flow system. In terms of changes in total pressure and total temperature between two points in a fluid flow system, we can compute entropy change by the equation  p02  T  s 2 − s1 = c P ln  02  − R   T01   p01 



(13.24)

For an adiabatic flow with T02 = T01, Equation 13.23 yields ∆s

−   p02 ∆p0 −1= =e  R −1 p01 p01

(13.25)

which relates the increase in entropy to the loss in total pressure between any two points. Using the static pressure and static temperature, the quantities directly available in CFD results, we can easily compute the entropy at any point in the flow by the equation s* =

s cP  T   p  − = ln R R  Tref   pref 

(13.26)

where we have arbitrarily assumed zero entropy at the reference pressure pref and reference temperature Tref. Thus, we can develop an entropy map of the entire flow system by post-processing the 3-D CFD results. Such a map provides an invaluable insight into the design space for local improvements of regions of excessive entropy production. Further details in this promising area of CFD application to design optimization are given in Naterer and Camberos (2008) and Sciubba (1997).

13.7  Turbomachinery Aerodynamic Design Process 13.7.1  3-D Flow Field All turbomachines feature complex turbulent shear flows, which are inherently unsteady in the mean with unsteady wake interactions due to the proximity of blade rows moving relative to each other. Additional complexities arise from the buildup of boundary layers on the casing and hub, the leakage flows around the blade tip and stator shroud, and the mixing

Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design

247

FIGURE 13.7 Blade-to-blade (S1) and hub-to-tip (S2) stream surfaces (after Wu, 1952).

of the low-momentum secondary flows with the high-momentum primary flow. During the pre-CFD era, Wu (1952) proposed to analyze the complex 3-D turbomachinery flow by analyzing the 2-D intersecting stream surfaces called S1 (blade-to-blade) and S2 (hub-totip), as shown in Figure 13.7. Even with Euler’s momentum equations, which neglect the effects of viscosity, obtaining the iteratively converged S1/S2 solutions was not trivial, and, therefore, the approach could not become a part of the prevailing turbomachinery design process. Currently, S1/S2 solutions are largely handled by fully 3-D CFD solutions with mixing planes. Some of the basic ideas of Wu’s proposal, however, remain integral to the stateof-the-art turbomachinery aerodynamic design, particularly in the detailed design phase, which involves 2-D axisymmetric throughflow analysis in the meridional (axial-radial) plane (S2,MEAN shown in Figure 13.7) and 2-D blade-to-blade analyses. Figure 13.8 shows a highly iterative process used for the aerodynamic design of turbomachines. Overall, the design is carried out in two phases—namely, preliminary design and detailed design. Each of these design phases is briefly discussed here. 13.7.2  Preliminary Design The goal of the preliminary design, which is essentially a meanline design, is to produce a safe, reliable, efficient, and economically competitive turbomachinery system. The meanline design, depicted in Figure 13.9a, involves solutions of 1-D forms of the conservation equations of mass, momentum, and energy at locations halfway between hub and casing, incorporating empirical correlations and prior product experience to account for various aerodynamic losses. This design phase ensures that the specified requirements of the engine cycle in terms of overall pressure ratio, mass flow rate, target efficiency, and so on are fully met by the final design. The key outputs of the preliminary design include the flowpath shape, number of stages and number of airfoils in each stage, meanline vector diagrams, and meanline 2-D geometry of airfoils. The initial configuration produced in a preliminary design is refined in the detailed design phase, which is discussed next.

248

Logan’s Turbomachinery

FIGURE 13.8 Turbomachinery aerodynamic design process.

13.7.3  Detailed Design As shown in Figure 13.8, with inputs from the preliminary design, the detailed design mainly consists of the throughflow design, which embodies a 2-D axisymmetric throughflow analysis along streamlines in the meridional (axial-radial) plane depicted in Figure 13.9b, and the airfoil design. Some of the design features evaluated in the detailed design phase include

FIGURE 13.9 (a) Meanline design, (b) throughflow analysis, and (c) blade-to-blade analysis.

Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design

249

blade lean, bow, scallop, sweep, fillet radius, midspan damper position, impeller splitter, blade position, and tip clearance. In addition to meeting the required aerodynamic efficiency, turbomachines are designed such that they have no significant aerodynamically excited vibrations, have good mechanical integrity and acceptable life, and can be manufactured with acceptable costs and turn-around times. However, these design aspects, shown within the dotted box in Figure 13.8, are handled by other specialists. 13.7.3.1  Throughflow Design The throughflow design uses 2-D axisymmetric analysis in the radial-axial (meridional) plane (S2 mean) and provides circumferentially averaged values of pressure, temperature, and density; velocity vector diagrams; and loading parameters. As shown in Figure 13.8, throughflow design is iterated with multiple 2-D blade-to-blade analyses at spanwise locations both to capture the 3-D variations and to factor in the smeared airfoil solidity (blockage) and blade forces in the throughflow analyses. Denton and Dawes (1999) point out that, even though 3-D calculations are used in the design procedure, the throughflow analysis remains an important tool for the design of turbomachines. Cumptsy (1989) and Lakshminarayana (1996) provide comprehensive details on the throughflow design. The most widely used method for throughflow design is the streamline curvature method (SCM), which is based on the radial equilibrium equation and a space-marching procedure, coupled to boundary layer calculations along the annulus endwalls. For turbomachinery design applications, Smith (1966) derived the most general form of the radial equilibrium equation, which is essentially the linear momentum equation in the radial direction. Novak (1967) and Marsh (1968) present the complete procedure and the related computer codes for performing a throughflow analysis. Spur (1980) presents a method that combines throughflow and blade-to-blade analyses to predict 3-D transonic flow in a turbomachine. 13.7.3.1.1  Streamline Curvature Method Although the actual flow field in the turbomachine is unsteady with rotating airfoils moving past stationary ones, SCM assumes that the flow is steady, adiabatic, compressible, and axisymmetric. This method solves the discrete equations of continuity, momentum, and energy along with the equation of state on the computational grid constructed in the meridional plane. Computation nodes are defined at the intersections of streamlines and quasi-orthogonal computation stations, which can be located at blade edges and inside blade passages. The throughflow analysis is essentially carried out in concentric annuli (streamtubes), which are bounded by meridional streamlines, thus allowing no spanwise mixing. Since the physical presence of the airfoils will change the slopes and curvatures of the streamlines in the axisymmetric throughflow analysis, this analysis is rerun but now with airfoil geometry present. The whole process shown in Figure 13.8 is then repeated in an iterative manner until the designer is satisfied with everything. In view of the multiple iterations needed to obtain a converged design solution, the throughflow analysis in an actual turbomachinery design is carried out using an in-house computer code. Appendix I presents a throughflow analysis using the simple radial equilibrium equation dp/dr = ρVu2/r in the annular gap with straight streamlines between the adjacent rows of airfoils. In this case, for axial-flow turbomachines, one can quickly obtain the converged solution using a spreadsheet—for example, MS Excel. To demonstrate this throughflow

250

Logan’s Turbomachinery

analysis technique and develop an overall understanding of the 3-D nature of turbomachinery flow and airfoil geometry, readers are encouraged to carry out the solutions for two cases, developed by Flack (1987), involving the axial-flow compressor design of a modern turbofan engine. 13.7.3.2  Airfoil Design As shown in Figure 13.8, the airfoil design is an iterative process intended to achieve the specified work transfer and vector diagrams obtained from the throughflow analysis, while ensuring the lowest aerodynamic loss at any given radius. Although the airfoils are not designed until the thermodynamic and aerodynamic properties are established, their presence is recognized by work input/output, flow input, flow-turning ability, blockage, aerodynamic loss, and force terms in the throughflow analysis. Both 2-D blade-to-blade analyses and 3-D flow analysis play a vital role in establishing the final design of 3-D airfoils. 13.7.3.2.1  2-D Blade-to-Blade Analysis This analysis at multiple radial locations is performed as a 2-D channel flow with periodic boundary conditions on lateral surfaces inclusive of airfoil pressure and suction surfaces. Considering the advances in grid generation technology, the overall capabilities of a commercial CFD code, and the available high-performance computing power, a 2-D blade-toblade analysis can be performed in minutes. The results of this analysis identify regions of excess entropy production in the flow, including the areas of flow separation, if present, on the airfoil surfaces. Aerodynamics engineers use these data to further improve the 2-D airfoil design, which was initially created from the velocity vectors obtained from the throughflow analysis. 13.7.3.2.2  3-D Flow Analysis When the 2-D airfoil sections are stitched (stacked) together, the airfoil becomes 3-D. The throughflow design does not permit spanwise mixing, as the streamtubes (concentric annuli) are bounded by meridional streamlines. For the design of a realistic airfoil, one must consider the spanwise (radial) mixing in turbomachinery flows. The following are various mechanisms that cause spanwise mixing in turbomachines: • Streamwise vorticity off the blades (stream surface twist), rendering the blade-toblade stream surfaces nonaxisymmetric • Secondary flows in the endwall boundary layers and in the blade boundary layers • Wake momentum transport downstream of the blade rows • Tip clearance flows with tip clearance vortices • Turbulent diffusion • Shock-boundary-layer interactions • Flow separations • Any other regions of high loss (entropy production) In the state-of-the art turbomachinery design system, all 3-D flow analyses (both steady and unsteady) are now carried out using a 3-D CFD tool—most likely commercial software with an advanced turbulence model and an accurate near-wall treatment.

Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design

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13.7.4  Role of 3-D CFD Based on the foregoing brief overview of the state-of-the-art turbomachinery design system, further improving machine efficiency, which is already around 90%, is inconceivable without the help of 3-D CFD technology, which quickly (with high-performance computing resources) provides a detailed distribution of all flow properties throughout the flowpath, highlighting areas in need of further design improvement. Some of the empirical correlations that have served past designs well are being replaced by the ability to compute flow fields with improved accuracy using CFD methodologies. Two-dimensional solutions of the flow through a blade row are gradually being supplemented, and even replaced, by 3-D ones. Denton (2014) presents various loss mechanisms found in turbomachines. These mechanisms are related to excess entropy production. As discussed in Section 13.6.2, 3-D CFD results are ideal for generating an entropy map of the complete flowpath toward making the design as isentropic as possible.

13.8  Concluding Remarks The state-of-the-art turbomachinery design system uses multifidelity modeling by iteratively comparing and converging the solutions from reduced-order 1-D and 2-D modeling with those from 3-D flow analyses. The reduced-order models are needed to ensure that the overall performance parameters of the final design meet and exceed the thermodynamic cycle requirements. For all 2-D and 3-D flow analyses for turbomachinery flowpath and airfoil designs, CFD has emerged as an indispensible tool. With the emergence of another disruptive technology—namely, additive manufacturing (3-D printing)—lighter and stronger airfoils of higher efficiency are on the horizon. In modern high-temperature gas turbines, the aerodynamic design of the turbine’s primary flowpath is also influenced by heat transfer and secondary flows used for cooling and sealing purposes. These effects are easily included in 3-D CFD analyses used in turbomachine design. In spite of impressive advances made in CFD technology over the last 30 years, aerodynamic efficiency predicted by a time-accurate 3-D CFD analysis using an advanced turbulence model could be more optimistic by 2%, as all the losses are not fully captured in the analysis. However, CFD plays a vital role in screening the good designs from the bad ones. Think about the cost and cycle-time implications of performing design optimization entirely by experimental means.

References Cumpsty, N.A. 1989. Compressor Aerodynamics. London: Longman. Denton, J.D. 2014. Loss mechanisms in turbomachines. Journal of Turbomachinery 115: 621–656. Denton, J.D., and W.N. Dawes. 1999. Computational fluid dynamics for turbomachinery design. Journal of Mechanical Engineering Science 213: 107–124. Durbin, P.A., and T.I-P. Shih. 2005. An overview of turbulence modeling. In Modeling and Simulation of Turbulent Heat Transfer (ed. B. Sunden and M. Faghri). Ashurst (Southampton), UK: WIT Press. Flack, R.D. 1987. Classroom analysis and design of axial flow compressors using a streamline analysis model. International Journal of Turbo and Jet Engines 4: 285–296.

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Lakshminarayana, B. 1996. Fluid Dynamics and Heat Transfer of Turbomachinery. New York: Wiley. Launder, B.E., and D.B. Spalding. 1974. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3: 269–289. Leschziner, M. 2016. Statistical Turbulence Modeling for Fluid Dynamics—Demystified: An Introductory Text for Graduate Engineering Students. London: Imperial College Press. Luo, J., E.H. Razinsky, and H-K. Moon. 2012. Three-dimensional RANS prediction of gas-side heat transfer coefficients on turbine blade and endwall. Journal of Turbomachinery 135(2): 1–11. Marsh, H. 1968. A Digital Computer Program for the Through-Flow Fluid Mechanics in an Arbitrary Turbomachine Using a Matrix Method (Reports and Memoranda No. 3509). London: Her Majesty’s Stationery Office. Naterer, G.F., and J.A. Camberos. 2008. Entropy-Based Design and Analysis of Fluids Engineering Systems. Boca Raton, FL: CRC Press. Novak, R.A. 1967. Streamline curvature computing procedures for fluid-flow problems. Journal of Engineering for Power 89(4): 478–490. Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Boca Raton, FL: CRC Press. Pletcher, R.H., J.C. Tannehill, and D.A. Anderson. 2012. Computational Fluid Mechanics and Heat Transfer, 3rd edition. Boca Raton, FL: CRC Press. Sciubba, E. 1997. Calculating entropy with CFD. Mechanical Engineering 119(10): 86–88. Smith, L.H. 1966. The radial-equilibrium equation of turbomachinery. Journal of Engineering for Power 88: 1–12. Spur, A. 1980. The prediction of 3D transonic flow in turbomachinery using a combined throughflow and blade-to-blade time marching method. International Journal of Heat and Fluid Flow 2(4): 189–199. Sultanian, B.K. 2015. Fluid Mechanics: An Intermediate Approach. Boca Raton, FL: Taylor & Francis. Sultanian, B.K., G.P. Neitzel, and D.E. Metzger. 1987. Turbulent flow prediction in a sudden axisymmetric expansion. In Turbulence Measurements and Flow Modeling (ed. C.J. Chen, L.D. Chen, and F.M. Holly). New York: Hemisphere. Thompson, J.F., B.K. Soni, and N.P. Weatherill (eds.). 1998. Handbook of Grid Generation. Boca Raton, FL: CRC Press. Wu, C.H. 1952. A General Theory of Three-Dimensional Flow in Subsonic und Supersonic Turbomachines of Axial-, Radial- and Mixed-Flow Types (NACA Tech. Note No. 2604). Washington, DC: National Advisory Committee for Aeronautics.

Bibliography Bradshaw, P. 1971. An Introduction to Turbulence. New York: Pergamon Press. Bradshaw, P. 1997. Understanding and prediction of turbulent flow—1996. International Journal of Heat and Fluid Flow 18: 45–54. Cummings, R.M., W.H. Mason, S.A. Morton, and D.R. McDaniel. 2015. Applied Computational Aerodynamics: A Modern Engineering Approach (Cambridge Aerospace Series). Cambridge: Cambridge University Press. Denton, J.D. 1978. Throughflow calculations for transonic axial flow turbines. Journal of Engineering for Power 100(2): 212–221. Denton, J.D. 1992. The calculation of three-dimensional viscous flow through multistage turbomachines. Journal of Turbomachinery 114(1): 18–26. Durbin, P.A. 1991. Near-wall turbulence modeling without damping functions. Theoretical and Computational Fluid Dynamics 3: 1–13. Durbin, P.A. 1993. A Reynolds stress model for near-wall turbulence. Journal of Fluid Mechanics 249: 465–498.

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Durbin, P.A., and G. Medic. 2007. Fluid Dynamics with a Computational Perspective. Cambridge: Cambridge University Press. Engelman, M.S. 1993. CFD—An industrial perspective. In Incompressible Computational Fluid Dynamics (ed. M.D. Gunzburger and R.A. Nicolaides). Cambridge: Cambridge University Press. Ferziger, J.H., and M. Peric. 2013. Computational Methods for Fluid Dynamics, 3rd edition. New York: Springer. Flack, R.D. 2010. Fundamentals of Jet Propulsion with Applications (Cambridge Aerospace Series). Cambridge: Cambridge University Press. Horlock, J.H., and J.D. Denton. 2005. A review of some early design practice using computational fluid dynamics and a current perspective. Journal of Turbomachinery 127(1): 5–13. Launder, B.E., and D.B. Spalding. 1972. Mathematical Models of Turbulence. New York: Academic Press. Lewis, R.I. 1996. Turbomachinery Performance Analysis. New York: Wiley. Piomelli, U. 1993. Application of large eddy simulations in engineering: An overview. In Large Eddy Simulation of Complex Engineering and Geophysical Flows (ed. B. Galperin and S.A. Orszag). Cambridge: Cambridge University Press. Pope, S.B. 2000. Turbulent Flows. Cambridge: Cambridge University Press. Rodi, W. 1980. Turbulence Models and Their Applications in Hydraulics—A State of the Art Review. Delft, Netherlands: International Association for Hydraulic Research. Sultanian, B.K. 1984. Numerical modeling of turbulent swirling flow downstream of an abrupt pipe expansion. PhD diss., Arizona State University. Sultanian, B.K., S. Nagao, and T. Sakamoto. 1999. Experimental and three-dimensional CFD investigation in a gas turbine exhaust system. Journal of Engineering for Gas Turbines and Power 121: 364–374. Sultanian, B.K., and D.A. Nealy. 1987. Numerical modeling of heat transfer in the flow through a rotor cavity. In Heat Transfer in Gas Turbine Engines (vol. 87, HTD Series, ed. D.E. Metzger), 11–24. New York: American Society of Mechanical Engineers. Sultanian, B.K., G.P. Neitzel, and D.E. Metzger. 1987. Comment on the flow field in a suddenly enlarged combustion chamber. AIAA Journal 25(6): 893–895. Tucker, P.G. 2016. Advanced Computational Fluid and Aerodynamics. Cambridge: Cambridge University Press.

Nomenclature A c p Cµ Cε 1, Cε 2 C p E h k  k k P k in K   m

Section area Specific heat at constant pressure Proportionality constant in the equation to define turbulent viscosity using k and ε Model constants in the ε equation Pressure recovery coefficient Constant in the logarithmic law of the wall Specific enthalpy Turbulent kinetic energy Thermal conductivity Turbulent kinetic energy at a near-wall node Mean turbulent kinetic energy at inlet Von Karman constant Turbulent length scale Inlet mixing length

254

 m p p ′ p0 pref p P Pf Pk Pr Prt q w R s s* S  S t T TP Tw Tu T01 T02 Tref u u i u j U U + U * U i U j U i U in U P U w x i x j y + Γ Φ δ ij ε ε P ε in κ µ µ t

Logan’s Turbomachinery

Mass flow rate Static pressure Fluctuating part of p Total pressure Reference pressure Time-average value of p(t) Near-wall point Jaytilleke’s P-function Production rate of k Prandtl number Turbulent Prandtl number Wall heat flux Gas constant Specific entropy Dimensionless specific entropy Source term in the common-form transport equation Source terms other than pressure gradient terms in the momentum equation Time Temperature Temperature at the near-wall node P Wall temperature Turbulence intensity Total temperature at location 1 Total temperature at location 2 Reference temperature Fluctuating part of U Fluctuating velocities in tensor notation Fluctuating velocities in tensor notation Velocity component along x direction Dimensionless velocity in a wall boundary layer Shear velocity Velocities in tensor notation Velocities in tensor notation Time-average value of U i (t) Mean inlet velocity Velocity at the near-wall node P Velocity at the wall (start of the linear log-law zone) Coordinates in tensor notation Coordinates in tensor notation Dimensionless transverse distance in a wall boundary layer Diffusivity for Φ Kronecker delta = 1 for i = j and = 0 for i ≠ j Dissipation rate of k Value of ε at a near-wall node Mean inlet value of ε Ratio of specific heats (κ = c p/c v ) Dynamic viscosity Turbulent (or eddy) viscosity

Computational Fluid Dynamics and Its Role in Turbomachinery Flowpath Design

v vt σ k σ ε ρ −ρu i u j ρ w τ w υ φ Φ Φ

255

Kinematic viscosity (v = µ/ρ) Kinematic eddy viscosity (v t = µ t /ρ) Turbulent Prandtl number for k Turbulent Prandtl number for ε Density Reynolds stresses Fluid density at wall temperature Wall shear stress Turbulent velocity scale Fluctuating part of Φ General transport variable of the common-form governing transport equation Time-average value of Φ(t)

Appendix A: Tables of Conversion Factors, Pump Efficiency, and Compressor Specific Speed TABLE A.1 Conversion Factors Quantity

Original Units 1.0 ft 2

Equivalent

2

Density

1.0 in 1.0 lbm/ft 3

0.0929 m 2 6.4516 × 10−4 m 2 16.018 kg/m 3

Energy

1.0 slug/ft 3 1.0 ft ⋅ lbf

515.379 kg/m 3 1.3558 J

Area

Flow Heat transfer coefficient Length

Mass Power Pressure

Rotational speed Specific energy Specific heat Specific speed Thermal conductivity Viscosity (dynamic) Viscosity (kinematic) Volume

1.0 Btu 1.0 ft 3 /s 1.0 Btu/(h ⋅ ft 2 ⋅ °F) 1.0 ft 1.0 in 1.0 mile 1.0 slug 1.0 hp 1.0 psi 1.0 Pa 1.0 atm 1.0 in water 1.0 rad/s 1.0 ft 2/s 2 1.0 Btu/lbm 1.0 Btu/(lbm ⋅ °F) 1.0 (unitless) 1.0 Btu/(h ⋅ ft ⋅ °F) 1.0 lbm /(ft ⋅ s) 1.0 centipoise 1.0 ft 2/s 1.0 ft 2/s 1.0 ft 3

1055.1 J 448.831 gpm 5.678 W/(m 2 ⋅ K) 0.3048 m 0.0254 m 1609.3 m 32.174 lbm 745.7 W 6894.8 Pa 1.0 N/m 2 101, 325 Pa 5.2 psf 9.5493 rpm 1.0 ft ⋅ lbf/slug 2326 J/kg 4188 J/(kg ⋅ K) 2732.9 rpm (gpm)1/2 /ft 3/4 1.731 W/(m ⋅ K) 1.488 (N ⋅ s)/m 2 0.001 (N ⋅ s)/m 2 0.0929 m 2/s 92.903 cs 0.02832 m 3

gpm = gallons/min; rpm = revolutions/min; cs = centistokes; psf = lbf /ft2; psi = lbf /in2.

257

258

Appendix A

TABLE A.2 Pump Efficiency as a Function of Specific Speed and Capacity Ns

Capacity Q (gpm)

Dimensional Specific Speed

30

50

100

200

300

500

1000

3000

10,000

 200  300  400  500  600  800 1000 1200 1400 1600 1800 2000 3000

0.21 0.31 0.38 0.43 0.47 0.52 0.55 0.57 0.59 0.60 0.60 0.60 0.60

0.24 0.34 0.42 0.47 0.50 0.56 0.60 0.62 0.63 0.64 0.65 0.65 0.65

0.29 0.40 0.48 0.54 0.57 0.63 0.66 0.68 0.70 0.70 0.71 0.71 0.71

– – – 0.59 0.62 0.67 0.70 0.72 0.73 0.74 0.74 0.75 0.75

– – – 0.61 0.65 0.70 0.73 0.75 0.76 0.77 0.77 0.77 0.77

– – – 0.64 0.67 0.72 0.76 0.78 0.79 0.80 0.80 0.81 0.81

– – – 0.67 0.70 0.75 0.78 0.81 0.82 0.83 0.83 0.84 0.84

– – – 0.70 0.73 0.78 0.82 0.84 0.85 0.86 0.86 0.87 0.87

– – – 0.73 0.76 0.82 0.85 0.87 0.87 0.88 0.88 0.89 0.89

Source: Constructed from graphs in I.J. Karassik, J.P. Messina, P. Cooper, and C.C. Heald. 2007, Pump Handbook. 4th edition. (New York: McGraw-Hill).

TABLE A.3 Compressor Specific Diameter as a Function of Specific Speed and Efficiency Compressor Efficiency

Dimensional Specific Speed

0.40

0.50

0.60

0.70

0.80

 50  60  65  70  80  85  90 100 110 120 130 140 150 160 170 180 190 200

2.42 1.94 1.77 1.66 1.44 1.36 1.30 1.16 1.07 1.00 0.91 0.87 0.83 0.80 0.80 0.80 0.79 0.79

2.65 2.14 1.92 1.82 1.55 1.48 1.39 1.25 1.14 1.06 1.00 0.96 0.94 0.91 0.91 0.91 0.91 0.91

2.91 2.26 2.02 1.89 1.63 1.53 1.43 1.29 1.17 1.10 1.03 1.00 1.00 1.00 1.00 1.00 1.01 –

– – 2.14 1.96 1.68 1.57 1.46 1.32 1.21 1.15 1.08 1.06 1.07 1.04 1.11 – – –

– – – – – 1.70 1.59 1.41 1.29 1.22 1.18 – – – – – – –

Source:  L.F. Scheel, 1972, Gas Machinery (Houston, TX: Gulf Publishing).

Appendix B: Derivation of Equation for Slip Coefficient The relative eddy (Figure B.1) at the impeller tip is located between adjacent blades, has diameter d, and rotates at the impeller rotation rate N but in the opposite direction to that of the impeller. The triangle ABC has one vertex at point A on the periphery of the impeller. The tangent AB to the blade forms the blade angle β2 with the tangent AC to the circular wheel periphery at point A. The length of the leg BC is approximated by the eddy diameter d, and the hypotenuse AC is taken as equal to the arc length AD. The slip Vu2 − Vu2′ or ΔWu is calculated as the speed Nd/2 of a point on the circumference of the relative eddy, in which only solid body rotation is present. The foregoing assumptions lead to the following formulations: sin β2 =    

BC dn B = AC 2 π r2

∆Wu n B Nr2 sin β2 = π   ∆Wu =     

π U 2 sin β2 nB

where n B is the number of impeller blades.

FIGURE B.1 Relative eddy at impeller tip.

259

260

Appendix B

FIGURE B.2 The effect of slip on the exit velocity diagram.

From Figure B.2, ΔWu = Vu2 − Vu2′. The slip coefficient is



µs =

Vu2′ ∆Wu = 1− Vu2 Vu2

and, finally,



µs = 1 −

π U 2 sin β2 n B Vu2

Appendix C: Formulation of Equation for Hydraulic Loss in Centrifugal Pumps with Backward-Curved Blades According to Csanady (1964), the hydraulic loss in a centrifugal pump can be represented by the relation gH L =



k d V22′ k r W22′ + 2 2

(C.1)

where V2′ and W2′ denote the absolute and relative velocities, as shown in the velocity diagram of Figure B.2, and k r and k d represent the loss coefficients for the rotor and diffuser, respectively. Referring to Figure B.2, it is seen that sin 2 β2′ =



Wm2 2 W22′

(C.2)

and that 2 W22′ = Wm2 + ( U 2 − Vu2′ )



2

(C.3)



Noting that the actual and ideal flow coefficients are equal, we can write ϕ 2 = ϕ 2′ =



Wm2 U2

(C.4)

Substitution of Equations C.3 and C.4 into Equation C.2 gives sin 2 β2′ =

ϕ 22 2 ϕ + (1 − Vu2′ /U 2 ) 2 2

(C.5)

The hydraulic efficiency is defined as



ηH =

H H − HL = in H in H in

(C.6)

It is clear from Equation C.6 that the maximum efficiency corresponds to the minimum value of H L /H in . Dividing Equation C.1 by gH in—by U 2 Vu2′,—we obtain



H L k d V22′ + k r W22′ = H in 2U 2 Vu2′

(C.7)

2 2 V22′ = Wm2 + Vu2 ′

(C.8)

From Figure B.2, we observe that

261

262

Appendix C

Substituting Equations C.2 and C.8 into Equation C.7 yields



2 2 2 2 + k d Vu2 H L k d Wm2 ′ + k r Wm2 /sin β2′ = H in 2U 2 Vu2′

(C.9)

Factoring, substituting Equation C.4 into Equation C.9, and rearranging give the final form:



2H L ϕ 22 = k d H in Vu2 ′ /U 2

 k r /k d  Vu2 ′  1 + sin 2 β 2  + U 2 ′



(C.10)

Using Csanady’s value of 1 3 for k r/k d allows evaluation of the parameter on the left-hand side of Equation C.10, which is to be minimized, for each pair of assumed values of ϕ 2 and Vu2′/U2. Determination of the optimum value of Vu2′/U2 is a straightforward task and is carried out in Problem 4.13. Optimum values of Vu2′/U2 are in the range between 0.5 and 0.55 for typical values of the flow coefficient. There is a corresponding optimum value of β2 for each flow coefficient; it varies from 5.7° for ϕ 2 = 0.05 to 23.5° at ϕ 2 = 0.20.

Reference Csanady, G. 1964. Theory of Turbomachines. New York: McGraw-Hill.

Appendix D: Viscous Effects on Pump Performance TABLE D.1 Viscosities of Liquids at 70°F Kinematic Viscosity (centistokes)

Liquid Ethylene glycol Water Gasoline Kerosene Engine oil Glycerin

17.8 1.0 0.67 2.69 165 648

TABLE D.2 Correction Factors for Oil with Kinematic Viscosity of 176 Centistokes Water Capacity     ( gpm)

Head (ft) Factor

15

100

600

 100

cE cQ cH

0.39 0.80 0.82

0.49 0.88 0.87

0.60 0.93 0.90

1000

cE cQ cH

0.60 0.93 0.90

0.67 0.96 0.92

0.72 0.97 0.93

5000

cE cQ cH

0.72 0.97 0.93

0.77 0.98 0.95

0.83 1.0 0.96

Source: I.J. Karassik, J.P. Messina, P. Cooper, and C.C. Heald 2007, Pump Handbook. 4th edition. (New York: McGraw-Hill).

263

Appendix E: Comparison of Formulas for Compressor Slip Coefficient, μs = Vu2′/Vu2 Originator

Reference

Balje

1

Busemann

6

Formula 0.75 π sin β2 nB 2.4 1− nB

1−

(β 2 = 90° only.) Eck

8

Pfleiderer

7

Stodola

2, 4, 5

Stanitz

2, 3, 4

  2 sin β 2 1 +    n 1 − D /D ( )  B 1S 2    8 ( k + 0.6 sin β 2 )   1 + 3n B   (0.55 < k < 0.68) 1− 1−

−1

−1

 π  sin β2   n B  1 − ϕ 2 cot β2 

 0.63 π  1   n B  1 − ϕ 2 cot β2 

References Balje, O.E. 1981. Turbomachines. New York: Wiley. Boyce, M.P. 1982. Gas Turbine Engineering Handbook. Houston, TX: Gulf Publishing. Saravanamutto, H.I.H., G.F.C. Rogers, H. Cohen, P.V. Straznicky, and A.C. Nix. 2017. Gas Turbine Theory, 7th edition. Harlow, UK: Pearson. Dixon, S.L. 1975. Fluid mechanics. In Thermodynamics of Turbomachinery. Oxford, UK: Pergamon Press. Ferguson, T.B. 1963. The Centrifugal Compressor Stage. London: Butterworths. Harman, R.T.C. 1981. Gas Turbine Engineering. New York: Wiley. Pfleiderer, C. 1949. Die Kreiselpumpen. Berlin: Springer-Verlag. Wilson, D.G. 1984. The Design of High-Efficiency Turbomachinery and Gas Turbines. Cambridge, MA: MIT Press.

265

Appendix F: Molecular Weight of Selected Gases TABLE F.1 Molecular Weight of Selected Gases Gas Air Oxygen Nitrogen Carbon monoxide Carbon dioxide Hydrogen Methane Ethane Propane n-Butane

Molecular Weight 28.966 32.00 28.016 28.010 44.010 2.016 16.042 30.068 44.094 58.120

Source:  L.F. Scheel, 1972, Gas Machinery (Houston, TX: Gulf Publishing).

267

Appendix G: Pressure and Temperature Changes in Isentropic Free and Vortex Vortices

G.1 Introduction Vortex flows, which are bulk circular motions of the entire flow field, are ubiquitous in turbomachines due to their rotation. Both streamlines and isobars (lines of constant static pressure) in a vortex are concentric circles. In contrast, for straight and parallel streamlines, the static pressure changes only along the streamlines and remains constant in the transverse direction. Free-vortex designs of gas-turbine airfoils are common in axial-flow turbomachinery. The flow in a rotating duct behaves like a forced vortex. The distinction between vortex and vorticity, which is another kinematic vector property of a flow field, is worth noting. The vorticity vector (ζ) at any point in a flow is obtained from the curl of the velocity vector (ζ = ∇ × V). The component of the vorticity vector along any coordinate direction represents twice the rate of local counterclockwise rotation of fluid particles about that coordinate direction. As the name implies, a free vortex is free from any external torque; as a result, its angular momentum about the axis of rotation remains constant. In such a vortex, tangential velocity at the inner radius is higher than that at the outer radius. Like a rotating solid body, a forced vortex has constant angular velocity everywhere. In this vortex, the tangential velocity increases linearly with radius. The vorticity remains constant everywhere in a forced vortex. In contrast, all the vorticity in a free vortex is confined to the singularity at the origin (r = 0). Away from the origin, the free-vortex flow has zero vorticity. Hence, it is called irrotational or potential flow; the latter name arises from the fact that the velocity field of a free vortex can be obtained from the gradient of a scalar potential—the curl of the gradient of any scalar is identically zero. Figure G.1a shows how a tiny floating rod moves with continuous change in its orientation (nonzero vorticity causes local rotation) on a circular streamline of a forced-vortex flow. Figure G.1b shows the circular streamline of a free vortex, in which the rod moves without changing its own orientation (zero vorticity does not cause local rotation).

G.2  Free and Forced Vortices Mathematically, we can express free and forced vortices as follows: Free vortex:

rVu = C1

(G.1)

Vu = ωf r

(G.2)

Forced vortex:

269

270

Appendix G

FIGURE G.1 Movement of a tiny floating rod in a vortex flow: (a) forced vortex and (b) free vortex.

where C1 is a constant and ω f is fluid angular velocity, which remains constant for a forced vortex. Figure G.2 shows the variation of tangential velocity with radius in both free and forced vortices. Sultanian (2015) discusses in detail free, forced, and Rankine vortices. We may combine Equations G.1 and G.2 into Equation G.3 to represent both free and forced vortices:

Vu r n = C2

(G.3)

where the exponent n = 1 and C2 = C1 for a free vortex and n = −1 and C2 = ω f for a forced vortex. Note that n has no physical significance. It is simply a mathematical artifice with binary values to combine Equations G.1 and G.2 into one equation.

G.3  Changes in Static Temperature and Pressure Change in entropy in a fluid flow is given by ds = c p



dp dT −R T p

dp dT −R ds = c p T p

FIGURE G.2 Variation of tangential velocity with radius in free and forced vortices.

(G.4)

271

Appendix G

For an isentropic (ds = 0) vortex flow, we can write dp c p dT = p R T



(G.5)

With no shear force and no change in the radial momentum flow across a differential control volume, the radial pressure gradient must balance the body force due to rotation. The resulting radial equilibrium equation (see Appendix I) becomes dp ρVu2 = dr r



(G.6)

Substituting for density in Equation G.6 using the equation of state p/ρ = RT, we obtain dp pVu2 = dr RTr



dp Vu2 dr = p RT r



which with the substitution for dp/p from Equation G.5 yields dT =

Vu2 dr cpr

Substituting for Vu from Equation G.3, we obtain dT =

C22 dr C22 dr = c p r 2n r c p r 2n + 1

(G.7)

Integrating Equation G.7 between two points in a vortex results in 2



∫ 1

C2 dT = 2 cp

T2 − T1 =

r2

∫r

dr 2n + 1

r1

2 2

(

−2 n −2 n C r2 − r1 c p (−2n)

)

(G.8)

Substituting Equation G.3 into Equation G.8 yields



  r 2n  2 T2 − T1 Vu1 = 1 −  1   T1 2nc p T1   r2  

(G.9)

Substituting c p/R = γ/( γ − 1) and M u1 = Vu1/ γ RT in Equation G.9 and rearranging the resulting expression, we finally obtain



2n T2 γ − 1  2   r1   M u1 1 −    = 1 +    2n  T1   r2  

(G.10)

272

Appendix G

which using the isentropic relation gives the pressure ratio as p2  T2  = p1  T1 



γ

γ −1

 ( γ − 1)   r  2n   = 1 + M 2u1 1 −  1    2n    r2   

γ

γ −1

(G.11)

G.3.1  A Simple Approach Using Equation G.3 in Equation G.8, we can alternatively write T2 − T1 = ∆T =

2 1  Vu2 V2  − u1   (− n)  2c p 2c p 

(G.12)

which is much more revealing and lends itself to a simpler approach to computing changes in static temperature and static pressure in free and forced vortices. For a free vortex (n = 1), Equation G.12 shows that the change in static temperature is equal to and opposite of the change in dynamic temperature resulting from the tangential velocity change. In contrast, these changes are equal for a forced vortex (n = −1). Let us suppose that we are given the vortex properties—namely, T1 , p1 , and Vu1 —at point 1, where r = r1. For each vortex, we present here a stepwise procedure to compute T2 and p2 at point 2, which corresponds to r = r2 . G.3.1.1  Forced Vortex (n = –1) 1. Compute Vu2 from the equation Vu2 =



r2 Vu1 r1

(G.13)

2. Use Equation G.12 to compute ( ∆T)forced vortex. ( ∆T)forced vortex



2 2 2 2   Vu2 Vu1 − Vu1  r2  = =   − 1  2c p 2c p  r1  

(G.14)

3. Compute T2. 4. Compute p2.



T2 = T1 + ( ∆T)forced vortex

T  p 2 = p1  2   T1 

γ

γ −1





(G.15)

(G.16)

G.3.1.2  Free Vortex (n = 1) 1. Compute Vu2 from the equation



Vu2 =

r1 Vu1 r2

(G.17)

273

Appendix G



2. Use Equation G.12 to compute ( ∆T)free vortex. ( ∆T)free vortex =



2 2 2 2   r  2 Vu1 V 2   r   V 2  r  − Vu2 = u1 1 −  1   = u1  2  − 1  1  2c p 2c p   r2   2c p  r1    r2 

(G.18)

2

r  ( ∆T)free vortex =  1  ( ∆T)forced vortex  r2 

where ( ∆T)forced vortex is obtained from Equation G.14. 3. Compute T2 .

T2 = T1 + ( ∆T)free vortex



(G.19)

4. Use Equation G.16 to compute p2.

G.4  Changes in Total Temperature and Pressure At any point in a vortex with axial, radial, and tangential velocity components, the total and static temperatures are related by the following equation: T0 = T +

Va2 V2 V2 V2 V2 + r + u =T+ m + u 2c p 2c p 2c p 2c p 2c p

(G.20) where Vm is the meridional velocity. If Vm does not change between two points in a vortex, the above equation yields  V2 V2  T02 − T01 = ( T2 − T1 ) +  u2 − u1   2c p 2c p 

(G.21) where the first parenthetical term on the right-hand side represents the change in static temperature and the second parenthetical term represents the change in dynamic temperature resulting from the change in the tangential velocity. As we noted in Section G.2, these two changes negate each other in an isentropic free vortex (n = 1), giving T02 = T01 and p02 = p01. Thus, both total temperature and total pressure remain constant in an isentropic free vortex. For a forced vortex (n = −1), as we noted in Section G.2, the change in static temperature equals the change in dynamic temperature between any two points. The change in total temperature between two radial locations is therefore twice the difference in static temperatures at these locations. Thus, for a forced vortex, we can write

T02 − T01 = ( ∆T0 )forced vortex = 2 ( T2 − T1 ) = 2( ∆T)forced vortex T02 = T01 + 2( ∆T)forced vortex

(G.22)

where ( ∆T)forced vortex is given by Equation G.14. From Equation G.20, we obtain T01 = T1 +

Vm2 V2 + u1 2c p 2c p

(G.23)

274

Appendix G

For an isentropic forced vortex, we first obtain p01 at point 1 from the equation

and then p02 from the equation



T  p01 = p1  01   T1 

T  p02 = p01  02   T01 

γ

γ −1

γ

γ −1



(G.24)

(G.25)

An alternate approach to computing the change in total temperature in an adiabatic forced vortex is to treat it as an “aerodynamic rotor” and apply Euler’s turbomachinery equation; assuming the fluid tangential velocity is equal to the rotor velocity at each location, this yields

2 2 h 02 − h 01 = Vu2 − Vu1

T02 − T01 =

2 2 Vu2 − Vu1 cp

(G.26)

   which can be verified to be identical to Equation G.22. Note that the forced vortex, viewed as an “aerodynamic rotor,” acts like a compressor for a radially outward flow ( r2 > r1 ) and like a turbine for a radially inward flow ( r1 > r2 ).

Reference Sultanian, B.K. 2015. Fluid Mechanics: An Intermediate Approach. Boca Raton, FL: Taylor & Francis.

Appendix H: Dimensionless Velocity Diagrams for Axial-Flow Compressors and Turbines

H.1 Introduction In this appendix, we derive equations that relate performance parameters—namely, the flow coefficient, blade loading coefficient, and degree of reaction of axial-flow compressors and turbines—in order to compute all parameters of the inlet and outlet velocity triangles (diagrams) along a streamline of constant radius and constant blade velocity. Normalizing all absolute and relative flow velocities by the constant blade velocity, we develop here a quick step-by-step method to draw dimensionless velocity diagrams directly from the knowledge of the three performance parameters. The resulting velocity diagram features blade inlet and outlet velocity triangles with a common apex. Once constructed, we can slide each velocity triangle along the tangential direction (horizontal) to obtain the composite inlet-outlet velocity diagram with a common base that corresponds to the dimensionless unit vector for the blade tangential velocity.

H.2  Performance Parameters Three key performance parameters of a turbomachinery stage are the flow coefficient (ϕ), blade loading coefficient (ψ), and degree of reaction or simply reaction (R). Here we will use the stage definitions for axial-flow compressors and turbines, as shown in Figure H.1. Note that, in both cases, the rotor inlet is designated by 1 and the outlet by 2. For axial-flow compressors and turbines with U1 = U 2 = U, the three performance parameters are defined as follows. H.2.1  Flow Coefficient The flow coefficient is defined as

ϕ=

Va U

(H.1)

∆h 0 U2

(H.2)

H.2.2  Loading Coefficient The loading coefficient is defined as

ψ=−

275

276

Appendix H

FIGURE H.1 (a) Axial-flow compressor stage and (b) axial-flow turbine stage.

which using Euler’s turbomachinery equation (see Chapter 2) can be expressed as ψ=−



( U 2 Vu2 − U1Vu1 ) U2

(H.3)



Since U 2 = U1 = U for an axial-flow turbomachine, Equation H.3 reduces to



ψ=−

( Vu2 − Vu1 ) = − ∆Vu U

(H.4)

U

which makes ψ negative for an axial-flow compressor and positive for an axial-flow turbine. Since Vu = Wu + U, we can rewrite Equation H.4 as

ψ=−

∆Vu ∆Wu ( Wu2 − Wu1 ) =− =− U U U

(H.5)

H.2.3  Stage Reaction The degree of reaction of a turbomachine is defined by the equation R=

∆h rotor ∆h rotor = ∆h stage ∆h stator + ∆h rotor

(H.6)

An intuitive understanding of reaction can be had by noting the changes in static pressure in the stator and rotor of a stage. For example, for the Pelton wheel shown in Figure H.2a, the entire change in static pressure occurs in the nozzle, and the static pressure remains constant in the bucket. According to Equation H.6, we have R = 0 or zero reaction in this case. For the flow situation shown in Figure H.2b, the change in static pressure occurs in both the nozzle and the blade passage. If these two changes are equal, this turbomachine will have R = 0.5 or 50% reaction. Finally, Figure H.2c shows a lawn sprinkler, where the entire change in static pressure occurs in each rotary arm; hence, it has R = 1.0 or 100% reaction. For the axial-flow compressor stage shown in Figure H.1a, we can write

R=

h2 − h1

( h3 − h2 ) + ( h2 − h1 )

(H.7)

277

Appendix H

FIGURE H.2 (a) Zero reaction (impulse turbine), (b) 50% reaction, and (c) 100% reaction.

Since the total (stagnation) enthalpy across the stator (adiabatic with no work transfer) remains constant, we obtain h 02 = h 03 h2 +



V22 V2 = h3 + 3 2 2

h3 − h2 =



(H.8)

V22 V32 − 2 2

From the constancy of rothalpy across the rotor (see Chapter 2), we write I1 = I 2



h1 +

W12 U12 W2 U2 − = h2 + 2 − 2 2 2 2 2

which for an axial-flow compressor with U1 = U 2 reduces to h2 − h1 =



W12 W22 − 2 2

(H.9)

Substituting Equations H.8 and H.9 into Equation H.7 yields R=

(

W12 − W22 V22 − V32 + W12 − W22

) (

)

which with equal absolute velocities at the compressor stage inlet and outlet ( V1 = V3 ) for a repeating stage reduces to R=

(

W12 − W22 V − V12 + W12 − W22 2 2

) (

)

(H.10)

Additionally, if we assume that the axial velocity remains constant at the rotor inlet and outlet ( Va1 = Va2 ), Equation H.10 becomes R=

(

2 2 − Wu2 Wu1 2 2 2 2 Vu2 − Vu1 + Wu1 − Wu2

) (

)

(H.11)

278

Appendix H

Substituting Vu1 = Wu1 + U and Vu2 = Wu2 + U into Equation H.11 yields R= R=

{( W

2 u2

) (

2 2 Wu1 − Wu2

2 + 2Wu2 U + U 2 − Wu1 + 2Wu1U + U 2

)} + ( W

2 u1

( Wu1 − Wu2 )( Wu1 + Wu2 ) 2U ( Wu2 − Wu1 )

R=−

2 − Wu2

) (H.12)

( Wu1 + Wu2 ) 2U



which using Wu1 = Vu1 − U and Wu2 = Vu2 − U can be expressed in terms of absolute tangential velocities at the rotor inlet and outlet as



R = 1−

( Vu1 + Vu2 ) 2U



(H.13)

Equations H.12 and H.13, which are derived here to compute the reaction for an axialflow compressor stage, shown in Figure H.1a, are also valid for an axial-flow turbine stage, shown in Figure H.1b.

H.3  Dimensionless Velocity Diagrams One can draw a composite inlet-outlet velocity diagram for axial-flow compressors and turbines with equal axial flow velocities in two ways: (1) using the blade tangential velocity U as the common base for the velocity triangles at the rotor inlet and outlet, as shown in Figure H.3a, and (2) using the point where the absolute and relative velocities join together as the common apex for the velocity triangles at the rotor inlet and outlet, as shown in Figure H.3b. In the first case, the distance between the peaks of the two triangles measures the magnitude of the loading coefficient times the blade tangential velocity. In the second case, the angle between the absolute velocities gives the flow turning angle ( α 1 − α 2 ) over the rotor blade.

FIGURE H.3 Composite inlet-outlet velocity diagrams: (a) common base and (b) common apex.

Appendix H

279

Figures H.4a and H.4b are the velocity diagrams for an axial-flow compressor and an axial-flow turbine, respectively, drawn with a common apex, where each velocity is made dimensionless by dividing it by the blade tangential velocity U. In these velocity diagrams, the dimensional blade velocity becomes unity. Each dimensionless velocity diagram features the three performance parameters—ϕ , ψ , and R—which we have presented in the foregoing. It is interesting to note that we obtain the velocity diagram of an axial-flow turbine from that of an axial-flow compressor, and vice versa, by simply exchanging the subscripts 1 and 2 of various quantities involved. This implies that the compressor outlet becomes the turbine inlet and the compressor inlet becomes the turbine outlet, both having the identical values of ϕ , ψ , and R.

FIGURE H.4 Dimensionless velocity diagrams showing performance parameters: (a) axial-flow compressor and (b) axialflow turbine.

280

Appendix H

H.3.1 Derivations of Equations to Compute Velocities and Angles of Dimensionless Velocity Diagram Before we present a stepwise method to quickly draw a dimensionless velocity diagram using the performance parameters ϕ , ψ , and R, let us first derive equations to compute the dimensionless absolute and relative velocities at the rotor inlet and outlet and their angles from the axial direction. We will carry out these derivations using the dimensionless velocity diagrams shown in Figure H.4. H.3.1.1  Absolute Velocity Angle at Rotor Inlet α1 From the inlet velocity triangle of Figure H.4a, we can write tan α 1 =



Vu1  Vu1  1 =  Va  U  ϕ

(H.14)

From the definition of the loading coefficient ψ given by Equation H.4, we obtain Vu2 V = −ψ + u1 U U



(H.15)

Substituting for Vu2 /U from Equation H.15 into Equation H.13, we obtain R = 1+



ψ Vu1 Vu1 ψ V − − = 1 + − u1 2 2U 2U 2 U

Vu1 ψ = −R+1 U 2



(H.16)

Substituting for Vu1/U from Equation H.16 into Equation H.14, we finally obtain tan α 1 =



ψ/2 + (1 − R) ϕ (H.17)

 ψ/2 + (1 − R)  α 1 = tan −1   ϕ       H.3.1.2  Absolute Velocity Angle at Rotor Outlet α2 From the outlet velocity triangle of Figure H.4a, we can write tan α 2 =



Vu2  Vu2  1 =  Va  U  ϕ

(H.18)

Equation H.4 yields

Vu1 V = ψ + u2 U U

(H.19)

281

Appendix H

Substituting for Vu1/U from Equation H.19 into Equation H.13, we obtain

R = 1−

ψ Vu2 Vu2 ψ V − − = 1 − − u2 2 2U 2U 2 U

Vu2 ψ = − + 1− R U 2

(H.20)

Substituting for Vu2 /U from Equation H.20 into Equation H.18, we finally obtain



tan α 2 = −

ψ/2 − (1 − R) ϕ

 (1 − R) − ψ/2  α 2 = tan   ϕ      

(H.21)

−1

H.3.1.3  Relative Velocity Angle at Rotor Inlet β1 From the inlet velocity triangle of Figure H.4a, we can write tan β1 =



Wu1  Wu1  1 =  U  ϕ Va

(H.22)

From the definition of the loading coefficient ψ given by Equation H.5, we obtain Wu2 W = −ψ + u1 U U



(H.23)

Substituting for Wu2 /U from Equation H.23 into Equation H.12, we obtain

R=

ψ Wu1 Wu1 ψ Wu1 − − = − 2 2U 2U 2 U

(H.24)

Wu1 ψ = −R U 2 Substituting for Wu1/U from Equation H.24 into Equation H.23, we finally obtain



tan β1 =

   

ψ/2 − R ϕ

 ψ/2 − R  β1 = tan −1   ϕ 

(H.25)

H.3.1.4  Relative Velocity Angle at Rotor Outlet β2 From the outlet velocity triangle of Figure H.4a, we can write

tan β 2 =

Wu2  Wu2  1 =  U  ϕ Va

(H.26)

282

Appendix H

Equation H.5 yields Wu1 W = ψ + u2 U U



(H.27)

Substituting for Wu1/U from Equation H.27 into Equation H.12, we obtain



R=−

ψ Wu2 Wu2 ψ W − − = − − u2 2 2U 2U 2 U

Wu2 ψ = − −R U 2

(H.28)

Substituting for Wu2 /U from Equation H.28 into Equation H.26, we finally obtain



tan β 2 = −

ψ/2 + R ϕ

  ψ/2 + R   β 2 = tan −1 −    ϕ   



(H.29)

H.3.1.5  Dimensionless Absolute Velocity at Rotor Inlet V1/U From the inlet velocity triangles for the axial-flow compressor and turbine shown in Figure H.4, we can write 2



2

2

 V1  =  Va  +  Vu1  = ϕ 2 +  Vu1          U U U  U 

2

which upon substituting for Vu1/U from Equation H.16 yields



V1 = ϕ 2 + (ψ/2 − R + 1)2 U

{

}

1

(H.30)

2



H.3.1.6  Dimensionless Absolute Velocity at Rotor Outlet V2/U From the outlet velocity triangles for the axial-flow compressor and turbine shown in Figure H.4, we can write 2



2

2

 V2  =  Va  +  Vu2  = ϕ 2 +  Vu2          U U U  U 

2

which upon substituting form Vu2 /U from Equation H.20 yields



V2 = ϕ 2 + (ψ/2 + R − 1)2 U

{

}

1

2



(H.31)

283

Appendix H

H.3.1.7  Dimensionless Relative Velocity at Rotor Inlet W1/U From the inlet velocity triangles for the axial-flow compressor and turbine shown in Figure H.4, we can write 2



2

2

 W1  =  Va  +  Wu1  = ϕ 2 +  Wu1          U U U  U 

2

which upon substituting for Wu1/U from Equation H.24 yields W1 = ϕ 2 + (ψ/2 − R)2 U

{



}

1

(H.32)

2



H.3.1.8  Dimensionless Relative Velocity at Rotor Outlet W2/U From the outlet velocity triangles for the axial-flow compressor and turbine shown in Figure H.4, we can write 2



2

2

 W2  =  Va  +  Wu2  = ϕ 2 +  Wu2          U  U U  U 

2

which upon substituting for Wu2 /U from Equation H.28 becomes



W2 = ϕ 2 + (ψ/2 + R)2 U

{

}

1

2



(H.33)

H.3.2 Using φ, ψ, and R to Quickly Draw a Dimensionless Velocity Diagram For drawing a dimensionless velocity diagram for axial-flow compressors and turbines using the performance parameters ϕ , ψ , and R, we present here a step-by-step procedure for the case of an axial-flow compressor with ϕ = 0.5, ψ = −0.5, and R = 0.5. Note that, in the dimensionless velocity diagram, the nondimensional blade tangential velocity (U/U) is always unity, which is presently represented by 10 cm for drawing purposes. H.3.2.1  Step 1: Flow Coefficient (φ = 0.5) As shown in Figure H.5, we first draw two dotted horizontal parallel lines that are separated by the given value of ϕ = 0.5, which scales to 5 cm. We also draw a dashed vertical line for the axial direction, intersecting both horizontal lines. The common apex of the velocity diagram will lie on the top horizontal line, and all the absolute and relative flow velocity vectors will end at the bottom horizontal line, which represents the tangential direction. H.3.2.2  Step 2: Stage Reaction (R = 0.5) We mark on the bottom horizontal line a point that is 5 cm, which corresponds to R = 0.5, from the dashed vertical line, as shown in Figure H.6. Connecting this point to the apex gives the mean relative flow velocity through the blade.

284

Appendix H

FIGURE H.5 Partially drawn dimensionless velocity diagram showing flow coefficient.

H.3.2.3 Step 3: Dimensionless Relative Flow Velocities W1/U and W2/U at Blade Inlet and Outlet On the bottom horizontal line, we mark two points, one on either side of the tip of the mean relative flow velocity, at a distance corresponding to half of the loading coefficient magnitude ( ϕ /2 = 0.25 ), as shown in Figure H.7. Connecting these points to the apex gives us the dimensionless relative flow velocities W1/U and W2 /U. Due to diffusion in a compressor rotor, we have W2 /U < W1/U . H.3.2.4 Step 4: Dimensionless Absolute Flow Velocities V1/U and V2/U at Blade Inlet and Outlet We now connect the tip of each dimensionless relative flow velocity vector on the bottom horizontal line with the tail of the dimensionless blade velocity (U/U = 1) vector. The line

FIGURE H.6 Partially drawn dimensionless velocity diagram showing flow coefficient, stage reaction, and mean relative flow velocity through blade.

Appendix H

285

FIGURE H.7 Partially drawn dimensionless velocity diagram showing flow coefficient, stage reaction, mean relative flow velocity through blade, and blade inlet and outlet relative flow velocities.

connecting the tip of this vector and the common apex gives the corresponding dimensionless absolute flow velocity, as shown in Figure H.8. H.3.2.5  Step 5: Absolute and Relative Flow Angles at Blade Inlet and Outlet Finally, we remove the extra lines and construction aids and mark the absolute and relative flow angles at the blade inlet and outlet, as shown in Figure H.9. Note that, according to the convention presented in Chapter 2, both the relative flow angles β1 and β 2 are negative in this case. For the velocity diagram shown in Figure H.8, we can scale the magnitudes of the dimensionless absolute and relative flow velocities at the blade inlet and outlet and measure the corresponding absolute and relative flow angles using a protractor. However, the

FIGURE H.8 Partially drawn dimensionless velocity diagram showing flow coefficient, stage reaction, mean relative flow velocity through blade, blade inlet and outlet relative flow velocities, and corresponding absolute flow velocities.

286

Appendix H

FIGURE H.9 Final dimensionless velocity diagram for the axial-flow compressor with the given performance parameters φ, ψ, and R.

equations developed in Section H.3.1 yield more accurate numerical values of these quantities. Using these equations, the following values are easily computed: Rotor inlet: V1 /U = 0.559 Rotor outlet: V2 /U = 0.901

W1 /U = 0.901 W2 /U = 0.559

α 1 = 26.57° α 2 = 56.31°

β1 = −56.31° β 2 = −26.57°

Since the given degree of reaction is 0.5, these computed values confirm that the velocity diagram is symmetric. Note that the velocity diagram for the axial-flow compressor, shown in Figure H.9, can be easily converted to represent that for an axial-flow turbine with identical performance parameters (ϕ = 0.5, ψ = 0.5, and R = 0.5 ) by simply switching subscripts 1 and 2 for all the quantities involved.

Appendix I: Throughflow Design with Simple Radial Equilibrium Equation

I.1 Introduction In Chapter 13, we briefly discussed the throughflow design using the streamline curvature method. This method, which is typically coded in a computerized design tool, uses the complete radial equilibrium equation—for example, that derived by Smith (1966)—which includes the forces from the rotor and stator airfoils present in the flow path. These airfoils are smeared circumferentially to maintain the 2-D axisymmetric nature of the throughflow analysis. In this appendix, we present the derivation of the simple radial equilibrium equation. This equation can be used in rotor-stator and interstage gaps for carrying out a simplified throughflow design and developing a good understanding of spanwise variations of all flow properties. For instructional reinforcement, we also present here two project cases, which readers should be able to carry out using MS Excel or Matlab. These project cases are adapted from Flack (1987).

I.2  Radial Equilibrium Equation For deriving the radial equilibrium equation in the gap between adjacent airfoil rows, let us consider a differential fluid control volume with center-point static pressure p, axial velocity Vx , radial velocity Vr , and tangential velocity Vu , as shown in Figure I.1. In the meridional plane, shown in Figure I.1b, the velocity Vs (resultant of Vx and Vr ) is tangent to the streamline, making an angle α s with the axial direction. For the derivation presented here, we assume that the flow is axisymmetric, inviscid, and adiabatic and that there are no radial shifts of the meridional streamlines. Under these assumptions, for the differential fluid element, we need to balance the pressure force in the radial direction against the inertial forces, which arise from (a) the centrifugal force associated with the tangential velocity Vu (shown in Figure I.1a), (b) the radial component of the centrifugal force associated with the curved streamline in the meridional plane, and (c) the radial component of the force required to produce the radial acceleration along the streamline. We express these inertial forces as follows:

mVu2 V2 = (ρ r dr dθ ) u r r

(I.1)

mVs2 V2 cos α s = (ρ r dr dθ ) s cos α s rs rs

(I.2)

F(a) = F(b) =

287

288

Appendix I

FIGURE I.1 Differential fluid control volume for deriving the radial equilibrium equation in the gap between adjacent rows of airfoils: (a) radial-tangential section and (b) axial-radial (meridional) section.



F(c) =

mdVs dV sin α s = (ρ r dr dθ ) s sin α s dt dt

(I.3)

Combining Equations I.1, I.2, and I.3, we obtain the total inertial force as



FI = F(a) + F(b) + F(c)   V2 V2 dVs FI = (ρ r dr dθ)  u + s cos α s + sin α s  r r dt   s

(I.4)

With reference to Figure I.1a, the net pressure force producing the inertia force on the fluid element can be expressed as Fp = (p + dp)(r + dr)dθ − p(rdθ) − 2 ( p + dp/2 ) dr sin(dθ/2) and upon simplifying, neglecting higher-order terms (the product of the three differentials), and setting sin(dθ/2) = dθ/2, this yields Fp = r dpdθ

(I.5) Equating Fp from Equation I.5 with FI from Equation I.4 yields the required radial equilibrium equation for the flow in gaps between adjacent airfoil rows:





1 dp Vu2 Vs2 dVs = + cos α s + sin α s ρ dr r rs dt

(I.6)

289

Appendix I

For most design purposes, the radius of curvature rs is large, and angle α s is small enough to be neglected. Under these assumptions, Equation I.6 reduces to



1 dp Vu2 = ρ dr r

(I.7)

which is known as the simple radial equilibrium equation and is valid for both compressible and incompressible flows. This equation shows that, in a flow with tangential (swirl) velocity, static pressure increases with radius, even if the streamline has no curvature in the meridional plane. In fact, in the spaces between airfoil rows, Vr is much smaller than either Vx or Vu , and it is therefore negligible. To get a qualitative understanding of radial variations of flow properties, let us consider an incompressible flow through a turbine stator-rotor stage, as shown in Figure I.2a. At the stator inlet (section 1, Figure I.2b), the flow has radially uniform static pressure, total velocity, and total pressure; their magnitude in each case is represented by the length of the arrow from the datum. Since the static pressure is radially uniform, the velocity is axial. At the stator outlet, or the rotor inlet, which is designated as section 2 (Figure I.2c), the flow has swirl velocity, which according to Equation I.7 results in a radially increasing static pressure distribution. For a potential flow with no loss in total pressure over the stator, the flow total velocity will increase radially inward po = p + ρV 2/2 —high pressure, low velocity. Figures I.2b and I.2c also show that the static pressure difference across the stator airfoil is the highest at the root and the lowest at the tip. Except for a free-vortex flow,

(

)

FIGURE I.2 (a) Axial-flow stator-rotor turbine stage, (b) flow properties at section 1, (c) flow properties at section 2, and (d) flow properties at section 3.

290

Appendix I

FIGURE I.3 Rotor inlet velocity diagrams at blade tip, midspan, and root.

which we will later show has radially uniform axial velocity and mass flux, an uneven pressure distribution generally results in radially varying velocity and mass flux. With zero swirl exit from the rotor (section 3, Figure I.2d), we return to uniform distributions of static pressure, velocity, and total pressure, which is lower than its value at section 1 due to work extraction by the rotor. Due to solid body rotation, the blade tangential velocity increases linearly with radius. In combination with the varying absolute velocity at the rotor inlet shown in Figure I.2c, the resulting velocity diagrams at the blade tip, midspan, and root are depicted in Figure I.3. This figure shows that, even with a constant absolute flow angle at the blade inlet, both the relative velocity and the relative flow angle, which the relative velocity makes with the axial direction, are increasing from tip to hub. Note that, according to the convention used in velocity diagrams (see Appendix H), the relative flow angle is shown to be negative at the blade tip inlet.

I.3  Vortex Energy Equation At any point in the flow in gaps at the airfoil inlet and outlet, the total (stagnation) enthalpy h 0 can be expressed as



h0 = h +

V2 1 = h + Vx2 + Vu2 2 2

(

)

(I.8)

where we have neglected the radial velocity component Vr . Using this equation, the variation of h 0 in the radial direction is given by



dh 0 dh dVx dVθ = + Vx + Vθ dr dr dr dr

(I.9)

291

Appendix I

From the thermodynamic relation Tds = dh − dp/ρ, we obtain dh ds 1 dp =T + dr dr ρ dr

which using Equation I.7 becomes

dh ds Vu2 =T + dr dr r



(I.10)

Substituting for dh/dr from Equation I.10 into Equation I.9 yields



dh 0 ds Vu2 dVx dVu =T + + Vx + Vu dr dr r dr dr

(I.11)

where the term T(ds/dr) represents the radial variation of loss across the annular gap. Neglecting this term gives us the following vortex energy equation:

dh 0 Vu2 dVx dVu = + Vx + Vu dr r dr dr

(I.12)

If we now assume that the total enthalpy remains radially uniform in the gaps at the blade inlet and outlet, it will entail constant specific work transfer at all radii. For this situation, Equation I.12 reduces to

Vu2 dVx dVu + Vx + Vu =0 r dr dr

(I.13)

I.4  Free-Vortex Design The specific angular momentum ( rVu ) remains constant in a free vortex, giving

rVu = C

C Vu = r  

(I.14)

which when substituted in Equation I.13 yields



C2 dVx C2 d  1  + V + x   =0 r3 dr r dr  r 

C2 dVx C2 + V − 3 =0 x 3 dr r     r

(I.15)

dVx Vx =0          dr which leads to the result that the axial velocity in a free-vortex design is radially uniform.

292

Appendix I

We can alternatively assume a uniform axial velocity and arrive at the result from Equation I.13 that the flow field in the gap is a free vortex. This deduction is left as an exercise for readers. I.4.1  Degree of Reaction in an Axial-Flow Turbomachine With reference to Figure I.2a, as derived in Appendix H, we can express the stage degree of reaction by the equation which with U = rω becomes

R = 1−

Vu2 + Vu1 2U

R = 1−

Vu2 + Vu1 2rω

V + Vu1 1 − R = u2 2rω

(I.16)

(I.17)

For a free-vortex flow in gaps both upstream and downstream of the rotor, we can write at any radius r

Vu2 =

C2 C and Vu3 = 3 r r

which when substituted in Equation I.17 yields 1− R =

C2 + C3 2 r2ω

(I.18) If at the midspan radius rm the degree of reaction is R m , we can use Equation I.18 to yield the following equation in order to compute the degree of reaction at any other radius in a free-vortex design:



2



1 − R  rm  =  1 − Rm  r 

(I.19)

I.5  General Vortex Design To solve Equation I.13, it is necessary to independently specify Vx as a function of r or Vu as a function of r or a relation between Vu and Vr . In the case of the free-vortex design discussed in Section I.4, we essentially specified how Vu or Vx varies with radius. In a general vortex design, a variation of swirl (tangential) velocity with radius is often specified as Vu = Kr n or, in terms of mean-section conditions, as

(I.20) n



Vu  r  = Vum  rm 

(I.21)

293

Appendix I

As presented by Whitney and Stewart (1972), substituting Equation I.21 and its differential form into Equation I.13 and then integrating between the limits of rm and r yields 1



2 2n    Vx n + 1   r   2 − 1  = 1 − tan α m    n   rm  Vxm     

(I.22)

where α m is the absolute flow angle at the mean radius. Note that Equation I.22 is not valid for the special case of n = 1 (constant Vu ). For this special case, the integration of Equation I.13 yields



 Vx  r  = 1 − 2tan 2 α m ln     rm   Vxm 

1

2

(I.23)

Another special case of design interest is where the absolute flow angle is radially constant. In this case, with Vu = Vx tan α , Equation I.13 integrates to



Vu V  r  = x =  Vum Vxm  rm 

− sin 2 α

(I.24)

Using Equations I.22, I.23, and I.24, it is instructive to find out how the swirl velocity, axial velocity, and absolute flow angle vary for different vortex designs.

I.6  Throughflow Design Project The throughflow design project presented here is based on Flack (1987). The project requires the application of the simple streamline analysis method for the throughflow design of multistage axial-flow fans and compressors for educational use in graduate and undergraduate courses in turbomachinery. To demonstrate the technique for a turbofan engine compressor design, the project considers two cases: (1) irrotational free-vortex flow and (2) specified radially varying absolute flow angles with radially varying efficiencies. This project, which can be completed with relative ease using a commercial math solver, such as MS Excel or Matlab, offers a powerful means to gain valuable insight into the 3-D nature of a turbomachinery flow. Additional details on this and similar design projects may also be found in Flack (2005). I.6.1  Axial-Flow Compressor Layout and Nomenclature Figure I.4 shows the geometry of a simplified axial-flow compressor with constant hub radius rh and tip radius rt , resulting in a constant annulus flow area for gaps between adjacent airfoil rows. The proposed design includes an inlet guide vane (IGV). The axial station number i = 1 starts at the IGV inlet. For the purpose of throughflow analysis, the passage is divided into N streamtubes, which are essentially annuli made of a bundle of streamlines; the mass flow rate entering each annulus must therefore remain constant throughout the passage. As an example, for

294

Appendix I

FIGURE I.4 Simplified axial-flow compressor geometry.

the sake of simplicity, Figure I.5 depicts the first rotor-stator stage with just three annuli. Note that the inner radius of the annulus, whose center node corresponds to j = 1, equals the hub radius. Similarly, the outer radius of the last annulus, whose center node corresponds to j = 3 in this simple example, equals the hub radius. It is worth noting that, at each axial station, the annulus outer radius is designated by the inner radius of the next annulus. For example, Figure I.5 shows that the radius r3,2 at axial station i = 3 corresponds to the inner radius of annulus j = 2, which is the same as the outer radius of annulus j = 1 at this axial station. Thus, ri,1 = rh and ri,N + 1 = rt . Further, according to this convention, the mean radius corresponding to the center node of annulus j at an axial station i will be computed as



ri,j =

ri,j + ri,j+ 1 2

(I.25)

and the blade tangential velocity at the center node is obtained by the equation

U i,j = ri,jω



where ω is the rotor angular velocity in rad/s.

FIGURE I.5 First rotor-stator stage with three annuli of streamline bundles.

(I.26)

295

Appendix I

FIGURE I.6 Blade velocity diagrams at rotor inlet and outlet.

For the annulus j, the meanline inlet and outlet velocity diagrams for the compressor blade rows between axial stations i and i + 1 are shown in Figure I.6. From the velocity diagram at the rotor inlet (station i), we can write the following trigonometric relations:

Vxi,j = Vi,j cos α i,j Vui,j = Vi,j sin α i,j tan β i,j =



(I.27)



(I.28)



Vui,j − U i,j Vxi,j

(I.29)

where U i,j is given by Equation I.26 and, as shown in Figure I.6, β i,j is negative by our sign convention. We can express similar trigonometric relation at the blade outlet (station i + 1) by simply replacing subscript i with i + 1 in Equations I.27, I.28, and I.29. I.6.2  Governing Equations For computing various flow properties at each annulus-center node (i, j) in the present compressor throughflow design, we use a quasi-2-D control volume method to derive the required conservation equations of mass, angular momentum, and energy under the following simplifying assumptions: 1. The flow is incompressible with constant density. 2. The axial compressor flowpath is an annulus of constant inner (hub) and outer (tip) radii. 3. The relative flow velocity is tangent to the blades at both the entrance and the exit of each stage; i.e., both the incidence and the deviation angles are zero.

296

Appendix I

I.6.2.1  Continuity Equation For the annulus j (streamtube), the conservation of mass yields the following equation between stations i and i + 1:

(

)

(

Vxi,j ri,2j+ 1 − ri,j2 = Vxi + 1, j ri2+ 1, j+ 1 − ri2+ 1,j



)

(I.30)

We can compute the constant volumetric flow rate through the annulus j as

(

Q j = Vxi,j π ri,j2 + 1 − ri,j2

)

(I.31)

which remains constant at each axial location. I.6.2.2  Angular Momentum Equation Balancing the aerodynamic torque with the change in the angular momentum flow over the control volume yields

 i + 1,j − H  i,j Γ i,j = H

(I.32)



where and

 i,j = ρQ j ri,j Vu H i,j

(I.33)



 i + 1,j = ρQ j ri + 1,j Vu H i + 1,j

(I.34)



I.6.2.3  Energy Equation For an incompressible flow, the total pressure may be viewed as total mechanical energy (flow work + kinetic energy) per unit volume. Accordingly, the mechanical power transfer in a compressor stage, which occurs only over each blade row, may be written as

(

)

− Pi,j = Q j p0 i + 1, j − p0 i,j / ηi,j



(I.35)

Note that the negative sign for the power term indicates that the work is being done on the fluid. In Equation I.35, ηi,j is the efficiency of conversion from aerodynamic to mechanical power. Using the rotor torque given by Equation I.32, we can also express the mechanical power as

− Pi,j = Γ i,jω

(I.36)



Because the angular velocity of the stator airfoils is zero, this equation shows that any torque produced in these airfoils by virtue of the change in the flow angular momentum across them yields zero power transfer. From Equations I.35 and I.36, we obtain

p0 i + 1,j = p0 i,j + ηi,j Γ i,jω/Q j



(I.37)

which indicates that, in the absence of any other loss, the total pressure must remain constant across the stator.

297

Appendix I

I.6.2.4  Simple Radial Equilibrium Equation In the foregoing, we have presented the complete set of conservation equations in the axial direction (flow direction). In the radial direction, with no radial velocity, we require that the throughflow satisfy the simple radial equilibrium equation. Thus, using discrete nodal values, Equation I.7 yields the following relations: 2

pi,j+ 1 − pi,j ρV ui,j = ri,j+ 1 − ri,j 0.5 ri,j + ri,j+ 1

(



pi,j+ 1 = pi,j +

ρV

2 u i,j

(

(r

i,j+ 1

)

− ri,j

0.5 ri,j + ri,j+ 1

)

(I.38)

)

where V ui,j =



Vui,j+ 1 + Vui,j 2

(I.39)



I.6.3  Case 1: Single-Stage Compressor with Potential Flow (Free-Vortex Design) This case is based on the assumption of incompressible potential (irrotational) flow through the single-stage compressor shown in Figure I.5. In the gap between adjacent rows of airfoils, the radial distribution of the flow tangential velocity is that of a free vortex, which implies constant angular momentum. Thus, we write the following equation: Vui,j ri,j = Vui,j+ 1 ri,j+ 1



(I.40)



As discussed in Section I.4, a free-vortex design leads to radially uniform axial velocity. Because both hub and tip radii are constant for this design and the flow is incompressible, the axial velocity must be uniform everywhere in the flow field to satisfy the continuity equation. We also know that, in a potential flow, the total pressure remains constant everywhere. This condition leads to the equation

(

)

(

poi,j = pi,j + ρVi,j2/2 = poi,j+ 1 = pi,j+ 1 + ρVi,j2 + 1/2  

(

pi,j+ 1 = pi,j + ρV i,j Vi,j − Vi,j+ 1



(

)

) (I.41)

)

where V i,j = Vi,j+ 1 + Vi,j /2 . Note that we can also obtain the radial distribution of static pressure using Equation I.38. It is left for the reader to establish the equivalence between the two approaches in a free-vortex design. This can also be used as a check for the accuracy of numerical results obtained in this design. I.6.3.1  Design Data For this case, divide the big annulus between the tip and hub radii at the IGV inlet into nine (N = 9) annuli, with each annulus having a constant difference between its outer and

298

Appendix I

inner radii, which will imply radially increasing flow area and flow rate. Additionally, the following data are specified: Inlet mass flow rate = 109 kg/s Inlet static pressure = 137,000 Pa Fluid density (ρ) = 1.70 kg/m 3 Rotor angular velocity = 382.227 rad/s Hub radius ( rh ) = 0.484 m Tip radius ( rt ) = 0.614 m

( )

Absolute flow angle at IGV inlet α 1,j = 0 At the exits of the IGV, rotor 1 blade row, and stator 1 vane row, the absolute flow angles at the midspan are given as α 2,5 = 21°, α 3,5 = 46°, and α 4,5 = 21°. Note that this case does not require an iterative solution, only the direct use of various equations given in this appendix. (Hint: At each axial station, start first with the solutions at the midspan annulus (j = 5), where the absolute flow angles are given, and then generate solutions for other annuli—that is, for the remaining values of j.) I.6.3.2  Results and Discussion Using MS Excel or Matlab, compute the radial distributions of all flow properties of the throughflow design at the rotor inlet (i = 2), rotor outlet (i = 3), and stator outlet (i = 4), and compare them with the results on absolute flow angle, relative flow angle, airfoil turning angle, static pressure, total pressure, and rotor reaction presented by Flack (1987), who uses the following equation to compute rotor reaction: R=

pR i + 1,j − pR i,j p0R i+1,j − p0R i,j

(I.42)

Discuss the reason for any differences in the computed results and those presented by Flack (1987). I.6.4  Case 2: Single-Stage Compressor with Specified Flow Angles and Efficiency Unlike case 1, discussed in Section I.6.3, this case does not assume a potential (irrotational) flow through the single-stage compressor shown in Figure I.5. Instead, radial distributions of absolute flow angles at the exits of the IGV, rotor 1 blade row, and stator 1 vane row are specified; see Table I.1, which also includes the data on stage efficiency. I.6.4.1  Design Data For this case, divide the big annulus between the tip and hub radii at the IGV inlet into nine (N = 9) annuli, with each annulus having the same mass flow rate. This will result in a radially decreasing difference between the annulus outer and inner radii. Additionally, the following data are specified: Inlet mass flow rate = 109 kg/s Inlet static pressure = 137,000 Pa

299

Appendix I

TABLE I.1 Radial Distributions of Exit Absolute Flow Angles and Stage Efficiency rˆ

α at i = 2 (degrees)

α at i = 3 (degrees)

α at i = 4 (degrees)

η

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

17.50 17.00 16.60 16.25 16.00 15.75 15.50 15.25 14.85 14.30 13.50

47.95 46.50 45.25 44.50 44.05 43.85 43.60 43.50 43.45 43.40 43.30

23.10 23.00 22.90 22.60 22.40 22.20 22.15 22.10 22.05 22.03 22.00

0.85 0.90 0.93 0.93 0.93 0.93 0.93 0.93 0.93 0.91 0.85

Fluid density (ρ) = 1.70 kg/m 3

Rotor angular velocity = 382.227 rad/s  Hub radius ( rh ) = 0.484 m Tip radius ( rt ) = 0.614 m Absolute flow angle at IGV inlet α 1, j = 0

( )

I.6.4.2  Iterative Solution Method At each axial station, we need to use an iterative solution method, which consists of an inner loop and an outer loop, to compute properties at each annulus-center node. For example, we outline here a stepwise solution method at the IGV exit (i = 2). Outer Iteration Loop Step 1: For the first annulus (j = 1), assume Vx2 ,1 and compute the outer annulus radius r2,2 to satisfy the constant mass flow rate through the annulus; then compute the annulus mean radius r2,1 and blade tangential velocity U 2,1. Step 2: Determine α 2,1 at r2,1 from Table I.1, using linear interpolation between adjacent values as needed; then compute V2,1 , Vu 2 ,1 , β 2,1 , and p2,1 , assuming zero loss in total pressure across the IGV. Inner Iteration Loops Step 3: For the next annulus, assume Vx2 ,2 and compute the outer annulus radius r2,3 to satisfy the continuity equation; then compute the mean annulus radius r2,2 and blade tangential velocity U 2,2 . Step 4: Determine α 2,2 at r2,2 from Table I.1, using linear interpolation between adjacent values as needed; then compute V2,2 , Vu 2 ,2 , β 2,2 , and p2,2 , assuming zero loss in total pressure across the IGV. Step 5: Now iterate on Vx2 ,2 such that p2,2 computed in Step 4 equals its value computed by the simple radial equilibrium equation (Equation I.38).

300

Appendix I

Step 6: Repeat Steps 3, 4, and 5 for all remaining annuli at i = 2. This completes all inner iteration loops for an outer iteration loop. Step 7: Check whether the outer radius R 2,10 of the last annulus (annulus 9) equals the given tip radius. If they are different, start another outer iteration loop with a new value of Vx2 ,1 , and repeat Steps 1 to 7 until they are equal, resulting in a converged solution for i = 2. To complete the throughflow design of the complete compressor stage, we need to repeat the foregoing iterative solution procedure demonstrated for the IGV exit (i = 2) at the rotor 1 exit (i = 3) and stator 1 exit (i = 4). I.6.4.3  Results and Discussion As project deliverables for this case, the results obtained in Section I.6.4.2 need to be compared with the results on the radial distributions of streamline, axial velocity, relative flow angle, airfoil turning angle, static pressure, total pressure, and rotor reaction presented by Flack (1987), including a discussion of the reason for any observed differences.

References Flack, R.D. 1987. Classroom analysis and design of axial flow compressors using a streamline analysis model. International Journal of Turbo and Jet Engines 4: 285–296. Flack, R.D. 2005. Fundamentals of Jet Propulsion with Applications (Cambridge Aerospace Series). Cambridge: Cambridge University Press. Smith, L.H. 1966. The radial-equilibrium equation of turbomachinery. Journal of Engineering for Power 88: 1–12. Whitney, W.J., and W.L. Stewart. 1972. Velocity diagrams. In Turbine Design and Applications (ed. A.J. Glassman), 1: 69–98. Washington, DC: NASA.

Nomenclature  H N p P Q r rˆ R U V W α

Angular momentum flow rate Number of annuli Static pressure Power Volumetric flow rate Radius Dimensionless radius = ( r − rh )/( rt − rh ) Degree of reaction Blade tangential velocity Absolute fluid velocity Relative (to blade) fluid velocity Angle between absolute total velocity and axial direction

Appendix I

β Γ η ρ

Angle between relative total velocity and axial direction Torque Efficiency Density

Subscripts 0 Stagnation (total) value h Hub i Axial stations j Radial stations R Rotor t Tip u Tangential (circumferential) direction x Axial direction

301

Appendix J: Review of Necessary Mathematics

J.1  Suffix Notation and Tensor Algebra In Cartesian coordinates, the unit vectors along the x, y, and z coordinate directions are ˆ respectively. In Cartesian tensor, we name the x, y, and z coordidenoted by ˆi, ˆj, and k, nates as x 1, x 2, and x 3, respectively, with the corresponding unit vectors denoted by eˆ 1, eˆ 2,  and eˆ 3. A typical vector a with components a1, a2, and a3 in Cartesian coordinates can be expressed as  a = a1 ˆi + a2 ˆj + a3 kˆ = a1eˆ 1 + a2 eˆ 2 + a3 eˆ 3



J.1.1  Summation Convention When an index is repeated precisely twice in a term, it implies summation over all possible values of the index, which has values 1, 2, and 3 in Cartesian coordinates. Accordingly, we  can express our vector a in the compact tensor notation as  a = ai eˆ i



Note that the summation convention possesses the commutative and distributive properties, as shown below. Commutative: a i bi = bi a i

Distributive:

a i ( bi + c i ) = a i bi + a i c i



J.1.2  Free and Dummy Indices In the term on the left-hand side of the following equation, the index j is repeated twice, implying summation. This index is called the dummy index. On the other hand, the index m can have any value and is called the free index:

amj x j = c m The above equation can be alternatively written as



akq x q = c k 303

304

Appendix J

Both these equations are a compact form (in tensor notation) of the following three equations:

a11 x 1 + a12 x 2 + a13 x 3 = c1



a21 x 1 + a22 x 2 + a23 x 3 = c 2



a31 x 1 + a32 x 2 + a33 x 3 = c 3

J.1.3  Two Special Symbols Kronecker delta: if i = j



 1 δ ij = eˆ i ⋅ eˆ j =  0 



 δ 11  δ ij =  δ 21  δ 31 

δ 13 δ 23 δ 33

δ 12 δ 22 δ 32

if i ≠ j   1   = 0   0 

0 1 0

0 0 1

   

The alternating symbol:

(

ε ijk = eˆ i ⋅ eˆ j × eˆ k

)

 1  =  −1  0 

if i, j, k are a cyclic permutation of 1, 2, and 3 if i, j, k are a anticyclic permutation of 1, 2, and 3 if any index is repeated

According to the above definition of εijk , we can write the following identities: ε ijk = ε kij = ε jki and ε jik = −ε ijk



J.2  Gradient, Divergence, Curl, and Laplacian J.2.1 Gradient Define the gradient using ∇=

∂ ˆ ∂ ˆ ∂ ˆ ˆ ∂ i+ j + k = ei ∂x ∂y ∂z ∂x i

For example, we can write the gradient of a scalar Φ as ∇Φ =

∂Φ ˆ ∂Φ ˆ ∂Φ ˆ ∂Φ ∂Φ ∂Φ ∂Φ i+ j+ k= eˆ 1 + eˆ 2 + eˆ 3 = eˆ i ∂x ∂y ∂z ∂x 1 ∂x 2 ∂x 3 ∂x i

305

Appendix J

In cylindrical coordinates, the above equation becomes ∇Φ =



∂Φ ˆ 1 ∂Φ ˆ ∂Φ ˆ er + eθ + ez ∂r r ∂θ ∂z

where eˆ r, eˆ θ, and eˆ z are the unit vectors along the r, θ, and z coordinate directions, respectively. J.2.2 Divergence

 We can write the divergence of a vector V = uˆi + vˆj + wkˆ = eˆ i u i as  ∂u ∂ v ∂ w ∂u 1 ∂u 2 ∂u 3 ∂u i ∇⋅V = + + = + + = ∂x ∂ y ∂ z ∂x 1 ∂x 2 ∂x 3 ∂x i



In cylindrical coordinates, the above equation becomes  1 ∂(ru r ) 1 ∂u θ ∂u z ∇⋅V = + + ∂z r ∂r r ∂θ

J.2.3 Curl

 We can write the curl of a vector V = uˆi + vˆj + wkˆ = eˆ i u i as





 ˆ  i   ∂ ∇×V=   ∂x  u 

  ˆ   e1 ∂   ∂ = ∂ z   ∂x 1  w   u1 kˆ

ˆj ∂ ∂y v

eˆ 2 ∂ ∂x 2 u2

eˆ 3 ∂ ∂x 3 u3

     

  ∂w ∂v  ˆi +  ∂u − ∂ w  ˆj +  ∂ v − ∂u  kˆ ∇×V= −    ∂x ∂ y  ∂ z ∂x   ∂ y ∂ z    ∂u ∂u k ∂u  ∂u  ∂u   ∂u  ∂u ∇ × V =  3 − 2  eˆ 1 +  1 − 3  eˆ 2 +  2 − 1  eˆ 3 = eˆ i ε ijk  ∂x 1 ∂x 2   ∂x 3 ∂x 1   ∂x 2 ∂x 3  ∂x j In cylindrical coordinates, the above equation becomes



  1 ∂u z ∂u θ   1 ∂(ru θ ) 1 ∂u r  ˆ  ∂u ∂u  ∇×V =  − eˆ r +  r − z  eˆ θ +  − ez   r ∂θ  ∂z  r ∂r ∂z  ∂r  r ∂u θ 

J.2.4 Laplacian We can write the Laplacian of a scalar Φ as ∇ 2 Φ = ∇ ⋅ ∇Φ =

∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂ ∂Φ + + = + + = ∂x 2 ∂ y 2 ∂ z 2 ∂x 12 ∂x 22 ∂x 23 ∂x i ∂x i

306

Appendix J

In cylindrical coordinates, the above equation becomes



∇2 Φ =

∂2 Φ 1 ∂Φ 1 ∂2 Φ ∂2 Φ 1 ∂  ∂Φ  1 ∂ 2 Φ ∂2 Φ + + + = + r + ∂r 2 r ∂r r 2 ∂θ2 ∂ z 2 r ∂r  ∂r  r 2 ∂θ2 ∂z2

J.3  Dyad in Total Derivative

 The total or substantial derivative of the velocity vector V can be written in vector form as     DV ∂ V = + (V ⋅ ∇)V Dt ∂t



 which makes it independent of the coordinate system. In the above equation, V ⋅ ∇ is called a dyad, which is easily handled in the tensor notation. Using the vector identity, we can express the dyad in the above equation as     1 (V ⋅ ∇)V = ∇  V 2  − V × (∇ × V) 2 



J.4  Total Derivative The total or substantial or material derivative following a particle (Lagrangian viewpoint) in a fluid flow can be written in various forms (vector, differential, and tensor), as given in the following. Cartesian coordinates:





    DV ∂ V = + (V ⋅ ∇)V Dt ∂t   ∂v DV  ∂u ∂u ∂u ∂u  ∂v ∂v ∂v  = +u +v + w  ˆi +  +u +v + w  ˆj Dt  ∂t ∂x ∂y ∂z  ∂x ∂y ∂z   ∂t  ∂w ∂w ∂w ∂w  ˆ + +u +v +w k ∂ t ∂ x ∂ y ∂ z   Cartesian tensor notation:



 ∂u DV ∂u i = + uj i ∂t ∂x j Dt

307

Appendix J

Cylindrical coordinates:  DV  ∂u r u2  ∂u r u θ ∂u r ∂u = + ur + + u z r − θ  eˆ r Dt  ∂t r ∂θ r  ∂r ∂z ∂u ∂u u ∂u θ uu   ∂u +  θ + ur θ + θ + u z θ + r θ  eˆ θ  ∂t ∂r ∂z r ∂θ r  ∂u z u θ ∂u z ∂u z   ∂u +  z + ur + + uz  eˆ z  ∂t ∂r ∂z  r ∂θ



J.5  Vector Identities

   In the following vector identities, Φ is a scalar and A, B, and C are vectors:

∇ × ∇Φ = 0  ∇ ⋅ (∇ × A) = 0    ∇ ⋅ (ΦA) = Φ(∇ ⋅ A) + A ⋅ ∇Φ    ∇ × (ΦA) = ∇Φ × A + Φ(∇ × A)   1     (A ⋅ ∇)A = ∇(A ⋅ A) − A × (∇ × A) 2    ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇ 2 A       ∇ ⋅ (A × B) = B ⋅ (∇ × A) − A ⋅ (∇ × B)           ∇ × (A × B) = A(∇ ⋅ B) − B(∇ ⋅ A) − (A ⋅ ∇)B + (B ⋅ ∇)A

For further details on the mathematical topics presented in this appendix and for additional topics, interested readers may refer to the references listed in the Bibliography section.

Bibliography Aris, R. 1962. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Englewood Cliffs, NJ: Prentice-Hall. Hughes, W.A., and E.W. Gaylord. 1964. Basic Equations of Engineering Science. New York: McGraw-Hill.

Index Note: Italicized page numbers refer to figures, bold page numbers refer to tables. A Absolute flow angles, 285–286 Absolute flow velocity, 65, 284–285. See also Relative flow velocity components, 26 dimensionless, 283 Pelton wheel, 197 reaction turbines, 194 at rotor inlet V1/U, 282 at rotor outlet V2/U, 282 at shroud, 94 Absolute gas angle, 108 Absolute temperature, 94 Absolute velocity angle at rotor inlet α1, 280 at rotor outlet α2, 280–281 Acoustic speed, 94 Actuator disk, 207 Actuator theory, 205–209 Aero-derivative gas turbines, 9 Aerodynamic power transfer, 23 Aerothermodynamics angular momentum equation, 20–25 axial-flow compressor, 29–30 axial-flow impulse turbine, 28–29 centrifugal pump, 30–31 compressible flow, 14–18 compressible mass flow functions, 16–18 energy equation, 18–19 hydraulic turbine, 32 incompressible flow, 14–15 linear momentum equation, 19–20 mass flow rate, 15 total temperature, 14–15 velocity diagram, 25–27 worked examples, 34–44 Ainley-Mathieson method, 171 Air, molecular weight of, 267 Air dynamic pressure, 39 Air jet area, 42 Air mass flow rate, 39 Air momentum flow rate, 39 Air specific heat at constant pressure, 42 Air total pressure, 42

Airfoil design, 250–251. See also Turbomachinery aerodynamic design 3-D flow analysis, 250 2-D blade-to-blade analysis, 250 Ancient Greece, 1 Angular momentum of exit flow, 76 specific, 291 in vaneless space, 13, 97 Angular momentum equation, 20–22 alternate form of Euler’s turbomachinery equation, 25 Euler’s turbomachinery equation, 22–23 rothalpy, 24 throughflow design, 296 Angular speed, 54 Area, 257 Area ratio, 173 Armengaud, Réné, 2 Assyria, 1 Axial momentum equation, 219, 222 Axial stream thrust coefficient, 228–229 Axial velocity, 122, 230 Axial-flow blower, 4 Axial-flow compressor, 29–30 blades, 116, 118, 130 compressor map, 133 degree of reaction in, 292 design, 127–131 dimensionless velocity diagrams for, 278–286 efficiency, 137–138 energy transfer, 116, 133, 135 flow coefficient, 275 four-stage, 137–138 layout and nomenclature, 293–295 loading coefficient, 275–276 performance, 131–133 performance parameters, 275–278 schematics of, 5 simplified geometry, 294 specific speed range, 57 stage pressure rise, 118–120 throughflow design, 293–295 vanes, 130 worked examples, 133–138

309

310

Axial-flow fan blades, 116 design, 125–127 Eck’s correlation, 126 optimal solidity for, 127 Axial-flow gas turbines, 167–183. See also Gas turbines basic theory, 170–175 blade profiles in, 168 design, 175–180 design features, 230 energy transfer, 175 sectional view of, 168 symmetrical velocity diagram, 169 velocity diagrams for, 170 worked examples, 180–183 Axial-flow impulse turbine, 28–29 Axial-flow pumps blades, 116 design, 121–125 energy transfer, 116 optimum number of blades, 122 specific speed range, 57 Axial-flow turbines basic features, 189 degree of reaction in, 292 dimensionless velocity diagrams for, 278–286 specific speed range, 57 Axial-flow turbomachines, 4 B Backward-curved blades, 261–262 Balje diagram, 155–157, 156 Balje equation, 265 Barber, John, 2 Bernoulli equation, 217 extended, 224 Blade(s), 22 angle, 76, 106, 178 backward-curved, 261–262 compressor, 116 deviation, 178 diffusion factor, 1 efficiency, 192 enthalpy, 151 incidence, 178 loading coefficient, 119, 136, 177 mean blade speed, 130, 134 mean radius, 182 motion, 117 optimum number of, 76

Index

optimum spacing, 179 preliminary design, 210 profile, 116 radial position, 210 turbine, 116 velocity, 116 Blade-to-blade analysis, 248, 250 Blade-to-blade stream surface, 247 Boundary conditions, 240–244 inlet, 240 outlet, 240 turbulence model, 244 wall, 241–244 Boussinesq hypothesis, 238 Brayton cycle, 147 Brown and Boveri, 2 Brown Boveri and Thyssen, 2 Buckingham pi theorem, 53 Busemann equation, 265 n-Butane, molecular weight of, 267 C Carbon dioxide, molecular weight of, 267 Carbon monoxide, molecular weight of, 267 Carter rule, 131 Cartesian coordinates, 306 Cartesian tensor notation, 306 Cascade blade, control volume of, 119 Casing, 65 Cavitation, 73, 202 Centistokes, 263 Centrifugal blower, 5 Centrifugal compressors, 91–108 design parameters, 96 diffuser design, 96–99 efficiency, 92, 95, 104–105 energy transfer, 91, 95, 106 examples, 102–108 eye velocity, 104–105 impeller design, 93–96 inlet density, 105–108 map, 99 performance, 99–101 schematics of, 5 single-stage, 102–103 specific speed range, 57 Centrifugal fans, 9, 33, 77, 78 Centrifugal pumps, 30–31, 65–77 with backward-curved blades, 261–262 design of, 74–77 dimensional specific speed for, 59 double-suction, 82–84

311

Index

efficiency, 68–70 equal-efficiency contours for, 72 head-capacity curves, 80 hydraulic efficiency, 78 hydraulic loss, 68, 69, 261–262 impeller flow, 65–67 impellers, 65 mechanical efficiency, 78 performance characteristics, 70–74, 71 with piping, 68 pump efficiency, 78 specific speed, 59 specific speed range, 57 volumetric efficiency, 78 volute of, 77 Centrifugal turbomachines, 4 China, 1 Choked divergent nozzles, 37–38 Choked flow rate, 100–101 Closure problem, 238 Combustion gas turbine, 2 Commutative properties, 303 Compressible flow. See also Incompressible flow gas-turbine exhaust diffusers, 228 versus incompressible flow, 14–18 mass flow functions, 16–18 mass flow rate, 15 pressure, 14–15 temperature, 14–15 Compressor(s), 1, 3 axial-flow, 29–30 blades, 26, 27, 29, 115, 116, 127, 131, 295 centrifugal, 91–108 efficiency, 92, 258 map, 56, 99 outlet temperature, 138 specific diameter, 258 specific speed, 258 Computational fluid dynamics (CFD), 233–251 airfoil design, 250–251 blade-to-blade analysis, 248 boundary conditions, 240–244 Boussinesq hypothesis, 238 closure problem, 238 commercial codes, 235, 235 common equation form, 236 defined, 233 detailed design, 248–250 entropy map, 246 factors in industrial use of, 233 inlet boundary conditions, 240 large control volume analysis, 245, 245 meanline design, 248

methodology, 234–235, 234 outlet boundary conditions, 240 physics-based post processing, 244–246 preliminary design, 247 Reynolds averaging, 237 Reynolds equations, 237–238 role of, 251 3-D, 251 3-D analysis, 245, 245 3-D flow analysis, 250 3-D flow field, 246–247 throughflow design, 248, 249–250 turbomachinery aerodynamic design, 246–251 turbulence modeling, 237–240 wall boundary conditions, 241–244 Constant-area duct, diffusion in, 218–221, 218 Continuity equation, 39, 40 computational fluid dynamics, 236 diffuser, 97 diffusion in constant-area duct, 219 dump diffuser, 221 throughflow design, 296 Convection term, 236 Convergent nozzles, 37–38 Conversion factors, 257 Cordier diagram, 57, 58, 121, 126, 200 Coriolis force, 67 Curl of a vector, 305 Curtis stage, 167, 193, 193 Curtis turbine, 167 Cycle thermal efficiency, 147 Cylindrical coordinates, 307 D De Laval, Gustaf, 2 Density, 257 Diffusers, 215–231 arrangement of, 97 axial stream thrust coefficient, 228–229 compressible flow, 228 design rules, 229–231 diffuser isentropic efficiency, 223–224 diffusion process, 217–222 dump, 221–222 flow physics of, 216–217, 216 h-s diagram, 223 ideal characteristics of, 215 incompressible flow, 227 isentropic efficiency, 223–224 with no boundary layer separation, 218

312

performance evaluation, 223–229 pressure rise coefficient, 224–227 roles of, 215–217 schematics of, 216 with separated boundary layers, 218 vaneless, 96–99 velocity components, 215 Diffusion, 217–222 axial momentum equation, 219 in constant-area duct, 218–221, 218 continuity equation, 219 loss of total pressure, 219–220 pressure recovery in dump diffuser, 221–222 Diffusion ratio, 108 Diffusion term, 236 Dimensional loss, 60 Dimensional specific speed, 59, 104, 258 Dimensionless loss, 60 Dimensionless quantities, 53–60 efficiency, 55 flow coefficient, 53 head coefficient, 53 power coefficient, 54 ratio of outlet pressure to inlet pressure, 55 ratio of outlet temperature to inlet temperature, 55 specific diameter, 54 specific speed, 53, 59 turbomachine variables, 53–55 worked examples, 58–60 Dimensionless specific speed, 59 Dimensionless velocity diagrams, 278–286. See also Velocity diagrams absolute and relative flow angles at blade inlet and outlet, 285–286 absolute velocity angle at rotor inlet α1, 280 absolute velocity angle at rotor outlet α2, 280–281 for axial-flow compressors and turbines, 278–286 dimensionless absolute velocity at rotor inlet V1/U, 282, 284–285 dimensionless absolute velocity at rotor outlet V2/U, 282, 284–285 dimensionless relative velocity at rotor outlet W2/U, 283–284 dimensionless relative velocity W1/U at rotor inlet, 283–284 flow coefficient, 283 relative velocity angle at rotor inlet β1, 281

Index

relative velocity angle at rotor outlet β2, 281–282 stage reaction, 276–278, 283 Direct numerical simulation (DNS), 237 Discharge nozzle, 77 Distributive properties, 303 Divergence of a vector, 305 Divergent nozzles, 37–38 Doomsday Book, 2 Double-suction centrifugal impeller, 74, 75 Double-suction centrifugal pumps, 82–84 Draft tube, 197 Drag coefficient, 120–121, 212 Drag force, 231 Drag-lift ratio, 124, 133 Dummy index, 303–304 Dump diffuser. See also Diffusers axial momentum equation, 222 continuity equation, 221 loss of total pressure, 222 pressure recovery in, 221–222 Dyad in total derivative, 306 Dynamic pressure, 39–40 Dynamic temperature, 14 Dynamic viscosity, 257 E Eck’s correlation, 126, 265 Efficiency blade, 192 defined, 55, 60 four-stage compressor, 138 hydraulic, 69, 78, 79, 198 impeller, 95 mechanical, 70, 78, 91 overall pump, 69, 78 polytropic, 138 stage, 120, 135–137, 195 total-to-static, 153 total-to-total, 154 volumetric, 69, 70, 78, 83 wind turbine, 208 Egypt, 1 Électricité de France, 3 Energy, 257 Energy conversion, 1 Energy equation, 18–19 computational fluid dynamics, 236 throughflow design, 296 Energy transfer axial-flow compressor, 133, 135 axial-flow gas turbines, 175

313

Index

axial-flow pumps/fans/compressors, 116 centrifugal compressors, 91, 95, 106 Francis turbine, 199 Pelton wheel, 198 per unit mass, 116 rate of, 116 reaction turbines, 195 single-stage centrifugal air compressor, 102–103 Enthalpy, 68, 91, 151 Enthalpy-entropy diagram, 92, 170 Entropy, 246 Ethane, molecular weight of, 267 Ethylene glycol, kinematic viscosity of, 263 Euler equation, 171 Euler turbine equation, 195 Euler’s momentum equation, 247 Euler’s turbomachinery equation, 22–25 alternate form of, 25 rothalpy, 24 Exhaust diffusers, 215–231 axial stream thrust coefficient, 228–229 compressible flow, 228 design rules, 229–231 diffuser isentropic efficiency, 223–224 diffusion process, 217–222 flow physics of, 216–217, 216 ideal characteristics of, 215 incompressible flow, 227 with no boundary layer separation, 218 performance evaluation, 223–229 pressure rise coefficient, 224–227 roles of, 215–217 schematics of, 216 with separated boundary layers, 218 velocity components, 215 Exit flow angle, 123 Exit velocity diagram, 260 Extended Bernoulli equation, 14, 224 F Fans, 1, 3 axial-flow, 116, 125–127 centrifugal, 77 Flow, 257 compressible, 14–18, 228 incompressible, 14–18 Flow coefficient, 53, 75 axial-flow compressor, 136, 275 axial-flow gas turbines, 177 centrifugal compressors, 96, 106 centrifugal pumps and fans, 71

dimensionless velocity diagrams, 283 double-suction centrifugal pumps, 83 Flow passage, 67 Flow path, 3 Flow work, 68 Forced vortex, 269–270, 270 static temperature and pressure changes, 270–274 total temperature and pressure changes, 273–274 Four-stage compressors, 137–138 Francis turbine, 148, 199–201. See also Hydraulic turbines energy transfer, 199 runner designs, 200 schematics of, 5 specific speed range, 57 velocity diagram for, 199 Free index, 303–304 Free vortex, 269–270, 270 condition, 178 degree of reaction in axial-flow turbomachine, 292 method, 125 single-stage compressor with potential flow, 297–298 specific angular momentum in, 291 static temperature and pressure changes, 270–274 throughflow design, 291–292 total temperature and pressure changes, 273–274 Friction factor, 81 G Gas angles, 178 Gas turbines, 3, 6–8 aero-derivative, 9 applications, 9 axial-flow, 167–183 basic, 4, 5 exhaust diffusers, 215–231 history, 2, 3 marine and industrial, 6 optimum parameters, 176 power generation, 7–8 propulsion, 6 radial-flow, 147–160 shaft-power, 6–8 turbofan engines, 6 turbojet engines, 6 Gases, molecular weight of, 267

314

Gasoline, kinematic viscosity of, 263 Gas-turbine exhaust diffusers, 215–231 axial stream thrust coefficient, 228–229 compressible flow, 228 design rules, 229–231 diffuser isentropic efficiency, 223–224 diffusion process, 217–222 flow physics of, 216–217, 216 ideal characteristics of, 215 incompressible flow, 227 with no boundary layer separation, 218 performance evaluation, 223–229 pressure rise coefficient, 224–227 roles of, 215–217 schematics of, 216 with separated boundary layers, 218 velocity components, 215 Gas-turbine power plants, 148 GE9X™ turbofan engine, 3 General vortex design, 292–293 Glauert’s theory, 210 Glycerin, kinematic viscosity of, 263 Gradient of a scalar, 304–305 H Harz Mountains (Germany), 2 Head coefficient, 53, 55, 71 Head loss, 80, 81 Head or energy transfer, 54 Head-capacity curves, 55, 55, 72–73 Heat recovery steam generator (HRSG), 215, 216 Heat transfer coefficient, 257 Hero of Alexandria, 2 High-Reynolds-number two-equation κ-ε model, 239–240 Holzwarth, Hans, 2 Horizontal-axis machines, 209–212 Horizontal-axis wind turbines, 206 efficiency, 208 Hub diameter, 95 Hub-shroud ration, 83 Hub-tip ratio, 76, 122, 171, 173 Hub-tip stream surface, 247 Hydraulic efficiency, 69, 78, 79, 198, 261 Hydraulic loss centrifugal pumps, 68, 69, 261–262 formulation of equation for, 261–262 in rotor flow passage, 59 Hydraulic turbines, 3, 32, 197–202. See also Velocity diagrams cavitation, 202

Index

Francis turbine, 199–201 Kaplan turbine, 201 Pelton wheel, 197–199 radial-flow, 32 Hydroelectric power plant, history, 2 Hydrogen, molecular weight of, 267 I Ideal head, 65 Impeller(s), 65 arrangement of, 97 design, 93–96 diameter, 75, 95 double-suction, 74, 75 efficiency, 95, 107 flow, 65–67 flow rate, 75, 83 number of blades, 259 pressure ratio, 107 relative eddy at tip of, 259 rotational speed, 79 throat section, 100 tip speed, 75, 79 tip width, 107 Impingement air cooling, 42–44 Impulse turbines, 4, 190–193 axial-flow, 28–29 efficiency and torque for, 192 energy transfer, 190 expansion process in, 191 velocity diagram for, 190 Incompressible dynamic pressure, 14 Incompressible flow, 227. See also Compressible flow versus compressible flow, 14–18 incompressible, 227 mass flow rate, 15 total pressure, 14–15 total temperature, 14–15 Incompressible fluid flow, 1, 3 Industrial gas turbine, 6 Inlet boundary conditions, 240 density, 104–105 dynamic pressure, 225 fluid temperature, 54 guide vane (IGV), 293 specific enthalpy, 54 temperature, 137–138 volume flow rate, 104 Inner iteration loops, 299–300 Input head, 67, 79

315

Index

Inward-flow radial (IFR) gas turbine, 150. See also Gas turbines design parameters, 157 relative eddies in, 158 Isentropic compression, 103 Isentropic efficiency, diffuser, 223–224 Isentropic flow tables, 18 Iterative solution, 299–300 J Jaytilleke’s P-function, 243 Junkers, 2 K Kaplan turbine, 201. See also Hydraulic turbines schematics of, 5 specific speed range, 57 velocity diagram for, 201 Kerosene, kinematic viscosity of, 263 Kinematic viscosity, 54, 73, 73, 81, 257 Kinetic energy, 68 Kolmogorov-Prandtl relation, 239 Kronecker delta, 238, 304 L Laplacian, 305–306 Large eddy simulation (LES), 237 Leibniz, Gottfried, 2 Lemale, Charles, 2 Length, 257 Lift coefficient, 97, 117, 180, 212 Linear momentum equation, 19–20 Ljungstrom turbine, 148, 189 Loading coefficient, 119, 275–276 Louis XIV, 2

static-pressure, 16–18 total-pressure, 16–17 Mass flow rate, 15, 159 through a rotating pipe, 42 Mass-weighted average dynamic pressure, 40 Matlab, 293, 298 Mean blade speed, 130 Mean radius, 134, 171 Mechanical efficiency, 70, 78, 91 Mechanical energy, 68 increase in, 68 loss of, 68 Mechanical energy equation, 224 Mellor charts, 127 Meridional plane, 4 Meridional velocity, 75, 83, 273 Methane, molecular weight of, 267 Microsoft Excel, 293, 298 Mixed flow, 3 Mixed-flow pump, 4, 5 Mixture dynamic pressure, 40 Molecular weight, of selected gases, 267 Mollier charts, 195 Momentum equation, 39, 41, 247 angular, 20–25, 296 computational fluid dynamics, 236 linear, 19–20 Moody diagram, 59, 81 Moss, Sanford, 2 N Navier-Stokes equations, 238 Net positive suction head (NPSH), 73, 80, 81, 202 Nitrogen, molecular weight of, 267 Nondimensional speed, 54 Nozzle loss coefficient, 155 Nozzles choked divergent, 37–38 convergent, 37–38

M Mach numbers, 14–15, 17–18 Madaras wind turbine, 213 Marine gas turbine, 6 Marly, France, 2 Mass flow coefficient, 54 Mass flow functions compressible, 16–18 isentropic flow tables for computing, 18

O One-dimensional flow analyses, 234 Outer iteration loop, 299 Outlet, boundary conditions, 240 Output head, 68, 79, 105 Overall pump efficiency, 69, 78 Overshot waterwheel, 2 Oxygen, molecular weight of, 267

316

P Parsons, Charlie, 2 Pelton wheel, 197–199, 277 bucket velocity diagram for, 198 energy transfer, 198 schematics of, 5 specific speed range, 57 Penstock, 197 Pfleiderer equation, 93, 265 Polytropic efficiency, 138 Potential energy, 68 Power, 56, 257 Power coefficient, 211 Power generation gas turbines combined-cycle, 8 simple cycle, 7 Power-plant cycle, thermal efficiency of, 189 Pratt & Whitney engines, 3 Premixing chambers, 38–39 Pressure, 14–15, 257 changes in isentropic free and vortex vortices, 270–274 stage pressure ratio, 135 stage pressure rise, 118–120, 134 stagnation, 14 static, 95, 96, 270–274 total, 14, 40, 120–121, 219–220, 228, 230, 273–274 Pressure ratio, 135 Pressure recovery axial momentum equation, 222 continuity equation, 221 in dump diffuser, 221–222 loss of total pressure and, 222 Pressure rise coefficient, 224–227 Pressure-compounded turbine, 167 Profile drag coefficient, 124 Profile loss coefficients, 129, 171, 172, 178, 180 for blading with zero inlet gas angle, 172 Propane, molecular weight of, 267 Propulsion gas turbines, 6, 7 Pump(s), 1, 3 centrifugal, 30–31 efficiency, 69, 78, 258 impeller, 4, 5, 65–67, 66 laws, 56 volute of, 77 R Radial equilibrium equation, 287–290, 297 Radial flow, 3

Index

Radial pressure gradient, 66 Radial velocity, 97–98 Radial-flow gas turbines, 147–160 basic theory, 148–157 design, 156 design parameters, 157 longitudinal section, 149 minimum number of blades, 157 relative eddies in, 158 thermodynamic processes in, 151 total-to-total efficiency of, 154 transverse section, 149 velocity diagrams for, 150 worked examples, 159–160 Radial-flow turbomachines, 4 Rankine cycle, 189 Rateau turbine, 167 Ratio of outlet pressure to inlet pressure, 55 Ratio of outlet temperature to inlet temperature, 55 Reaction turbines, 4, 167, 193–195 energy transfer, 195 stator of, 194 thermodynamic processes in, 194 velocity diagram for, 194 Reciprocating steam engine, 2 Relative air angle, 134–135, 136 Relative eddy, 259 Relative flow angles, 285–286 Relative flow velocity, 65, 284. See also Absolute flow velocity at blade inlet W1/U, 283–284 at blade outlet W2/U, 283–284 components, 26 dimensionless, 284 reaction turbines, 194 at shroud, 94 Relative total pressures, 173 Reynolds averaging, 237 Reynolds equations, 237–238 Reynolds number, 60, 81, 191, 241 Rolls-Royce, 3 Roman Empire, 1 Rotating pipe, 41–42 Rotation, 13 Rotational speed, 79 Rothalpy, 24 Rotor adjacent blades, 22–23 diameter, 54 loss coefficient, 155 profile loss coefficient, 171 secondary-flow loss coefficient, 173

317

Index

Rotor blades, 22 deviation, 178 diffusion factor, 1 enthalpy, 151 incidence, 178 optimum spacing, 179 preliminary design, 210 radial position, 210 Rotor temperature ratio, 152 Rural Electric Administration Act of 1936, 2 S Savonius wind turbine, 213 Scaling laws, 59 Secondary-flow loss coefficient, 171, 172, 172, 173, 180–181 Seine River, 2 Shaft torque, 74 design of, 74–77 Shaft work, 1 Shaft-power gas turbines, 6–8, 8 Shear velocity, 241, 242 Shroud diameter, 75, 83, 94 Shroud-tip ratio, 96 Similitude, 55–58 Single-stage centrifugal air compressor, 102–103 Single-stage compressor, 287–300 energy transfer, 102–103 with potential flow, 297–298 with specified flow angles and efficiency, 298–300 Slip coefficient, 67 centrifugal compressors, 92, 102, 106 centrifugal pumps, 76, 79 comparison of formulas for, 265 derivation of equation for, 259 Specific diameter, 54 Specific energy, 257 Specific heat, 257 Specific kinetic energy, section-average, 228 Specific speed, 53, 257 assumed, 80 critical, 73 dimensional, 83, 106, 258 suction, 81 Spouting velocity, 153, 176 Stage efficiency, 120, 135–137, 170, 176, 195 Stage pressure ratio, 135 Stage pressure rise, 118–120, 134 Stagnation pressure, 14–15; see also Total pressure

Stagnation temperature, 14; see also Total temperature Stanitz equation, 92–93, 265 Static pressure, 14–15 forced vortex, 270–274 free vortex, 270–274 Static temperature, 14 forced vortex, 270–274 free vortex, 270–274 section-average, 228 Stator loss coefficient, 171, 174, 181 mean absolute gas angle for, 173 reaction turbines, 194 Steam power plant, 8–9 history, 2 Ljungstrom turbine in, 148 Steam turbines, 3, 8–9, 189–195 design, 195 history, 2 impulse turbines, 190–193 reaction turbines, 193–195 schematics of, 5 Stodola equation, 265 Stolze, Franz, 2 Streamline curvature method (SCM), 249–250 Suction specific speed, 81 Suffix notation, 303 Sulzer Brothers, 2 Sumation convention, 303 T Tail race, 197 Tangential velocity, 102, 106 Temperature, 14–15 absolute, 94 changes in isentropic free and vortex vortices, 270–274 dynamic, 14 exhaust, 182 stagnation, 14 static, 14 total, 14–15 Temperature-entropy diagram, 190 Tensor algebra, 303 Tesla, Nikola, 148 Tesla turbine, 148 Thermal conductivity, 257 Thermal energy, 246 Thickness-to-chord ratio, 171 Thoma’s cavitation parameter, 202

318

Throughflow design, 248, 249–250 angular momentum equation, 296 axial-flow compressor layout and nomenclature, 293–295 continuity equation, 296 energy equation, 296 free-vortex design, 291–292 general vortex design, 292–293 performance parameters, 279 radial equilibrium equation for, 287–290 relative velocity at rotor inlet W1/U, 283 rotor inlet velocity diagrams, 290 simple radial equilibrium equation, 296–297 single-stage compressor with potential flow, 297–298 vortex energy equation, 290–291 Tip clearance ratio, 135–136 Tip diameter, 106, 159 Tip radius, 134 Tip speed, 71, 75, 79, 102, 106 Tip-clearance loss coefficient, 171, 173, 182 Torque, 212 Total derivative, 306–307 Total drag force, 40–41 Total head, 69 Total momentum flow rate, 39 Total pressure, 14, 40 forced vortex, 273–274 free vortex, 273–274 loss of, 120–121, 219–220, 230 section-average, 228 Total temperature, 14–15, 134–135 forced vortex, 273–274 free vortex, 273–274 Total-to-static efficiency, 153, 170, 171 Total-to-static pressure ratio, 15, 35–36 Total-to-static temperature ratio, 15 Total-to-total efficiency, 154, 170 Transient term, 236 Transonic blades, 133 Turbine(s), 1, 3, 212 axial-flow, 167–183 blades, 26, 27, 29, 115, 116, 170, 201, 211 hydraulic, 192–202, 197–202 radial-flow, 147–160 steam, 189–195 wind, 205–213 Turbochargers, 147 history, 2 Turbofan engine, 3, 6, 7 Turbojet engine, 2, 6, 7, 293

Index

Turbomachinery, 1 Turbomachinery aerodynamic design, 246–251, 248 airfoil design, 250–251 blade-to-blade analysis, 248 detailed design, 248–250 meanline design, 248 preliminary design, 247 3-D flow analysis, 250 3-D flow field, 246–247 throughflow design, 248, 249–250 Turbomachines defined, 1 degree of reaction of, 276 by functions, 3 gas turbines, 6–8 general classifications of, 3–4, 3 history of, 1–3 by types of fluid used, 3, 3 variables, 53–55 Turboprop engine, 6, 7 Turbulence modeling, 237–240. See also Computational fluid dynamics (CFD) boundary conditions, 244 Boussinesq hypothesis, 238 closure problem, 238 high-Reynolds-number two-equation κ-ε model, 239–240 Reynolds averaging, 237 Reynolds equations, 237–238 Turbulent kinetic energy, 238 Turbulent length scale, 238, 239 Turbulent Prandtl number, 239, 243 Turbulent velocity scale, 238–239 Turbulent viscosity, 238 U Undershot waterwheel, 2 V Vaneless diffuser, 96–99 Vavra method, 182–183 Vector identities, 307 Velocity coefficients, 15, 191, 194, 198 Velocity diagrams, 25–27 for axial-flow compressors and turbines, 278–286 for axial-flow gas turbines, 176 composite inlet-outlet, 278, 278 for compressor stage, 126

319

Index

dimensionless, 278–286 for Francis turbines, 199 for horizontal-axis machines, 209 at impeller exit, 93 at impeller inlet, 94 for impulse turbine, 190 at inlet and outlet, 66 for Kaplan turbine, 201 for radial-flow gas turbines, 150 for reaction turbines, 194 at rotor inlet, 123 at rotor outlet, 123 sign convention, 27 for vertical-axis machines, 213 Velocity triangles, 55 Velocity-compounded turbine, 167 Vertical-axis machines, 212–213 Vertical-axis wind turbines, 206 Virtual head, 65 Viscosity correction factors, 263 dynamic, 257 kinematic, 73, 73, 81, 257, 263 of liquids, 263 Volumetric efficiency, 69, 70, 78, 83 Von Ohain, Hans, 2 Vortex energy equation, 290–291 Vortices defined, 269 forced, 269–270 free, 269–270 static temperature and pressure changes, 270–274 total temperature and pressure changes, 273–274

W Wall alternative near-wall treatment, 243–244 boundary conditions, 241–244 logarithmic law of, 241–242 modified logarithmic law for velocity parallel to, 242–243 specifications of κp-εp model, 243 temperature, 243 Wall-function treatment, 241–244 Water, kinematic viscosity of, 81, 263 Water pumps, 1 Waterwheels, 1–2 Whirl component of velocity, 133 Whittle, Frank, 2 William the Conqueror, 2 Wind power, history, 2 Wind turbines, 205–213 actuator disk, 207 actuator theory, 205–209 efficiency, 208 as extended turbomachine, 205 Glauert’s theory, 210 history, 2 horizontal-axis, 206, 209–212 torque per blade, 213 vertical-axis machines, 206, 212–213 Wind-electric plants, history, 2 Windmills, 1 actuator theory, 206 history, 2 Z Zero-reaction turbines, 4