Turbomachinery Flow Physics and Dynamic Performance 9783642246746, 9783642246753, 3642246745

Table of contents :
Title
Preface
Contents
Nomenclature
Part I: Turbomachinery Flow Physics
Introduction, Turbomachinery, Applications, Types
Turbine
Compressor
Application of Turbomachines
Power Generation, Steam Turbines
Power Generation, Gas Turbines
Aircraft Gas Turbines
Diesel Engine Application
Classification of Turbomachines
Compressor Types
Turbine Types
Working Principle of a Turbomachine
References
Kinematics of Turbomachinery Fluid Motion
Material and Spatial Description of the Flow Field
Material Description
Jacobian Transformation Function and Its Material Derivative
Spatial Description
Translation, Deformation, Rotation
Reynolds Transport Theorem
References
Differential Balances in Turbomachinery
Mass Flow Balance in Stationary Frame of Reference
Incompressibility Condition
Differential Momentum Balance in Stationary Frame of Reference
Relationship between Stress Tensor and Deformation Tensor
Special Case: Euler Equation of Motion
Some Discussions on Navier-Stokes Equations
Energy Balance in Stationary Frame of Reference
Mechanical Energy
Thermal Energy Balance
Total Energy
Entropy Balance
Differential Balances in Rotating Frame of Reference
Velocity and Acceleration in Rotating Frame
Continuity Equation in Rotating Frame of Reference
Equation of Motion in Rotating Frame of Reference
Energy Equation in Rotating Frame of Reference
References
Integral Balances in Turbomachinery
Mass Flow Balance
Balance of Linear Momentum
Balance of Moment of Momentum
Balance of Energy
Energy Balance Special Case 1: Steady Flow
Energy Balance Special Case 2: Steady Flow, Constant Mass Flow
Application of Energy Balance to Turbomachinery Components
Application: Accelerated, Decelerated Flows
Application: Combustion Chamber
Application: Turbine, Compressor
Irreversibility and Total Pressure Losses
Application of Second Law to Turbomachinery Components
Flow at High Subsonic and Transonic Mach Numbers
Density Changes with Mach Number, Critical State
Effect of Cross-Section Change on Mach Number
Compressible Flow through Channels with Constant Cross Section
The Normal Shock Wave Relations
The Oblique Shock Wave Relations
Detached Shock Wave
Prandtl-Meyer Expansion
References
Theory of Turbomachinery Stages
Energy Transfer in Turbomachinery Stages
Energy Transfer in Relative Systems
General Treatment of Turbine and Compressor Stages
Dimensionless Stage Parameters
Relation between Degree of Reaction and Blade Height for a Normal Stage Using Simple Radial Equilibrium
Effect of Degree of Reaction on the Stage Configuration
Effect of Stage Load Coefficient on Stage Power
Unified Description of a Turbomachinery Stage
Unified Description of Stage with Constant Mean Diameter
Generalized Dimensionless Stage Parameters
Special Cases
Case 1, Constant Mean Diameter
Case 2, Constant Mean Diameter and Meridional Velocity Ratio
Increase of Stage Load Coefficient, Discussion
References
Turbine and Compressor Cascade Flow Forces
Blade Force in an Inviscid Flow Field
Blade Forces in a Viscous Flow Field
The Effect of Solidity on Blade Profile Losses
Relationship between Profile Loss Coefficient and Drag
Optimum Solidity
Optimum Solidity, by Pfeil
Optimum Solidity, by Zweifel
Generalized Lift-Solidity Coefficient
Lift-Solidity Coefficient for Turbine Stator
Turbine Rotor
References
Part II: Turbomachinery Losses, Efficiencies, Blades
Losses in Turbine and Compressor Cascades
Turbine Profile Loss
Viscous Flow in Compressor Cascade
Calculation of Viscous Flows
Boundary Layer Thicknesses
Boundary Layer Integral Equation
Application of Boundary Layer Theory to Compressor Blades
Effect of Reynolds Number
Stage Profile Losses
Trailing Edge Thickness Losses
Losses Due to Secondary Flows
Vortex Induced Velocity Field, Law of Bio -Savart, Preparatory
Calculation of Tip Clearance Secondary Flow Losses
Calculation of Endwall Secondary Flow Losses
Flow Losses in Shrouded Blades
Losses Due to Leakage Flow in Shrouds
Exit Loss
Trailing Edge Ejection Mixing Losses of Gas Turbine Blades
Calculation of Mixing Losses
Trailing Edge Ejection Mixing Losses
Effect of Ejection Velocity Ratio on Mixing Loss
Optimum Mixing Losses
Stage Total Loss Coefficient
Diffusers, Configurations, Pressure Recovery, Losses
Diffuser Configurations
Diffuser Pressure Recovery
Design of Short Diffusers
Some Guidelines for Designing High Efficiency Diffusers
References
Efficiency of Multi-stage Turbomachines
Polytropic Efficiency
Isentropic Turbine Efficiency, Recovery Factor
Compressor Efficiency, Reheat Factor
Polytropic versus Isentropic Efficiency
Incidence and Deviation
Cascade with Low Flow Deflection
Conformal Transformation
Flow Through an Infinitely Thin Circular Arc Cascade
Thickness Correction
Optimum Incidence
Effect of Compressibility
Deviation for High Flow Deflection
Calculation of Exit Flow Angle
References
Simple Blade Design
Conformal Transformation, Basics
Joukowsky Transformation
Circle-Flat Plate Transformation
Circle-Ellipse Transformation
Circle-Symmetric Airfoil Transformation
Circle-Cambered Airfoil Transformation
Compressor Blade Design
Low Subsonic Compressor Blade Design
Compressors Blades for High Subsonic Mach Number
Transonic, Supersonic Compressor Blades
Turbine Blade Design
Graphic Design of Camberline
Camberline Coordinates Using Bèzier Curve
Alternative Calculation Method
Assessment of Blades Aerodynamic Quality
References
Radial Equilibrium
Derivation of Equilibrium Equation
Application of Streamline Curvature Method
Step-by-Step Solution Procedure
Compressor Examples
Turbine Example, Compound Lean Design
Blade Lean Geometry
Calculation of Compound Lean Angle Distribution
Special Cases
Free Vortex Flow
Forced Vortex Flow
Flow with Constant Flow Angle
References
Part III: Turbomachinery Dynamic Performance
Nonlinear Dynamic Simulation of Turbomachinery Components and Systems
Theoretical Background
Preparation for Numerical Treatment
One-Dimensional Approximation
Time Dependent Equation of Continuity
Time Dependent Equation of Motion
Numerical Treatment
References
Generic Modeling of Turbomachinery Components and Systems
Generic Component, Modular Configuration
Plenum as Coupling Module
Group1 Modules: Inlet, Exhaust, Pipe
Group 2: Heat Exchangers, Combustion Chamber, After- Burners
Group 3: Adiabatic Compressor and Turbine Components
Group 4: Diabatic Turbine and Compressor Components
Group 5: Control System, Valves, Shaft, Sensors
System Configuration, Nonlinear Dynamic Simulation
Configuration of Systems of Non-linear Partial Differential Equations
References
Modeling of Inlet, Exhaust, and Pipe Systems
Unified Modular Treatment
Physical and Mathematical Modeling of Modules
Example: Dynamic Behavior of a Shock Tube
Shock Tube Dynamic Behavior
References
Modeling of Recuperators, Combustion Chambers, Afterburners
Modeling Recuperators
Recuperator Hot Side Transients
Recuperator Cold Side Transients
Coupling Condition Hot, Cold Side
Recuperator Heat Transfer Coefficient
Modeling Combustion Chambers
Mass Flow Transients
Temperature Transients
Combustion Chamber Heat Transfer
Example: Startup and Shutdown of a Combustion Chamber-Preheater System
Modeling of Afterburners
References
Modeling the Compressor Component, Design and Off-Design
Compressor Losses
Profile Losses
Diffusion Factor
Generalized Maximum Velocity Ratio for Cascade, Stage
Compressibility Effect
Shock Losses
Correlations for Boundary Layer Momentum Thickness
Influence of Different Parameters on Profile Losses
Compressor Design and Off-Design Performance
Stage-by-Stage and Row-by-Row Adiabatic Compression Process
Generation of Steady State Performance Map
Inception of Rotating Stall
Degeneration of Rotating Stall into Surge
Compressor Modeling Levels
Module Level 1: Using Performance Maps
Module Level 2: Row-by-Row Adiabatic Calculation Procedure
Module Level 3: Row-by-Row Diabatic Compression
References
Turbine Aerodynamic Design and Off-Design Performance
Stage-by-Stage and Row-by-Row Adiabatic Design and Off-Design Performance
Stage-by-Stage Calculation of Expansion Process
Row-by-Row Adiabatic Expansion
Off-Design Efficiency Calculation
Behavior under Extreme Low Mass Flows
Example: Steady Design and Off-Design Behavior of a Multi-Stage Turbine
Off-Design Calculation Using Global Turbine Characteristics Method
Modeling the Turbine Module for Dynamic Performance Simulation
Module Level 1: Using Turbine Performance Characteristics
Module Level 2: Row-by-Row Adiabatic Expansion Calculation
Module Level 3: Row-by-Row Diabatic Expansion
References
Gas Turbine Engines, Design and Dynamic Performance
Gas Turbine Steady Design Operation, Process
Gas Turbine Process
Improvement of Gas Turbine Thermal Efficiency
Non-linear Gas Turbine Dynamic Simulation
State of Dynamic Simulation, Background
Engine Components, Modular Concept, Module Identification
Levels of Gas Turbine Engine Simulations, Cross Coupling
Non-linear Dynamic Simulation Case Studies
Case Study 1: Compressed Air Energy Storage Gas Turbine
A Byproduct of Dynamic Simulation: Detailed Efficiency Calculation
Summary Part III, Further Development
References
Part IV: Turbomachinery CFD-Essentials
Basic Physics of Laminar-Turbulent Transition
Transition Basics: Stability of Laminar Flow
Laminar-Turbulent Transition, Fundamentals
Physics of an Intermittent Flow
Intermittent Behavior of Statistically Steady Flows
Turbulent/Non-turbulent Decisions
Intermittency Modeling for Flat Plate Boundary Layer
Physics of Unsteady Boundary Layer Transition
Experimental Simulation of the Unsteady Boundary Layer
Ensemble Averaging High Frequency Data
Intermittency Modeling for Periodic Unsteady Flow
Implementation of Intermittency into Navier Stokes Equations
Reynolds-Averaged Navier-Stokes Equations (RANS)
Conditioning RANS for Intermittency Implementation
References
Turbulent Flow and Modeling in Turbomachinery
Fundamentals of Turbulent Flows
Type of Turbulence
Correlations, Length and Time Scales
Spectral Representation of Turbulent Flows
Spectral Tensor, Energy Spectral Function
Averaging Fundamental Equations of Turbulent Flow
Averaging Conservation Equations
Equation of Turbulence Kinetic Energy
Equation of Dissipation of Kinetic Energy
Turbulence Modeling
Algebraic Model: Prandtl Mixing Length Hypothesis
Algebraic Model: Cebeci-Smith Model
Baldwin-Lomax Algebraic Model
One-Equation Model by Prandtl
Two-Equation Models
Grid Turbulence
Examples of Two-Equation Models
Internal Flow, Sudden Expansion
Internal Flow, Turbine Cascade
External Flow, Lift-Drag Polar Diagram
Case Study: Flow Simulation in a Rotating Turbine
Results, Discussion
Rotating Turbine RANS- URANS-Shortcomings, Discussion
References
Introduction into Boundary Layer Theory
Boundary Layer Approximations
Exact Solutions of Laminar Boundary Layer Equations
Zero Pressure Gradient Boundary Layer
Non-zero Pressure Gradient Boundary Layer
Polhausen Approximate Solution
Boundary Layer Theory, Integral Method
Boundary Layer Thicknesses
Boundary Layer Integral Equation
Turbulent Boundary Layers
Universal Wall Functions
Velocity Defect Function
Boundary Layer, Differential Treatment
Solution of Boundary Layer Equations
Measurement of Boundary Flow, Basic Techniques
Experimental Techniques
Examples: Calculations, Experiments
Steady State Velocity Calculations
Periodic Unsteady Inlet Flow Condition
Application of 􀈛-􀈦 Model to Boundary Layer
Parameters Affecting Boundary Layer
Parameter Variations, General Remarks
Effect of Periodic Unsteady Flow
References
A Vector and Tensor Analysis in Turbomachinery Fluid Mechanics
Tensors in Three-Dimensional Euclidean Space
Index Notation
Vector Operations: Scalar, Vector and Tensor Products
Scalar Product
Vector or Cross Product
Tensor Product
Contraction of Tensors
Differential Operators in Fluid Mechanics
Substantial Derivatives
Differential Operator
Operator Applied to Different Functions
Scalar Product of and V
Vector Product
Tensor Product
Scalar Product of and a Second Order Tensor
Eigenvalue and Eigenvector of a Second Order Tensor
References
B Tensor Operations in Orthogonal Curvilinear Coordinate Systems
Change of Coordinate System
Co- and Contravariant Base Vectors, Metric Coefficients
Physical Components of a Vector
Derivatives of the Base Vectors, Christoffel Symbols
Spatial Derivatives in Curvilinear Coordinate System
Application of to Zeroth Order Tensor Functions
Application of to First and Second Order Tensor Functions
Application Example 1: Inviscid Incompressible Flow Motion
Equation of Motion in Curvilinear Coordinate Systems
Special Case: Cylindrical Coordinate System
Base Vectors, Metric Coefficients
Christoffel Symbols
Introduction of Physical Components
Application Example 2: Viscous Flow Motion
Equation of Motion in Curvilinear Coordinate Systems
Special Case: Cylindrical Coordinate System
References
Index

Citation preview

Turbomachinery Flow Physics and Dynamic Performance

Meinhard T. Schobeiri

Turbomachinery Flow Physics and Dynamic Performance Second and Enhanced Edition

With 433 Figures

ABC

Author Prof. Dr.-Ing. Meinhard T. Schobeiri Department of Mechanical Engineering Texas A&M University USA

ISBN 978-3-642-24674-6 e-ISBN 978-3-642-24675-3 DOI 10.1007/978-3-642-24675-3 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012935425 c Springer-Verlag Berlin Heidelberg 2005, 2012  This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Second Edition

The first edition of this book appeared in 2005 and was concentrated on turbomachinery flow physics, design, off-design and nonlinear dynamics performance. As I mentioned in the preface to the first edition, in dealing with three-dimensional turbulent flows through a turbomachinery component, for the reasons I outlined, I refrained from presenting a multitude of existing turbulence models. In the meantime, only a few of those models have survived exhibiting the standard working tools implemented in almost all computational tools dealing with turbomachinery flow simulations. Experienced turbomachinery aerodynamicist knowledgeable of the underlying physics of the models in existing commercial codes is able to appropriately interpret the results. Without this knowledge, however, the user of commercial codes may tend to uncritically accept computational results as infallible. This issue has been addressed in the current second edition. While the unifying character of the book structure is maintained, all chapters have undergone a rigorous update and enhancement. Accounting for the need of the turbomachinery community in general and my students in particular, I added three chapters, Chapters 19, 20 and 21, that deal with laminar turbulent transition, turbulence and boundary layer. In these chapters, I tried to provide the reader with the physics underlying the computational tools of these subject areas. Since the turbulence and transition models are still far from being complete, I strongly recommend the readers to stay updated on these issues. With this edition, the readers have a solid source of information for designing state-of-the art turbomachinery components and systems at hand. The current edition has four parts. While Parts I through III contain substantial revisions and enhancements, Part IV, which contains Chapter 19, 20 and 21, is dedicated to computational aspects of turbomachinery design. In each of these new chapters, several case studies are presented that help the reader better understand the physics of computational tools. Here, as in the first edition, typing numerous equations, errors may occur. I tried hard to eliminate typing, spelling and other errors, but I have no doubt that some remain to be found by readers. In this case, I sincerely appreciate the reader notifying me of any mistakes found; the electronic address is given below. I also welcome any comments or suggestions regarding the improvement of future editions of the book. My students and numerous colleagues around the world have encouraged me to write this current edition. For their encouragement and suggestions, I would like to express my sincere thanks. College Station, May 2011

Meinhard T. Schobeiri [email protected]

Preface to the First Edition

Turbomachinery performance, aerodynamics, and heat transfer have been subjects of continuous changes for the last seven decades. Stodola was the fist to make a successful effort to integrate the treatment of almost all components of steam and gas turbines in a single volume monumental work. Almost half a century later, Traupel presented the third and last edition of his work to the turbomachinery community. In his two-volume work, Traupel covered almost all relevant aspects of turbomachinery aerodynamics, performance, structure, rotordynamics, blade vibration, and heat transfer with the information available up to the mid-seventies. In the meantime, the introduction of high speed computers and sophisticated computational methods has significantly contributed to an exponential growth of information covering almost all aspects of turbomachinery design. This situation has lead to a growing tendency in technical specialization causing the number of engineers with the state-of-the-art combined knowledge of major turbomachinery aspects such as, aerodynamics, heat transfer, combustion, rotordynamics, and structure, to diminish. Another (not minor) factor in the context of specialization is the use of “black boxes” in engineering in general and in turbomachinery in particular. During my 25 years of turbomachinery R&D experience, I have been encountering engineers who can use commercial codes for calculating the complex turbomachinery flow field without knowing the underlying physics of the code they use. This circumstance constituted one of the factors in defining the framework of this book, which aims at providing the students and practicing turbomachinery design engineers with a solid background in turbomachinery flow physics and performance. As a consequence, it does not cover all aspects of turbomachinery design. It is primarily concerned with the fundamental turbomachinery flow physics and its application to different turbomachinery components and systems. I have tried to provide the turbomachinery community, including students and practicing engineers, with a lasting foundation on which they can build their specialized structures. I started working on this book when I left Brown Boveri, Gas Turbine Division in Switzerland (1987) and joined Texas A&M University. A part of my lecture notes in graduate level turbomachinery constitutes the basics of this book, which consists of three parts. Part I encompassing Chapters 1 to 6 deals with turbomachinery flow physics. Explaining the physics of a highly three-dimensional flow through a turbine or a compressor stage requires adequate tools. Tensor analysis is a powerful tool to deal with this type of flow. Almost all graduate level fluid mechanics and computational fluid dynamics (CFD) courses at Texas A&M University, and most likely at other universities, make use of tensor analysis. For the reader, who might not be familiar with the tensor analysis, I tailored the quintessential of tensor analysis in Appendixes A and B to the need of the reader who is willing to spend a few hours on this topic.

VIII

Preface

Chapters 2 and 3 deal with the Kinematics and Differential Balances in turbomachinery in three-dimensional form. In dealing with three-dimensional turbulent flows through a turbomachinery component, I refrained from presenting a multitude of existing turbulence models for three reasons. Firstly, the models are changing continuously; the one which is thought today to be the ultimo ratio may be obsolete tomorrow. Secondly, there is a vast amount of papers and publications devoted to turbulence modeling and numerical methods in turbomachinery. In my view, summarizing these models and methods without getting into details is not appropriate for a text. Thirdly, the emergence of the direct numerical simulation (DNS) as the ultimate solution method, though not very practical at the time being, will in the near future undoubtedly eliminate the problems that are inherently associated with Reynolds averaged Navier Stokes equations (RANS). Considering this situation, I focused my effort on briefly presenting the RANS-equations and introducing the intermittency function. The resulting equations containing the product of the intermittency and the Reynolds stress tensor clearly illustrate how the accuracy of these two models affect the numerical results. As a logical followup, Chapter 4 extensively discusses the integral balances in turbomachinery. Chapters 5 and 6 deal with a unified treatment of energy transfer, stage characteristics, and cascade and stage aerodynamics. Contrary to the traditional approach that treats turbine and compressor stages of axial or radial configuration differently, I have tried to treat these components using the same set of equations. Part II of this book containing Chapters 7 through 11 starts with the treatment of cascade and stage efficiency and loss determination from a physically plausible point of view. I refrained from presenting recipe-types of empirical formulas that have no physical foundation. Chapters 10 and 11 deal with simple designs of compressors and turbine blades and calculations of incidence and deviation. Radial equilibrium is discussed in Chapter 11, which concludes Part II. Part III of the book is entirely dedicated to dynamic performance of turbomachinery components and systems. Particular attention is paid to gas turbine components, their individual modeling, and integration into the gas turbine system. It includes Chapter 12 to 18. Chapter 12 introduces the non-linear dynamic simulation of turbomachinery systems and its theoretical background. Starting from a set of general four dimensional partial differential equations in temporal-spatial domain, two-dimensional equation sets are derived that constitute the basis for component modeling. The following Chapter 13 deals with generic modeling of turbomachinery components and systems followed by Chapters 14, 15, 16, and 17 in which individual components ranging from the inlet nozzle to the compressor, combustion chamber, turbine, and exhaust diffuser are modeled. In modeling compressor and turbine components, non-linear adiabatic and diabatic expansion and compression calculation methods are presented. Chapter 18 treats gas turbine engines, design and dynamic performance. Three representative case studies conclude this chapter. In preparing Part III, I tried to be as concrete as possible by providing examples for each individual component. In typing several thousand equations, errors may occur. I tried hard to eliminate typing, spelling and other errors, but I have no doubt that some remain to be found by readers. In this case, I sincerely appreciate the reader notifying me of any mistakes

Preface

IX

found; the electronic address is given below. I also welcome any comments or suggestions regarding the improvement of future editions of the book. My sincere thanks are due to many fine individuals and institutions. First and foremost, I would like to thank the faculty of the Technische Universität Darmstadt, from whom I received my entire engineering education. I finalized major chapters of the manuscript during my sabbatical in Germany where I received the Alexander von Humboldt Prize. I am indebted to the Alexander von Humboldt Foundation for this Prize and the material support for my research sabbatical in Germany. My thanks are extended to Prof. Bernd Stoffel, Prof. Ditmar Hennecke, and Dipl. Ing. Bernd Matyschok for providing me with a very congenial working environment. I truly enjoyed interacting with these fine individuals. NASA Glenn Research Center sponsored the development of the nonlinear dynamic code GETRAN which I used to simulate cases in Part III. I wish to extend my thanks to Mr. Carl Lorenzo, Chief of Control Division, Dr. D. Paxson, and the administration of the NASA Glenn Research Center. I also thank Dr. Richard Hearsey for providing me with a three-dimensional compressor blade design example that he obtained from his streamline curvature program. I also would like to extend my thanks to Dr. Arthur Wennerstom for providing me with the updated theory on the streamline curvature method. I am also indebted to the TAMU administration for partially supporting my sabbatical that helped me in finalizing the book. Special thanks are due to Mrs. Mahalia Nix, who helped me in cross-referencing the equations and figures and rendered other editorial assistance. Last but not least, my special thanks go to my family, Susan and Wilfried for their support throughout this endeavor. College Station, Texas, September 2003

Meinhard T. Schobeiri [email protected]

Contents

I Turbomachinery Flow Physics 1 Introduction, Turbomachinery, Applications, Types . . . . . . . 3 1.1 1.2 1.3

Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Application of Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Power Generation, Steam Turbines . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Power Generation, Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Aircraft Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 Diesel Engine Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Classification of Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Compressor Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.2 Turbine Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Working Principle of a Turbomachine . . . . . . . . . . . . . . . . . . . . . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Kinematics of Turbomachinery Fluid Motion . . . . . . . . . . . 15 2.1

Material and Spatial Description of the Flow Field . . . . . . . . . . . . . . 2.1.1 Material Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Jacobian Transformation Function and Its Material Derivative 2.1.3 Spatial Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Translation, Deformation, Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 20 21 25 27

3 Differential Balances in Turbomachinery . . . . . . . . . . . . . . . . 29 3.1 Mass Flow Balance in Stationary Frame of Reference . . . . . . . . . . . . 3.1.1 Incompressibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Differential Momentum Balance in Stationary Frame of Reference . 3.2.1 Relationship between Stress Tensor and Deformation Tensor 3.2.2 Navier-Stokes Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Special Case: Euler Equation of Motion . . . . . . . . . . . . . . . . . 3.3 Some Discussions on Navier-Stokes Equations . . . . . . . . . . . . . . . . . 3.4 Energy Balance in Stationary Frame of Reference . . . . . . . . . . . . . . . 3.4.1 Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Thermal Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 31 32 34 36 38 41 42 42 45

XII

Contents

3.4.3 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Differential Balances in Rotating Frame of Reference . . . . . . . . . . . . 3.5.1 Velocity and Acceleration in Rotating Frame . . . . . . . . . . . . . 3.5.2 Continuity Equation in Rotating Frame of Reference . . . . . . . 3.5.3 Equation of Motion in Rotating Frame of Reference . . . . . . . . 3.5.4 Energy Equation in Rotating Frame of Reference . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 49 50 50 52 53 55 57

4 Integral Balances in Turbomachinery . . . . . . . . . . . . . . . . . . . . . 59 4.1 Mass Flow Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Balance of Moment of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Balance of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.1 Energy Balance Special Case 1: Steady Flow . . . . . . . . . . . . . 77 4.4.2 Energy Balance Special Case 2: Steady Flow, Constant Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Application of Energy Balance to Turbomachinery Components . . . 78 4.5.1 Application: Accelerated, Decelerated Flows . . . . . . . . . . . . . 79 4.5.2 Application: Combustion Chamber . . . . . . . . . . . . . . . . . . . . . 80 4.5.3 Application: Turbine, Compressor . . . . . . . . . . . . . . . . . . . . . 81 4.5.3.1 Uncooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5.3.2 Cooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.3.3 Uncooled Compressor . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Irreversibility and Total Pressure Losses . . . . . . . . . . . . . . . . . . . . . . 84 4.6.1 Application of Second Law to Turbomachinery Components . . 85 4.7 Flow at High Subsonic and Transonic Mach Numbers . . . . . . . . . . . 87 4.7.1 Density Changes with Mach Number, Critical State . . . . . . . . . 88 4.7.2 Effect of Cross-Section Change on Mach Number . . . . . . . . . 93 4.7.3 Compressible Flow through Channels with Constant Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.7.4 The Normal Shock Wave Relations . . . . . . . . . . . . . . . . . . . . 109 4.7.5 The Oblique Shock Wave Relations . . . . . . . . . . . . . . . . . . . . 115 4.7.6 The Detached Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.7.7 Prandtl-Meyer Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Theory of Turbomachinery Stages . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 5.2 5.3 5.4 5.5 5.6

Energy Transfer in Turbomachinery Stages . . . . . . . . . . . . . . . . . . . Energy Transfer in Relative Systems . . . . . . . . . . . . . . . . . . . . . . . . General Treatment of Turbine and Compressor Stages . . . . . . . . . . Dimensionless Stage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation between Degree of Reaction and Blade Height . . . . . . . . Effect of Degree of Reaction on the Stage Configuration . . . . . . . .

123 124 125 129 131 134

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XIII

Effect of Stage Load Coefficient on Stage Power . . . . . . . . . . . . . . Unified Description of a Turbomachinery Stage . . . . . . . . . . . . . . . 5.8.1 Unified Description of Stage with Constant Mean Diameter . 5.8.2 Generalized Dimensionless Stage Parameters . . . . . . . . . . . . 5.9 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Case 1, Constant Mean Diameter . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Case 2, Constant Mean Diameter and Meridional Velocity Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Increase of Stage Load Coefficient, Discussion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136 137 137 138 140 141

5.7 5.8

141 140 144

6 Turbine and Compressor Cascade Flow Forces . . . . . . . . . . 145 6.1 6.2 6.3 6.4 6.5

Blade Force in an Inviscid Flow Field . . . . . . . . . . . . . . . . . . . . . . . Blade Forces in a Viscous Flow Field . . . . . . . . . . . . . . . . . . . . . . . The Effect of Solidity on Blade Profile Losses . . . . . . . . . . . . . . . . Relationship Between Profile Loss Coefficient and Drag . . . . . . . . Optimum Solidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Optimum Solidity, by Pfeil . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Optimum Solidity, by Zweifel . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Generalized Lift-Solidity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Lift-Solidity Coefficient for Turbine Stator . . . . . . . . . . . . . . 6.6.2 Turbine Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 150 156 156 158 158 158 162 164 168 171

II Turbomachinery Losses, Efficiencies, Blades 7 Losses in Turbine and Compressor Cascades . . . . . . . . . . . . 175 7.1 7.2

7.3 7.4

7.5 7.6

Turbine Profile Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscous Flow in Compressor Cascade . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Calculation of Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Boundary Layer Integral Equation . . . . . . . . . . . . . . . . . . . . . 7.2.4 Application of Boundary Layer Theory to Compressor Blades 7.2.5 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Stage Profile Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trailing Edge Thickness Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Losses Due to Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Vortex Induced Velocity Field, Law of Bio-Savart . . . . . . . . 7.4.2 Calculation of Tip Clearance Secondary Flow Losses . . . . . . 7.4.3 Calculation of Endwall Secondary Flow Losses . . . . . . . . . . Flow Losses in Shrouded Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Losses Due to Leakage Flow in Shrouds . . . . . . . . . . . . . . . . Exit Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176 178 178 179 181 182 186 186 186 192 194 197 200 204 204 211

XIV

Contents

7.7

Trailing Edge Ejection Mixing Losses of Gas Turbine Blades . . . . 7.7.1 Calculation of Mixing Losses . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Trailing Edge Ejection Mixing Losses . . . . . . . . . . . . . . . . . . 7.7.3 Effect of Ejection Velocity Ratio on Mixing Loss . . . . . . . . . 7.7.4 Optimum Mixing Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Stage Total Loss Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Diffusers, Configurations, Pressure Recovery, Losses . . . . . . . . . . . 7.9.1 Diffuser Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Diffuser Pressure Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Design of Short Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.4 Some Guidelines for Designing High Efficiency Diffusers . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212 212 217 217 219 219 220 221 222 225 228 229

8 Efficiency of Multi-stage Turbomachines . . . . . . . . . . . . . . . . 231 8.1 Polytropic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Isentropic Turbine Efficiency, Recovery Factor . . . . . . . . . . . . . . . 8.3 Compressor Efficiency, Reheat Factor . . . . . . . . . . . . . . . . . . . . . . . 8.4 Polytropic versus Isentropic Efficiency . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 234 237 239 240

9 Incidence and Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1

Cascade with Low Flow Deflection . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Flow Through an Infinitely Thin Circular Arc Cascade . . . . . 9.1.3 Thickness Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Optimum Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Effect of Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Deviation for High Flow Deflection . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Calculation of Exit Flow Angle . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 250 256 256 258 259 261 263

10 Simple Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1 Conformal Transformation, Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Joukowsky Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Circle-Flat Plate Transformation . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Circle-Ellipse Transformation . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Circle-Symmetric Airfoil Transformation . . . . . . . . . . . . . . . 10.1.5 Circle-Cambered Airfoil Transformation . . . . . . . . . . . . . . . . 10.2 Compressor Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Low Subsonic Compressor Blade Design . . . . . . . . . . . . . . . 10.2.2 Compressors Blades for High Subsonic Mach Number . . . . 10.2.3 Transonic, Supersonic Compressor Blades . . . . . . . . . . . . . . 10.3 Turbine Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Graphic Design of Camberline . . . . . . . . . . . . . . . . . . . . . . . .

265 267 267 268 269 271 272 273 279 280 281 282

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10.3.2 Camberline Coordinates Using Bèzier Curve . . . . . . . . . . . . . 10.3.3 Alternative Calculation Method . . . . . . . . . . . . . . . . . . . . . . . 10.4 Assessment of Blades Aerodynamic Quality . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 285 287 290

11 Radial Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 11.1 Derivation of Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Application of Streamline Curvature Method . . . . . . . . . . . . . . . . . 11.2.1 Step-by-step solution procedure . . . . . . . . . . . . . . . . . . . . . . . 11.2 Compressor Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Turbine Example, Compound Lean Design . . . . . . . . . . . . . . . . . . . 11.3.1 Blade Lean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Calculation of Compound Lean Angle Distribution . . . . . . . . 11.3.3 Example: Three-Stage Turbine Design . . . . . . . . . . . . . . . . . 11.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Free Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Forced vortex flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Flow with constant flow angle . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

292 300 302 306 309 310 311 313 316 316 317 318 319

III Turbomachinery Dynamic Performance 12 Dynamic Simulation of Turbomachinery Components . . . 323 12.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Preparation for Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . 12.3 One-Dimensional Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Time Dependent Equation of Continuity . . . . . . . . . . . . . . . . 12.3.2 Time Dependent Equation of Motion . . . . . . . . . . . . . . . . . . . 12.3.3 Time Dependent Equation of Total Energy . . . . . . . . . . . . . . 12.4 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

324 330 331 331 333 334 339 340

13 Generic Modeling of Turbomachinery Components . . . . . 341 13.1 Generic Component, Modular Configuration . . . . . . . . . . . . . . . . . . 13.1.1 Plenum as Coupling Module . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Group 1: Modules: Inlet, Exhaust, Pipe . . . . . . . . . . . . . . . . . 13.1.3 Group 2: Recuperators, Combustion Chambers, Afterburners 13.1.4 Group 3: Adiabatic Compressor and Turbine Components . . 13.1.5 Group 4: Diabatic Turbine and Compressor Components . . . 13.1.6 Group 5: Control System, Valves, Shaft, Sensors . . . . . . . . . 13.2 System Configuration, Nonlinear Dynamic Simulation . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

342 343 345 346 348 350 352 352 356

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14 Modeling of Inlet, Exhaust, and Pipe Systems . . . . . . . . . . . . 357 14.1 Unified Modular Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Physical and Mathematical Modeling of Modules . . . . . . . . . . . . . . 14.3 Example: Dynamic behavior of a Shock Tube . . . . . . . . . . . . . . . . 14.3.1 Shock Tube Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357 357 360 361 365

15 Modeling of Recuperators, Combustors, Afterburners . . . 367 15.1 Modeling Recuperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Recuperator Hot Side Transients . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Recuperator Cold Side Transients . . . . . . . . . . . . . . . . . . . . . 15.1.3 Coupling Condition Hot, Cold Side . . . . . . . . . . . . . . . . . . . . 15.1.4 Recuperator Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . 15.2 Modeling Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Mass Flow Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Temperature Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . 15.3 Example: Startup and Shutdown of a Combustion Chamber . . . . . . 15.4 Modeling of Afterburners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

368 369 369 370 371 372 373 374 376 378 381 382

16 Modeling of Compressor Component, Design, Off-Design 383 16.1 Compressor Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Profile Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Diffusion Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 Generalized Maximum Velocity Ratio for Cascade, Stage . . 16.1.4 Compressibility Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.5 Shock Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.6 Correlations for Boundary Layer Momentum Thickness . . . . 16.1.7 Influence of Different Parameters on Profile Losses . . . . . . . 16.1.7.1 Mach Number Effect . . . . . . . . . . . . . . . . . . . . . . . . 16.1.7.2 Reynolds Number Effect . . . . . . . . . . . . . . . . . . . . . 16.2 Compressor Design and Off-Design Performance . . . . . . . . . . . . . . 16.2.1 Stage-by-Stage and Row-by-Row Compression Process . . . 16.2.1.1 Stage-by-Stage Calculation of Compression Process 16.2.1.2 Row-by-Row Adiabatic Compression . . . . . . . . . . . 16.2.1.3 Off-Design Efficiency Calculation . . . . . . . . . . . . . 16.3 Generation of Steady State Performance Map . . . . . . . . . . . . . . . . . 16.3.1 Inception of Rotating Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Degeneration of Rotating Stall into Surge . . . . . . . . . . . . . . . 16.4 Compressor Modeling Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Module Level 1: Using Performance Maps . . . . . . . . . . . . . . 16.4.1.1 Quasi-dynamic Modeling Using Performance Maps 16.4.1.2 Simulation Example: . . . . . . . . . . . . . . . . . . . . . . . .

384 385 387 391 393 397 406 407 407 408 409 409 409 411 415 418 420 422 423 424 426 427

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16.4.2 Module Level 2: Row-by-Row Adiabatic Compression . . . . 16.4.2.1 Active Surge Prevention by Adjusting the Stator Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2.2 Simulation Example: Surge and Its Prevention . . . . 16.4.3 Module Level 3: Row-by-Row Diabatic Compression . . . . . 16.4.3.1 Description of Diabatic Compressor Module . . . . . 16.4.3.2 Heat Transfer Closure Equations: . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429 430 431 436 437 439 442

17 Turbine Aerodynamic Design, Performance . . . . . . . . . . . . . 445 17.1 Stage-by-Stage and Row-by-Row Design . . . . . . . . . . . . . . . . . . . . . 17.1.1 Stage-by-Stage Calculation of Expansion Process . . . . . . . . 17.1.2 Row-by-Row Adiabatic Expansion . . . . . . . . . . . . . . . . . . . . 17.1.3 Off-Design Efficiency Calculation . . . . . . . . . . . . . . . . . . . . . 17.1.4 Behavior under Extreme Low Mass Flows . . . . . . . . . . . . . . 17.1.5 Example: Steady Design and Off-Design Behavior . . . . . . . 17.2 Off-Design Calculation Using Global Turbine Characteristics . . . . 17.3 Modeling of Turbine Module for Dynamic Performance Simulation 17.3.1 Module Level 1: Using Turbine Performance Characteristics 17.3.2 Module Level 2: Row-by-Row Expansion Calculation . . . . . 17.3.3 Module Level 3: Row-by-Row Diabatic Expansion . . . . . . . . 17.3.3.1 Description of Diabatic Turbine Module, First Method: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3.2 Description of Module, Second Method . . . . . . . . . 17.3.3.3 Heat Transfer Closure Equations . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 448 449 454 456 459 461 462 463 464 465 467 469 471 472

18 Gas Turbine Engines, Design and Dynamic Performance 473 18.1 Gas Turbine Steady Design Operation, Process . . . . . . . . . . . . . . . 18.1.1 Gas Turbine Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Improvement of Gas Turbine Thermal Efficiency . . . . . . . . . 18.2 Non-Linear Gas Turbine Dynamic Simulation . . . . . . . . . . . . . . . . . 18.2.1 State of Dynamic Simulation, Background . . . . . . . . . . . . . . 18.3 Engine Components, Modular Concept, Module Identification . . . . 18.4 Levels of Gas Turbine Engine Simulations, Cross Coupling . . . . . . 18.5 Non-Linear Dynamic Simulation Case Studies . . . . . . . . . . . . . . . . 18.5.1 Case Study 1: Compressed Air Energy Storage Gas Turbine . 18.5.1.1 Simulation of Emergency Shutdown . . . . . . . . . . . . 18.5.2 Case Study 2: Power Generation Gas Turbine Engine . . . . . . 18.5.3 Case Study 3: Simulation of a Multi-Spool Gas Turbine . . . 18.6 A Byproduct of Dynamic Simulation: Detailed Calculation . . . . . . 18.7 Summary Part III, Further Development . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475 477 483 485 486 487 493 494 495 497 499 504 507 510 511

XVIII

Contents

IV Turbomachinery CFD-Essentials 19 Basic Physics of Laminar-Turbulent Transition . . . . . . . . . . 515 19.1 Transition Basics: Stability of Laminar Flow . . . . . . . . . . . . . . . . . 19.2 Laminar-Turbulent Transition, Fundamentals . . . . . . . . . . . . . . . . . 19.3 Physics of an Intermittent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 Intermittent Behavior of Statistically Steady Flows . . . . . . . 19.3.2 Turbulent/Non-turbulent Decisions . . . . . . . . . . . . . . . . . . . . 19.3.3 Intermittency Modeling for Flat Plate Boundary Layer . . . . 19.4 Physics of Unsteady Boundary Layer Transition . . . . . . . . . . . . . . . 19.4.1 Experimental Simulation of the Unsteady Boundary Layer . 19.4.2 Ensemble Averaging High Frequency Data . . . . . . . . . . . . . 19.4.3 Intermittency Modeling for Periodic Unsteady Flow . . . . . . . 19.5 Implementation of Intermittency into Navier Stokes Equations . . . . 19.5.1 Reynolds-Averaged Navier-Stokes Equations (RANS) . . . . . 19.5.2 Conditioning RANS for Intermittency Implementation . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

515 515 518 519 520 524 525 527 530 533 536 536 540 542

20 Turbulent Flow and Modeling in Turbomachinery . . . . . . . 545 20.1 Fundamentals of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.1 Type of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.2 Correlations, Length and Time Scales . . . . . . . . . . . . . . . . . . 20.1.3 Spectral Representation of Turbulent Flows . . . . . . . . . . . . . 20.1.4 Spectral Tensor, Energy Spectral Function . . . . . . . . . . . . . . 20.2 Averaging Fundamental Equations of Turbulent Flow . . . . . . . . . . 20.2.1 Averaging Conservation Equations . . . . . . . . . . . . . . . . . . . . 20.2.1.1Averaging the Continuity Equation . . . . . . . . . . . . . 20.2.1.2 Averaging the Navier-Stokes Equation . . . . . . . . . . 20.2.1.3 Averaging the Mechanical Energy Equation . . . . . . 20.2.1.4 Averaging the Thermal Energy Equation . . . . . . . . 20.2.1.5 Averaging the Total Enthalpy Equation . . . . . . . . . 20.2.1.6 Quantities Resulting from Averaging to Be Modeled 20.2.2 Equation of Turbulence Kinetic Energy . . . . . . . . . . . . . . . . . 20.2.3 Equation of Dissipation of Kinetic Energy . . . . . . . . . . . . . . . 20.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Algebraic Model: Prandtl Mixing Length Hypothesis . . . . . . 20.3.2 Algebraic Model: Cebeci-Smith Model . . . . . . . . . . . . . . . . . 20.3.3 Baldwin-Lomax Algebraic Model . . . . . . . . . . . . . . . . . . . . . 20.3.4 One- Equation Model by Prandtl . . . . . . . . . . . . . . . . . . . . . . 20.3.5 Two-Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.5.1 Two-Equation k-J Model . . . . . . . . . . . . . . . . . . . . . 20.3.5.2 Two-Equation k-Ȧ-Model . . . . . . . . . . . . . . . . . . . . 20.3.5.3 Two-Equation k-Ȧ-SST-Model . . . . . . . . . . . . . . . . 20.4 Grid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

545 547 548 555 558 560 561 561 561 562 563 565 568 570 576 577 578 584 585 586 587 587 589 590 592

Contents

XIX

20.5 Examples of Two-Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1 Internal Flow, Sudden Expansion . . . . . . . . . . . . . . . . . . . . . . 20.5.2 Internal Flow, Turbine Cascade . . . . . . . . . . . . . . . . . . . . . . . 20.5.3 External Flow, Lift-Drag Polar Diagram . . . . . . . . . . . . . . . . 20.5.4 Case Study: Flow Simulation in a Rotating Turbine . . . . . . . 20.5.5 Results, Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.6 Rotating Turbine RANS, URANS-Shortcomings, Discussion References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

594 594 595 596 597 605 614 615

21 Introduction into Boundary Layer Theory . . . . . . . . . . . . . . . 619 21.1 Boundary Layer Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Exact Solutions of Laminar Boundary Layer Equations . . . . . . . . . 21.2.1 Zero Pressure Gradient Boundary Layer . . . . . . . . . . . . . . . 21.2.2 Non-zero Pressure Gradient Boundary Layer . . . . . . . . . . . . . 21.2.3 Polhausen Approximate Solution . . . . . . . . . . . . . . . . . . . . . . 21.3 Boundary Layer Theory, Integral Method . . . . . . . . . . . . . . . . . . . . 21.3.1 Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Boundary Layer Integral Equation . . . . . . . . . . . . . . . . . . . . . 21.4 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.1 Universal Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 Velocity Defect Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Boundary Layer, Differential Treatment . . . . . . . . . . . . . . . . . . . . . 21.5.1 Solution of Boundary Layer Equations . . . . . . . . . . . . . . . . . 21.6 Measurement of Boundary Flow, Basic Techniques . . . . . . . . . . . . 21.6.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.1.1 HWA Operation Modes, Calibration . . . . . . . . . . . . 21.6.1.2 HWA Averaging, Sampling Data . . . . . . . . . . . . . . 21.7 Examples: Calculations, Experiments . . . . . . . . . . . . . . . . . . . . . . . 21.7.1 Steady State Velocity Calculations . . . . . . . . . . . . . . . . . . . . 21.7.1.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . 21.7.1.2 Heat Transfer Calculation, Experiment . . . . . . . . . . 21.7.2 Periodic Unsteady Inlet Flow Condition . . . . . . . . . . . . . . . . 21.7.2.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . 21.7.2.2 Heat Transfer Calculation, Experiment . . . . . . . . . . 21.7.3 Application of ț-Ȧ Model to Boundary Layer . . . . . . . . . . . . 21.8 Parameters Affecting Boundary Layer . . . . . . . . . . . . . . . . . . . . . . 21.8.1 Parameter Variations, General Remarks . . . . . . . . . . . . . . . . 21.8.2 Effect of Periodic Unsteady Flow . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

620 623 624 626 629 631 631 633 637 640 643 648 653 653 653 654 655 657 657 659 660 661 664 666 667 667 668 672 680

A Vector and Tensor Analysis in Turbomachinery . . . . . . . . . 685 A. 1 Tensors in Three-Dimensional Euclidean Space . . . . . . . . . . . . . . . 685 A.1.1 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 A.2 Vector Operations: Scalar, Vector and Tensor Products . . . . . . . . . 687

XX

Contents

A.2.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Vector or Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Contraction of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Differential Operators in Fluid Mechanics . . . . . . . . . . . . . . . . . . . . A.4.1 Substantial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.2 Differential Operator / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Operator / Applied to Different Functions . . . . . . . . . . . . . . . . . . . A.5.1 Scalar Product of / and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.2 Vector Product / × V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.3 Tensor Product of / and V . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.4 Scalar Product of / and a Second Order Tensor . . . . . . . . . . A.5.5 Eigenvalue and Eigenvector of a Second Order Tensor . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

687 688 688 689 690 690 691 693 694 695 695 696 700 702

B Tensors in Orthogonal Curvilinear Coordinate Systems . . 703 B.1 Change of Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Co- and Contravariant Base Vectors, Metric Coefficients . . . . . . . . . B.3 Physical Components of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Derivatives of the Base Vectors, Christoffel Symbols . . . . . . . . . . . B.5 Spatial Derivatives in Curvilinear Coordinate System . . . . . . . . . . . B.5.1 Application of / to Zeroth Order Tensor Functions . . . . . . . . B.5.2 Application of / to First and Second Order Tensor Functions B.6 Application Example 1: Inviscid Incompressible Flow Motion . . . . . B.6.1 Equation of Motion in Curvilinear Coordinate Systems . . . . . B.6.2 Special Case: Cylindrical Coordinate System . . . . . . . . . . . . . B.6.3 Base Vectors, Metric Coefficients . . . . . . . . . . . . . . . . . . . . . B.6.4 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.5 Introduction of Physical Components . . . . . . . . . . . . . . . . . . . B.7. Application Example 2: Viscous Flow Motion . . . . . . . . . . . . . . . . . B.7.1 Equation of Motion in Curvilinear Coordinate Systems . . . . . B.7.2 Special Case: Cylindrical Coordinate System . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

703 703 706 707 708 708 709 710 711 712 712 713 714 715 715 716 716

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Nomenclature

A Ac Ah b c c c cax CD Cf Ci CL CL* cp, cv d D D D D De Dh Dm e ei E E E(k) EȜ f fC fS fT f F F g

acceleration, force vector cold side blade surface area hot side blade surface area trailing edge thickness blade chord length complex eigenfunction, c = cr + ici speed of sound blade axial chord length cascade drag coefficient friction coefficient constants blade lift coefficient camberline lift coefficient specific heat capacities projected trailing edge thickness d = b/sinĮ2 diffusion factor dimensionless trailing edge thickness material or substantial differential operator deformation tensor equivalent diffusion factor hydraulic diameter modified diffusion factor specific total energy orthonormal unit vector source (+), sink (-) strength total energy energy spectrum Planck’s spectral emissive power slot thickness/trailing edge thickness ratio f = s/b reheat factor for multistage compressors sampling frequency recovery factor for multistage turbines recovery factor for turbines with infinite number of stages auxiliary function force vector blade geometry function

XXII

gi, gi gij, gij G G Gi Gi h h H H H12 H13 i I I I1, I2, I3 J k k K l, m lm L Lij(x,t) m m ˙ m ˙c M M Ma n n n Nu p, P P p+ Pr Pre Prt q qc qh

Nomenclature

co-, contravariant base vectors in orthogonal coordinate system co-, contravariant metric coefficients blade geometry parameter circulation function auxiliary functions transformation vector height specific static enthalpy specific total enthalpy immersion ratio form factor, H12 = į1/į2 form factor, H32 = į3/į2 incidence angle intermittency function moment of inertia principle invariants of deformation tensor Jacobian transformation function thermal conductivity wave number vector specific kinetic energy coordinates introduced for radial equilibrium specific shaft work stage power turbulence length scale mass mass flow cooling mass flow Mach number vector of moment of momentum axial vector of moment of momentum number of stations polytropic exponent normal unit vector Nusselt number static, total pressure total pressure P = p + ȡV2/2 deterministic pressure fluctuation dimensionless pressure gradient random pressure fluctuation Prandtl number effective Prandtl number turbulent Prandtl number specific thermal energy (heat per unit of mass) cold side specific heat rejected from stage hot side specific heat transferred to the stage

Nomenclature

q/ q// q˙ Q ʖ Q ʖ Q c ʖ Q h ʖ/ Q ʖ // Q r ri r R R R R Re Rw R(x, t, r, IJ) s s s Si St Str t t t T T T To Tc Th TW Tr u u uIJ u+

specific heat transferred to stator blades specific heat transferred to rotor blades heat flux vector thermal energy (heat) thermal energy flow (heat flow) thermal energy flow (heat flow), blade cold side thermal energy flow (heat flow), blade hot side thermal energy flow (heat flow) to/from stator blades thermal energy flow (heat flow) to/from rotor blades degree of reaction radius of stage stream surface radius vector radius in conformal transformation density ratio, R = ȡ3/ȡ2 radiation radius of the mean flow path Reynolds number thermal resistance second order correlation tensor specific entropy slot thickness blade spacing cross section at station i Stanton number Strouhal number time thickness tangential unit vector static temperature tangential force component stress tensor, T = eiejIJij stagnation or total temperature static temperature on blade cold side static temperature on blade hot side blade material temperature trace of second order tensor specific internal energy velocity wall friction velocity dimensionless wall velocity, u+= u/uIJ time averaged wake velocity defect

U, V, W v V

rotational, absolute, relative velocity vectors specific volume volume

XXIII

XXIV

Nomenclature

mean velocity vector random velocity fluctuating vector co- and contravariant component of a velocity vector Vmax W

ensemble averaged velocity vector maximum velocity on suction surface mechanical energy mechanical energy flow (power) shaft power

X y+ xi Z

state vector dimensionless wall distance, y+= uIJ y/Ȟ coordinates stage profile loss

Greek Symbols Į Į, ȕ Įst Ȗ Ȗ Ȗ Ȗ Ȗmin, Ȗmax, ī ī īijk į įij į į1, į2, į3, ǻ1, ǻ2 J J J Jh Jm Jijk İ/ İ// ȗ ȗ Ș

heat transfer coefficient absolute and relative flow angles stagnation angle in conformal transformation blade stagger angle intermittency factor specific circulation function shock angle minimum, maximum intermittency circulation strength circulation vector Christoffel symbol deviation angle Kronecker delta Kronecker tensor, į = eiej įij boundary layer displacement, momentum, energy thickness dimensionless displacement, momentum thickness loss coefficient ratio convergence tolerance turbulence dissipation eddy diffusivity eddy viscosity permutation symbol dimensionless parameter for stator blades heat transfer dimensionless parameter for rotor blades heat transfer Kolmogorov’s length scale total pressure loss coefficient efficiency

Nomenclature

Ș ș Ĭ Ĭ Ĭ Ĭij(k1, t) ț ț Ȝ Ȝ ȁ ȝ ȝ ȝ,Ȟ,ij Ȟ Ȟm ȟ ʌ Ȇ ȡ ȡij(x,t,r.IJ) ı IJ IJo, IJW ȣ ij ĭ ĭ, ȥ ĭ(k.t) ȋ ȥ Ȍ Ȧ ȍ

velocity ratio segment angle blade flow deflection angle shock expansion angle temperature ratio one-dimensional spectral function isentropic exponent ratio of specific heats stage load coefficient wave length load function mass flow ratio absolute viscosity velocity ratios kinematic viscosity straight cascade stagger angle distance ratio ȟ = x/b pressure ratio stress tensor, Ȇ = eiejʌ density dimensionless correlation coefficient cascade solidity ı = c/s temperature ratio wall sear stress Kolmogorov’s velocity scale stage flow coefficient dissipation function potential, stream function spectral tensor complex function isentropic stage load coefficient stream function angular velocity Rotation tensor

Subscripts and Superscripts a, t c c C C, S, R ex F F

axial, tangential compressible cold side compressor cascade, stator, rotor exit flame fuel

XXV

XXVI

Fi G h in max P, S s s t w S   * + /, //

Nomenclature

film combustion gas hot side inlet maximum blade pressure, suction surface isentropic shock turbulent wall time averaged random fluctuation deterministic fluctuation dimensionless wall functions stator, rotor

Part I: Turbomachinery Flow Physics

1

Introduction, Turbomachinery, Applications, Types

Turbomachines are devices within which conversion of total energy of a working medium into mechanical energy and vice versa takes place. Turbomachines are generally divided into two main categories. The first category is used primarily to produce power. It includes, among others, steam turbines, gas turbines, and hydraulic turbines. The main function of the second category is to increase the total pressure of the working fluid by consuming power. It includes compressors, pumps, and fans.

1.1

Turbine

Turbines serve as power producing devices. Figure 1.1 exhibits a large steam turbine used for base load power generation. It consists of an integrated high pressure (HP), an intermediate pressure (IP), and two identical low pressure (LP) components.

Fig 1.1: A large steam turbine for base load power generation with an integrated high pressure, an intermediate pressure part (left), and two identical double inflow low pressure turbine components (Brown Boveri & Cie design) Each component incorporates a number of stages consisting of stator- and rotor-blading. Steam at a given level of total energy, enters the HP-turbine component shown in Fig. 1.2, passes through the first stator row, and undergoes a certain degree of deflection, which is necessary to provide appropriate inlet conditions for the rotor row that follows. The stator blades are attached to the inner casing, which is under higher pressure than the outer one, Fig. 1.2. During the course of deflection, the working fluid is accelerated. As a result, the potential energy of steam is partially converted into kinetic energy, which is used in the following rotor blading. Due to the rotational motion of the rotor, a part of the total energy is converted into mechanical energy, generating shaft power. This process is repeated in the following stages until the exit

4

1 Introduction, Turbomachinery, Applications, Types

conditions are reached. The HP-turbine component is characterized by a relatively small blade height compared to the LP-component. As seen in Figs. 1.1, the cross sections of the integrated HP- and IP-part, experience only a moderate change due to a moderate increase in specific volume. In contrast, the specific volume changes drastically within the LP-part, requiring an excessive opening of the cross section to accommodate large blade heights.

Fig. 1.2: A typical HP-steam turbine with the first control stage, inner and outer casings and labyrinth seal packets, (ABB Power Generation)

Fig. 1.3: A three stage high pressure research turbine at TPFL. The blades are cylindrical with tip shroud to reduce the tip leakage losses.

1 Introduction, Turbomachinery, Applications, Types

5

Figure 1.2 shows an HP-turbine with a control stage and a multi-stage arrangement attached to a shaft that consists of three welded discs. It also reveals the labyrinths installed on both shaft ends to seal the high pressure steam against the atmospheric pressure. HP-turbine blades may have cylindrical (2-D) or three-dimensionally (3-D) bowed blades. Figures 1.3 and 1.4 show two different rotors with the 2-D cylindrical and 3-D bowed blade. The cylindrical blades are more cost effective and easier to manufacture than the 3-D bowed blades, however, their efficiency is below the 3-D blade efficiency.

Fig. 1.4: A three-stage high pressure research turbine with bowed blades and tip shroud to reduce secondary flow and tip clearance losses, TPFL Details of a typical LP-turbine component are shown in Fig. 1.5. Characteristic features of this component are the symmetric configuration of the shaft and the blading, as well as the steep increase in blade height. The symmetric configuration is used to cancel the axial thrust on the bearings. In contrast to the HP-components with a high back pressure, the back pressure of LP-component corresponds to the condenser pressure, which is a fraction of the environmental pressure and strongly depends on environmental temperature. Figure 1.6 exhibits the details of a typical labyrinth seal. The kinetic energy of the jets through the labyrinth clearance is dissipated within the labyrinth chambers preventing the leakage mass flow to become excessive. While the HP-labyrinths reduce the steam leakage to the atmosphere, the LP-seals prevent the atmospheric air to penetrate into the LP-turbine, disrupting the condenser vacuum.

6

1 Introduction, Turbomachinery, Applications, Types

Fig. 1.5: A double inflow low pressure turbine component (ABB Power generation

Fig. 1.6: HP-turbine labyrinth seals

1 Introduction, Turbomachinery, Applications, Types

1.2

7

Compressor

The function of a compressor is to elevate the total pressure of the working fluid. According to the conservation law of energy, this total pressure increase requires external energy input which must be added to the system in the form of mechanical energy.

Fig. 1.7: A multi-stage high pressure compressor (Sulzer)

Fig. 1.8: A high pressure centrifugal compressor with an exit pressure of up to 900 bar and a volume flow rate of up to 300,000 m3/h (MANNESMANN-DEMAG)

8

1 Introduction, Turbomachinery, Applications, Types

The compressor rotor which is driven either by an electric motor or a turbine, exerts forces on the fluid medium and therefore increases its pressure. Compressors are utilized, among others, in pipeline systems, chemical industry, steel production for blowing oxygen and oxygen enrichment of blast furnace air, biological sewage treatments, and gas liquefaction. In power generation and aircraft gas turbines the compressor component provides the pressure ratio essential for maintaining the process of power or thrust generation. Pumps are a special type of turbo compressor with liquids as working fluids. Pumps have a wide application field. In base load power generation they serve as the feed water pumps and main condenser pumps to elevate the steam pressure from condenser pressure to boiler pressure. Pumps are also major components in rocket engines.

Fig. 1.9: Rotor unit of a heavy duty gas turbine with multi-stage compressor and turbine, compressor pressure ratio 15:1, (ABB-GT13E2) Figure 1.7 shows a high pressure axial compressor with a carefully designed inlet nozzle and exit diffuser. To balance the axial thrust originating from the pressure difference, the pressures on both sides of the shaft are balanced through a connecting pressure equalizing pipe. Figure 1.8 shows major design features of a high-pressure compressor with a wide variety of applications. The area of application for these types of compressors include chemical and petrochemical industries, amonia plants, urea and methanol synthetics, gas pipeline, and nuclear reactors. Figure 1.9 shows the integration of the compressor component onto the shaft of a heavy duty power

1 Introduction, Turbomachinery, Applications, Types

9

generation gas turbine. The compressor component provides the optimum process pressure ratio of 15:1 and a design flow of 525 kg/s that are essential for generating a power output of 161 MW. Significantly higher pressure ratios can be obtained by utilizing centrifugal compressors.

1.3

Application of Turbomachines

1.3.1 Power Generation, Steam Turbines There is a wide range for application of turbomachines; a few of them are already mentioned above . The most important application field is electric power generation. Demand of electrical energy is covered by large (up to 1300 MW) and medium size (up to 400 MW) steam turbines, Fig. 1.1. Steam turbines of small size are used in the chemical and paper industry as well as in transportation systems. Hydraulic turbines are used primarily for electric power generation.

1.3.2 Power Generation, Gas Turbines Another important application field of turbomachines is gas turbine technology. A gas turbine engine incorporates compressor and turbine components. As power generating units, gas turbines are used to cover the electric energy demand during peak load periods. The thermal energy of exhaust gases of a gas turbine can be used to generate steam for additional power generation. This is accomplished by a combined cycle which exhibits an efficient device for base load power generation with an overall thermal efficiency of over 55%. Figure 1.9 exhibits a conventional single-spool power

Fig. 1.10: Single-spool power generation gas turbine engine with a multi-stage compressor, a combustion chamber, and multi-stage turbine, (GT-9, ABB)

10

1 Introduction, Turbomachinery, Applications, Types

Fig. 1.11: Schematic cross- section of GT-24 gas turbine engine with a single stage reheat turbine and a second combustion chamber, ABB generation gas turbine engine with a thermal efficiency close to 33%. To substantially improve the thermal efficiency without a significant increase in turbine inlet temperature, a well-known reheat principle as a classical method for thermal efficiency augmentation is applied. This standard efficiency improvement method is routinely applied in steam turbine power generation, and for the first time, in 1948, it was applied to a power generation gas turbine plant in Beznau, Switzerland, Frutschi [1]. The plant is still operational after almost half a century, has a turbine inlet temperature of 600 oC, and an efficiency of 30%, which is remarkably high for this very low turbine inlet temperature. Despite the predicted high efficiency at the conventional turbine inlet temperature, the sequential combustion did not find its way into the aircraft and power generation gas turbine design. The reason was the inherent problem of integrating two combustion chambers into a conventionally designed gas turbine engine. This issue raised a number of unforeseeable design integrity and operational reliability concerns. ABB (former Brown Boveri & Cie) was the first to develop a gas turbine engine with a single shaft, two combustion chambers, and a reheat turbine stage followed by a multi-stage turbine, Fig. 1.11.

1.3.3 Aircraft Gas Turbines Besides power generation, gas turbines play an important role in transportation. Aircraft gas turbines are the main propulsion systems of large, medium, and small size aircrafts. As an example, a high bypass ratio aircraft gas turbine engine is shown in Fig. 1.12.

1 Introduction, Turbomachinery, Applications, Types

Fig. 1.12: Twin-spool, high bypass ratio aircraft engine (Pratt & Whitney)

Fig. 1.13: Schematic of a twin-spool core engine with its derivatives

11

12

1 Introduction, Turbomachinery, Applications, Types

It consists of a high pressure and a low pressure spool. The low pressure spool carries the fan stage and the LP-turbine component. The HP- turbine drives the HPcompressor via the connecting shaft. The two spools running at two different rotational speed are connected aerodynamically with each other. Using a single spool or a twin spool core engine, a variety of derivatives can be designed to perform different functions as Fig. 1.13 suggests.

1.3.4 Diesel Engine Application Turbochargers, which are small gas turbines, are applied to small and large Diesel engines, to increase the effective mean piston pressure and therefore improve the thermal efficiency of Diesel engine. As an example, typical turbocharger for large Diesel engines is shown in Fig. 1.14. It consists of an air filter, inlet nozzle, a radial compressor stage, driven by a single stage axial turbine. Exhaust gas from Diesel engine enters the turbocharger turbine side and drives the turbine. The turbine drives the compressor stage that sucks air from the environment and delivers it to the piston, thereby substantially increasing the mean effective piston pressure and thermal efficiency of the engine. High efficiency of 45% are achieved by turbocharging large large engines. The compression process is accomplished by a single radial impeller.

Fig. 1.14: Turbocharger for a large Diesel engine of 10 MW power, compressor pressure ratio 4.0:1 (ABB)

1 Introduction, Turbomachinery, Applications, Types

1.4

13

Classification of Turbomachines

Turbomachines are classified into different categories according to their specific applications. Following parameters determine the type of the turbomachines: (1) Working fluid, liquid or gaseous, (2) required power generation for turbines, (2) required pressure ratio and mass flow for compressors.

1.4.1

Compressor Types

Radial compressors are generally used for higher pressure ratios and small mass flow rates. For pressure ratios up to 35 and higher mass flow rates, axial compressors are used. For liquid working fluids, radial compressors (pumps) are common.

Fig. 1. 15: Single stage radial compressor with an inlet guide vane and exit diffuser

Fig. 1.16: Radial compressor rotor, 3-D image, front view, cross section

14

1 Introduction, Turbomachinery, Applications, Types

Figure 1.15 exhibits the cross section of a typical radial compressor rotor with the exit diffusers to reduce the exit kinetic energy of the working medium. A 3-D image of such a rotor is shown in Fig. 1.16.

1.4.2 Turbine Types A significant criterion for selecting gas or steam turbine types is the mass flow. For small mass flow rates, both axial and radial turbines can be used. However, for higher mass flow rates the application of the axial type is the common practice. Hydraulic turbine types depends on the head and capacity of water sources. The axial type (Kaplan turbine) is especially applied to low head, high capacity; while the radial type (Francis turbine) is for medium head. For very high hydraulic head, a special water wheel (Pelton wheel) is used.

1.5

Working Principle of a Turbomachine

The conversion of total energy into shaft work or vice versa, can also be established with simple reciprocating (piston-cylinder) engines. Why should a turbomachine be applied? The answer to this question involves the limitation of power and mass flow associated with the reciprocating engines. The reciprocating engine, which works entirely on the displacement principle, is not able to transfer large amount of mass flow or mechanical energy. The largest operating Diesel engine has a power output of about 20 MW, whereas a large steam power plant may produce up to 1300 MW. Unlike the reciprocating engines the working principle of a turbomachine is based on exchange of momentum between the blading and the working fluid.

References 1. Frutschi, H.U.: Advanced Cycle System with new GT24 and GT26 Gas Turbines, Historical Background. ABB Review 1/94 (1994)

2

2.1

Kinematics of Turbomachinery Fluid Motion

Material and Spatial Description of the Flow Field

2.1.1 Material Description The turbomachinery aerodynamic design process has been experiencing continuous progress using the Computational Fluid Dynamics (CFD) tools. The use of CFD-tools opens a new perspective in simulating the complex three-dimensional (3-D) turbomachinery flow field. Understanding the details of the flow motion and the interpretation of the numerical results require a thorough comprehension of fluid mechanics laws and the kinematics of fluid motion within the turbomachinery component. Kinematics is treated in many fluid mechanics texts. Aris [1] and Spurk [2] give excellent accounts of the subject. In the following sections, a compact and illustrative treatment is given to cover the basic needs of the reader. The kinematics is the description of the fluid motion and the particles without taking into account how the motion is brought about. It disregards the forces that cause the fluid motion. As a result, in the context of kinematics, no conservation laws of motion will be dealt with. Consequently, the results of kinematic studies can be applied to all types of fluids and exhibit the ground work that is necessary for describing the dynamics of the fluid. The motion of a fluid particle with respect to a

Fig. 2.1: Material description of a fluid particle at different instants of time

16

2 Kinematics of Turbomachinery Fluid Motion

reference coordinate system is in general given by a time dependent position vector x(t), Fig. 2.1. To identify a particle or a material point at a certain instance of time t = t0 = 0, we introduce the position vector ȟ = x(t0 ). Thus, the motion of the fluid is described by the vector: (2.1) with xi as the components of vector x , as explained in A1.1. Equation (2.1) describes the path of a material point that has an initial position vector ȟ that characterizes or better labels the material point at t = t0. We refer to this description as the material description. It should be mentioned that the term Lagrangian description is also used, but is lacking descriptive quality and is forgotten or confused by students. Considering another material point with different ȟ- coordinates their paths are similarly described by Eq.(2.1). If we assume that the motion is continuous and single valued, then the inversion of Eq.(2.1) must give the initial position ȟ or material coordinate of each fluid particle which may be at any position x and any instant of time t; that is, (2.2) The necessary and sufficient condition for an inverse function to exist is that the Jacobian transformation function (2.3) does not vanish. Because of the significance of the Jacobian transformation function to fluid mechanics, we derive this function in the following section.

Fig. 2.2: Deformation of a differential volume at different instance of time

2 Kinematics of Turbomachinery Fluid Motion

17

2.1.2 Jacobian Transformation Function and Its Material Derivative We consider a differential material volume at the time t = 0, to which we attach the reference coordinate system ȟ1, ȟ2, ȟ3, as shown in Fig. 2.2. At the time t = 0, the reference coordinate system is fixed so that the undeformed differential material volume dV0 (Figs. 2.2 and 2.3) can be described as: (2.4) Moving through the space, the differential material volume may undergo certain deformation and rotation. As deformation takes place, the sides of the material volume initially given as dȟi would be convected into a non-rectangular, or curvilinear form. The changes of the deformed coordinates are then: (2.5) Using the index notation for the position vector rearranged in the following way:

, Eq.(2.5) may be

(2.6)

Fig. 2.3: Transformation of dȟ1, dȟ2, dȟ3 into dx1, dx2, and dx3 using G-transformation

18

with the vector

2 Kinematics of Turbomachinery Fluid Motion

as (2.7)

where Gi is a transformation vector function that transforms the differential changes dȟi into dxi. Figure 2.3 illustrates the deformation of the material volume and the transformation mechanism. The new deformed differential volume is obtained by: (2.8) Introducing Eq.(2.6) into (2.8) leads to: (2.9) Inserting Gi from Eq.(2.7) into Eq.(2.9) and considering A2.2, we arrive at: (2.10) Now we replace the vector product and the following scalar product of the two unit vectors in Eq.(2.10) with the permutation symbol and the Kronecker delta: (2.11) Applying the Kronecker delta to the terms with the indices l and n, we arrive at:

(2.12) The expression in first parenthesis in Eq.(2.12) represents the Jacobian function J. (2.13)

The second parenthesis in Eq.(2.12) represents the initial infinitesimal material volume in the undeformed state at t = 0, described by Eq.(2.4). Using these terms, Eq.(2.12) is rewritten as: (2.14)

2 Kinematics of Turbomachinery Fluid Motion

19

where dV represents the differential volume in the deformed state and dV0 the differential volume in the undeformed state at the time t = 0. The transformation function J is also called the Jacobian functional determinant. Performing the permutation in Eq.(2.13), the determinant is given as:

(2.15)

With the Jacobian functional determinant, we now have a necessary tool to directly relate any time dependent differential volume dV = dV(t) to its fixed reference volume dV0 at the reference time t=0 as shown in Fig. 2.4. The Jacobian transformation function and its material derivative are the fundamental tools to understand the conservation laws using the integral analysis in conjunction with control volume method. To complete this section, we briefly discuss the material derivative of the Jacobian function. As the volume element dV follows the motion from to it changes and, as a result, the Jacobian transformation function undergoes a time change. To calculate this change, we determine the material derivative of J (see A4.1); that is,

(2.16) Inserting Eq.(2.13) into (2.16), we obtain: (2.17)

Let us consider an arbitrary element of the Jacobian determinant, for example . Since the reference coordinate is not a function of time and is fixed, the differentials with respect to t and ȟ2, can be interchanged resulting in: (2.18)

20

2 Kinematics of Turbomachinery Fluid Motion

Performing similar operations for all elements of the Jacobian determinant and noting that the second expression on the right-hand side of Eq.(2.17) identically vanishes, we arrive at: (2.19) The expression in the parenthesis of Eq.(2.19) is the well known divergence of the velocity vector. Using vector notation, Eq.(2.19) becomes: (2.20)

Fig. 2.4: Jacobian transformation of a material volume, change of states

2.1.3 Spatial Description The material description we discussed in the previous section deals with the motion of the individual particles of a continuum, and is used in continuum mechanics. In turbomachinery fluid dynamics, we are primarily interested in determining the flow quantities such as velocity, acceleration, density, temperature, pressure, and etc., at fixed points in space. For example, determining the three-dimensional distribution of temperature, pressure and shear stress helps engineers design turbines, compressors, combustion engines, etc. with higher efficiencies. For this purpose, we introduce the spatial description, which is also called the Euler description. The independent variables for the spatial descriptions are the space characterized by the position vector x and the time t. Consider the transformation of Eq.(2.1), where ȟ is solved in terms of x:

2 Kinematics of Turbomachinery Fluid Motion

21

(2.21) The position vector ȟ in the velocity of the material element (2.21):

is replaced by Eq.

(2.22) For a fixed x, Eq.(2.22) exhibits the velocity at the spatially fixed position x as a function of time. On the other hand, for a fixed t, Eq.(2.22) describes the velocity at the time t. With Eq.(2.22), any quantity described in spatial coordinates can be transformed into material coordinates provided the Jacobian transformation function J, which we discussed in the previous section, does not vanish.

2.2

Translation, Deformation, Rotation

During a general three-dimensional motion, a fluid particle undergoes a translational and rotational motion which may be associated with deformation. The velocity of a particle at a given spatial, temporal position ( x + dx, t ) can be related to the velocity at (x, t) by using the following Taylor expansion: (2.23) Inserting in Eq.(2.23) for the differential velocity change (2.23) is re-written as:

, Eq.

(2.24) The first term on the right-hand side of Eq.(2.24) represents the translational motion of the fluid particle. The second expression is a scalar product of the differential displacement dx and the velocity gradient . We decompose the velocity gradient, which is a second order tensor, into two parts resulting in the following identity: (2.25) The superscript T indicates that the matrix elements of the second order tensor /VT are the transpositions of the matrix elements that pertain to the second order tensor . The first term in the right-hand side represents the deformation tensor, which is a symmetric second order tensor:

22

2 Kinematics of Turbomachinery Fluid Motion

(2.26) with the components:

(2.27)

The second term of Eq.(2.24) is called the rotation or vorticity tensor, which is antisymmetric and is given by: (2.28) The components are:

(2.29)

Inserting Eq.(2.26) and (2.28) into Eq.(2.24), we arrive at: (2.30) Equation (2.30) describes the kinematics of the fluid particle, which has a combined translational and rotational motion and undergoes a deformation. Figure 2.5 illustrates the geometric representation of the rotation and deformation, [3]. Consider the fluid particle with a square-shaped cross section in the x1-x2 plane at the time t. The position of this particle is given by the position vector x = x(t). By moving through the flow field, the particle experiences translational motion to a new position x + dx. This motion may be associated with a rotational motion and a deformation. The deformation is illustrated by the initial and final state of diagonal A-C, which is stretched to A'-C' and the change of the angle 2Ȗ to 2Ȗ'. The rotational motion can be appropriately illustrated by the rotation of the diagonal by the angle dij3 = Ȗ' + dĮ Ȗ, where Ȗ' can be eliminated using the relation 2Ȗ' + dĮ + dȕ = 2Ȗ. As a result, we obtain the infinitesimal rotation angle: (2.31) where the subscript 3 denotes the direction of the rotation axis, which is parallel to x3. Referring to Fig. 2.5, direct relationships between dĮ, dȕ and the velocity gradients can be established by:

2 Kinematics of Turbomachinery Fluid Motion

23

Fig. 2.5: Translation, rotation and deformation details of a fluid particle

(2.32)

A similar relationship is given for:

(2.33)

Substituting Eqs. (2.32) and (2.33) into Eq.(2.31), the rotation rate in the x3 -direction is found: (2.34) Executing the same procedure, the other two components are: (2.35)

24

2 Kinematics of Turbomachinery Fluid Motion

The above three terms in Eqs. (2.34) and (2.35) may be recognized as one-half of the three components of the vorticity vector Ȧ, which is:

(2.36)

Examining the elements of the rotation tensor matrix,

(2.37)

The diagonal elements of the above antisymmetric tensor are zero and only three of the six non-zero components are distinct. Except for the factor of one-half, these three distinct components are the same as those making up the vorticity vector. Comparing the components of the vorticity vector Ȧ given by Eq.(2.36) and the three distinct terms of Eq.(2.37), we conclude that the components of the vorticity vector, except for the factor of one-half, are identical with the axial vector of the antisymmetric tensor, Eq.(2.37). The axial vector of the second order rotation tensor ȍ is the double scalar product of the third order permutation tensor with ȍ: (2.38) Expanding Eq.(2.38) results in: (2.39)

Comparing Eq.(2.39) to (2.36) shows that the right-hand side of Eq.(2.39) multiplied with a negative sign is exactly equal the right-hand side of Eq.(2.36). This indicates that the axial vector of the rotation tensor is equal to the negative rotation vector and can be expressed as:

2 Kinematics of Turbomachinery Fluid Motion

25

(2.40)

The existence of the vorticity vector Ȧ and therefore, the rotation tensor ȍ, is a characteristic of a viscous flow that in general undergoes a rotational motion. This is particularly true for boundary layer flows, where the fluid particles move very close to the solid boundaries. In this region, the wall shear stress forces (friction forces) cause a combined deformation and rotation of the fluid particle. In contrast, for inviscid flows, or the flow regions, where the viscosity effect may be neglected, the rotation vector Ȧ may vanish if the flow can be considered isentropic. This ideal case is called potential flow, where the rotation vector vanishes in the entire potential flow field ( ).

2.3

Reynolds Transport Theorem

The conservation laws in integral form are, strictly speaking, valid for closed systems, where the mass does not cross the system boundary. In turbomachinery fluid mechanics, however, we are dealing with open systems, where the mass flow continuously crosses the system boundary. To apply the conservation laws to the turbomachinery components, we briefly provide the necessary mathematical tools. In this section, we treat the volume integral of an arbitrary field quantity f(X,t) by deriving the Reynolds transport theorem. This is an important kinematic relation that we will use in Chapter 4. The field quantity f(X,t) may be a zeroth, first or second order tensor valued function, such as mass, velocity vector, and stress tensor. The time dependent volume under consideration with a given time dependent surface moves through the flow field and may experience dilatation, compression and deformation. It is assumed to contain the same fluid particles at any time and therefore, it is called the material volume. The volume integral of the quantity f(X,t): (2.41) is a function of time only. The integration must be carried out over the varying volume v(t). The material change of the quantity F(t) is expressed as: (2.42) Since the shape of the volume v(t) changes with time, the differentiation and integration cannot be interchanged. However, Eq.(2.42) permits the transformation of the time dependent volume v(t) into the fixed volume v0 at time t = 0 by using the Jacobian transformation function:

26

2 Kinematics of Turbomachinery Fluid Motion

(2.43) With this operation in Eq.(2.43), it is possible to interchange the sequence of differentiation and integration: (2.44) The chain differentiation of the expression within the parenthesis results in (2.45) Introducing the material derivative of the Jacobian function, Eq.(2.19) into Eq. (2.45) yields:

(2.46) Equation (2.46) permits the back transformation of the fixed volume integral into the time dependent volume integral: (2.47) According to A4.1, the first term in the parenthesis can be written as: (2.48) Introducing Eq.(2.48) into (2.47) results in: (2.49)

The chain rule applied to the second and third term in Eq.(2.49) yields: (2.50)

2 Kinematics of Turbomachinery Fluid Motion

27

The second volume integral in Eq.(2.50) can be converted into a surface integral by applying the Gauss' divergence theorem: (2.51) where V represents the flux velocity and n the unit vector normal to the surface. Inserting equation (2.51) into Eq.(2.50) results in the following final equation, which is called the Reynolds transport theorem

(2.52) Equation (2.52) is valid for any system boundary with time dependent volume V(t) and surface S(t) at any time, including the time t = t0 , where the volume V = VC and the surface S = SC assume fixed values. We call VC and SC the control volume and control surface.

References 1. Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publication, New York (1962) 2. Spurk, J.: Fluid Mechanics. Springer, New York (1997) 3. White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (1974)

3

Differential Balances in Turbomachinery

In this and the following chapter, we present the conservation laws of fluid mechanics that are necessary to understand the basics of flow physics in turbomachinery from a unified point of view. The main subject of this chapter is the differential treatment of the conservation laws of fluid mechanics, namely conservation law of mass, linear momentum, angular momentum, and energy. These subjects are treated comprehensively in a recent book by Schobeiri [1]. In many engineering applications, such as in turbomachinery, the fluid particles change the frame of reference from a stationary frame followed by a rotating one. The absolute frame of reference is rigidly connected with the stationary parts, such as casings, inlets, and exits of a turbine, a compressor, a stationary gas turbine or a jet engine, whereas the relative frame is attached to the rotating shaft, thereby turning with certain angular velocity about the machine axis. By changing the frame of reference from an absolute frame to a relative one, certain flow quantities remain unchanged, such as normal stress tensor, shear stress tensor, and deformation tensor. These quantities are indifferent with regard to a change of frame of reference. However, there are other quantities that undergo changes when moving from a stationary frame to a rotating one. Velocity, acceleration, and rotation tensor are a few. We first apply these laws to the stationary or absolute frame of reference, then to the rotating one. The differential analysis is of primary significance to all engineering applications such as compressor, turbine, combustion chamber, inlet, and exit diffuser, where a detailed knowledge of flow quantities, such as velocity, pressure, temperature, entropy, and force distributions, are required. A complete set of independent conservation laws exhibits a system of partial differential equations that describes the motion of a fluid particle. Once this differential equation system is defined, its solution delivers the detailed information about the flow quantities within the computational domain with given initial and boundary conditions.

3.1

Mass Flow Balance in Stationary Frame of Reference

The conservation law of mass requires that the mass contained in a material volume , must be constant: (3.1) Consequently, Eq. (3.1) requires that the substantial changes of the above mass must disappear:

30

3 Differential Balances in Turbomachinery

(3.2) Using the Reynolds transport theorem (see Chapter 2), the conservation of mass, Eq. (3.2), results in:

(3.3) Since the integral in Eq. (3.3) is zero, the integrand in the bracket must vanish identically. As a result, we may write the continuity equation for unsteady and compressible flow as: (3.4) Equation (3.4) is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:

(3.5) Expanding Eq. (3.5), we get:

(3.6)

For an orthogonal curvilinear coordinate system, the continuity equation (3.6) is written as (see Appendix A.34a): (3.7) Applying Eq. (3.7) to a cylindrical coordinate system with the Christoffel symbols, Eq. (3.8), from Appendix (B.61)

(3.8)

3 Differential Balances in Turbomachinery

31

and introducing the physical components, Eq. (3.7) becomes: (3.9) Equation (3.9) is valid only for cylindrical coordinate systems. To apply the continuity balance to any arbitrary orthogonal coordinate system, one has to determine first the Christoffel symbols as outlined in Appendix B and then find the continuity equation.

3.1.1 Incompressibility Condition The condition for a working medium to be considered as incompressible is that the substantial change of its density along the flow path vanishes. This means that:

(3.10) Inserting Eq. (3.10) into (3.4) and performing the chain differentiation of the second term in Eq. (3.4) namely, , the continuity equation for an incompressible flow reduces to: (3.11) In a Cartesian coordinate system, Eq. (3.11) can be expanded as written in (3.12):

(3.12)

In an orthogonal, curvilinear coordinate system, the continuity balance for an incompressible fluid is, (B.33b): (3.13) Inserting the Christoffel symbols into Eq.(3.13) and the physical components for cylindrical coordinate systems, we obtain the continuity equation in terms of its physical components (3.14):

(3.14)

32

3.2

3 Differential Balances in Turbomachinery

Differential Momentum Balance in Stationary Frame of Reference

In addition to the continuity equation we treated above, the detailed calculation of the entire flow field through different turbomachinery components requires the equation of motion in differential form. In the following, we provide the equation of motion in differential form in a four-dimensional time-space coordinate.

Fig. 3.1: Surface and gravitational forces acting on a volume element

We start from Newton's second law of motion and apply it to an infinitesimal fluid element shown in Fig. 3.1, with mass dm for which the equilibrium condition is written as: (3.15) The acceleration vector A is the well known material derivative (see Chapter 2): (3.16) In Eq. (3.15), A is the acceleration vector and dF the vector sum of all forces exerting on the fluid element. In the absence of magnetic, electric or other extraneous effects,

3 Differential Balances in Turbomachinery

33

the force dF is equal to the vector sum of the surface force dFs acting on the particle surface and the gravity force dmg as shown in Fig. 3.1. Inserting Eq. (3.16) into Eq. (3.15), we arrive at: (3.17) Consider the fluid element shown in Fig. 3.1 with sides dx1, dx2, dx3 parallel to the axis of a Cartesian coordinate system. The stresses acting on the surfaces of this element are represented by the stress tensor Ȇ which has the components ʌij that produce surface forces. The first index i refer to the axis, on which the fluid element surface is perpendicular, whereas the second index j indicates the direction of the stress component. Considering the stress situation in Fig. 3.1, the following resultant forces are acting on the surface dx2dx3 perpendicular to the x1 axis: (3.18) The total resulting forces acting on the entire surface of the element are obtained by adding the nine components that result in Eq. (3.19):

(3.19)

Since the stress tensor Ȇ and the volume of the fluid element are written as: (3.20) It can be easily shown that Eq. (3.19) is the divergence of the stress tensor expressed in Eq. (3.20) (3.21) The expression is the scalar differentiation of the second order tensor Ȇ and is called divergence of the tensor field Ȇ which is a first order tensor or a vector. Inserting Eq. (3.21) into Eq. (3.17) and divide both sides by dm, results in the following Cauchy equation of motion: (3.22)

34

3 Differential Balances in Turbomachinery

The stress tensor in Eq. (3.22) can be expressed in terms of deformation tensor, as we will see in the following section.

3.2.1 Relationship between Stress Tensor and Deformation Tensor Since the surface forces resulting from the stress tensor cause deformation of fluid particles, it is obvious that one might attempt to find a functional relationship between the stress tensor and the velocity gradient: (3.23) As we saw in Chapter 2, the velocity gradient in Eq. (3.23) can be decomposed into an symmetric part called deformation tensor and an anti-symmetric part, called rotation or vorticity tensor: (3.24) Consequently, the stress tensor may be set: (3.25) with the deformation tensor as: (3.26) and the rotation tensor, which is antisymmetric, and is given by Eq. (2.27): (3.27)

Since the stress tensor Ȇ in Eq. (3.25) is a frame indifferent quantity, it remains unchanged or invariant under any changes of frame of reference. Moving from an absolute frame (for example stator blade channel) into a relative one (for example rotor blade channel) exhibits such a change in frame of reference. Thus, the stress tensor Ȇ satisfies the principle of frame indifference, also called the principle of material objectivity. To satisfy the objectivity principle, the arguments of the functional f must also be frame indifferent quantities. This is true only for the first argument D in Eq. (3.25). The second argument ȍ in Eq. (3.25) is not a frame indifferent quantity. As a consequence, the stress tensor is a function of deformation tensor D only. (3.28) A general form of Eq. (3.28) may be a polynomial in D as suggested in [2]:

3 Differential Balances in Turbomachinery

35

(3.29) with I = eiejįij as the unit Kronecker tensor. To fulfill the frame indifference requirement, the functions fi must be invariant. This means they depend on either the thermodynamic quantities, such as pressure, or the following three-principal invariant of the deformation tensor: (3.30) (3.31) (3.32) Of particular interest is the category of those fluids for which there is a linear relationship between the stress tensor and the deformation tensor. Many working fluids used in turbomchinery applications, such as air, steam, and combustion gases, belong to this category. They are called the Newtonian fluids for which Eq. (3.29) is reduced to: (3.33) where the functions f1 and f2 are given by: (3.34) with ȝ as the absolute viscosity and ȝ as the bulk viscosity. Introducing Eq. (3.34) into (3.29) results in the Cauchy-Poisson law: (3.35) In Eq. (3.35), the terms with the coefficients involving the viscosity are grouped together leading to a pressure tensor and a friction stress tensor that reads: (3.36) The first term on the right-hand side of Eq. (3.35) associated with the unit Kronecker tensor, pI, represents the contribution of the thermodynamic pressure to the normal stress. The second term, , exhibits a normal stress contribution caused by a volume dilatation or compression due to the compressibility of the working medium. For an incompressible medium, this term identically vanishes. The coefficient Ȝ related to the coefficient of shear viscosity ȝ and the bulk viscosity ȝ is given in

36

3 Differential Balances in Turbomachinery

Eq. (3.34) as . For most of the fluids used in turbomachinery applications, the bulk viscosity may be approximated as leading to . This relation, frequently called the Stokes’ relation, is valid for monoatomic gases [3]. Finally, the last term expresses a direct relationship between the shear stress tensor and the deformation tensor. For an incompressible fluid, Eq. (3.35) reduces to: (3.37)

3.2.2 Navier-Stokes Equation of Motion Inserting Eq. (3.35) into (3.22), we obtain: (3.38) This is the Navier-Stokes equation for a compressible fluids. In Eq. (3.38), the coefficient Ȝ can be expressed in terms of shear viscosity ȝ. This, however, requires rearranging the second and third term in the bracket by using the index notation. Assuming that the viscosities ȝ, and thus Ȝ are not varying spatially, for we may write:

(3.39)

We apply the same procedure to

:

(3.40)

Introducing Eqs.(3.39) and (3.40) into Eq. (3.38), we arrive at (3.41) For

, Eq. (3.41) results in:

3 Differential Balances in Turbomachinery

37

(3.42) For incompressible flows with constant shear viscosity, Eq. (3.38) reduces to: (3.43) Performing the differentiation on the right-hand side and dividing by ȡ leads to: (3.44) with Ȟ = ȝ/ȡ as the kinematic viscosity and as the Laplace operator. Equation (3.38) or its special case, Eq. (3.44) with the equation of continuity and energy, exhibit a system of partial differential equations. This system describes the flow field completely. Its solution yields the detailed distribution of flow quantities. In many turbomachinery applications, with the exception of hydro-power generation, the contribution of the gravitational term compared to the other terms is negligibly small. Equation (3.44) in Cartesian index notation is written as:

(3.45)

Using the Einstein summation convention, the three components of Eq. (3.45) are:

(3.46)

To obtain the components of the Navier-Stokes equation in an orthogonal curvilinear coordinate system, we use metric coefficients, Christoffel symbols, and the index notation outlined in Appendix B, Eq. (B.74)

38

3 Differential Balances in Turbomachinery

(3.47)

Using the Christoffel symbols and the physical components for a cylindrical coordinate system as specified in Appendix B, we arrive at the component of NavierStokes equation in r-direction:

(3.48)

in ș-direction,

(3.49)

and in z-direction:

(3.50)

3.2.3 Special Case: Euler Equation of Motion For the special case of steady inviscid flow (no viscosity), Eq. (3.44) is reduced to: (3.51) This equation is called Euler equation of motion. Its index notation is:

3 Differential Balances in Turbomachinery

39

(3.52) Replacing the convective term in Eq. (3.51) by the following vector identity: (3.53) we find that the convective acceleration is expressed in terms of the gradient of the kinetic energy , and a second term which is a vector product of the velocity and the vorticity vector . If the flow field under investigation allows us to assume a zero vorticity within certain flow regions, then we may assign a potential to the velocity field that significantly simplifies the equation system. This assumption is permissible for the flow region outside the boundary layer and is discussed more in detail in Chapter 21. Before proceeding with the conservation of energy, in context of the Euler equation that describes the motion of inviscid flows, it is appropriate to present the Bernoulli equation, which exhibits a special integral form of Euler equations. For this purpose, we first rearrange the gravitational acceleration vector by introducing a scalar surface potential z, whose gradient has the same direction as the unit vector in -direction. Furthermore, it has only one component that points in the negative -direction. As a result, we may write . Thus, the Euler equation of motion assumes the following form: (3.54) Equation (3.54) shows that despite the inviscid flow assumption, it contains vorticities that are inherent in viscous flows and cause additional entropy production. This can be expressed in terms of the second law of thermodynamics, , where the changes of entropy, enthalpy, and static pressure, ds, dh, dp, or other thermodynamic properties are expressed in terms of the product of their gradients and a differential displacement as shown in Appendix A by Eq. (A.24) . Replacing the quantity Q by the following properties, we obtain: (3.55) with s as the specific entropy, h as the specific static enthalpy and p the static pressure. Inserting the above property changes into the first law of thermodynamics, , we find: (3.56) The expression in the parentheses on the left-hand side of Eq. (3.56) is the total enthalpy . In the absence of mechanical or thermal energy addition or rejection, H remains constant meaning that its gradient vanishes. Furthermore, for steady flow cases and Eq. (3.56) reduces to: (3.57)

40

3 Differential Balances in Turbomachinery

Equation (3.57) is an important result that establishes a direct relation between the vorticity and the entropy production in inviscid flows. In a flow field with discontinuities as a result of the presence of shock waves, there are always jumps in velocities across the shock front causing vorticity production and therefore, changes in entropy. x

Time t + dt

2

V + dX

B'

d's s+ 'h, d h+

dX *'

B

dX* X+ dX *

T+

, d'p p+ d't,

V

Time t

A

dX= Time t

Vdt

A'

V Time t + dt

T+dt, p+dp, h+dh, s+ds

X

p T,

,s ,h

x

1

3

Fig. 3.2: Fluid particles at different thermodynamic state conditions The Bernoulli equation can be obtained as a scalar product of the Euler differential equation (3.51) and a differential displacement vector. Figure 3.2 shows different displacement vectors that, in principle, may be used. The vector shows the differential distance between two neighboring fluid particles located at positions A and B at the same time t with the thermodynamic states shown in Fig. 3.2. The particles move along their flow paths and at t+dt, they occupy the positions A` and B`. The distance AA` is denoted by . The thermodynamic conditions at A`, denoted by , indicate that the changes this particle has undergone are different from those of particle B. Thus, the vector is the appropriate vector which we choose to multiply with the Euler equation of motion. For steady flow cases, the differential distance along the particle path is identical with a distance along a streamline. Thus, the multiplication of Euler equation (3.56) with the differential displacement , gives: (3.58) The terms in Eq. (3.58) must be rearranged as follows. The first term starts with a scalar product of two vectors, eliminating the Kronecker delta and utilizing the Einstein summation convention results in: (3.59)

3 Differential Balances in Turbomachinery

41

For rearrangement of the second and third term, we use Eq. (A. 24) . The fourth term in Eq.(3.58) identically vanishes because the vector V is perpendicular to the vector . As a result, we obtain: (3.60) Integrating Eq. (3.60) results in: (3.61) Integrating Eq. (3.60) from the initial point A to the final point B, we arrive at: (3.62) For an unsteady, incompressible flow, the integration of Eq. (3.60) delivers: And finally, for a steady, incompressible flow, Eq. (3.63) is reduced to: (3.63)

(3.64) which is the Bernoulli equation.

3.3

Some Discussions on Navier-Stokes Equations

The flow in a turbomachinery component is characterized by a three-dimensional, highly unsteady motion with random fluctuations due to the interactions between the stator and rotor rows. Considering the flows within the blade boundary layer, based on the blade geometry and pressure gradient, three distinctly different flow patterns are identified: 1) laminar flow (or non-turbulent flow) characterized by the absence of stochastic motions, 2) turbulent flow, where flow pattern is determined by a fully stochastic motion of fluid particles, and 3) transitional flow characterized by intermittently switching from laminar to turbulent at the same spatial position. Of the three patterns, the predominant one is the transitional flow pattern. The Navier-Stokes equations presented in this section generally describe the steady and unsteady flows through a variety of turbomachinery components. Using a direct numerical simulation (DNS) approach delivers the most accurate results . however, as extensively discussed in [1], the application of DNS, for the time being, is restricted to simple flows at low Reynolds numbers. For calculating the complex flow field within a reasonable time

42

3 Differential Balances in Turbomachinery

frame, the Reynolds averaged Navier-Stokes (RANS) can be used. This issue is discussed in Chapter 19, where the laminar-turbulent transition, Reynolds- and ensemble averaging and the implementation of transition models into Navier-Stokes is treated extensively

3.4

Energy Balance in Stationary Frame of Reference

For the complete description of flow process, the total energy equation is presented. This equation includes mechanical and thermal energy balances.

3.4.1 Mechanical Energy The mechanical energy balance is obtained by the scalar multiplication of the equation of motion, Eq. (3.22), with the local velocity vector: (3.65) The product

is modified using the following vector identity: (3.66)

The expression is a double-scalar multiplication of the stress tensor and the velocity gradient which is a second order tensor. The velocity gradient / V can be decomposed into deformation D and rotation ȍ part as shown in Eq. (3.24):

(3.67) With this operation, the trace of the second order tensor in Eq. (3.66) is calculated from: (3.68) Since the second term on the right-hand side of Eq. (3.68) vanishes identically, Eq. (3.66) reduces to: (3.69) As a result, the mechanical energy balance, Eq. (3.65), becomes:

3 Differential Balances in Turbomachinery

43

(3.70) Incorporating Eq. (3.35) for Newtonian fluids into (3.70) leads to:

(3.71)

with

. For incompressible flow, Eq. (3.71) is reduced to

(3.72)

The index notation of (3.72) is:

(3.73)

The sum of the last two terms in the second bracket of Eq. (3.71) is called the dissipation function: (3.74) The dissipation function Eq. (3.74) is identical with the double scalar product between the friction stress tensor and the deformation tensor: (3.75) The index notation of (3.74) is (3.76)

44

3 Differential Balances in Turbomachinery

Expanding (3.76) results in:

(3.77)

In Eq. (3.77) the coefficient Ȝ can be replaced by an incompressible flow, Eq. (3.74) reduces to:

from Eq. (3.34). For

(3.78) and as a result, Eq. (3.77) is written as:

(3.79)

The dissipation function indicates the amount of mechanical energy dissipated as heat, which is due to the deformation caused by viscosity. Consider a viscous flow along a flat plate, an aircraft wing, a compressor stator or turbine rotor blade or any other turbomchinery surfaces exposed to a flow. Close to the wall in the boundary layer region, the velocity experiences a high deformation because of the no-slip condition. By moving outside the boundary layer, the rate of deformation decreases leading to lower dissipation. To analyze the individual terms in the equation of energy and to demonstrate the role of shear stress and its effect on the dissipation of mechanical energy, we introduced the friction stress tensor, Eq. (3.36) (3.80) The off-diagonal elements of this tensor represent the shear stress components and characterize the shear-deformative behavior of this tensor. The diagonal elements of this tensor

3 Differential Balances in Turbomachinery

45

(3.81) exhibit additional contributions caused by the volume dilatation or compression due to the compressibility of the working medium as mentioned before. This contribution is added to the normal stress components of the pressure tensor . For an incompressible flow with , these terms identically disappear. Inserting Eq. (3.80) into (3.71), we arrive at:

(3.82)

Equation (3.82) exhibits the mechanical energy balance in differential form. The first term on the right-hand side represents the rate of mechanical energy due to the change of pressure acting on the volume element. The second term is the rate of work done by viscous forces on the fluid particle. The third term represents the rate of irreversible mechanical energy due to the friction stress. It dissipates as heat and increases the internal energy of the system. This term corresponds to the dissipation function defined by Eq. (3.78). Finally, the forth term represents the mechanical energy necessary to overcome the gravity force acted on the fluid particle. Equation (3.82) exhibits the general differential form of mechanical energy balance for a viscous flow. For a steady, incompressible, inviscid flow, Eq. (3.82) is simplified as: (3.83) where the vector g is replaced by to the Bernoulli equation of energy:

. Integration of the above equation leads

(3.84) This equation is easily derived by multiplying the Euler equation of motion with a differential displacement.

3.4.2 Thermal Energy Balance The thermal energy balance is described by the first law of thermodynamics which is postulated for a closed “thermostatic system”. For this system, properties, such as temperature, pressure, entropy, internal energy, etc., have no spatial gradients. Since in an open system the thermodynamic properties undergo temporal and spatial changes, the classical first law must be formulated under open system conditions. To do so, we start from an open system within which a steady flow process takes place and replaces the differential operator d from classical thermodynamics by the

46

3 Differential Balances in Turbomachinery

substantial differential operator D. This operation implies the requirement that the thermodynamic system under consideration be at least in a locally stable equilibrium state. Starting from the first law for an internally irreversible system: (3.85) where u as the internal energy, and Q1 as the specific thermal energy added to (or removed from) the system, p the thermodynamic pressure, v the specific volume, and įwf the part of mechanical energy dissipated as heat by the internal friction. The subscript f refers to the irreversible nature of the process caused by internal friction. Applying the differential operator D: (3.86) where is the rate of thermal energy added to or removed from the open system per unit mass and time. It can be expressed as the divergence of the thermal energy flux vector . The rate of the mechanical energy dissipated as heat is identical to the third term T:D/ȡ in Eq. (3.82): (3.87) The negative sign of is introduced to account for a positive heat transfer to the system. Furthermore, since the first term on the left-hand side is per unit mass and time, the divergence of the heat flux vector as well as the dissipation term T:D, had to be divided by the density to preserve the dimensional integrity. Multiplying both sides of Eq. (3.87) with ȡ, we obtain (3.88) In Eq. (3.88), first we replace the specific volume v by 1/ȡ and consider the continuity equation (3.89) then we insert Eq. (3.89) into (3.88) and arrive at: (3.90)

1

Usually Q (kJ) represents the thermal energy and q (kJ/kg) the specific thermal energy. However, in order to avoid confusion that may arise using the same symbol for specific thermal energy and the heat flux vector , we use Q for the specific thermal energy. We will use q, wherever there is no reason for confusion

3 Differential Balances in Turbomachinery

47

In Eq. (3.90) the internal energy can be related to the temperature by the thermodynamic relation with cv as the specific heat at constant volume. The heat flux vector can also be expressed in terms of temperature using the Fourier heat conduction law. For an isotropic medium, the Fourier law of heat conduction is written as: (3.91) with k (kJ/msec K) as the thermal conductivity. Introducing Eq. (3.91) into (3.90), for an incompressible fluid we get: (3.92) For a steady flow, Eq. (3.92) can be simplified as: (3.93) The thermal energy equation can equally be expressed in terms of enthalpy (3.94) Following exactly the same procedure that has lead to Eq. (3.90), we find

(3.95) Introducing the temperature in terms of

:

(3.96) The index notation of Eq. (3.96) reads: (3.97) and taking

from Eq. (3.79), we can expand (3.97) to arrive at:

48

3 Differential Balances in Turbomachinery

(3.98)

3.4.3 Total Energy The combination of the mechanical and thermal energy balances, Eqs. (3.82) and Eq. (3.90), results in the following total energy equation: (3.99) We may rearrange the second and third term on the right-hand side of Eq. (3.99) (3.100) The argument inside the parenthesis within the bracket exhibits the stress tensor , thus, Eq. (3.100) can be written as: (3.101)

The second term on the right-hand side of Eq. (3.101) constitutes the mechanical energy necessary to overcome the surface forces. The heat flux vector in Eq. (3.101) can be replaced by the Fourier equation (3.91) that gives (3.102) Equation (3.101) may be written in different forms using different thermodynamic properties. Since in an open system enthalpy is used, we replace the internal energy by the enthalpy and find (3.103)

and with the Fourier equation (3.91)

3 Differential Balances in Turbomachinery

49

(3.104) The expression in the parenthesis on the left-hand side is called the total enthalpy which is defined as . With this definition, the re-arrangement of Eq. (3.104) gives (3.105) and its index notation reads (3.106)

3.4.4 Entropy Balance The second law of thermodynamics expressed in terms of internal energy is: (3.107) The infinitesimal heat added to or rejected from the system may include the heat generated by the irreversible dissipation process. Replacing the differential d by the material differential operators, we arrive at: (3.108) The right-hand side of Eq. (3.108) is expressed by Eq. (3.90) as: (3.109) replacing the left-hand side of Eq. (3.109) by Eq. (3.108) results in (3.110) The second term on the right-hand side, which includes the second order friction tensor T, is the dissipation function Eq. (3.74) (3.111)

50

3 Differential Balances in Turbomachinery

This equation shows clearly that the total entropy change generally consists of two parts. The first part is the entropy change due to a reversible heat supply to the system (addition or rejection) and may assume positive, zero, or negative values. The second term exhibits the entropy production due to the irreversible dissipation and is always positive. Thus, Eq. (3.111) may be modified as:

(3.112)

with

and

. In context of the turbomachinery

aerodynamics, in uncooled turbine or compressor blade channels, the profile losses (see Chapter 7 for more details) are partly generated by the dissipation.

3.5

Differential Balances in Rotating Frame of Reference

3.5.1 Velocity and Acceleration in Rotating Frame We consider now a rotating frame of reference that is attached to the rotor, thus, turns with an angular velocity Ȧ about the machine axis. From a stationary observer point of view, a fluid particle that travels through a rotation frame has at an arbitrary time t, the position vector r, and a relative velocity W. In addition, it is subjected to the inherent rotation of the frame, causing the fluid particle to rotate with the velocity . Thus, the observer located outside the rotating frame observes the velocity (3.113) Inserting Eq. (3.113) into Eq. (3.16), the substantial acceleration is found

(3.114) We multiply Eq. (3.114) out and find

(3.115)

3 Differential Balances in Turbomachinery

51

Investigating the terms in Eq. (3.115), we begin with the second term on the righthand side

(3.116) since the absolute radius vector in the first term on the right-hand side of Eq. (3.116) does not change with time, . Furthermore, the last three terms of Eq. (3.115) are:

(3.117)

The proof of correctness of Eq.(3.117) is provided by first using the index notation and considering the identity with as the position vector. Considering Eqs. (3.116) and (3.117), Eq. (3.115) becomes

(3.118) The first term on the right-hand side expresses the local acceleration of the velocity field within the relative frame of reference. In the second term, is the angular velocity acceleration. It is non-zero during any dynamic operation of turbines and compressors such as startup, load change and shutdown, where the shaft speed experience changes. The third term, , constitutes the convective term within the relative frame of reference. The forth term is the centrifugal force. Finally, the last term in Eq. (3.118), , is called the Coriolis acceleration. It can be equal zero only if the relative velocity vector W and the angular velocity vector Ȧ are parallel. As shown in Eqs. (3.117), two terms contributed to producing the Coriolis acceleration. The first term originates from the spatial changes of W because of the rotation. The second one from the changes in circumferential velocity, , in the direction of W. For the case that Ȧ and W are parallel, both terms become zero. The centrifugal acceleration and Coriolis accelerations are fictitious forces that are produced as a result of transformation from absolute into a relative frame of reference. Figure 3.3 shows the direction of the Coriolis force which is perpendicular to the plane described by the two vectors Ȧ and W. The force vector is perpendicular to the plane described by the vectors and and points towards the axis of rotation. The direction of the radius vector er is expressed in terms of the radius gradient .

3 Differential Balances in Turbomachinery

Z

Coriolis acceleration

2 Z xW

'

52

W

R

Z F T

Z xr

Centripetal acceleration



Z tio rec Di

no

f ro

R

xr

Z x Z x r

i on ta t

Z xZ

Tangential blade force

r

Z O

Z

Fig. 3.3: Coriolis and centripetal forces created by the rotating frame of reference

3.5.2 Continuity Equation in Rotating Frame of Reference Inserting the velocity vector from Eq. (3.113) into the continuity equation for absolute frame of reference, Eq. (3.4), we obtain: (3.119) When we expand the second term in Eq. (3.119), we find: (3.120) After a simple rearrangement, Eq. (3.121) leads to: (3.121)

It is necessary to discuss the individual terms in Eq. (3.120) before rearranging them. The first term indicates the time rate of change of density at a fixed station in an absolute (stationary) frame of reference. The second term involves the spatial change of density registered by a stationary observer. Combining the first and second terms expresses the time rate of change of the density within the rotating frame of reference: (3.122)

3 Differential Balances in Turbomachinery

53

From Eq. (3.122), it becomes clear that in cases where the local change of the density in an absolute frame might be zero, , in a rotating frame of reference, it will become a function of time . Since the product exhibits the circumferential change of the density in the rotating frame, it can vanish only if the flow within the rotating frame is considered axisymmetric. Since the last term in Eq. (3.120), , identically vanishes, the equation of continuity in a rotating frame reduces to: (3.123) Equation (3.123) has the same form as Eq. (3.4), however, the spatial operator / as well as the time derivative refer to the relative frame of reference. Since the flow in the rotor is understood exclusively with reference to a relative frame of reference, from now on it is unnecessary to differentiate between the operators and the time derivatives.

3.5.3 Equation of Motion in Rotating Frame of Reference Replacing the acceleration in Eq. (3.22) by the expression obtained in (3.118): (3.124) and replacing stress tensor Ȇ by Eq. (3.35), becomes:

, Eq. (3.124)

(3.125)

Combining the last two terms in the bracket as setting for , we re-arrange Eq. (3.125) as:

, and

(3.126) The friction force f was given a negative sign, since it opposes the flow motion and causes energy dissipation. Using the Clausius entropy relation, the pressure gradient can be expressed in terms of enthalpy and entropy gradients: (3.127) The thermodynamic properties s, h, and p are uniform continuous scalar point functions whose changes are expressed as:

54

3 Differential Balances in Turbomachinery

(3.128) with as the differential displacement along the path of the fluid particle. We replace the quantities in Eq. (3.127) by those in Eq. (3.128) and arrive at: (3.129) Since the differential displacement in Eq. (3.129) , bracket must vanish

, the vector sum in the

(3.130) Replacing the pressure gradient term in Eq. (3.126) by Eq. (3.130), we find

(3.131) Further treatment of Eq. (3.131) requires a re-arrangement of few terms. As Fig. 3.3 shows, the centrifugal acceleration points in the negative direction of the gradient of the radius vector and can be written as: (3.132) With Eq. (3.132), the equation of motion in a relative frame of reference becomes: (3.133) Using the vector identity, (3.134) Equation (3.133) is modified as:

(3.135)

For a constant rotational speed and with

, we find,

3 Differential Balances in Turbomachinery

55

(3.136) We introduce now the concept of the relative total enthalpy: (3.137)

which is used in the following section.

3.5.4 Energy Equation in Rotating Frame of Reference The energy equation for rotating frame of reference is simply obtained by multiplying the equation of motion with a differential displacement along the path of a particle that moves within a rotating frame of reference. It is given by,

(3.138)

Multiplying out and re-arranging the terms, we find:

(3.139)

In Eq. (3.139), dR denotes the changes in a relative frame of reference. Since the vectors and are perpendicular to W, their scalar products with W are zero. As a result, Eq. (3.139) reduces to: (3.140)

Multiplying out the right-hand side and considering the identity (3.140) is modified as:

, Eq.

(3.141)

56

3 Differential Balances in Turbomachinery

2 The term is identified as heat that consists of two contributions. The first contribution comes from heat supplied or removed from a fluid particle that moves along its path within the relative frame of reference. We call this contribution the reversible part, . The second contribution is the irreversible part due to the internal friction and dissipation of mechanical energy into heat, which is identical with the friction work, . We summarize the above statement in the following relation:

(3.142) A simple re-arrangement of Eq. (3.142) yields: (3.143) We insert Eq. (3.143) into Eq. (3.141) and obtain: (3.144) With Eq. (3.144), the changes of relative total enthalpy in a relative frame of reference along the path of a fluid particle is expressed as: (3.145)

Only for adiabatic steady flow inside the rotating frame of reference the enthalpy change is zero resulting in: (3.146) It should be pointed out that Eq (3.146) is strictly valid along the path of a fluid particle. If the flow within the relative frame can be approximated as steady, then Eq. (3.146) is also valid along the streamline. Its value changes however, by moving from one streamline to the next. The energy equation for a stationary frame can be obtained by setting . In conjunction with Eq. (3.113) with , thus Eq. (3.146) becomes: (3.147) For a turbine or a compressor stator, Eq. (3.147) is written as:

2

Since in this section the heat flux vector is not involved we revert to conventional nomenclature with q (kJ/kg) as the thermal energy per unit mass.

3 Differential Balances in Turbomachinery

57

(3.148) with subscripts 1 and 2 referring to stator inlet and exit stations as shown in Fig. 3.4. For a turbine or a compressor rotor row under the above assumption, Eq. (3.146) is written as: (3.149) where the subscripts 2 and 3 in Eq. (3.149) refer to the inlet and exit station of the rotor row. The topic of energy transfer with a turbomachinery stage consisting of a stator and a rotor row is extensively discussed in Chapter 5. (b)

(a)

3 2

3

1

Stator

3

W

U

V3

W2 V2

V1

V2

V3

W

2

Z xr

W

3

U V1

Rotor

Rotor

3

Stator

Z xr 3

2

1

Fig. 3.4: Energy transfer in stationary (stator) and rotating (rotor) frame of reference

References 1. Schobeiri, M.T.: Fluid Mechanics. A Graduate Level Textbook. Springer, New York (2010) 2. Truesdell, C., Noll, W.: Handbuch der Physik. Editor S. Flügge. Springer, Berlin (1965) 3. Truesdell, C.: J. of Rational Mech. Analysis 1, 125 (1952)

4

Integral Balances in Turbomachinery

In the following sections, we summarize the conservation laws in integral form essential for applying to the turbomachinery flow situations. Using the Reynolds transport theorem explained in Chapter 2, we start with the continuity equation, which will be followed by the equation of linear momentum, angular momentum, and the energy. Vavra [1] utilized an alternative approach by directly integrating the differential balances. Both approaches are valid and lead to the same results.

4.1

Mass Flow Balance

We apply the Reynolds transport theorem by substituting the function f(X,t) in Chapter 2 by the density of the flow field: (4.1) where the density generally changes with space and time. To obtain the integral formulation, Reynolds transport theorem from Chapter 2 is applied. The requirement that the mass be constant leads to:

(4.2)

If the density does not undergo a time change (steady flow), the above equation is reduced to: (4.3)

For practical purposes, a fixed control volume is considered, where the integration must be carried out over the entire control surface:

(4.4)

60

4 Integral Balances in Turbomachinery V In 1

n n

In 1

Wall

t In 1

t Wall

V Out 1

V Out 2

V Out 3 n

t In 2 n In 2 V In 2

Out i

t Out i

t Wall n

Wall

V Wall

e 2

e

1

Fig. 4.1: Control volume, unit normal and tangential vectors The control surface may consist of one or more inlets, one or more exits, and may include porous walls, as shown in Fig. 4.1. For such a case Eq. (4.4) is expanded as:

(4.5)

As shown in Fig. 4.1 and by convention, the normal unit vectors,

,

point away from the region bounded by the control surface. Similarly, the tangential unit vectors, , point in the direction of shear stresses. A representative example where the integral over the wall surface does not vanish, is a film cooled turbine blade with discrete film cooling hole distribution along the blade suction and pressure surfaces as shown in Fig. 4.2. To establish the mass flow balance through a turbine or cascade blade channel, the control volume should be placed in such a way that it includes quantities that we consider as known, as well as those we seek to find. For the turbine cascade in Fig. 4.2, the appropriate control surface consists of the surfaces AB, BC, CD, and DA. The two surfaces, BC and DA, are portions of two neighboring streamlines. Because of the periodicity of the flow through the cascade, the surface integrals along these streamlines will cancel each other. As a result, the mass flow balance reads:

4 Integral Balances in Turbomachinery St re

V

am

lin e

St re am A

lin e

B

S

61

Uniform inlet velocity

D A

n

B

Asymmetric wake velocity

CAS-FIL

D

C

Uniform velocity after wake mixing

D V

Fig. 4.2: Flow through a rectilinear turbine cascade with discrete film cooling holes

(4.6) The last surface integral accounts for the mass flow injection through the film cooling holes. If there is no mass diffusion through the wall surfaces, the last integral in Eq. (4.6) will vanish leading to:

(4.7)

4.2

Balance of Linear Momentum

The momentum equation in integral form applied to a control volume determines the integral flow quantities such as blade lift and drag forces, average pressure, temperature, and entropy. The motion of a material volume is described by the Newton’s second law of motion, which states that mass times acceleration is the sum of all external forces acting on the system. In the absence of electrodynamic, electrostatic, and magnetic forces, the external forces can be summarized as the surface forces and the gravitational forces:

62

4 Integral Balances in Turbomachinery

(4.8) Equation (4.8) is valid for a closed system with a system boundary that may undergo deformation, rotation, expansion or compression. In a turbomachinery component, however, there is no closed system with a defined system boundary. The mass is continuously flowing from one point within a turbomachinery component to another point. Thus, in general, we deal with mass flow rather than mass. Consequently, Eq. (4.8) must be modified in such a way that it is applicable to a predefined control volume, with mass flow passing through it. This requires applying the Reynolds transport theorem to a control volume, as we already discussed in the previous section. For this purpose, we prepare Eq. (4.8) before proceeding with the Reynolds transport theorem. In the following steps, we add a zero-term to Eq. (4.8): (4.9) Adding this term to Eq. (4.8) leads to: (4.10) Using the Leibniz’s chain rule of differentiation, Eq. (4.10) can be rearranged: (4.11) Applying the Reynolds transport theorem to the left-hand side of Eq. (4.11), we arrive at:

(4.12) and replace the second volume integral by a surface integral using the Gauss conversion theorem (see Chapter 2): (4.13) We consider now the surface and gravitational forces acting on the moving material volume under investigation. The first term on right-hand side of Eq. (4.10), represents the resultant surface force acting on the entire control surface. It can be written as the

4 Integral Balances in Turbomachinery

63

integral of a scalar product of the normal unit vector with the total stress tensor acting on the surface element ds:

(4.14)

The product of the normal vector and the stress tensor gives a stress vector, which can be decomposed into a normal and a shear stress force (4.15) with n as the normal unit vector that points away from the surface and t as the tangential unit vector. The negative signs of n and t have been chosen to indicate that the pressure p and the shear stress IJ are exerted by the surroundings on the surface S. Thus, the surface force acting on a differential surface is: (4.16) Inserting Eq. (4.16) into Eq. (4.11) and considering Eq. (4.12), we arrive at:

(4.17) Since the control volume does not change with time (fixed), Eq. (4.17) becomes with ȡdv = dm:

(4.18)

In Eq. (4.18) the integration must be carried out over the entire control surface. For a control surface consisting of inlet, exit, and wall surfaces, the second integral on the left-hand side gives:

(4.19)

64

4 Integral Balances in Turbomachinery

Fig. 4.3: Control volume, single inlet, single and outlet and porous wall Evaluating the integrands on the right-hand side of Eq. (4.19) by considering the directions of the unit vectors shown in Fig. 4.3, we find for the single inlet cross section: (4.20) In the case of a control volume with multiple inlets, as Fig. 4.1 shows, we need to integrate over the entire inlet cross sections. For the exit cross section we obtain: (4.21) And finally for the wall: (4.22) Inserting the Eqs. (4.20) through (4.22) into Eq. (4.19) and the results into Eq. (4.18), we obtain a relation that includes the mass flow through the control volume: (4.23)

4 Integral Balances in Turbomachinery

65

The first term expresses the total momentum exchange of all particles contained in the region (control volume) under consideration, at the time t, because of velocity changes produced by a non-steady flow. For a steady flow, it vanishes. The second and third integral are leaving and entering velocity momenta. The fourth term exhibits the velocity momentum through the wall. This term is different from zero if the wall is porous (permeable) or has perforations or slots that may be used for different purposes such as cooling turbine blades, Fig. 4.2, boundary layer suction, etc. For a solid wall, this term, of course, vanishes identically. The first and the second integral on the right-hand side of Eq. (4.23) are momentum contributions due to the action of static pressure and the shear stresses. These integrals must be taken over the entire bounding surface that includes inlet, exit and wall surfaces, Fig. 4.3:

(4.24)

According to the convention in Fig. 4.1, the direction of unit normal vectors nin, nout, and nwall point away from the region bounded by the control surface SC. The last two integrals in Eq. (4.24) determine the reaction forces. To demonstrate the physical significance of the reaction force, we consider a rectilinear turbine cascade, Fig. 4.4. St re

am

lin e

St re am A

lin e

B

1

Fig. 4.4: Reaction force on a turbine blade with F and D as the lift, drag forces, and R the resultant force

66

4 Integral Balances in Turbomachinery

The reaction force R which is exerted by the flow on the surface SCw, that is, on the turbine blade wall between the stations (1) and (2) and the body, is therefore:

(4.25)

As Eq. (4.25)indicates, the flow force equals the negative value of the two last integrals. Considering an steady flow and implementing Eq. (4.25) into (4.23), the reaction forces can be determined using the relationship:

(4.26)

The vector equation (4.26) can be decomposed into three components. An order of magnitude estimation suggests that the shear stress terms at the inlet and outlet are, in general, very small compared to the other terms. It should be pointed out that the wall shear stress is already included in the resultant force FR.

4.3

Balance of Moment of Momentum

To establish the conservation law of moment of momentum for a time dependent material volume, we start from the second law of Newton, Eq. (4.18): (4.27) The moment of the force given by Eq. (4.27) is then (4.28) with X as the position vector originating from a fixed point. To rearrange Eq. (4.28) for further analysis, its left-hand side is extended by adding the following zero-term identities: (4.29) and:

4 Integral Balances in Turbomachinery

67

(4.30) Introducing the identities (4.29) and (4.30) into Eq. (4.28), we arrive at:

(4.31)

Using the Leibniz’s chain differential rule, a simple rearrangement of Eq. (4.31) allows the application of the Reynolds transport theorem as follows:

(4.32)

Since:

, Eq. (4.32) can be written as:

(4.33)

We apply the Reynolds transport theorem and the Gauss conversion theorem (Chapter 2) to the left-hand side of Eq. (4.33) and arrive at:

(4.34)

We now interchange the sequence of multiplication for the vector product inside the parenthesis of the second integral in Eq. (4.34), and obtain:

(4.35)

Introducing the mass flow

, Eq. (4.35) results in:

68

4 Integral Balances in Turbomachinery

(4.36) The surface integral has to be carried out over the entire control surface SC.

(4.37) Now we consider the moment of momentum of all other forces on the right-hand side of Eq. (4.33):

(4.38) Since the right side of Eq. (4.33) is equal to the right side of Eq. (4.38), the equation of moment of momentum can be presented in a more compact form that contains the contributions of velocity, pressure and shear stress momenta:

(4.39)

The integration of the first two integrals on the right-hand side have to be performed over SCin, SCout and SCW.

(4.40)

Similar to the expression for the reaction force, the last two integrals on the righthand side of Eq. (4.40) determine the reaction moment, M0.

4 Integral Balances in Turbomachinery

69

Fig. 4.5: A mixed flow compressor with control surfaces This reaction moment is exerted by the flow on the solid boundary SW of the system with respect to a fixed point such as coordinate origin shown in Fig. 4.5. This figure exhibits the flow through a mixed axial-radial compressor stage, where the flow undergoes a change in radial direction associated with certain deflection from inlet at station 1 to the exit at station 2. A fixed control volume is placed on the rotor that includes a compressor blade. The normal unit vectors at the inlet and exit are used to establish the mass flow balances at station 1 and 2. The wall surface SW represents one blade surface (pressure or suction surface) that is projected on the drawing plane. The reaction moment consists of the moment by the surface shear and pressure forces: (4.41) From Eq. (4.41) it is seen that the last two integrals of Eq. (4.40) are equal to - M0. Therefore, from Eqs. (4.40) and (4.41), the moment M0 exerted by the flow on the solid boundary SW with respect to a fixed point using the station numbers in Fig. 4.5 is:

(4.42)

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4 Integral Balances in Turbomachinery

Equation (4.42) describes the moment of momentum in general form. The first integral on the right-hand side expresses the angular momentum contribution due to the unsteadiness. The second and third term represents the contribution due to the velocity momenta at the inlet and exit. The forth and sixth term are formally the contributions of pressure momenta at the inlet and exit. The shear stress integrals, the fifth and seventh terms, representing the moment due to shear stresses at the inlet and exit, are usually ignored in practical cases. For applications to turbomachines, Eq. (4.42) can be used to determine the moment that the flow exerts on a turbine or compressor cascade. Of practical interest is the axial moment M = Ma which acts on the cascade with respect to the axis of rotation. The moment M = Ma is equal to the component of the moment vector parallel with the axis of surfaces of revolution.

D

M

e 2

F axial R

Di e 2

M0

M

axial

r

o on i t ec

n tio a t o fr

Z

e2

e3

e1 Fig. 4.6: Illustration of the axial moment by projecting the reaction moment in axial direction e2 As shown in Fig. 4.6, the axial moment is: (4.43) Neglecting the contribution of the shear stress terms at the inlet and exit but not along the wall surfaces, SW, and performing the above scalar multiplication, the pressure contributions vanish identically. Furthermore, the moment contribution of gravitational force will vanish. With this premise, Eq. (4.43) reduces to:

4 Integral Balances in Turbomachinery

71

(4.44) with Vu as the absolute velocity component in circumferential direction. For steady flow, Eq. (4.44) reduces to: (4.45) For the case where the velocity distributions at the inlet and exit of the channel are fully uniform and the turbomachine is rotating with the angular velocity Ȧ, the power consumed (or produced) by a compressor (or by a turbine) stage is calculated by: (4.46)

V

m

Although the application of conservation laws are extensively discussed in the following chapters, it is found necessary to present a simple example of how the moment of momentum is obtained by utilizing the velocity diagram of a single-stage axial compressor. Figure 4.7a represents a single stage axial compressor with the constant hub and tip diameters.

V

V

m2

W

W

V

E

V2

D

V1

E

U

U2

V

U1

Rotor

D

Stator

U2

U1

W2

W1

Compressor stage

STAG-TDC-N

Fig. 4.7: A single-stage axial compressor (a), velocity diagram (b). The circumferential velocity difference is responsible for the power consumption.

72

4 Integral Balances in Turbomachinery

We consider the flow situation at the mid- section. The flow is first deflected by the stator row. Entering the rotor row, the fluid particle moves through a rotating frame, where the rotational velocity U is superimposed on the relative velocity W (for details see Chapters 5). The constant radii at the inlet and exit of the mid-section results in simplifying Eq. (4.46) to . The expression in the parenthesis, , is shown in the velocity diagram, Figure 4.7b. It states that the compressor power consumption is related to the flow deflection expressed in terms of the circumferential velocity difference. The larger the difference is, the higher the pressure ratio that the compressor produces. However, for each type of compressor design (axial, radial, subsonic, super sonic) there is always a limit to this difference, which is dictated by the flow separation, as we will see later. For the case where no blades are installed inside the channel and the axial velocity distributions at the inlet and exit of the channel are fully uniform, Eq. (4.45) is reduced to: (4.47) This is the so called free vortex flow.

4.4

Balance of Energy

The conservation law of energy in integral form, that we discuss in the following sections, is based on the thermodynamic principals, primarily first law of thermodynamics for open systems and time dependent control volume. It is fully independent from conservation laws of fluid mechanics. However, it implicitly contains the irreversibility aspects described by the dissipation process in the previous chapter. The contribution of the irreversibility is explicitly expressed by using the Clausius-Gibbs entropy equation, known as the second law of thermodynamics. The energy equation is applied to a variety of turbomachine components and describes a chain of energy conversion process that usually takes place in a thermo-fluid dynamics system. As an example, Fig. 4.8 shows a high performance gas turbine engine with several components to which we apply the results of our derivations. In this chapter, we apply the conservation law of energy to a material volume with a system boundary that moves through the space where it may undergo deformation, rotation and translation. The first law of thermodynamics in integral form states that if we add thermal energy (heat) Q and mechanical energy ( work) W to a closed system, the total energy of the system E experiences a change from initial state E1 to the final state E2. Expressing in terms of energy balance, we have: (4.48) The total energy E is the sum of internal, kinetic and potential energies, (4.49)

4 Integral Balances in Turbomachinery

73

Fig. 4.8: A modern power generation gas turbine engine with a single shaft, two combustion chambers, a multi-stage compressor, a single-stage reheat turbine and a multi-stage turbine In order to apply the conservation law of energy to a control volume, we divide Eq. (4.49) by the mass m to arrive at the specific total energy, (4.50) with e as the specific total energy. Similar to the conservation laws of mass, momentum, and moment of momentum, we ask for substantial change of the total energy, i.e.: (4.51)

with

as the thermal and mechanical energy flow, respectively. Since , Eq. (4.51) yields:

(4.52) To apply the the conservation of energy to a control volume, we use the Reynolds transport theorem. Using the Jacobi-transformation function , and introducing the time fixed volume v0, we arrive at: (4.53)

74

4 Integral Balances in Turbomachinery

We now introduce

for the substantial derivative of the Jacobian

(4.54) Developing the first integral term:

(4.55) Application of the chain rule to the second and third term yields:

(4.56)

With Gauss-Divergence Theorem:

(4.57)

The above equation is valid for any volume including , which might be a fixed control volume. In Eq. (4.57), the integration must be carried out over the entire control surface. For a control surface consisting of inlet, exit, and wall surfaces, (Fig. 4.4) the second integral on the left-hand side gives:

(4.58)

Evaluating the integrands on the right-hand side of Eq. (4.58) by considering the directions of the unit vectors shown in Fig. 4.4, , we find for the inlet cross section:

(4.59)

4 Integral Balances in Turbomachinery

75

For the exit cross section we obtain:

(4.60) Inserting the Eqs. (4.59) and (4.60) into Eq. (4.57), we obtain the energy equation for a control volume:

(4.61) with the specific internal energy,

,

(4.62)

For uniform velocity distribution, Eq. (4.62) is reduced to:

(4.63)

The mechanical energy flow consists of the shaft power and the mechanical energy flow which is needed to overcome the shear and normal stresses at the system or control volume boundaries: (4.64) The shaft power is the sum of the net shaft power and the power dissipated by the bearings .

76

4 Integral Balances in Turbomachinery

The second term is the power required to overcome the normal and shear stress forces at the inlet and exit of the system. It is the product of the flow force vector F and the displacement vector dX

(4.65)

Consider a turbine component, Fig. 4.9, where the working fluid (gas or steam) enters the inlet station. To force the differential mass dm into the turbine, which is under high pressure, a force is required that must compensate the pressure and the shear stress forces at the inlet. Figure. 4.9 exhibits a simplified schematic of one of the turbine components in Fig.4.8. It shows the directions of the forces and the displacements. At the inlet, the force vector F is expressed in terms of pressure and the inlet area and is oriented toward negative e1-direction. The displacement vector dX has the positive direction. As a result, the product: (4.66) is negative. The differential volume can be expressed as the product of the specific volume and the differential mass. Replacing, in Eq. (4.66), dV with vdm ( ) and dividing the result by dt, we arrive at: (4.67)

S 1 Q

2

n

'x dm

W

Shaft

S

f

Fig. 4.9: Explanation of flow forces, sketch of a turbine component

4 Integral Balances in Turbomachinery

77

Inserting Eq. (4.67) into (4.65), the integration from initial state to final state and assuming a constant mass flow, we find:

(4.68)

To eliminate the internal energy from the equation of energy for open systems, we introduce the thermodynamic property enthalpy into Eq. (4.63).

(4.69)

For a fixed control volume, the volume integral can be rearranged as:

(4.70) We set

, and since

can only change with time, the partial

derivative is replaced by the ordinary one,

. As a result, we obtain: (4.71)

Equation (4.71) exhibits the general form of energy equation for an open system with a fixed control volume. For technical applications, several special cases are applied which we will discuss in the following.

4.4.1 Energy Balance Special Case 1: Steady Flow If a power generating machine such as a turbine, or a power consuming machine such as a compressor, operates in a steady design point, the first term on the right-hand side of Eq. (4.71) disappears, , which leads to: (4.72)

78

4 Integral Balances in Turbomachinery

Equation (4.72) is the energy balance for a turbomachine with thermal energy (heat) addition or rejection (kJ/s) per unit if time and the shaft power is supplied or consumed .

4.4.2 Energy Balance Special Case 2: Steady Flow, Constant Mass Flow In many applications, the mass flow remains constant from the inlet to the exit of the machine. Examples are uncooled turbines and compressors where no mass flow is added during the compression or expansion process. In this case, Eq. (4.72) reduces to: (4.73)

Now, we define the specific total enthalpy (4.74) and insert it into Eq. (4.73), from which we get: (4.75) In Eq. (4.73) or (4.75), the contribution of compared to and is negligibly small. Using the above equation, the energy balance for the major components of the gas turbine engine shown in Fig. 4.8 can be established as detailed in the following section.

4.5

Application of Energy Balance to Turbomachinery Components

The gas turbine engine shown in Fig. 4.8 consists of a variety of components, to which the energy balance in different form can be applied. These components can be categorized in three groups. The first group entails all those components that serve either the mass flow transport from one point of the engine to another or the conversion of kinetic energy into the potential energy and vice versa. Pipes, diffusers, nozzles, and throttle valves are representative examples of the first group. Within this group no thermal or mechanical energy (shaft work) is exchanged with the surroundings. We in thermodynamic sense these components are assumed adiabatic. The second group contains those components, within which thermal energy is generated or exchanged with the surroundings. Combustion chambers or heat exchangers are typical examples of these components. Thermodynamically speaking,

4 Integral Balances in Turbomachinery

79

in this case we are dealing with diabatic systems. Finally, the third group includes components within which thermal and mechanical energy is exchanged. In the following sections, each group is treated individually.

4.5.1 Application: Accelerated, Decelerated Flows Nozzles, turbine stator cascades, throttle and, to some extent, pipes are examples of accelerated flows. Diffuser and compressor cascades decelerate the flow. For these devices, Eq. (4.75) reduces to: (4.76) The h-s-diagrams of the pipe, nozzle, and diffuser are shown in Fig. 4.10. Accelerated flow

h

P1

P2

p dp/dx 0

p

p

2

2

1

Diffuser

Compressor cascade V1

Decelerated Flow dp/dx >0, dV/dx 0.4, Eq. (5.15) should be used. Considering the integration of Eq. (5.23) for station 1 from an arbitrary diameter R to the mean diameter Rm yields, (5.36)

134

5 Theory of Turbomachinery Stages

At station (2) we have, (5.37) and finally, at station (3) we arrive at:

(5.38)

with

. Introducing Eqs. (5.36) , (5.37) and

(5.38) into (5.35), we finally arrive at a simple relationship for the degree of reaction: (5.39) From a turbine design point of view, it is of interest to estimate the degree of reaction at the hub and tip radius by inserting the corresponding radii into Eq. (5.39). As a result, we find: (5.40)

Equation (5.40) represents a simple radial equilibrium condition which allows the calculation of the inlet flow angle in radial direction by integrating Eq. (5.32): (5.41) This leads to determination of the inlet flow angle in a spanwise direction, (5.42)

5.6

Effect of Degree of Reaction on the Stage Configuration

The distribution of degree of reaction can also be obtained by simply using the velocity ratio for r. If, for example, the degree of reaction at the mean diameter is set at r = 50%, Eq. (5.40) immediately calculates r at the hub and tip for the simple radial equilibrium condition as presented above. It should be mentioned that, for a turbine, a negative degree of reaction at the hub may lead to flow separation and

5 Theory of Turbomachinery Stages

135

is not desired. Likewise, for the compressor, r should not exceed the value of 100%. The value of r has major design consequences. For turbine blades with r = 0, as shown in Fig. 5.9(a) and 5.10, the flow is deflected in the rotor blades without any enthalpy changes. As a consequence, the magnitude of the inlet and exit velocity vectors are the same and the entire stage static enthalpy difference is partially converted within the stator row. Note that the flow channel cross section remains constant. For r = 0.5, shown in Fig. 5.9(b), a fully symmetric blade configuration is established. Figure 5.9(c) shows a turbine stage with r > 0.5. In this case, the flow deflection inside the rotor row is much greater than the one inside the stator row. Figure 5.10 shows the flow deflection within a high speed rotor cascade. In the past, mainly two types of stages were common designs in steam turbines. The stage with a constant pressure across the rotor blading called action stage, was used frequently. This turbine stage was designed such that the exit absolute velocity vector was swirl free. It is most appropriate for the design of single stage turbines and as the last

Fig. 5.9: Effect of degree of reaction on the stage configuration

Fig. 5.10: Flow through a high speed turbine rotor with a degree of reaction , note that and

136

5 Theory of Turbomachinery Stages

stage of a multi-stage turbine. As we shall see in Chapter 8, the exit loss, which corresponds to the kinetic energy of the exiting mass flow, becomes a minimum by using a swirl free absolute velocity. The stage with r = 0.5 is called the reaction stage.

5.7

Effect of Stage Load Coefficient on Stage Power

The stage load coefficient Ȝ defined in Eq. (5.13) is an important parameter which describes the stage capability to generate/consume shaft power. A turbine stage with low flow deflection, thus, low specific stage load coefficient Ȝ, generates lower specific mechanical energy. To increase the stage mechanical energy lm , blades with higher flow deflection are used that produce higher stage load coefficient Ȝ. The effect of an increased Ȝ is shown in Fig. 5.11, where three different bladings are plotted. D E2 2

r = 0.5

D3

Vm2 U3 3

U

3

3

2

U3

O

I V m3 U3

U

U3

U2

O 

W

V2

W

E3 V3

W2

U3

U3 U3

VU + V 2

U3

U3

I

W2 O 

O 

I

W2

O  I

O  O 

Fig. 5.11: Dimensionless stage velocity diagram that explains the effect of stage load coefficient Ȝ on flow deflection and blade geometry The top blading with the stage load coefficient Ȝ = 1 has a low flow deflection. The middle blading has a moderate flow deflection and moderate Ȝ = 2 which delivers a stage power twice as high as the top blading. Finally, the bottom blading with Ȝ = 3, delivers three times the stage power as the first one. In the practice of turbine design, among other things, two major parameters must be considered. These are the specific load coefficients and the stage polytropic efficiencies. Lower deflection generally yields higher stage polytropic efficiency, but many stages are needed to produce the required turbine power. However, the same turbine

5 Theory of Turbomachinery Stages

137

power may be established by a higher stage flow deflection and, thus, a higher Ȝ at the expense of the stage efficiency. Increasing the stage load coefficient has the advantage of significantly reducing the stage number, thus, lowering the engine weight and manufacturing cost. In aircraft engine design practice, one of the most critical issues besides the thermal efficiency of the engine, is the thrust/weight ratio. Reducing the stage numbers may lead to a desired thrust/weight ratio. While a high turbine stage efficiency has top priority in power generation steam and gas turbine design, the thrust/weight ratio is the major parameter for aircraft engine designers.

5.8

Unified Description of a Turbomachinery Stage

The following sections treat turbine and compressor stages from a unified standpoint. Axial, mixed flow, and radial flow turbines and compressors follow the same thermodynamic conservation principles. Special treatments are indicated when dealing with aerodynamic behavior and loss mechanisms. While the turbine aerodynamics is characterized by negative (favorable) pressure gradient environments, the compression process operates in a positive (adverse) pressure gradient environment. As a consequence, partial or total flow separation may occur on compressor blade surfaces leading to partial stall or surge. On the other hand, with the exception of some minor local separation bubbles on the suction surface of highly loaded low pressure turbine blades, the turbine operates normally without major flow separation or breakdown. These two distinctively different aerodynamic behaviors are due to different pressure gradient environments. Turbine and compressor cascade aerodynamics and losses are extensively treated in Chapters 7 and 16. In this section, we will first present a set of algebraic equations that describes the turbine and compressor stages with constant mean diameter and extend the approach to general cases where the mean stage diameter changes.

5.8.1

Unified Description of Stage with Constant Mean Diameter

For a turbine or compressor stage with constant mean diameter (Fig. 5.7), we present a set of equations that describe the stage by means of the dimensionless parameters such as stage flow coefficient ij, stage load coefficient Ȝ, degree of reaction r, and the flow angles. From the velocity diagram with the angle definition in Fig. 5.7, we obtain the flow angles:

(5.43)

Similarly, we find the other flow angles, thus, we summarize:

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5 Theory of Turbomachinery Stages

(5.44)

(5.45)

(5.46)

(5.47) The stage load coefficient can be calculated from: (5.48) The velocity diagram of the last stage of a compressor or a turbine differs considerably from the normal stage. As mentioned in the previous section, to minimize the exit losses, the last stage usually has an exit flow angle of . Figure 5.12 compares the velocity diagram of a normal stage with the one in the last stage of the same turbine component. As shown, by changing the exit flow angle to , substantial reduction in exit velocity vector and, thus, the exit kinetic energy can be achieved. This subject is treated in Chapter 7 in detail.

5.8.2 Generalized Dimensionless Stage Parameters In this chapter, we extend the foregoing consideration to compressor and turbine stages where the diameter, circumferential velocities, and meridional velocities are not constant. Examples are axial flow turbine and compressors, Fig. 5.5 and 5.6, radial inflow (centripetal) turbines, Figs. 5.13, and centrifugal compressors, 5.14.

Fig. 5.12: Turbine stage velocity diagrams, (a) for a normal stage, (b) for the last stage of a multi-stage turbine

5 Theory of Turbomachinery Stages Radial inflow turbine

Impeller cross section

139 Velocity diagram

Fig. 5.13: A centripetal turbine stage with cross section and velocity diagram Centrifugal Compressor with vaned diffuser

Impeller cross section

Velocity diagram

V V

V W

W

U V

Labyrinth seals G=Axial gap

V U

Inlet guide vane

Fig. 5.14: A centrifugal compressor stage with cross section and velocity diagram

In the following, we develop a set of unifying equations that describe the above axial turbine and compressor stages, as well as the centripetal turbine and centrifugal compressor stages shown in Figs. 5.13 and 5.14. We introduce new dimensionless parameters: (5.49) as the specific static enthalpy with Vm, U from the velocity diagrams and differences in rotor and stator. The dimensionless parameter ȝ represents the meridional velocity ratio for the stator and rotor respectively, Ȟ the circumferential velocity ratio, ij the stage flow coefficient, Ȝ the stage load coefficient and r the degree of reaction. Introducing these parameters into the equations of continuity, moment of momentum, and the relation for degree of reaction, the stage is completely defined by a set of four equations: (5.50)

140

5 Theory of Turbomachinery Stages

(5.51) The relation for the specific load coefficient is obtained from (5.52) and the degree of reaction is: (5.53) The four Eqs. (5.50), (5.51), (5.52) and (5.53) contain nine unknown stage parameters. To find a solution, five parameters must be guessed. Appropriate candidates for the first guess are: the diameter ratio , the stator and rotor exit angles , the exit flow angle and the stage degree of reaction r. In addition, the stage flow coefficient ij can be estimated by implementing the information about the mass flow and using the continuity equation. Likewise, the stage load coefficient Ȝ can be estimated for turbine or compressor stages by employing the information about the compressor pressure ratio or turbine power. Once the five parameters are guessed, the rest of the four parameters are determined by solving the above equation system. In this case, the four parameters calculated fulfill the conservation laws for the particular compressor or turbine blade geometry for which five stage parameters were guessed. This preliminary estimation of stage parameters, however, is considered the first iteration toward an optimum design. A subsequent loss and efficiency calculation, presented in Chapter 8, will clearly show if the guessed parameters were useful or not. In fact, few iterations are necessary to find the optimum configuration that fulfills the efficiency requirement set by the compressor or turbine designer. All stage parameters in Eqs. (5.50), (5.51), (5.52), and (5.53) can be expressed in terms of the flow angles , which lead to a set of four nonlinear equations:

(5.54)

5.9

Special Cases

Equations (5.50) through (5.54) are equally valid for axial, radial, and mixed flow turbine and compressor stages. Special stages with corresponding dimensionless parameters are described as special cases as listed below.

5 Theory of Turbomachinery Stages

141

5.9.1 Case 1, Constant Mean Diameter In this special case, the diameter remains constant leading to the circumferential velocity ratio of .The meridional velocity ratio . The flow angles expressed in terms of other dimensionless parameters are:

(5.55)

The stage load coefficient is calculated from:

(5.56)

5.9.2 Case 2, Constant Mean Diameter and Meridional Velocity Ratio In this special case, circumferential and meridional velocities are equal leading to: . The flow angles are calculated from:

(5.57)

The stage load coefficient is calculated from: (5.58) In the following, we summarize the generalized stage load coefficient for different ȝ, Ȟ- cases,

142

5 Theory of Turbomachinery Stages

(5.59)

5.10 Increase of Stage Load Coefficient, Discussion Following the discussion in Section 5.3 regarding the increase of the specific stage mechanical energy and the subsequent discussion in Section 5.8, we proceed with Eq. (5.52), where the stage load parameter Ȝ is expressed in terms of ȝ, Ȟ, Į2 and ȕ3: (5.60) This equation is of particular significance for the first estimation of the stage load coefficient which is responsible for the stage power of a turbine or a compressor. While the stage flow coefficient ij can be estimated from the continuity equation and the velocity ratios ȝ and Ȟ can be guessed for the first design iteration, the angles Į2 and ȕ3 follow closely the exit metal angles of stator and rotor. The effect of flow deflection on stage load coefficient of axial flow turbines was already discussed in Section 5.8. As we saw, turbine blades can be designed with stage load coefficients Ȝ as high as 2 or more. In turbine blades with high Ȝ and Reynolds numbers , the governing strong negative pressure gradient prevents the flow from major separation. However, if the same type of blade operates at lower Reynolds numbers, flow separation may occur that results in a noticeable increase in profile losses. For high pressure turbines (HP-turbines), the strong favorable pressure gradient within the blade channels prevents the flow from major separation. However, low pressure turbine (LPT) blades, particularly those of aircraft gas turbine engines that operate at low Reynolds numbers (cruse condition up to Re = 120,000), are subjected to laminar flow separation and turbulent reattachment. While axial turbine blades can be designed with relatively high positive Ȝ, the flow through axial compressor blade channels is susceptible to flow separation, even at relatively low Ȝ. This is primarily due to a positive pressure gradient in the streamwise direction that causes the boundary layer to separate once a certain deflection limit or a diffusion factor (see Chapter 16) has been exceeded. To achieve a higher Ȝ and, thus, a higher stage pressure ratio, a smaller diameter ratio can be applied. Using a moderate diameter ratio range of results in a mixed flow configuration. At a lower range of Ȟ such as , centrifugal compressor configuration is designed. Figure 5.15 shows, schematically, three centrifugal compressors with three different exit flow angles and the corresponding velocity diagrams. Figure 5.15a exhibits a centrifugal impeller with the trailing edge portion that is forward leaned

5 Theory of Turbomachinery Stages

143

Fig. 5.15: Centrifugal compressor stage with velocity diagrams, a) forward lean, b) radial zero lean, c) backward lean and has a negative lean angle of . The reference configuration, Fig. 5.15 b, shows the impeller with a radial exit flow angle and, thus, . Finally, Fig. 5.15c shows the impeller with backward leaned trailing edge portion with a positive lean angle . All three impellers have the same diameter ratio Ȟ and the same rotational speed Ȧ. The Ȝ-behavior of these impellers is shown in Fig. 5.16, where the relative exit flow angle is varied in the -0.6

Backward

-0.7

Q = 0.5, P = 1.0, I = 1.0

-0.8 10 5 0

O

-0.9

10 0 0

-1

95 0 E3 = 90: Rad

-1.1 -1.2

85

-1.3

Forward

ial

80

-1.4 80

85

90

95

100 0 2

105

110

115

120

D

Fig. 5.16: Influence of lean angle on Ȝ: Forward lean with lean , Zero lean

, Backward

144

5 Theory of Turbomachinery Stages

range of . As shown, forward lean results in higher deflection Ĭ, larger and, thus, higher negative Ȝ, which is associated with a higher profile loss. Backward lean, however, reduces the flow deflection Ĭ and . As a result, the stage load coefficient Ȝ reduces. For the comparison, the radial exit case with is plotted. In calculating the stage load coefficient Ȝ, the influence of the radius ratio on the stage load coefficient becomes clear.

References 1. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, New York (1960) 2. Traupel, W.: Thermische Turbomaschinen, vol. I. Springer, New York (1977) 3. Horlock, J.H.: Axial Flow Compressors. Butterworth, London (1966) 4. Horlock, J.H.: Axial Flow Turbine London. Butterworth, London (1966)

6

Turbine and Compressor Cascade Flow Forces

The last chapter was dedicated to the energy transfer within turbomachinery stages. The stage mechanical energy production or consumption in turbines and compressors were treated from a unified point of view by introducing a set of dimensionless parameters. As shown in Chapter 4, the mechanical energy, and therefore the stage power, is the result of the scalar product between the moment of momentum acting on the rotor and the angular velocity. The moment of momentum in turn was brought about by the forces acting on rotor blades. The blade forces are obtained by applying the conservation equation of linear momentum to the turbine or compressor cascade under investigation. In this chapter, we first assume an inviscid flow for which we establish the relationship between the lift force and circulation. Then, we consider the viscosity effect that causes friction or drag forces on the blading.

6.1

Blade Force in an Inviscid Flow Field

Starting from a given turbine cascade with the inlet and exit flow angles shown in Fig.6.1, the blade force can be obtained by applying the linear momentum principles

Fig. 6.1: Inviscid flow through a turbine cascade. The blades are assumed to have zero trailing edge thickness

146

6 Turbine and Compressor Cascade Flow Forces

to the control volume with the unit normal vectors and the coordinate system shown in Fig.6.1. As in Chapter 4, the blade force is:

(6.1) with h as the blade height that can be assumed as unity. The relationship between the control volume normal unit vectors and the unit vectors pertaining to the coordinate system is given by: . The inviscid flow force is obtained by the linear momentum equation where no shear stress terms are present:

(6.2) The subscript i refers to inviscid flow. The above velocities can be expressed in terms of circumferential as well as axial components;

(6.3) With

and

from Fig. 6.1, Eq. (6.3) is rearranged as:

(6.4) with the circumferential and axial components

(6.5) The above static pressure difference is obtained from the Bernoulli equation , where the static pressure difference is obtained:

(6.6) Inserting the pressure difference along with the mass flow into Eq. (6.5) and the blade height , we obtain the axial as well as the circumferential components of the lift force:

6 Turbine and Compressor Cascade Flow Forces

147

(6.7) From Eq.(6.2) and considering (6.7), the lift force vector for the inviscid flow is:

(6.8)

This means that the direction of the blade force is identical with the direction of the vector within the brackets. In the next step, we evaluate the expression in the bracket. First, we calculate the mean velocity vector V :

(6.9) and the circulation around the profile shown in Fig. 6.1 is:

(6.10) where is a differential displacement along the closed curve and is the velocity vector. The closed curve is placed around the blade profile so that it consists of two streamlines that are apart by the spacing s. Performing the contour integral around the closed curve c, we find: (6.11)

Since the following integrals cancel each other:

(6.12)

We obtain the circulation and thus the circulation vector:

148

6 Turbine and Compressor Cascade Flow Forces

(6.13) We multiply the circulation vector with the mean velocity vector (6.14) and compare Eq.(6.14) with (6.8) to arrive at the inviscid flow force: (6.15)

*

This is the well-known Kutta-Joukowsky lift-equation for inviscid flow. Figure 6.2 exhibits a single blade taken from a turbine cascade with the velocities , , as well as the circulation vector , and the force vector . As seen, the inviscid flow force vector is perpendicular to the plane spanned by the mean velocity vector and the circulation vector .

F i

Fi

V 1

V

h

*

V 1

D (a)

V 2

V

(b)

V 2 Fig. 6.2: (a) A turbine blade in an inviscid flow field with velocity, circulation and lift force vectors, (b) inviscid lift force vector perpendicular to the plane by and

Equation (6.15) holds for any arbitrary body that might have a circulation around it regardless of the body shape. It is valid for turbine and compressor cascades and exhibits the fundamental relation in inviscid flow aerodynamics. The magnitude of the force vector is obtained from:

6 Turbine and Compressor Cascade Flow Forces

149

(6.16) Expressing Fi in terms of

, the inviscid lift force for a turbine cascade is: (6.17)

Figure 6.3 exhibits the inviscid flow forces acting on a turbine and a compressor cascade. V

Vax1

D

D

D

V

V V

F

V

Vu1 Vu2

Turbine Cascade, Inviscid Flow Forces

Vu2 F

D V

V

8

V

D Vax1

D

Compressor Cascade, Inviscid Flow Forces

Vu1

8

Vu2 Vu

Fig. 6.3: Turbine (top) and compressor (bottom) cascade inviscid flow forces

The flow deflection through the cascades is shown using the velocity diagram for accelerated flow (turbine) and decelerated flow (compressor). As shown in Fig. 6.3, the inviscid flow force (inviscid lift) is perpendicular to the mean velocity vector . The lift force can be non-dimensionalized by dividing Eq. (6.17) by a product of the exit dynamic pressure and the projected area with the height h assumed . Thus, the lift coefficient is obtained from:

150

6 Turbine and Compressor Cascade Flow Forces

(6.18)

As shown above, the first equation was expressed in terms of the cascade flow angles and the geometry.

6.2

Blade Forces in a Viscous Flow Field

The working fluids in turbomachinery, whether air, gas, steam or other substances, are always viscous. The blades are subjected to the viscous flow and undergo shear stresses with no-slip condition on the suction and pressure surfaces, resulting in boundary layer developments on both sides of the blades. Furthermore, the blades have certain definite trailing edge thicknesses. These thicknesses together with the boundary layer thickness, generate a spatially periodic wake flow downstream of each cascade as shown in Fig. 6.4. The presence of the shear stresses cause drag forces that reduce the total pressure. In order to calculate the blade forces, the momentum Eq. (6.1) can be applied to the viscous flows. As seen from Eq. (6.5) and Fig. (6.4), the circumferential component remains unchanged. The axial component, however, changes in accordance with the pressure difference as shown in the following relations:

(6.19)

The blade height h in Eq. (6.19) may be assumed as unity. For a viscous flow, the static pressure difference cannot be calculated by the Bernoulli equation. In this case, the total pressure drop must be taken into consideration. We define the total pressure loss coefficient:

(6.20)

6 Turbine and Compressor Cascade Flow Forces

151

Fig. 6.4: Viscous flow through a turbine cascade. Station Å has a uniform velocity distribution. At station Æ wakes are generated by the trailing edge- and boundary layer thickness and are mixed out at Ç. with and as the averaged total pressure at stations 1 and 2. Inserting for the total pressure the sum of static and dynamic pressure, we get the static pressure difference as: (6.21) Incorporating Eq. (6.21) into the axial component of the blade force in Eq. (6.19) yields:

(6.22) with as the axial force component per unit height. We introduce the velocity components into Eq. (6.22) and assume that for an incompressible flow the axial components of the inlet and exit flows are the same. As a result, Eq. (6.22) reduces to:

152

6 Turbine and Compressor Cascade Flow Forces

(6.23) The second term on the right-hand side exhibits the axial component of the drag force per unit height accounting for the viscous nature of a frictional flow shown in Fig. 6.5. Thus, the axial projection of the drag force is obtained from: (6.24) Figure 6.5 exhibits the turbine and compressor cascade flow forces, including the lift and drag forces on each cascade for viscous flow, where the periodic exit velocity distribution caused by the wakes and shown in Fig. 6.4, is completely mixed out resulting in an averaged uniform velocity distribution, Fig. 6.5.

Viscous Flow Forces V1

D2

8

V2

D1

D

V

V

8

Fiax

Fi

FR

D

V2

ax

D

Fiu FRu

Turbine Cascade

F

CAS-FVIS

V Fiu FRu Fi

Dax Fiax

FR FRax

F

D1 8

D D 2 D

V V1

V

8

Compressor Cascade

V2

Fig. 6.5: Viscous flow forces on turbine cascade (top) and compressor cascade (bottom). The resultant force is decomposed into lift and drag component.

6 Turbine and Compressor Cascade Flow Forces

153

With Eq. (6.24), the loss coefficient is directly related to the drag force:

(6.25) Since the drag force D is in the direction of V , its axial projection Dax can be written as:

(6.26) Assuming the blade height h = 1, we define the drag and lift coefficients as:

(6.27) introduce the drag coefficient CD into Eq. (6.25), and obtain the loss coefficient (6.28) The magnitude of the lift force is the projection of the resultant force FR on the plane perpendicular to V : (6.29) Using the lift coefficient defined previously and inserting the lift force, we find

(6.30) Introducing the cascade solidity ı = c/s into Eq. (6.30) results in:

(6.31) The lift-solidity coefficient is a characteristic quantity for the cascade aerodynamic loading. Using the flow angles defined in Fig. 6.3, the relationship for the lift-solidity coefficient becomes:

154

6 Turbine and Compressor Cascade Flow Forces

Fig. 6.6: Lift-solidity coefficient as a function of inlet flow angle Į1 with the exit flow angle Į2 as parameter for turbine and compressor cascades

(6.32) with: (6.33) For a preliminary design consideration, the contribution of the second term in Eq. (6.32) compared with the first term can be neglected. However, for the final design, the loss coefficient ȗ needs to be calculated from Chapter 7 and inserted into Eq. (6.32). Figure 6.6 shows the results as a function of the inlet flow angle with the exit flow angle as the parameter for turbine and compressor cascades. As an example, a turbine cascade with an inlet flow angle of Į1 = 132o, and an exit flow of Į2 = 30o resulting in a total flow deflection of Ĭ = 102o, has a positive lift-solidity coefficient of CLı = 2.0. This relatively high lift coefficient is responsible for generating high

6 Turbine and Compressor Cascade Flow Forces

155

blade forces and thus, a high blade specific mechanical energy for the rotor. In contrast, a compressor cascade with an inlet flow angle of Į1 = 60o and an exit flow of Į2 = 80o which result in a total compressor cascade flow deflection of only Ĭ = 20o, has a lift-solidity coefficient of CLı = -0.8. This leads to a much lower blade force and thus, lower specific mechanical energy input for the compressor rotor. The numbers in the above example are typical for compressor and turbine blades. The high lift-solidity coefficient for a turbine cascade is representative of the physical process within a highly accelerated flow around a turbine blade, where, despite a high flow deflection, no flow separation occurs. On the other hand, in case of a compressor cascade, a moderate flow deflection, such as the one mentioned above, may result in flow separation. The difference between the turbine and compressor cascade flow behavior is explained by the nature of boundary layer flow around the turbine and compressor cascades. In a compressor cascade, shown in Fig. 6.7 the boundary layer flow is subjected to two co-acting decelerating effects, the wall shear stress dictated by the viscous nature of the fluid and the positive pressure gradient imposed by the cascade geometry.

p

Outside boundary layer

y

> 0, U=U(x)

V

u y

>0

u y

=0

u y

90o, the deflection becomes large, leading to higher ȗsf -values. Once the combined secondary and friction coefficient are determined from (7.83), the corresponding stage loss coefficient is calculated from: (7.84)

204

7.5

7 Losses in Turbine and Compressor Cascades

Flow Losses in Shrouded Blades

To reduce the clearance losses, shrouds with integrated labyrinth seals can be applied to the rotor tip and stator hub. This is a common practice in HP- and IP- steam turbine design. The purpose of the labyrinths is to reduce the pressure difference across each labyrinth, thus, reducing the clearance mass flow by dissipating the kinetic energy of the leakage mass flow through the clearance. Figure 7.19 shows a turbine stage with the labyrinth seals.

Fig. 7.19: a) A turbine stage with stator and rotor shrouds with labyrinth seals, b) reduction of pressure across a labyrinth by means of dissipation of kinetic energy

Since the shrouded blading has no tip clearances, there is no secondary flow vortices due to tip clearance, but there are still secondary flows which originate from the boundary layer development on hub and casing walls. Similar to the case treated in the previous section, the secondary flow loss as well as the endwall friction loss determine the flow loss situation close to the blade hub and tip. For their estimation, Eqs. (7.82) and (7.83) are used.

7.5.1 Losses Due to Leakage Flow in Shrouds Consider a shrouded turbine stage, Fig. 7.20, upstream of the stator blade, the stage mass flow is . Immediately at the inlet of the stator, a fraction of the this mass flow, , penetrates into the stator labyrinth. In the axial gap, there is a mixing process between the remaining mass flow and the leakage flow which is supposed to be completed immediately upstream of the rotor blading. After the mixing at station (2a), a small portion, , flows through the rotor labyrinth and, therefore, does not contribute to power generation. Similar to the procedure used in [15], for the calculation of , and their dependencies upon the pressure difference across the labyrinth tooth, the conservation laws discussed in Chapter 4 are used.

7 Losses in Turbine and Compressor Cascades

2

1

2a

205

3 s"

m h m'

m-m'

m"

m-m"

m'

m" m

D D''

Dm

D D'

s'

Fig. 7.20: Flow through a labyrinth seal with mixing process within the axial clearance. Quantities with “ ” and “ ” refer to stator and rotor, respectively. At station 3, mixing of and the remaining takes place. This mixing process is repeated for all stator and rotor rows and causes a total pressure loss that can be calculated by using the conservation equations. The continuity equation at station 2 in Fig. 7.20 yields: (7.85) with as the mass flow through the stator labyrinth. Assuming that at station 2a the mixing process has been completed, the mass flow is obtained from: (7.86) Assuming an axisymmetric flow, the momentum equation in circumferential direction is: (7.87) and in axial direction (7.88) For the stator, the total pressure loss due to the mixing is then: (7.89)

206

7 Losses in Turbine and Compressor Cascades

Incorporating the above continuity and the momentum equations into Eq.(7.89), we find for the stator: (7.90) and for the rotor: (7.91) where and represent the mass flows through the stator and rotor labyrinths. To find the mass flows and we consider the flow through a labyrinth seal which consists of n teeth, Fig. 7.21.

Fig. 7.21: Expansion and dissipation process through a labyrinth tooth and chamber. (a) Expansion from the labyrinth chamber with the area aA through the tooth clearance c with the area aB and the following dissipation in the labyrinth chamber. (b) Polytropic expansion, where the velocity inside the chamber is negligible compared with the tooth velocity. We assume that the flow passing through the labyrinth clearance is isentropic. This assumption is not quite accurate as the real polytropic jet expansions in Fig. 7.21(b) obviously show. However, the entropy increase associated with the sharp expansion compared to the one due to the quasi-isobaric dissipation is so small that it can be neglected. This allows applying the Bernoulli equation to the expansion process inside the clearance: (7.92) where the subscript A refers to the cross sectional area within the labyrinth chamber and B refers to the clearance area between the shaft and the labyrinth tooth. After entering the labyrinth chamber, its kinetic energy is isobarically dissipated as heat. Since the area aB > V A. Therefore VA can be neglected compared to VB. As a result, we get:

7 Losses in Turbine and Compressor Cascades

207

(7.93) since

we find: (7.94)

Introducing Eq. (7.94) into Eq. (7.93) yields: (7.95) For better understanding, the pressure distribution along a labyrinth is shown in Fig. 7.21. For an equal pressure drop in each labyrinth, the pressure difference in a stator labyrinth can be expressed as: (7.96) where n1 is the number of stator labyrinths. With Eq. (7.95), we find: (7.97)

Equation (7.97) delivers the mass flow fraction through the stator labyrinth as: (7.98) For an incompressible flow through the stator, the density is approximated as . This approximation allows the introduction of the enthalpy difference: (7.99) Inserting Eq. (7.99) into (7.98), the stator leakage mass flow is calculated as: (7.100) We introduce relative mass flows by using the main mass flow and the stage flow coefficient:

208

7 Losses in Turbine and Compressor Cascades

(7.101) The relative mass flow through the stator labyrinth is presented in terms of dimensionless stage parameters followed by the relative mass flow through the rotor labyrinth. The following step-by-step re-arrangement establishes relationships between the relative mass flow through the stator- and rotor-labyrinths and the stage parameters:

(7.102) with the

, the clearance area , and the stator tooth clearance labyrinth, we find:

, the blade cross section from Fig. 7.21. Similarly for the rotor

(7.103) The values of the labyrinth flow contraction coefficients Į1 and Į2 depend strongly on the labyrinth shape. Typical values are 0.8-0.93. For a stage with the given ij, Ȝ, r, and labyrinth geometry, Figs. 7.22(a) and (b) exhibit the relative mass flows through stator- and rotor-labyrinths and respectively with the degree of reaction r as parameter. For the stator, Fig. 7.22(a), the largest leakage occurs at r = 0. Increasing the degree of reaction leads to smaller relative mass flow through the labyrinths. This corresponds to the pressure difference across the stator labyrinths. For r = 0 , the largest pressure difference occurs across the stator blade and, thus, stator labyrinths, while the pressure difference across the rotor is almost zero. Increasing the degree of reaction to r = 0.5 reduces the pressure difference across the stator blades and, thus, the stator labyrinths to 50%, resulting in smaller relative mass flow through the stator labyrinths. The relative mass flow through the rotor labyrinths presented in Fig. 7.22(b), however, shows the opposite picture. Largest leakage flow occurs at r = 0.5, since it has the largest pressure difference across the rotor blading and, thus, the rotor labyrinths. Reducing the degree of reaction lowers the pressure difference across the rotor blading and, thus, the labyrinth causing the relative leakage mass flow to decrease. The stage loss coefficient due to mass flow loss is:

7 Losses in Turbine and Compressor Cascades

209

(7.104) with the specific stage mechanical energy ǻH = lm. 0.015 (a) Stator 0.1

0.01

MStator

0.3

r = 0.0 0.2

0.4

r=0.5

0.005

0 0

1

O

2

3

Fig. 7.22a: Relative mass flow through the stator as function of stage load coefficient Ȝ with degree of reaction r as parameter with , , and the clearance ration , see with

from Fig. 7.21(a).

0.009 (b) Rotor

0.4

0.006

0.003

0

r = 0.0

0

1

O

2

3

Fig. 7.22b: Relative mass flow through the rotor-labyrinths as function of stage load coefficient Ȝ with degree of reaction r as parameter with , , and .

210

7 Losses in Turbine and Compressor Cascades

The stage clearance leakage loss is comprised of stator and rotor mixing losses with the mass flow ratios given by Eqs. (7.102) and (7.103) and rotor mass flow loss (7.104). It is summarized as and is detailed as: (7.105) Equation (7.105) has been evaluated and is presented in Fig. 7.23 as a function of the degree of reaction r with the stage load coefficient Ȝ as parameter. As seen, the lowest leakage loss is found at r = 0. Keeping the stage load coefficient Ȝ constant and increasing the degree of reaction results in higher stage clearance leakage losses. Steeper increases are shown at higher Ȝ- values.

0.05

0 O=4. 3.5

ZLC

0.04

0.03

0.02

0

0.1

3 2.5

0.2

2

1.5

0.3

r

1 0.4

0.5

0.6

Fig. 7.23: Stage clearance leakage loss coefficient ZLC for a shrouded stage as a function of degree of reaction with the stage load coefficient as parameter

7.6

Exit Loss

The exit loss represents the total energy loss caused by the kinetic energy of the mass flow that exits the turbine or compressor component. For multi-stage turbomachines, it occurs only at the last stage. For a single stage machine, the exit loss has a significant influence on the overall drop of the stage efficiency. The exit loss coefficient is defined as the ratio of the exit kinetic energy with respect to specific mechanical energy of the stage: (7.106)

7 Losses in Turbine and Compressor Cascades

211

Expressing the exit velocity vector V3 in terms of axial velocity component and using the stage dimensionless parameter from Chapter 5, we get: (7.107) Replacing the exit flow angle Į3 by the stage parameters from Chapter 5 yields:

(7.108) The stage flow coefficient ij can be expressed as a function of the stator exit flow angle Į2, which is known as one of the stage design parameters. (7.109) Introducing Eq. (7.109) into Eq. (7.108), the result is: (7.110) Equation (7.110) shows the influence of the stage load coefficient Ȝ, the stator exit flow angle Į2 and the degree of reaction r on the exit loss coefficient ZE. The results are presented in Fig. 7.24. For each pair of stator exit flow angle and degree of reaction r, there is a minimum exit loss coefficient at a particular Ȝ. Since the exit

1 D2 = 40

r=0

0.8

o

0.25

zE

0.6

0.5

0.4 0.25

0.2 0

D2 = 20 r=0

0.5

0

o

1

2

3

4

5

O Fig. 7.24: Exit loss coefficient ZE as a function of stage load coefficient Ȝ with degree of reaction r as a parameter

212

7 Losses in Turbine and Compressor Cascades

kinetic energy of the last stage in a multi-stage turbine does not contribute to generation of shaft power, it must be kept as small as possible. For a stator exit flow angle of , a degree of reaction of yields a minimum at . The rotor row with this stage characteristic has an absolute exit flow angle of .

7.7

Trailing Edge Ejection Mixing Losses of Gas Turbine Blades

Increasing the thermal efficiency of power or thrust generating gas turbine engines requires high turbine inlet temperatures. For conventional turbine blade materials, cooling of the front stages allows an increase of the turbine inlet temperature. The required cooling mass flow is injected partially or entirely through the trailing edge slots into the downstream axial gap where the cooling and main mass flows are mixed. The trailing edge ejection affects the flow regime downstream to the cooled blade, especially the losses associated with the mixing of the cooling mass flow and the main mass flow. The ejection velocity ratio, cooling mass flow ratio, slot-width ratio, and ejection angle affect the mixing losses and, therefore, the efficiency of cooled blades. Improper selection of these parameters results in higher mixing losses that reduce the efficiency of the cooled turbine stage. Experimental research work by Prust [16], [17] on a two-dimensional turbine stator cascade shows that the trailing edge ejection significantly affects the blade efficiency. Analytical and experimental investigations by Schobeiri [18], [19], and Schobeiri and Pappu [20] aim at identifying and optimizing the crucial design parameters to significantly reduce the aerodynamic losses that originate from trailing edge ejection mixing processes.

7.7.1

Calculation of Mixing Losses

Starting from the conservation laws presented in Chapter 4, relations are derived that accurately describe the influence of the above parameters on the flow field downstream of the cooled turbine blade. Figure 7.25 shows the stations upstream of the blade ±, immediately at the trailing edge plane ² at the mixing plane ³. The conservation laws are applied to the control volume ABCDA. The continuity equation can be written as: (7.111) The momentum equation in y-direction is:

(7.112) The momentum equation in x-direction is:

7 Losses in Turbine and Compressor Cascades

213

(7.113) In Eqs.(7.111) through (7.113), the parameter f = s/b represents the ratio of slot width s and the trailing edge thickness b; Į2 and Įc are the exit flow angle and ejection angle, respectively, of the cooling mass flow; V2 and Vc are the blade exit velocity and mean ejection velocity; p2(y) is the static pressure distribution along the spacing t. The angles Į2 and Įc in Fig. 7.25 are assumed to be constant.

Fig. 7.25: Trailing edge ejection and mixing downstream of a cooled gas turbine blade Outside the trailing edge region, the static pressure p2 (y) = p20 can be assumed as constant. As shown by Sieverding [21], inside the trailing edge region, p2 has a nonlinear distribution whose average might differ from p20. Since this pressure difference occurs only in a relatively small trailing edge region, its contribution compared with the other terms in Eq. (7.113), can be neglected. At the exit of the slot, the pressure p2(y) is determined by the static pressure pc of the cooling mass flow. By

214

7 Losses in Turbine and Compressor Cascades

presuming pc = p 20 and considering the above facts, the static pressure can be approximated over the entire spacing by . For further treatment of Eqs.(7.111) through (7.113), the boundary layer parameters, namely displacement thickness, momentum thickness and the shape parameter, are introduced: (7.114)

The dimensionless thicknesses are found by:

(7.115)

where the indices S and P refer to the suction and pressure surfaces. Introducing the dimensionless parameters (7.115) into the dimensionless versions of Eqs. (7.111) through (7.113), the continuity relation yields: (7.116) The momentum equation in y-direction is:

(7.117)

The momentum equation in x-direction determines the static pressure difference:

(7.118)

with:

as velocity, temperature, and density ratio.

7 Losses in Turbine and Compressor Cascades

215

To define the energy dissipation due to the mixing process, it is necessary to consider not only the contribution of the main flow, but also the cooling mass flow. For this purpose, we combine the mechanical and thermal energy balances and arrive at a relationship that represents the change of total pressure within a system where an exchange of thermal and mechanical energy with the surroundings takes place (for detail see Chapter 12). For an adiabatic steady flow with no mechanical energy exchange, this equation is reduced to:

(7.119) where

is the total pressure, T the shear stress tensor, and ț the

ratio of the specific heats. Eq. (7.119) states that the rate of work done on fluid per unit volume by viscous forces causes a defect of total pressure work per unit volume. For an inviscid flow, Eq. (7.119) reduces to a Bernoulli equation. Integrating Eq. (7.119) over the control volume and converting the volume integrals into the surface integrals by means of Gauss’ divergence theorem leads to.

(7.120)

The second and third integral in Eq. (7.120) are carried out over the entire inlet and exit surfaces under consideration. Introducing the individual mass flows , and substituting the inlet total pressure by the total pressure of the potential core at station 2, the total energy dissipation is obtained from:

(7.121)

With respect to the exit kinetic energy, the loss coefficient is defined as:

(7.122)

Incorporating the mass flow ratios:

216

7 Losses in Turbine and Compressor Cascades

(7.123)

into Eq. (7.122), using Eqs. (7.116) through (7.118) and introducing the following auxiliary functions:

(7.124)

The loss coefficient ȗ is completely described by:

(7.125)

where all the significant parameters determining the influence of trailing edge ejection are present. With this relation, it is possible to predict the energy dissipation due to the trailing edge thickness, the boundary layer thickness at the trailing edge, and the trailing edge ejection. In this connection, the contribution of the trailing edge ejection to the energy dissipation is of particular interest. For this purpose, a typical gas turbine cascade is considered where its geometry, exit flow angle Į2, and the boundary layer parameters are known. To show the effect of trailing edge ejection, the loss coefficient ȗ is calculated from Eq. (7.125) as a function of cooling mass flow ratio .

7 Losses in Turbine and Compressor Cascades

217

The loss coefficient ȗ includes the profile losses that are caused by viscosity effects. To eliminate this contribution, a cascade with the same flow conditions and boundary layer parameters, but with an infinitesimally thin trailing edge and no ejection is considered. This cascade has almost the same pressure distribution on the suction and pressure sides and also the same profile loss coefficient. The difference (ȗ-ȗo) in this case illustrates the effect of trailing edge ejection for a given trailing edge thickness.

7.7.2 Trailing Edge Ejection Mixing Losses Equation (7.125) includes three major ejection parameters that influence the mixing loss coefficient ȗ. These are the cooling velocity ratio , the cooling mass flow ratio , the slot-width ratio , and the temperature ratio . The effects of these parameters were theoretically investigated by Schobeiri [18], [19] and experimentally verified by Schobeiri and Pappu [20] for two different turbines, namely the Space Shuttle Main Engine (SSME) turbine and an industrial gas turbine. For both the cascades, the cooling mass flow ratio was varied from 0.0 to 0.04 that corresponds to the cooling velocity ratio range of

.

7.7.3 Effect of Ejection Velocity Ratio on Mixing Loss Figure 7.26 shows the total pressure mixing loss coefficient as a function of the velocity ratio, , with the non-dimensional slot-width ratio f as the parameter for the SSME blade. The symbols represent results from experimental measurements and the solid lines represent results from theoretical evaluations. For no injection, , there is a pressure loss due to finite thickness of the trailing edge and also from the boundary layer development along the pressure and suction surfaces of the blade. With increasing the cooling jet velocities, the losses initially increase until a maximum is reached. Further increase in cooling jet velocities results in the decrease of to a minimum, and then increasing thereafter. For , the losses due to injection of cooling mass flow are higher than for the no injection case. This is caused by the low momentum of the cooling jet being unable to sustain the strong dissipative nature of the wake flow downstream of the trailing edge. So, the main mass flow entrains the cooling mass flow resulting in a complete dissipation of the energy of the jet. Therefore, higher mixing losses occur at low injection velocities until a maximum is reached, which occurs around beyond which begins to decrease. For injection velocity ratios , the momentum of a cooling jet is sufficient to overcome the wake flow without being dissipated completely. Owing to this phenomenon, a significant reduction in mixing losses is evident from Fig. 7.26. This reduction proceeds until reaches a minimum, which is around . Further increase of increases the losses again for similar reasons explained earlier.

218

7 Losses in Turbine and Compressor Cascades

0.06

0.7

0.055

f = 0.35

]

0.5

0.05

0.3 0.1

0.045

0.04

0

0.2

0.4

0.6

0.8

P

1

1.2

1.4

Fig. 7.26: Trailing edge mixing loss coefficient ȗ as a function of velocity ratio ȝ with slot-width ratio f as a parameter for SSME blade, from [19] and [20]. Experiments shown as full squares. 0.06

]

0.06

0.3

0.05

0.04

0

0.01

0. 5

0.02

7

9 0.

0.1

0.05

0.

0.03

0.04

35 0. f=

0.05

0.06

(mscVc)/ms2V2) Fig. 7.27: Loss coefficient ȗ as a function of momentum ratio with slot-width ratio f as parameter for SSME blades from [19] and [20]

Figure 7.27 shows the mixing loss coefficient ȗ as a function of cooling jet momentum ratio with slot- width ratio f as the parameter for the Space Shuttle Main Engine turbine blade. Also, it is evident from this figure that for a given f, the mixing loss coefficient has a pronounced optimum. Taking into

7 Losses in Turbine and Compressor Cascades

219

consideration the wall thicknesses on each side of the slot at the trailing edge exit plane, the slot-width ratios of 0.3 to 0.35 seem to be a quite reasonable estimation for a cooled turbine with trailing edge ejection.

7.7.4 Optimum Mixing Losses Of particular interest to the turbine blade aerodynamicists and designers is how small the slot-width ratio f should be for a given cooling mass flow which is dictated by the heat transfer requirements to meet the optimum conditions for . Figures 7.28 provides this crucial information. This figure shows the mixing loss coefficient ȗ as a function of sloth-width ratio with the mass flow ratio as the parameter. Optimum mixing loss can easily be found by choosing the appropriate slot-width ratio.

]

2.7%

0.06

msc/ms2 =3.6%

1.8%

0.9%

0.07

Optimum slot ratio

0.05 0.0% 0.04

0

0.25

0.5

f

0.75

1

Fig. 7.28: Loss coefficient ȗ as a function of slot- width ratio f for SSME blade from [19] and [20]

7.8

Stage Total Loss Coefficient

After calculating the individual stage loss coefficients Zi, the total stage coefficient is calculated as: (7.126) where index i represents the individual stage losses, for example, profile losses, secondary losses, etc. We define the total isentropic stage loss coefficient Zs: (7.127)

220

7 Losses in Turbine and Compressor Cascades

where ǻhloss represents all the enthalpy losses due to the different loss mechanisms discussed in this chapter, and ǻHs the available stage isentropic enthalpy difference. For the turbine stage, Eq. (7.127) is written as: (7.128) and for compressor stage: (7.129) The isentropic efficiency for turbine stage is defined as the ratio of actual total enthalpy difference, which is identical with stage mechanical energy and the isentropic stage total enthalpy difference. The isentropic stage efficiency in terms of Zs is: (7.130) and in terms of Z: (7.131) Equation (7.131) represents the isentropic turbine stage efficiency in terms of the actual total stage loss coefficient Z. Similar relations are derived for compressor stage in terms of Zs: (7.132) and in terms of Z: (7.133)

7.9

Diffusers, Configurations, Pressure Recovery, Losses

Diffusers are attached to the exit of turbomachinery components such as compressors and turbines to partially convert the kinetic energy of the working medium into the potential energy. In the case of a power generation gas turbine with a subsonic exit Mach number, the diffuser back pressure is identical with the ambient pressure, thus, it cannot be increased. As a consequence, the diffuser reduces its inlet pressure below the ambient pressure to establish its design pressure ratio. Thus, installing a diffuser at the turbine exit causes the stage back pressure to drop. As a result, the specific total enthalpy difference of the turbine component will raise leading to an increased specific power at a higher efficiency. At the same time, the diffuser reduces the exit kinetic energy which is associated with a decrease in the exit loss of the engine. In

7 Losses in Turbine and Compressor Cascades

221

2 Compressor

mD

p

2D

H 2D

H

P1

D 2

2

1

p

1

2

pRef

2

2

2

h

2

2

2

2

H

3

2

2

l

lm

mD

pRef

l

lm

2

2

h

1

2

h p

2

2

2

1

2

H 2D

2

H

D

h

Turbine

1

2

H

3

P1

2

h

2

steam turbine power plants, the diffuser attached to the last stage of the low pressure turbine has a back pressure which is dictated by the condenser pressure. The condenser pressure, however, is determined by the environmental temperature (river temperature, cooling tower air temperature), thus, is considered constant. In the case of a compressor with a predefined exit pressure, the installation of a diffuser leads to a decrease of specific mechanical energy input to the compressor. The thermodynamic working principles of a diffuser installed at the exit of a turbine and a compressor component is shown in Fig. 7.29.

h

p

1

2D

S

S

Fig. 7.29: Interaction of diffuser-thermodynamics with the a turbine component (left) and a compressor component (right). Quantities with subscript D refer to the cases with diffuser. As seen, the reference pressure pref in case of a gas turbine or a steam turbine is identical with the ambient pressure and the condenser pressure, respectively. In the case of a compressor, it represents the desired compressor exit pressure.

7.9.1 Diffuser Configurations To effectively convert the kinetic energy of the exiting fluid into potential energy, axial, radial and mixed flow diffusers are used. The choice of a particular diffuser configuration primary depends on the design constraints dictated by the compressor or turbine sizes and their exit geometries. For gas turbine applications, axial flow diffusers are used that may have different geometries. As shown in Fig. 7.30(a,b), they may have outer conical casings and conical shaped or constant diameter center bodies. Multi-channel short diffusers may also be used to allow larger opening angles to avoid flow separation. While the short diffusers allow a more compact design, the additional surfaces pertaining to the separators increase the friction losses causing a decrease in diffuser efficiency. For steam turbines, based on specific design requirements, axial as well as radial diffusers are applied, Fig. 7.31.

222

7 Losses in Turbine and Compressor Cascades

L T /2

A1

(a)

To

T in

A2

(b)

(c)

Fig. 7.30: Different diffuser configurations for axial turbines: (a) Diffuser with constant inner diameter, (b) with variable inner and outer diameter, (c) a multichannel diffuser Radial diffuser

Steam collector to condenser LPT-last atage

Multi-channel diffuser for LP-steam turbine

Fig. 7.31: Radial diffuser configurations for axial compressors and low pressure steam turbines For industrial compressors, radial diffusers are usually applied that facilitate an effective kinetic energy conversion from the compressor exit to the collector, as shown in Fig. 7.31.Diffusers of radial compressors may be either vaneless or have vanes based on the absolute exit flow angle from the rotor blades.

7.9.2 Diffuser Pressure Recovery Performing an optimum conversion of the exiting kinetic energy from the last stage of a compressor or a turbine component into potential energy requires adequate composition of the following diffuser parameters: (a) area ratio, (b) length/height ratio (L/h), (c) diffuser angle ș, which is directly related to the axial length ratio (Lax/L), (d) inlet swirl parameter, (e) inlet turbulence intensity, (g) Re-number and (f) Mach

7 Losses in Turbine and Compressor Cascades

L

T

h1

223

L h1

A1

A1

A2

L ax

Short diffuser:

A2

A

A

2 ___ 2 =( ) ( ___ ) A A1 1 L ___ = 2.7 h1

Short

Long

Long diffuser:

L ___ = 16.9 h1

Fig. 7.32: A short and a long diffuser with the same area ratio but different opening angles number. In addition, the boundary layer thickness at the inlet as well as blockage factors play a substantial role in separation-free operation of diffusers. As Fig.7.32 indicates, two diffusers may be designed with the same area ratio but have two different axial lengths. While the long diffuser has a smaller opening angle that makes it less susceptible to flow separation, the short one has the advantage of performing the energy conversion within a much shorter length resulting in a significantly reduced axial length of a turbomachine. A short diffuser, if designed properly, can produce the same or even higher degree of pressure recovery than a long one. Therefore, the objective of an appropriate diffuser design is to achieve a near-optimum pressure recovery at an acceptably short axial length (opening angle ș). An optimum pressure recovery can be achieved if the diffuser operates close to the separation point. At this point, the wall shear stress is causing a reduction of total pressure loss coefficient ȗ, thus, an increase in pressure recovery. Earlier classical research work by Kline, et. al.[22] was conducted on straight walled diffusers to expedite the basic design parameters. Sovran and Klomp [23] conducted a systematic experimental research to determine the optimum geometries for diffusers with rectangular, conical and annular cross-sections. From a multitude of experiments working diagrams were generated that allow extracting pressure recovery values for given area ratios and length/inlet height ratio . Averaging the flow quantities at the inlet and exit, the total pressure balance at any arbitrary position in x-direction reads: (7.134) Since the loss coefficient ȗ is directly related to the wall shear stress, for an incompressible flow and , Eq.(7.134) with approaches the Bernoulli equation

224

7 Losses in Turbine and Compressor Cascades

leading to a maximum pressure recovery. Considering the averaged quantities in Eq.(7.134), we define the pressure recovery factor:

(7.135)

In Eq.(7.135) cpR exhibits the real pressure recovery factor, the width at station 1 and x, respectively as shown in Fig. 7.33.

W P ___u

Ux

x

G1

p ___ >0 x

, b1 and bx

U1 sx

bx

b

x

b1

b

s1

y

T/ 2

L

y 1

x

x

Fig. 7.33: Schematics of velocity and boundary layer distributions As Eq. (7.135) indicates, the calculation of cpR requires the knowledge of the entire distribution of p1 and px over the width. Applying the Prandtl assumption that the pressure on the wall and within the boundary is determined by the pressure outside the boundary layer, Eq.(7.135) can be simplified as: (7.136) with p1 and px as the wall static pressures. Equation (7.136) can be applied for diffuser performance assessment as long as the boundary layer is fully attached. In case of a flow separation, the boundary layer thickness becomes large and the above Prandtl assumption loses its validity. Therefore, a different recovery factor needs to be defined that accounts for adverse changes of the boundary layer including separation. A physically similar criterion that also considers the adverse effects of boundary growth in case of flow separation is the conversion coefficient Ȝ defined as: (7.137)

7 Losses in Turbine and Compressor Cascades

225

with U1 and Ux as the velocities outside the boundary layer, as shown in Fig. 7.33. Equation (7.137) indicates that increasing the boundary layer thickness as a result of an adverse pressure gradient caused by a large ș, reduces the size of the core flow, thus, increases its velocity and reduces Ȝ. Increasing ș beyond its limit leads to a massive flow separation. In this case, the fluid exits the diffuser as a jet with a velocity and . The flow separation produces a strongly vortical unsteady flow regime that occupies a major portion of the diffuser volume, Fig. 7.34. The exiting jet may oscillate between the walls or be attached to one of the side walls, as shown in Fig. 7.34. Separated flow regime

Separated flow regime

U1

T / 2 > Tlim/ 2

Attached jet

Fig. 7.34: Unsteady exiting jet oscillating between the two side walls for ș > șlim

7.9.3 Design of Short Diffusers While the specific net power of gas turbines continuously increase, making them more compact, their exit diffusers still occupy a substantial volume. To reduce the volume, short diffusers may be designed that fulfill the following criteria: (a) the diffuser contour should be configured such that its boundary layer is less susceptible to separation, (b) measures should be taken to prevent/suppress the boundary layer separation by influencing the inlet flow condition of a short diffuser within which the boundary would separate. The above issues were addressed in an experimental and theoretical study by Schobeiri [24]. In case (a), a systematic conformal transformation was conducted, where the geometry of diffusers with straight, concave and convex walls were described, Fig. 7.35. To calculate the degree of susceptibility with regard to flow separation, in [25], a laminar flow was assumed which is more sensitive to separation than the turbulent one. For the above geometries, the Navier Stokes equations were transformed into the respective body fitted orthogonal coordinate systems and the corresponding solutions for different diffuser geometries were found. Given the same area ratio and the inlet flow condition for all three geometries (convex, straight, concave), the concave walled diffuser showed to be less sensitive to the flow separation as shown in Fig.7.35.

226

7 Losses in Turbine and Compressor Cascades

L

L

T /2 A1

T /2

A2

(a) Straight wall

L

A1

T /2

A2

(b) Convex wall

A1

A2

(c) Concave wall

Fig. 7.35: Optimization of the diffuser contour: (a) reference configuration flow close to separation ș = șlim , (b) convex wall: flow already separated, (c) concave wall: separation delayed In case (b) the boundary layer separation is prevented by generating vortices to control the flow separation as reported in [25]. Experimental investigations were performed by varying the opening angle Ĭ at three different diffuser length ratios , Fig.7.36. Straight walled diffuser with vortex generator edge (VGE) Vortex generator edge Details

Vortex generator edge

h VGE

A

L

22 W B A

mm

82

B

h VGE

h = 0, 3, 4, 6 mm

Fig. 7.36: Straight walled diffuser with vortex generator (VGE) to avoid early flow separation. To suppress or delay the boundary layer separation, a system of two vortex filaments were generated by two vortex generator edges (VGE) installed upstream of the diffuser inlet. Two types of VGE were applied, Fig. 7.36 A and B. The first VGE had a straight edge, whose immersion height was varying from 0.0 to 6.0 mm. The second VGE had a nozzle shape with an optimized angle of 82o and an immersion of 6.0 mm. The flow visualization, Fig. 7.37, shows the generation of a strong vortex filament by a VGE. This vortex generates downstream of the edge an induced velocity field that is described by the Bio-Sawart law, which we discussed in Section 7.4.1. Referring to Section 7.4.1, Fig. 7.15 schematically shows the working principle of the BioSawart law in context of diffuser flow stability. As seen, an infinitesimal section of a vortex filament dȟ with the vortex strength ī induces at a distance a velocity field

7 Losses in Turbine and Compressor Cascades

227

Fig. 7.37: Visualization of a straight vortex filament by a VGE which is perpendicular to the plane described by the normal unit vector at the unit vector of the distance . Integrating over the entire vortex length determines the induced velocity that adds a component to the flow velocity pushing the fluid particle to the diffuser wall thereby suppressing the onset of a separation. Besides the stabilizing effect through the induced velocity component, the presence of VGEs contributes to energizing the boundary layer by increasing the turbulence intensity, thereby increasing the exchange of lateral momentum with the boundary layer fluid. The effect of VGEs on conversion coefficient Ȝ is shown in Fig. 7.38 for a diffuser with l/w = 6 for four different VGE-configurations, A,B,C and D. In Diagram A, where the VGE-height h = 0, the conversion factor Ȝ for ș = 0, has a decreasing tendency which is equivalent to accelerating the flow within the diffuser because of the blockage due to the boundary layer development. Increasing ș causes an increase in Ȝ. Up to ș/2 = 5o, the entire diffuser operates separation free. Further increasing ș leads to separation. Introducing straight VGEs of height, 3mm and 4 mm respectively (Fig. 7.38 B and C), allows larger opening angles at higher Ȝ. A substantial increase in Ȝ is obtained by introducing a VGE with the height of 6 mm and an angle of 82o as shown in 7.38B. It should be emphasized that application of VGEs is associated with a total pressure loss. However, the pressure gain compensates for this loss [25].

228

7 Losses in Turbine and Compressor Cascades

Fig. 7.38: Effect of Vortex generator edge on diffuser conversion coefficient from [25]

7.9.4 Some Guidelines for Designing High Efficiency Diffusers Diffusers may improve the efficiency of gas turbines, steam turbines, compressors and turbochargers by up to 4 % if designed properly. As mentioned above, the axial length ratio and the area ratio are the major parameters defining the conversion coefficient Ȝ. Equally important is taking into consideration the distribution of the velocity and angle of the mass flow exiting the last stage of the turbomachinery

7 Losses in Turbine and Compressor Cascades

229

component. Here, particular attention must be paid to designing multi-channel diffusers with struts and positioning the latter in such a way that they are not exposed to adverse flow incidence. The following steps may help designing a diffuser that operates at a high Ȝ with adequate efficiency: 1. First, accurately determine the distribution of flow velocity and angle at the exit of the last stage. This can be done using the streamline curvature method for stage aerodynamic design and is treated in Chapter 10. Using commercial CFDpackages may help in qualitatively estimating these distributions. For details refer to Chapter 20. 2. Consider a plane diffuser with a total limiting angle șlim of 12o to 16o, and a given length ratio as the reference diffuser. 3. For the diffuser to be designed, convert the area ratio and the axial length ratio such that they correspond to the opening angle and the area ratio of the reference plane diffuser in step (2). 4. If a multi-channel design is used, each single channel should be treated individually as an independent diffuser following step (3). Also important is the design and the radial distribution of the struts to ensure incidence free flow.

References 1. Schobeiri, M.T.: Fluid Mechanics for Engineers. A Graduate Text Book. Springer, Heidelberg (2010) 2. Traupel, W.: Thermische Turbomaschinen, vol. I. Springer, New York (1977) 3. Pfeil, H.: Verlustbeiwerte von optimal ausgelegten Beschaufelungsgittern. Energie und Technik 20, Jahrgang, Heft 1, L.A. Leipzig Verlag Düsseldorf (1968) 4. Kirchberg, G., Pfeil, H.: Einfluß der Stufenkenngrössen auf die Auslegung von HD-Turbinen. Zeitschrift Konstruktion, 23. Jahrgang, Heft 6 (1971) 5. Speidel, L.: Einfluß der Oberflächenrauhigkeit auf die Strömungsverluste in ebenen Schaufelgittern. Zeitschrift Forschung auf dem Gebiete des Ingenieurwesens, vol. 20(5) (1954) 6. NASA SP-36 NASA Report (1965) 7. Prandtl, L.: Zur Berechnung von Grenzschichten. ZAMM 18, 77–82 (1938) 8. von Kármán, T.: Über laminare und turbulente Reibung. ZAMM 1(1921), 233–253 (1921) 9. Ludwieg, H., Tillman, W.: Untersuchungen über die Wandschubspannung in turbulenten Reibungsschichten. Ingenieur Archiv 17, 288–299, Summary and translation in NACA-TM-12185 (1950) 10. Lieblein, S., Schwenk, F., Broderick, R.L.: Diffusions factor for estimating losses and limiting blade loadings in axial flow compressor blade elements. NACA RM E53D01 June (1953) 11. Schobeiri, M.T.: A New Shock Loss Model for Transonic and Supersonic Axial Compressors with Curved Blades. AIAA, Journal of Propulsion and Power 14(4), 470–478 (1998) 12. Berg, H.: Untersuchungen über den Einfluß der Leistungszahl auf Verluste in Axialturbinen. Dissertation, Technische Hochschule Darmstadt, D 17 (1973)

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7 Losses in Turbine and Compressor Cascades

13. Wolf, H.: Die Randverluste in geraden Schaufelgittern. Dissertation, Technische Universität Dresden (1960) 14. Schichting, H.: Boundary Layer Theory. McGraw-Hill series. McGraw-Hill, New York (1979) 15. Pfeil, H.: Zur Frage der Spaltverluste in labyrinthgedichteten Hochdruckstufen von Dampfturbinen. Zeitschrift Konstruktion 23(4), 140–142 (1971) 16. Prust, H.: Cold-air study of the effect on turbine stator blade aerodynamic performance of coolant ejection from various trailing-edge slot geometries. NASA-Reports I: TMX 3000 (1974) 17. Prust, H.: Cold-air study of the effect on turbine stator blade aerodynamic performance of coolant ejection from various trailing-edge slot geometries. NASAReports II: TMX 3190 (1975) 18. Schobeiri, T.: Einfluß der Hinterkantenausblasung auf die hinter den gekühlten Schaufeln entstehenden Mischungsverluste. Forschung im Ingenieurwesen 51(1), 25–28 (1985) 19. Schobeiri, M.T.: Optimum Trailing Edge Ejection for Cooled Gas Turbine Blades. ASME Transaction, Journal of Turbo machinery 111(4), 510–514 (1989) 20. Schobeiri, M.T., Pappu, K.: Optimization of Trailing Edge Ejection Mixing Losses Downstream of Cooled Turbine Blades: A theoretical and Experimental Study. ASME Transactions, Journal of Fluids Engineering, 1999 121, 118–125 (1999) 21. Sieverding, C.H.: The Influence of Trailing Edge Ejection on the Base Pressure in Transonic Cascade. ASME Paper: 82-GT-50 (1982) 22. Kline, S.J., Abbot, D., Fox, R.: Optimum design of straight-wall diffusers. ASME, Journal of Basic Engineering 81, 321–331 (1959) 23. Sovran, G., Klomp, E.D.: Experimentally determined optimum geometries for rectilinear diffusers with rectangular, conical and annular cross-section. Fluid Dynamics of Internal Flow. Elsevier Publishing Co, Amsterdam (1967) 24. Schobeiri, M.T.: Theoretische und experimentelle Untersuchungen laminarer und turbulenter Strömungen in Diffusoren. Dissertation, Technische Universität Darmstadt, D17 (1979)

8

Efficiency of Multi-stage Turbomachines

In Chapter 7, we derived the equations for calculation of different losses that occur within a stage of a turbomachine. As shown, the sum of those losses determines the stage efficiency which shows the capability of energy conversion within the stage. The stage efficiency, however, is not fully identical with the efficiency of the entire turbomachine. In a multi-stage turbomachine, the expansion or compression process within individual stages causes an entropy production which is associated with a temperature increase. For an expansion process, this temperature increase leads to a heat recovery and for a compression process, it is associated with a reheat ash shown in Fig. 8.1. As a consequence, the turbine efficiency is higher and the compressor efficiency lower than the stage efficiency. The objective of this chapter is to describe this phenomenon by means of classical thermodynamic relations. The approach is adopted by many authors, among others, Traupel [1] and Vavra [2].

8.1

Polytropic Efficiency

For an infinitesimal expansion and compression, shown in Fig. 8.1, we define the small polytropic efficiency: (8.1) For an infinitesimal expansion, cp can be considered as constant:

(8.2)

For a polytropic process it generally holds pvn = const. Inserting Eq. (8.2) into Eq. (8.1):

(8.3)

232

8 Efficiency of Multi-stage Turbomachines

After expanding the expressions in the parentheses in Eq. (8.3): (8.4) and neglecting the higher order terms, we find: (8.5) Now, we introduce Eq. (8.5) into (8.3) and obtain the small polytropic efficiency: h B

p1

1

p

s

h*

dh* s

dh dh s

a

h

A

p-dp

p2

2

2s S

S

h C

N

D p p

2s

h*

dh* s

s

dh s

p+dp

dh

h

Ns

p 1

Fig. 8.1: A) Heat recovery in a turbine with Nstage=, B) Heat recovery in a turbine with Nstage=4, C) Reheat process in a compressor with Nstage=, D) Reheat process in a compressor with Nstage=4

8 Efficiency of Multi-stage Turbomachines

233

(8.6)

For the expansion process, Eq. (8.6) reduces to: (8.7) likewise for a compression process we find: (8.8) with n as the polytropic and ț as isentropic exponents. Equations (8.7) and (8.8) express the direct relation between the polytropic efficiency Șp and the polytropic exponent n for infinitesimally small expansion and compression, where n and ț are considered constant. In a multi-stage environment, both exponents change from the start of expansion or compression to the end. As a result, Eqs. (8.7) and (8.8) are not valid for finite expansion or compression processes. However, if the end points of these processes are known, averaged polytropic efficiency for multi-stage compression and expansion can be found. For a process with known beginning and end points, we may use the polytropic relation: (8.9) and the following enthalpy relation: (8.10) where we find:

are the averaged polytropic and isentropic exponents. From Eq. (8.9),

(8.11)

Inserting Eq. (8.11) into Eqs. (8.7) and (8.8), we find

234

8 Efficiency of Multi-stage Turbomachines

(8.12)

As mentioned previously, calculating the above polytropic efficiencies require the knowledge of the process end points, which are at the beginning of a design process, unknown, but can be determined after the individual losses are calculated. The averaged isentropic exponent in Eq. (8.12) is calculated from: (8.13)

8.2

Isentropic Turbine Efficiency, Recovery Factor

Assuming the turbine has an infinite number of stages, we start from an infinitesimal expansion to define the isentropic stage efficiency, Fig. 8.1:

(8.14) Using the definitions in Eq. (8.1), we find the enthalpy differential: (8.15) As shown in Fig. 8.1, for an infinitesimally small cycle with the area a, the isentropic enthalpy differentials are interrelated by the first law as: . Using Eq. (8.15), we arrive at: (8.16) Integration of Eq. (8.15) and (8.16) yields: (8.17) and: (8.18)

8 Efficiency of Multi-stage Turbomachines

235

with A as the area 122'1, and Ș s and Ș p as the mean isentropic turbine and mean polytropic efficiencies. Equating (8.17) and (8.18) results in:

(8.19)

We introduce the heat recovery factor for a turbine with the stage number N = : (8.20)

With this definition, Eq. (8.19) becomes:

(8.21) To find the recovery factor, we divide Eq. (8.17) by (8.18):

(8.22)

The integral in the numerator must be performed along the polytropic expansion, whereas the integral expression in the denominator must be performed along the isentropic expansion. We insert in Eq. (8.22) for the specific volume, the relationship for polytropic and isentropic processes:

(8.23)

and arrive at:

236

8 Efficiency of Multi-stage Turbomachines

(8.24)

The polytropic exponent in Eq. (8.23) can be expressed in terms of and Șp using Eq. (8.8). With this modification, the recovery factor is obtained from:

(8.25)

For a multi-stage turbine with N stages, Fig. 8.1B, the following geometrical approximations may be made:

(8.26)

with ǻm as the area that represents an average of all triangular areas ǻi. With this approximation, we obtain and define the ratio:

(8.27)

Since A and ǻm are similar, we may approximate (8.28) As a result, Eq. (8.27) reduces to: (8.29)

8 Efficiency of Multi-stage Turbomachines

237

Using the same procedure that has lead to Eq. (8.22) for a turbine component with a finite number of stages, we find: (8.30) Since fT is > 0, the turbine isentropic efficiency is greater than the polytropic efficiency. This is a consequence of stage entropy production that results in heat recovery. For a turbine component with a pressure ratio , the recovery factor is plotted in Fig. 8.2, where the averaged turbine polytropic efficiency is varied from 0.7 to 0.95. As seen, the lowest efficiency of causes the highest dissipation that leads to a high recovery factor. Increasing the polytropic efficiency to causes a much less dissipative expansion. As a result, the heat recovery for the turbine with with .

0.12

0.08

is much lower than the previous one

Kp=0.75 0.75

ffT

0.80 0.85

0.04

0

0.9 0.95 0

0.25

0.5

ST = p2/p1

0.75

1

Fig. 8.2: Recovery factor of a multi-stage turbine for ț = 1.4

8.3

Compressor Efficiency, Reheat Factor

To obtain the reheat factor for a multi-stage compressor, we apply a similar procedure as outlined in Section 8.2. The isentropic efficiency of an infinitely small compression, shown in Fig. 8.1, is defined as:

238

8 Efficiency of Multi-stage Turbomachines

(8.31) Polytropic efficiency is: (8.32)

For a compressor with an infinite number of stages, the mean isentropic efficiency is: (8.33) where the expression 1 + fC is the reheat factor and the subscript C refers to a compressor. The reheat factor is obtained from:

(8.34)

For a multi-stage high pressure compressor component with a pressure ratio , the reheat factor is plotted in Fig. 8.3 where the averaged compressor polytropic efficiency is varied from 0.7 to 0.95. Similar to the turbine component discussed in Section 9.2, the lowest efficiency of causes the highest dissipation that leads to a high reheat factor. Increasing the polytropic efficiency to causes much less dissipation. As a result, the reheat for the compressor with is much lower than the previous one with . Using the same procedure discussed in the previous section, we find the following relationship between for a multi-stage compressor with N stages:

(8.35)

Once the reheat factor for a compressor with an infinite number of stages

is

determined, the reheat factor of a compressor with a finite number of stages is calculated using Eq.(8.35).

8 Efficiency of Multi-stage Turbomachines

0.2

ffC

0.16

239

0.75 KP=0.7

0.8

0.12 0.85 0.08

0.9

0.04 0

0.95 5

10

SC = p2/p1

15

20

Fig. 8.3: Reheat factor of a multistage compressor for ț = 1.4

8.4

Polytropic versus Isentropic Efficiency

As shown in Eqs. (8.18) and (8.34), the polytropic efficiency accounts for the energy dissipation within a turbine and a compressor component. Furthermore, the polytropic efficiency can be used to compare turbine or compressor components that are operating under different initial conditions or have different working media. In these cases, both polytropic and isentropic efficiencies change. However, the rate of change for polytropic efficiency is much smaller than that of an isentropic one. It is indeed possible for two compressor or turbine components to have the same polytropic efficiency if certain similarity conditions are met. This is not the case for isentropic efficiency. Therefore, the isentropic efficiency is not appropriate for comparison purposes. Another major advantage of polytropic efficiency is the use of rowefficiencies to calculate the row-by-row expansion or compression process. Calculating the polytropic efficiency, however, requires the knowledge of expansion or compression end points which are generally unknown at the stage of design. The isentropic efficiency, on the other hand, exhibits a simple and practical tool, since the isentropic enthalpy difference is calculated from the inlet and exit conditions at constant entropy. As we saw in this chapter, these two efficiencies are related to each other. Figure 8.4 exhibits, for a multi-stage turbine component, the dependency of the isentropic efficiency upon the polytropic efficiency with the pressure ratio as the parameter. As shown in Fig. 8.4, the isentropic efficiency is always greater than the polytropic one. The difference diminishes by increasing the polytropic efficiency, which reflects a reduction in dissipation. Figure 8.5 exhibits an opposite tendency for a multi-stage compression process. In this case, the polytropic efficiency is always greater than the isentropic one.

240

8 Efficiency of Multi-stage Turbomachines

1

Turbine recovery: Kp > Ks 0.9

KS

p1/p2= 10

0.8

0.7

p1/p2 = 2, 'S=1.0

0.7

0.75

0.8

0.85

0.9

0.95

1

KP Fig. 8.4: Turbine isentropic efficiency as a function of polytropic efficiency with turbine pressure ratio as parameter for ț = 1.4

1

Compressor reheat: Ks > Kp 0.9

KS

p2/p1 = 2, 'S=1.0

0.8

p2/p1= 10 0.7

0.7

0.75

0.8

0.85

0.9

0.95

1

KP Fig. 8.5: Compressor isentropic efficiency as a function of polytropic efficiency with compressor pressure ratio as a parameter for ț = 1.4

References 1. Traupel, W.: Thermische Turbomaschinen, vol. I. Springer, New York (1977) 2. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960)

9

Incidence and Deviation

Up to this point, the relationships developed for a turbomachinery stage have been strictly correct for given velocity diagrams with known inlet and exit flow angles. We assumed that the flow is fully congruent with the blade profile. This assumption implies that the inlet and exit flow angles coincide with the camber angles at the leading and trailing edges. Based on the operation condition and the design philosophy, there might be a difference between the camber and flow angle at the leading edge, which is called the incidence angle. The difference between the blade camber angle and the flow angle at the exit is termed the deviation angle. Since the incidence and deviation affect the required total flow deflection, the velocity diagram changes. If this change is not predicted accurately, the stage operates under a condition not identical with the optimum operation condition for which the stage is designed. This situation affects the efficiency and performance of the stage and thus the entire turbomachine. In order to prevent this, the total flow deflection must be accurately predicted. The compressor and the turbine flows react differently to a change of incidence. For instance, a slight change of incidence causes a partial flow separation on the compressor blade suction surface that can trigger a rotating stall; a turbine blade is less sensitive even to greater incidence change. To obtain the incidence and deviation angle for compressor and turbine blades, we use two different calculation methods. The first method deals with the application of conformal transformation to cascade flows with low deflection as in compressor blades. The second method concerns the calculation of deviation in high loaded cascades as in turbine blades.

9.1

Cascade with Low Flow Deflection

9.1.1 Conformal Transformation Conformal transformation is one of the areas in classical complex analysis. Its application to fluid dynamics and cascade aerodynamics can be found, among others, in [1], [2], and [3]. Weinig [4] applied the conformal transformation method to turbomchinery blading and developed a simple procedure for calculating the exit flow angle of cascades with low flow deflection such as compressor cascades. Further treatment of cascade aerodynamics using method of conformal transformation by Weinig is found in [5], [6], [7], and [8]. Starting from the simplest case, consider the physical plane where a cascade consisting of identical straight-line profiles is located (Fig. 9.1).

242

9 Incidence and Deviation

Fig. 9.1: Conformal transformation of a straight cascade. The cascade in z-plane is mapped onto the unit circle in ȗ-plane. The profiles are staggered at an angle Ȟm and have a spacing s. The flow is parallel to the cascade profiles. In other words, the profiles themselves are stream lines. Furthermore, assume that the flow originates from -  and flows to + . Since the flow from one cascade strip to another remains the same, it is sufficient to map only one of the profiles into the complex plane ȗ on the unit circle. This unit circle exhibits one Riemann's sheet in the complex plane. The ȗ-plane consists of an infinite number of Riemann's sheets, which correspond to the number of straight-line profiles within the cascade. The origin and end of the flow field, located ±  in z-plane, must be mapped onto the real axis of ȗ-plane at ± R (Fig.9.1). The complex potential in z-plane may be written as: (9.1) where ĭ and Ȍ are the potential and stream functions. The function X(z) is holomorphic if it is on the closed curve ABA continuous and differentiable, in other words, it must be an analytic function with the derivative (9.2) that satisfies the Cauchy-Riemann's condition: (9.3) Inserting Eqs. (9.3) into Eq. (9.2), we find (9.4)

9 Incidence and Deviation

243

If the velocity in z-plane is V, its components in x- and y-directions are: (9.5) Introducing Eq. (9.5) into (9.4) yields: (9.6) The integration of Eq. (9.6) leads to : (9.7) which is: (9.8)

Comparing Eq. (9.8) with (9.1) delivers:

(9.9)

with ĭ in Eq. (9.9) as the complex velocity potential (potential function) and Ȍ the stream function. Moving from one profile to another, as shown in Fig. 9.1, the potential and stream functions experience the following changes: at at and

244

9 Incidence and Deviation

Using Eq. (9.9), the potential difference can be expressed as: (9.10) and the stream function difference is: (9.11) The stream function difference, Eq. (9.11), represents the volume flow through a strip of the cascade with spacing s and height h=1. This volume flow originates from a source at z = - and has a source strength Q with Q = ǻȌ = Vs cosȞ. To generate such ǻȌ and ǻĭ in z-plane, the corresponding conformal transformation in ȗ-plane at -R must contain a source and a vortex with the strengths Q and ī. Similarly at +R, a sink and a vortex with the strengths -Q and -ī must be placed that correspond to the station + in z-plane. Thus, a system of sources, sinks, and vortices is arranged on the real ȗ-axis outside (exterior) the unit circle. This system fulfills the boundary condition at +  and also ± R, but the most important requirement that the unit circle must itself be a streamline, is not fulfilled. This problem can be solved simply by reflecting the singularities onto the unit circle. The locations of the reflected singularities are then ± 1/R inside (interior) the unit circle. The next step is to find the corresponding transformation function in ȗ-plane.

Fig. 9.2: Singularities source and circulation distribution with stream function Ȍ and potential function ĭ The complex potential for a source located at ȗ= 0 and has a strength Q is (Fig. 9.2): (9.12) In Eq. (9.12) the complex function ȗ is:

9 Incidence and Deviation

245

(9.13)

Introducing Eq. (9.13) into (9.12) and decomposing the result into its real and imaginary parts ĭ, Ȍ respectively, we find

(9.14)

with the radius in Eq.(9.14) as (9.15) By moving around the center of the source along a closed curve, ș in Eq. (9.15) will change from 0 to 2ʌ and as a result, the change of stream function is: (9.16) which corresponds to the source strength. For a source located at ȗ = R, the complex potential is: (9.17) where for sources Q > 0 and for sinks Q < 0. The potential lines of source, at ȗ = 0 are concentric circles, Fig. 9.2. The vortex flow can be established by generating stream lines from these concentric circles, inserting Q = iī in Eq. (9.14): (9.18) The variables in the parentheses of Eq. (9.18) are expressed in terms of (9.19)

246

9 Incidence and Deviation

Using ĭ and Ȍ as the potential and stream function from Eq. (9.19), the velocity can be obtained from Eq. (9.20) (9.20) where the differential arc length ds = rdș. By moving around the vortex center from ș = 0 to 2ʌ, the potential difference is , where ī is the circulation. For a vortex located at ȗ = R, the complex potential is: (9.21) If the source is located at an arbitrary point ȗo, Eq. (9.21) can be written as: (9.22) For a vortex flow, Eq. (9.22) yields: (9.23) The required complex potential consists of sinks, sources and vortices located at ±R and ±1/R which we summarize as X(ȗ) = *X(ȗ)sources + *X(ȗ)sinks + *X(ȗ)vortices is given as

(9.24)

Rearranging Eq. (9.24):

(9.25)

9 Incidence and Deviation

Setting

247

and equating Eq. (9.7) and (9.25) yields:

(9.26) resulting in:

(9.27)

With Eq. (9.27), the transformation function is completely defined. As shown, the complex potential X(ȗ) fulfills the streamline requirement on the unit circle. The contributions of all the singularities outside (exterior) the circle are exactly equal to the contributions of the singularities inside (interior) the circle. The complex potential on the unit circle that characterizes the equilibrium state between the singularities is expressed as:

(9.28)

where the subscript c in Eq. (9.28) refers to unit circle contour with ȗc = 1.eiĮ. Decomposing Eq. (9.28) into its real and imaginary parts, the real part is:

(9.29)

The velocity distribution on the unit circle can be obtained by differentiating Eq. (9.29) around the circle:

(9.30)

248

9 Incidence and Deviation

The stagnation points on the unit circle requires that in Eq. (9.30) the velocity vanishes, V = 0. This requirement leads to:

(9.31)

In Eq. (9.31) Įs is the first stagnation angle. To find the potential difference between the stagnation points Įs and Įs + ʌ, the angle Į in Eq. (9.29) is replaced successively by Įs and Įs + ʌ. In ȗ-plane: (9.32)

In z-plane:

(9.33)

As a result, we obtain the potential difference from Eq. (9.33):

(9.34)

Equating the potential differences in ȗ- and z-plane from Eq. (9.32) and (9.34) yields: (9.35) and with Eq. (9.29) we have:

(9.36)

9 Incidence and Deviation

249

3

2

2.7

s/c

2.3 1.9

1

1.5 R=1.1

0

30

Q m>R@

60

90

Fig. 9.3: Spacing/chord ration s/c as a function of the cascade stagger angle Ȟm with the mapping parameter R as parameter

90

'R=0.2

DSt [o]

60

1.5

30 1.3

0

15

30

45

Q m [R@

R=1.1

60

75

90

Fig. 9.4: Stagnation angle Įst as a function of the cascade stagger angle Ȟm with the mapping parameter R as parameter

250

9 Incidence and Deviation

The cascade parameter Ȟm and c/s from Eq. (9.36) that belong to z-plane correspond to the Įs and R in ȗ-plane. Equations (9.31) and (9.36) can be computed easily. The result of such a computation is shown in Figs. 9.3 and 9.4. As Fig. 9.3 shows, to each given pair of s/c and Ȟm only one value is allocated to the transformation parameter R. Taking this parameter and the stagger angle Ȟm, the stagnation angle ĮSt is determined from Fig. 9.4. With the aid of Eqs. (9.24) to (9.36), it is possible to calculate exactly the potential flow through cascades that consist of straight-line profiles. Since the turbomachinery blading has always curved camber lines, the derived conformal transformation methods must be extended to the profiles with curved camber lines. Since the compressor blade profiles have usually low deflection, their camber line can be approximated by a circular arc. In the following section, we replace the straight-line cascade by circular arc one.

9.1.2 Flow Through an Infinitely Thin Circular Arc Cascade Consider a potential flow through a cascade consisting of a number of circular arcs. The potential flow condition requires that the contour of the circular arc follow the stream lines. This requirement implies that the flow direction must be identical to the contour direction Ȟ of the circular arc profiles under consideration. To simplify the problem, replace the circular arc cascade by a straight cascade, EA, with prescribed flow directions that correspond to the contour direction Ȟ as shown in Fig. 9.5. To establish the corresponding analytical relations, we start from the conformal transformation relations derived in the previous section and extend them to an

Fig. 9.5: Prescribed velocity directions on straight line cascade

9 Incidence and Deviation

251

infinitely thin cascade with variable direction. For the cascade with the prescribed direction Ȟ, the complex potential, Eq (9.7) is rewritten as: (9.37) with V as contour velocity and Ȟ its direction. Differentiating Eq. (9.37), its logarithm yields: (9.38) Since X(z) in Eq. (9.38) is an analytic function, its derivative and the logarithm of its derivative must be analytic functions also. It follows that: (9.39) where (9.40) Similar to Eq. (9.9), the velocity potential from Eq. (9.40) at an arbitrary point B, shown in Fig. 9.5 can be found: (9.41) The position vector r in Eq. (9.41) shown in Fig. 9.5 is obtained from

(9.42)

Considering Eqs. (9.39) and (9.42), we find : (9.43) Equation (9.43) states that the stream function Ȍ which according to Eq. (9.39) exhibits the flow direction, is up to the additive constant Ȟm proportional to the potential function ĭ. It follows that the complex potential X(z) is also up to an additive constant directory proportional to ĭ resulting in:

252

9 Incidence and Deviation

(9.44)

for V = 1 and (ln V - iȞm) = const. = C, Eq. (9.44) is reduced to

(9.45) The new complex potential is now mapped into the ȗ-plane. The potential ĭ represents in ȗ-plane vortices, which are located at ±R and reflected on the unit circle ±1/R. Since on the unit circle, the contribution of the vortices outside the circle is exactly equal to the contribution of those vortices located inside the circle, the new complex potential Eq. (9.45) can be written as:

(9.46)

The imaginary part of Eq. (9.46) corresponds to the imaginary part of Eq. (9.39) that contains the angle Ȟ. After decomposition of Eq. (9.46), we find:

(9.47)

Along the unit circle we set in Eq. (9.47) ȗ = 1 and replace the left-hand side of (9.47) by Eq. (9.45) and consider Eq. (9.42) that contains Ȟ ’, we find

(9.48)

In Eq. (9.48) we replace

from Eq. (9.42) and arrive at

(9.49)

9 Incidence and Deviation

253

For a circular arc, the chord-curvature radius ratio c/ȡ in Eq. (9.49) can be approximated by the flow deflection . As a result, Eq. (9.49) is modified as (9.50) For the second term on the right-hand side of Eq. (9.50), we now introduce the following auxiliary function:

(9.51)

1 V1

D 1c

D1

D 1c

s c

0.8

Jm

D2c

T

D2

V2

P

0.6 90 o

0.4

4 o 30 o 0 20 o

0.2 o

o

Jm=10 - 90 , 'Jm=10

0

0

0.5

o

1 s/c

10 o

1.5

Fig. 9.6: Weinig ȝ-factor as a function of s/c with Ȗm as parameter

2

254

9 Incidence and Deviation

For a straight cascade with the given cascade solidity (c/s) and stagger angle Ȟm, the transformation parameter R in Eq. (9.51) is completely defined by Eqs. (9.31) and (9.36). Thus, in the auxiliary function, Eq. (9.51), ȝ is uniquely determined. Figure 9.6 exhibits the values for ȝ as a function of space/chord ratio with the cascade stagger angle Ȟm as a parameter. Using Į2, and Ȗm instead of Ȟ1, Ȟ2, and Ȟm for a profile, the result for a compressor cascade or decelerated flow is: (9.52) and for accelerated flow: (9.53) Factor A is a correction factor that was introduced by Traupel [6] and accounts for a non-zero incidence. It depends on the cascade parameter s/c and the stagger angle Ȗm as is shown in Fig.9.7. 1 0.9

600

0.8

0

A

400

0.7

300

0.6 0.5

90

Jm=200 0

0.5

1

s/c

1.5

2

Fig. 9.7: Correction factor A as a function of s/c with Ȗm as parameter For A = 1, we have : (9.54) with (+) and (-) signs for compressor and turbine cascade, respectively. For an infinitely thin blade, the exit flow angle is identical to the camber angle Į2c since ȝ = 1. It is calculated from:

9 Incidence and Deviation

255

(9.55) The deviation angle įĮ2 is obtained as follows: (9.56) with Į2c as the exit camber angle, the correction factor A from Fig. 9.7, and ȝ from Fig. 9.5. Once the deviation angle is calculated from Eq. (9.56), the exit flow angle Į2 is calculated as the difference Table 9.1: Cascade parameters Ȗm

c/s

AĮ

At

Ȗm

c/s

AĮ

At

30

0.5 1.0 1.5 2.0

1.096 1.130 0.849 0.663

-0.084 -0.351 -0.448 -0.443

45

0.5 1.0 1.5 2.0

0.982 0.814 0.602 0.451

-0.076 -0.211 -0.245 -0.233

30

0.5 1.0 1.5 2.0

1.052 0.948 0.657 0.489

-0.17 -0.399 -0.368 -0.310

45

0.5 1.0 1.5 2.0

0.942 0.720 0.506 0.367

-0.154 -0.242 -0.222 -0.182

60

0.5 1.0 1.5 2.0

0.901 0.672 0.484 0.365

-0.056 -0.115 -0.135 -0.128

75

0.5 1.0 1.5 2.0

0.845 0.605 0.430 0.327

-0.027 -0.056 -0.060 -0.060

60

0.5 1.0 1.5 2.0

0.872 0.620 0.434 0.321

-0.125 -0.145 -0.133 -0.109

75

0.5 1.0 0.5 2.0

0.830 0.581 0.408 0.307

-0.090 -0.088 -0.069 -0.055

A purely empirical correlation for deviation angle known as the Carter’s rule is: (9.57) with Ĭ as the difference between the and inlet and exit camber angles. Carter [9] modified the correlation for compressor cascades as with m as an empirical functions of stagger angle.

256

9 Incidence and Deviation

9.1.3 Thickness Correction For a profile with finite thickness t, the stagger angle is corrected by:

(9.58)

with At and AĮ from Table 9.1. For compressor cascades, the range of thickness/chord ratio is

9.1.4 Optimum Incidence To obtain the optimum incidence angle iopt for an arbitrary blade thickness, Johnsen and Bullock [10] introduced an empirical correlation that relates the optimum incidence angle iopt to the incidence angle i10 of a NACA-type blade, Fig. 9.8, with the maximum thickness t/s=0.1.

Fig. 9.8: Incidence and deviation for a DCA-profile

9 Incidence and Deviation

257

Taking i10 from Fig. 9.9, iopt can be estimated from: (9.59) where i10 is the optimum incidence for a profile with t/c = 0.1. The coefficients Kp and Kth account for the profile type and profile thickness. For DCA and NACA profile types, the following empirical values for NACA profiles and for DCA profiles are suggested. Figure 9.9 shows the variation of i10 and n with the inlet flow angle Į1. Figure 10 shows the thickness factor Kth as a function of thickness-chord ratio t/c.

10 8

c/s=2 1.6

io

6 4

1.2 0.8

2 0 20

0.4 30

40

50

40

50

D1

60

70

80

90

60

70

80

90

0.5 0.4 0.3

0.4 0.8

k4

1.2

0.2

1.6

0.1

c/s=2

0 20

30

D1

Fig. 9.9: Coefficients i0 and Kș as functions of inlet flow angle Į1 with c/s as parameter

258

9 Incidence and Deviation

1.2

Kth

1 0.8 0.6 0.4 0.2 0

0

0.02

0.04

0.06

t/c

0.08

0.1

0.12

0.14

Fig. 9.10: Coefficients Kth as a function of the relative thickness

9.1.5 Effect of Compressibility For intermediate to high subsonic inlet flow with M = 0.6 - 0.8, the incidence and the deviation angle are affected by the compressibility. The deviation angle of a compressible flow may be related to the deviation angle of an incompressible flow by using the following correlation proposed by Wennerström [11]: (9.60) with Kc as the compressibility function from Fig. 9.11.

1 0.8

Kc

0.6 0.4 0.2 0 0.6

0.7

0.8

0.9

1

M1

1.1

Fig. 9.11: Influence of mach number on deviation factor KC

1.2

1.3

1.4

9 Incidence and Deviation

259

6 5 'D=20

' ic

4

o

3 20

2

o

1 0 0

0.2

0.4

0.6

0.8

M1

1

1.2

1.4

Fig. 9.12: Influence of Mach number on optimum incidence factors ǻiC The compressibility effect causes an increase in incidence angle, which can be corrected using the empirical correlation suggested in [6] and [11]: (9.61) with

9.2

from Eq. (9.59) and ǻic from Fig. 9.12.

Deviation for High Flow Deflection

The accelerated flow through the turbine blades generally undergoes higher deflection than the decelerated flow through the compressor blades. The major parameter affecting the behavior of an accelerated flow with respect to the change of inlet flow direction is the Mach number, which determines the cascade flow geometry. If the flow has a low subsonic Mach number, such as in high pressure or intermediate pressure steam turbines, then the flow behavior is affected by the change of incidence if the blade profile has a small leading edge thickness. This fact enables the application of relatively thicker profiles that are less sensitive to changes of the inlet flow direction. In the rear stages of a low pressure steam turbine or a modern gas turbine, however, the inlet flow in the tip region is high subsonic that requires relatively thin profiles. Fig. 9.13 depicts the blade sections at the hub and the tip of a typical gas turbine rotor blade. Since the flow at the hub is low subsonic with high deflection, a thick profile with a relatively large leading edge radius is applied, Fig. 9.13. Its corresponding profile loss coefficient is plotted in Fig. (9.14) against the inlet flow angle.

260

9 Incidence and Deviation

Fig. 9.13: Hub (a) and tip (b) section of a turbine blade Starting from the optimum flow angle Į1opt, Fig. 9.14, curve (a) shows that the loss coefficient is not affected by the change of the inlet flow angle within an incidence range of -15o < i < +15o. Thus the change of incidence for a high deflected turbine cascade with a relatively large leading edge diameter that operates at a low subsonic mach number does not significantly increase the profile loss at moderate off-design operation conditions. The situation changes drastically towards the tip where the blade profile is subjected to a high subsonic even transonic inlet flow condition. For the profile at the tip section, Fig. 9.14b, the loss coefficient in Fig. 9.14, curve b changes considerably if the inlet flow direction departs from the design point. As seen in Fig.9.14b, the flow has a comparatively low deflection, so that the calculation procedure derived in Chapter 9.1 can be applied. Based on the above facts for an accelerated flow, only the changes of deviation angles need to be considered. Since a high deflected turbine cascade cannot be approximated by a straight-line cascade, the conformal transformation method discussed in Chapter 9.1 can not be used for the

12 10

]p(%)

8 6

(b)

(a)

4 2 0

D1opt 0

20

40

60

D1opt 80

D1

100

120

140

160

Fig. 9.14: Profile loss coefficient as a function of inlet flow angle for (a) hub section and (b) tip section

9 Incidence and Deviation

261

calculation of exit flow angle. Based on conservation laws, Traupel [6] derived the following simple method that allows an accurate prediction of the deviation angle for cascades with high flow deflection.

9.2.1 Calculation of Exit Flow Angle Consider the pressure distribution on the suction and pressure surface of a turbine cascade shown in Fig. 9.15.

Fig. 9.15: On the estimation of pressure momentum along the throat CD and the suction surface blade portion DE by Traupel method [6]

The pressure difference: (9.62)

With the approximation described by Eq. (9.62), the momentum balance in circumferential direction yields:

(9.63)

262

9 Incidence and Deviation

From the continuity equation we find: (9.64) where h2 and ha in Eq. (9.64) are the blade heights at stations 2 and a. For an isentropic flow, Eqs. (9.63) and (9.64) are written as: (9.65) Further, we assume an isentropic expansion process between a and 2 and calculate the isentropic enthalpy difference: (9.66)

introducing the exit Mach number and the static enthalpy with c as the speed of sound at the exit, Eq. (9.66) becomes:

(9.67)

Introducing Eq. (9.63) into (9.67):

(9.68)

The solution of Eq. (9.68) for pressure ratio is: (9.69)

Incorporating Eq. (9.69) into (9.65):

(9.70)

9 Incidence and Deviation

263

For a Mach number approaching unity and a constant blade height h2 = ha, the exit flow angle ȕ2 approaches ȕa, which leads to

(9.71)

For incompressible flow, į2 can be obtained from: (9.72) where the exit flow angle ȕ2 is: (9.73) The deviation is obtained from the difference (9.74) with ȕ2c as the camber angle at the trailing edge.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Gostelow, J.P.: Cascade Aerodynamics. Pergamon Press, Oxford (1984) Scholz, N.: Aerodynamik der Schaufelgitter. Brwon-Verlag, Karlsruhe (1965) Betz, A.: Konforme Abbildung. Springer, Berlin (1948) Weinig, F.: Die Strömung um die Schaufeln von Turbomaschinen. Barth-Verlag, Leipzig (1935) Eckert, B., Schnell, E.: Axial-und Radialkompressoren, 2nd edn. Springer, Berlin (1961) Traupel, W.: Thermische Turbomaschinen, vol. I. Springer, Berlin (1977) Lakschminarayana, B.: Fluid Dynamics and Heat Transfer of Turbomachinery. John Wiley and Sons, Chichester (1995) Cumpsty, N.A.: Compressor Aerodynamics. Longman Group, New York (1989) Carter, A.D.: The Low Speed Performance of Related Airfoils in Cascades, Aeronautical Research Council (1950) Johnsen, I.A., Bullock, R.O.: Aerodynamic Design of Axial Flow Compressors. NASA SP 36 (1965) Wennerström, A.: Simplified design Theory of highly loaded axial Compressor Rotors and Experimental Study of two Transonic Examples (1965), Diss. ETHZürich

10 Simple Blade Design

Flow deflection in turbomachines is established by stator and rotor blades with prescribed geometry that includes inlet and exit camber angles, stagger angle, camber line, and thickness distribution. The blade geometry is adjusted to the stage velocity diagram which is designed for specific turbine or compressor flow applications. Simple blade design methods are available in the open literature (see References). More sophisticated and high efficiency blade designs developed by engine manufacturers are generally not available to the public. An earlier theoretical approach by Joukowsky [1] uses the method of conformal transformation to obtain cambered profiles that can generate lift force. The mathematical limitations of the conformal transformation do not allow modifications of a cambered profile to produce the desired pressure distribution required by a turbine or a compressor blade design. In the following, a simple method is presented that is equally applicable for designing compressor and turbine blades. The method is based on (a) constructing the blade camber line and (b) superimposing a predefined base profile on the camber line. With regard to generating a base profile, the conformal transformation can be used to produce useful profiles for superposition purposes. A brief description of the Joukowsky transformation explains the methodology of symmetric and a-symmetric (Cambered) profiles. The transformation uses the complex analysis which is a powerful tool to deal with the potential theory in general and the potential flow in particular. It is found in almost every fluid mechanics textbook that has a chapter dealing with potential flow. While they all share the same underlying mathematics, the style of describing the subject to engineering students differ. A very compact and precise description of this subject matter is found in an excellent textbook by Spurk [2].

10.1 Conformal Transformation, Basics Before treating the Joukowsky transformation, a brief description of the method is given below. We consider the mapping of a circular cylinder from the z-plane onto the ȗ-plane, Fig.10.1. Using a mapping function, the region outside the cylinder in the z-plane is mapped onto the region outside another cylinder in the ȗ-plane. Let P and Q be the corresponding points in the z- and ȗ-planes respectively.

10 Simple Blade Design

266

]

z-plane

-plane

iK

iy P

Q x

[

Fig. 10.1: Conformal transformation of a circular cylinder onto an airfoil

The potential at the point P is (10.1) The point Q has the same potential, and we obtain it by insertion of the mapping function (10.2) Taking the first derivative of Eq. (10.2) with respect to ȗ, we obtain the complex conjugate velocity in the ȗ plane from (10.3) Considering z to be a parameter, we calculate the value of the potential at the point z. Using the transformation function ȗ = f(z), we determine the value of ȗ which corresponds to z. At this point ȗ, the potential then has the same value as at the point z. To determine the velocity in the ȗ plane, we form

(10.4)

after introducing Eq. (10.3) into (10.4) and considering rearranged as

, Eq. (10.4) is

10 Simple Blade Design

267

(10.5) Equation (10.5) expresses the relationship between the velocity in ȗ-plane and the one in z-plane. Thus, to compute the velocity at a point in the ȗ- plane, we divide the velocity at the corresponding point in the z-plane by dȗ/dz. The derivative dF/dȗ exists at all points where dȗ/dz g 0. At singular points with dȗ/dz = 0, the complex conjugate velocity in the ȗ plane becomes infinite if it is not equal to zero at the corresponding point in the z-plane.

10.1.1 Joukowsky Transformation The conformal transformation method introduced by Joukowski allows mapping an unknown flow past a cylindrical airfoil to a known flow past a circular cylinder. Using the method of conformal transformation, we can obtain the direct solution of the flow past a cylinder of an arbitrary cross section. Although numerical methods of solution of the direct problem have now superseded the method of conformal mapping, it has still retained its fundamental importance. In what follows, we shall examine several flow cases using the Joukowsky transformation function: (10.6)

10.1.2 Circle-Flat Plate Transformation Decomposing Eq. (10.6) into its real and imaginary parts, we obtain: (10.7) The function f(z) maps a circle with radius r = a in the z-plane onto a “slit” in the ȗ plane. Equation (10.7) delivers the coordinates: (10.8) with ȟ as a real independent variable in the ȗ-plane. As the point P with the angle ș moves in z-plane from 0 to 2ʌ, Fig. 10.2, its image p’ moves from +2a to -2a in the ȗ -plane with the complex potential F(z). The potential F(z) is:

(10.9)

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268

z-plane iy z = ae

]-plane iK

2

iT

] = z + az

a

P P'

T

x -2a

[

+2a

Fig. 10.2: Transformation of a circle onto a slit (straight line section)

Setting R = a, the Joukowski transformation function directly provides the potential in the ȗ-plane as (10.10)

10.1.3 Circle-Ellipse Transformation For this transformation, the circle center is still at the origin of the z-plane. Now, if we map a circle with radius b which is smaller or larger than the mapping constant a, we obtain an ellipse. Replacing r by b (b ga), Eq. (10.7) becomes (10.11) Eliminating ș from Eq. (10.11), we find: (10.12) and with

, we obtain the equation of ellipse as: (10.13)

Equation (10.13) describes an ellipse, plotted in Fig. 10.3, with the major and minor axes which are given as the denominators in Eq. (10.13). In Fig. 10.3, b > a, however any ellipse may be constructed by varying the ratio b/a.

10 Simple Blade Design

z-plane iy

]-plane iK

2

z = be

iT

269

] = z + az

a 1. The incoming supersonic flow impinges on the sharp leading edge and forms a weak oblique shock, followed by an expansion fan. Passing through the shock front, the Mach number, although smaller, remains supersonic. Expansion waves are formed along the suction surface (convex side) of the blade from the leading edge L to the point e where the subsequent Mach wave at point e intersects the adjacent blade leading edge. Regarding the transonic compressors, the research efforts resulted in the design of controlled diffusion blades that almost eliminate shock losses [5], [6] and [7].

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M



M



Expansion Waves

Leading Edge Shock

Et L

e

L

e 4

Passage Shock Fig. 10.14: Schematic of a supersonic compressor cascade the supersonic inlet mach number, expansion wave and passage shocks

For compressors operating in the supersonic Mach range, the so called S-profiles with sharp leading edges are used. The pressure and suction surfaces consist of multicircular arcs with very small curvatures (large radii). Particular attention must be paid to the shock losses, as discussed in Chapters 7 and [8]. The flow decelerates within the convergent part by a system of oblique shock waves followed by a normal shock. Further deceleration is achieved by diffusion within the divergent part. As seen, these profiles have a slightly convergent inlet tangent to a throat which is followed by a slightly divergent channel.

10.3 Turbine Blade Design Different turbine blade designs are discussed in literature among others, [9], [10], [11], [12], and [13]. Given the stage velocity diagram with the flow velocity vectors and angles, the corresponding blade profiles are constructed such that the flow angles at the inlet and exit are realized. This includes the incorporation of the design incidence and deviation into the design process as discussed in Chapter 9. Figure 10.15 schematically shows the velocity diagram and the corresponding velocity stator and rotor blades. Analogous to Section 10.2, the task here is designing the camberline on which a base profile will be superimposed. The camberline can be constructed from the inlet and exit flow angles under consideration of the incidence and deviation discussed in Chapter 9. This can be done graphically or numerically. Both methods are discussed in the following Section.

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282

V1

D E2 2 D3

E3

V2 W2

V1

W2

W3

V3

U2

U3

W2

Fig. 10.15: Allocation of a set of stator and rotor blades to a given velocity diagram

10.3.1 Graphic Design of Camberline Given a chord length c =1 and stagger angle Ȗ, the tangents to the leading- and trailing-edge of the camberline to be determined are constructed by the inlet and exit metal angles as shown in Fig. 10.16. The latter is found from the stage velocity diagram under consideration of incidence and deviation angles that lead to Į1c and Į 2c, Fig. 10.17. V 1c

1c

I

I SJD1c

D

1c

SJ

c

Cascade front

J

I SJD2c 2c

I

2c

D2c V2c

Fig. 10.16: Camber angle definitions: Į1c, Į2c = camber angles at the cascade inlet and exit, ij1c, ij2c = auxiliary angles.

10 Simple Blade Design

n-1

en t

P1

TE - Ta nge

LE -T an g

1

nt

K max

P0 [

V

P2

max

[

1c

[ 

[= x C

V

I 2c

C=1

I

283

K= y C

[ 

Fig. 10.17: Nomenclature of the quantities for constructing the camberline The tangents intersect each other at the point P1 and are subdivided into n-1 equal distances as shown in Fig. 10.17. Starting with the first point 1 at the leading edge (LE) tangent, a line is drawn to intersect the trailing edge (TE) tangent at point n-1. The following lines start at the second point on the LE-tangent and intersects the TEtangent at n-2 and so on. The same procedure is repeated with first point at the TEtangent starting with P2. The envelope of the inner region of the connecting lines is the camberline, as shown in Fig.10.17.

10.3.2 Camberline Coordinates Using Bèzier Curve A similar camberline is constructed using Bézier polynomial presented, among others, in [14]. The general Bézier polynomial of degree n is given as:

(10.37) with Pi as the given points, t  [0,1] the curve parameter, and B(i) the position vector of the Bézier curve at the curve parameter t. For the purpose of designing the camberline, a quadratic Bézier curve is quite appropriate for a conventional turbine blade design. It reads: (10.38)

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284

P0

LE -T an ge nt

P1

[

V

1 B

I

K

TE -Ta nge nt

P2

BK

1

V 2

I 2c

[ C=1

1c

[ 

[= x C

K= y C

[ 

Fig. 10.18: Dimensionless quantities for calculating the Bèzier curve Before re-writing the Bèzier formulation in terms of our nomenclature, we resort to Fig. 10.18, where the camber and Bèzier quantities are shown. Equation (10.38) reads

(10.39)

Considering the nomenclature in Fig. 10.18, the boundary conditions are:

(10.40)

With Eq. (10.39), the coordinates of the camberline are calculated by varying the dimensionless variable ȟ from 0 to 1. A smooth camberline is obtained by introducing a small increment ȟ = 0.01 or less.

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285

10.3.3 Alternative Calculation Method Although a Bèzier curve provides a reasonable solution for camberline design, the position of the maximum thickness (ȟmax, Șmax) may or may not be accurate. In this case, the following simpler method offers an appropriate alternative. The camber line can be described by a polynomial: (10.41)

The coefficients aȞ are obtained from the following boundary conditions: BC1 BC2 BC3

where

and

are given. With the boundary conditions BC1 to BC3, five

coefficients can be determined. The corresponding polynomial is of fourth order:

(10.42) or (10.43) From BC1, it follows immediately that a0 = 0. It is important to ensure that there is no discontinuity on the profile surfaces. For this purpose, the final profile should be smoothed mathematically. For a highly deflected turbine cascade, the application of the above procedure may result in an inflection point. To prevent this, the camberline may be subdivided into two or more lower degree polynomials that tangent each other at the same slope. It should be pointed out that, for subsonic turbine blades with r >0.0%, the area between two adjacent blades must continuously converge, enabling the flow to accelerate continuously. For zero-degree reaction, the blade channel area must be constant throughout. Once the camberline is constructed, a turbine base profile is superimposed and the profile is generated, Fig.10.19.

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286

Fig. 10.19: Superposition of a base profile on the camber line

Very low flow deflection

Low flow deflection

V1 V1

V1

V1

V2

V2

V2

V2

Moderate flow deflection

Large flow deflection V1

V1 V1

V2 V2

V1

V2

Fig. 10.20: An example of a systematic turbine blade design for different flow deflections

10 Simple Blade Design

287

Applying the methods discussed above, four different conventional blade profiles are presented in Fig. 10.20 for different flow deflections. Moderate flow deflection blades are used in steam turbine design to keep the profile losses as small as possible. In contrast, the gas turbine engines, particularly the aircraft engines, use high deflection blades to keep the weight/thrust ratio as low as possible.

10.4 Assessment of Blades Aerodynamic Quality The objective of a turbine stator or rotor blade design is to accomplish a certain flow deflection that is required to produce a prescribed amount of power per stage. Given the inlet and the exit flow angles resulting in a given flow deflection, an infinite number of blades with different shapes can be constructed that realize the above flow deflection. These blades, however, will have different boundary layer distributions on suction and pressure surfaces and, thus, different profile loss coefficients. Among the numerous possible designs, only one has the lowest loss coefficient, i.e, the optimum efficiency at the design operating point only. Figure 10.21 shows the pressure distribution along the suction and pressure surfaces of a high efficiency turbine blade at the design point. This blade type is sensitive to incidence changes caused by off-design operation. If the operation conditions vary that result in a change of incidence angle, the losses may increase substantially. Thus, the operation condition of a turbine plays a crucial role in designing the blade. If the mass flow or the pressure ratio of a turbine component undergo frequent changes, the blades must be designed in such a way that their profile losses do not experience significant increases. Parameters determining

-0.40 o

-0.20

Inlet Angle D1= 104 Exit Angle D2= 16o o Stagger Angle J = 33.4

SS

0.00 Cp

V1

0.20 V2

0.40 0.60

PS 0.00

0.25

0.50 x/cax 0.75

1.00

Fig. 10.21: Pressure distribution along the suction and pressure surfaces of a high efficiency turbine blade, TPFL-Class Project 2010

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288

the efficiency and performance behavior of a turbine as a result of the boundary layer development on the suction and pressure surface are, among others, Reynolds number, Mach number, pressure gradient, turbulence intensity and unsteady inlet flow conditions. Of the above parameters, the geometry defines the pressure gradient which is, in the first place, responsible for the boundary layer development, separation and re-attachment. Since the pressure distribution is determined by the geometry and vice versa, the question arises as to how the blade geometry should be configured in order to establish a desired pressure distribution. This question has been the subject of numerous papers dealing with the inverse blade design, particularly for subcritical and supercritical compressor cascades [15], [16], and [17]. In context of this Chapter, we treat the question of the pressure distribution versus geometry from a simple physical point of view to establish a few criterions as a guideline for turbine blade design. The following example should demonstrate what criteria should be used in order to design a turbine blade that operates at a wide range of incidence changes without a substantial decrease of its efficiency: 1. 2. 3.

With exception of the leading edge portion, the pressure gradient on the pressure and suction surfaces should not experience sign changes over a range of 60% to70% of the blade axial chord. Over the above range, the pressure gradient should be close to zero. This means that flow on both surfaces neither accelerates nor decelerates. Over the rest of 30% to 40% of the blade surface, the flow should strongly accelerate.

Applying the criteria (1) through (3) associated with a moderately thick leading edge radius provides a turbine blade that is insensitive to adverse off-design operation conditions at a relatively high efficiency, Fig. 10.22. Figure 10.23 shows the numerical calculation of the pressure coefficient versus the dimensionless axial chord. As seen, with exception of the leading edge portion the pressure distributions do not change over a wide range of incidence from -15o to +21o. This numerical result is verified experimentally and presented in Figs. 10.24. +i

D1

D2

V1

V2

Fig. 10.22: A turbine cascade that fulfils the criteria (1) through (3)

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289

4.0

Numerical Simulation

PS

0.0 -4.0

cp

Variation of Incidence i = - 5.2oo i = -10.8o i = -15.3 i = + 9.5o o i = +15.2o i = +21.1

-8.0

-12.0 -16.0 -20.0

0

SS

0.2

0.4

x/cax

0.6

0.8

1

Fig. 10.23: Pressure distribution as a function of dimensionless axial chord with the incidence angle as parameter

4.0

Experimental Results 0.0

cp

-4.0 -8.0

o

Exp: i = -5.2 o Exp: i = -10.8 o Exp: i = -15.3 o Exp: i = +9.5 Exp: i = +15.2o Exp: i = +21.0o

-12.0 -16.0 -20.0

0

0.2

0.4

x/cax

0.6

0.8

1

]p(%)

2.5

2.0

1.5 -15

-10

-5

0

5

Incidence i

o

10

15

20

Fig. 10.24: Experimental pressure distribution as a function of dimensionless surface, total pressure loss coefficient as a function of incidence angle

290

10 Simple Blade Design

Figure 10.24 shows the experimental verification of the numerical results presented in Fig. 10.23. It also shows the profile loss coefficient ȗp as a function of the incidence angle. Here as in Fig. 10. 23, the pressure distributions do not change over a wide range of incidences from -15o to +21o. The results are in reasonable agreement with the numerical results, resulting in an approximate constant loss coefficient of 2%.

References 1. Joukowsky, N.E.: Vortex Theory of Screw Propeller. I-IV, I-III, The forth paper published in the Transactions of the Office for Aerodynamic Calculations and Essays of the Superior Technical School of Moscow (in Russian). Also published in Gauthier-Villars et Cie (eds). Théorie Tourbillonnaire de l’Hélice Propulsive, Quatrième Mémoire. 123–146: Paris, in French (1918) 2. Spurk, J.H.: Fluid Mechanics. Springer, New York (1997) 3. NASA SP-36 NASA Report (1965) 4. Cumpsty, N.A.: Compressor Aerodynamics. Longman Group, New York (1989) 5. Hobson, D.E.: Shock Free Transonic Flow in Turbomachinery cascade. Ph.DThesis, Cambridge University Report CUED/A Turbo/65 (1979) 6. Schmidt, J.F.: Redesign and Cascade tests of a Supercritical Controlled Diffusion stator Blade Section. AIAA Paper 84-1207 7. Lakschminarayana, B.: Fluid Dynamics and Heat Transfer of Turbomachinery. John Wiley and Sons, Chichester (1995) 8. Schobeiri, M.T.: A New Shock Loss Model for Transonic and Supersonic Axial Compressors With Curved Blades. AIAA, Journal of Propulsion and Power 14(4), 470–478 (1998) 9. Teufelberger, A.: Choice of an optimum blade profile for steam turbines. Rev. Brown Boveri 2, 126–128 (1976) 10. Kobayashi, K., Honjo, M., Tashiro, H., Nagayama, T.: Verification of flow pattern for three-dimensional-designed blades. ImechE paper C423/0l5 (1991) 11. Jansen, M., Ulm, W.: Modern blade design for improving steam turbine efficiency. VDI Ber. 1185 (1995) 12. Emunds, R., Jennions, I.K., Bohn, D., Gier, J.: The computation of adjacent blade-row effects in a 1.5 stage axial flow turbine. ASME paper 97-GT-81, Orlando, Florida (June 1997) 13. Dunavant, J.C., Erwin, J.R.: Investigation of a Related Series of Turbine-blade Profiles in Cascade. NACA TN-3802 (1956) 14. Gerald, F.: Curves and Surfaces for Computer-aided Geometric Design, 4th edn. Elsevier Science & Technology Books, Amsterdam (1997) 15. Bauer, F., Garabedian, P., Korn, D.: Supercritical Wing Sections III. Springer, New York (1977) 16. Dang, T., Damle, S., Qiu, X.: Euler-Based Inverse Method for Turbomachine Blades: Part II—Three Dimensions. AIAA Journal 38(11) (2000) 17. Medd, A.J.: Enhanced Inverse Design Code and Development of Design Strategies for Transonic Compressor Blading. Ph.D. dissertation, Department ofMechanical Engineering, Syracuse University (2002)

11 Radial Equilibrium

In Chapter 5, we briefly described a simple radial equilibrium condition necessary to determine the radial distribution of the stage parameters such as, ij, Ȝ, r, Įi, and ȕi.. Assuming an axisymmetric flow with constant meridional velocity and total pressure distributions, we arrived at free vortex flow as a simple radial equilibrium condition with In practice, from an aerodynamics design point of view, a constant meridional velocity component or constant total pressure may not be desirable. As an example, consider the flow field close to the hub or tip of a stage where secondary flow vortices predominate. As discussed in Chapter 6, these secondary vortices induce drag forces leading to secondary flow losses which reduce the efficiency of the stage. To reduce the secondary flow losses, specific measures can be taken that are not compatible with the simple radial equilibrium condition. In this case, the simple radial equilibrium method needs to be replaced by a general one. Wu [1] proposed a general theory for calculating the three-dimensional flow in turbomachines. He introduced two sets of surfaces: Blade to blade surfaces called S1i and hub-to-tip surfaces labeled with S2j. Utilizing S1i and S2j surfaces, Wu [1] proposed an iterative method to solve the three-dimensional inviscid flow field in turbomachinery stages. Coupling both surfaces, however, is associated with computational instabilities that gives rise to replacing the technique by complete Euler or Navier-Stokes solver, [2]. A computationally more stable method for solving the flow field is the streamline curvature technique. This method is widely used in turbomachinery industry and is the essential tool for generating the basic design structure necessary to start with CFD application. The streamline curvature method can be used for design, off-design, and analysis. Vavra [3] presented the theoretical structure for inviscid axisymmetric flow in turbomachines that can be used to derive the streamline curvature equations. A thorough review of the streamline curvature method can be found in Novak and Hearsey [4], Wilkinson [5], and Lakshminarayana [2]. Wennerstrom [6] presented a concise description of this technique which is given in this section in its original form. Rapid calculation procedures used in the turbomachinery industry determining the distribution of flow properties within the turbomachinery assume steady adiabatic flow and axial symmetry. The more sophisticated of these procedures include calculation stations within blade rows as well as the more easily treated stations at the blading, leading and trailing edges. The assumption of axial symmetry in the bladed region implies an infinite number of blades. The blade forces acting between the blades and the fluid are taken into consideration by body force terms in equation of motion. The streamlines are not straight lines as could be supposed. They usually have certain curvatures that are maintained if the forces

292

11 Radial Equilibrium

exerted on the fluid particle are in an equilibrium state described by the equilibrium equation. This equation includes various derivatives in the meridional plane and is solved in this plane along the computing stations which are normal to the average meridional flow direction. The equilibrium equation in its complete form cannot be solved analytically, therefore, numerical calculation methods are necessary. For the application of the equilibrium equation to the stage flow with streamline curvature, Hearsey [7] developed a comprehensive computer program which is successfully used in the turbomachinery industry for design of advanced compressors and turbines. Since the treatment of the corresponding numerical method for the solution of this equation is given in [7], this chapter discusses the basic physical description of the calculation method. As a result, the streamline curvature method can be used as a design tool and for the post design analysis. An advanced compressor design is presented followed by a brief discussion of special cases.

11.1 Derivation of Equilibrium Equation The starting point for this derivation is the formulation of the equations of motion for an axisymmetric flow in intrinsic coordinates fixed in space along streamlines as shown in Fig.11.1.

Fig. 11.1: Flow through axial compressor, streamline, directions are: n = normal, m = meridional, r = radial, z = axial, and l = computing station

11 Radial Equilibrium

(a)

(b)

Meridional streamline en

l

J

IJ

293

Meridional streamline L

r

H

B

em

I

m

Fp u

eu

em

Computational station

n en

r

E Fs

Axis of rotation

z

Fig. 11.2: (a) Coordinate directions in the meridional plane, (b) orientation of vectors with respect to m, u, n orthogonal coordinate system, redrawn from [6]

The momentum equation is in meridional direction: (11.1)

in stream surface normal direction: (11.2) and in circumferential direction: (11.3) with

and

as the components of a field force in meridional, normal and

circumferential directions. This field force is thought of as the real blade force that causes the flow to deflect from the leading edge to trailing edge of a turbine or a compressor. The thermodynamic equations required are the energy equations for steady adiabatic flow: (11.4) and the Clausius entropy relation: (11.5)

294

11 Radial Equilibrium

Since, in intrinsic coordinates there is no velocity component in the n-direction (by definition), (11.6) Equations (11.4), (11.5), and (11.6) are combined to give: (11.7) Since the computation station lies in the space between the stator and the rotor, usually not normal to the meridional stream surface, the derivatives must be defined with respect to the l-direction along a computing station, Fig.11.2: (11.8) Rearranging Eq.(11.8) to eliminate the derivatives with respect to the normal direction:

(11.9) Introducing Eq.(11.7) into Eq.(11.1) and (11.2), eliminating derivatives with respect to n with Eq.(11.9), the resulting equations are combined to eliminate

(11.10) The equation obtained after algebraic rearrangement and trigonometric substitution is:

(11.11)

In some cases, it is more convenient to work in a similar system of coordinates which is stationary with respect to a rotating blade row. As shown in Chapter 3, Eq.(11.11) is easily transformed into the rotating frame of reference by introducing the substitution:

11 Radial Equilibrium

295

(11.12) for the circumferential velocity component. The corresponding relative total enthalpy1 Hr from Eq. (3.104) is: (11.13) Combining Eqs. (11.12), and (11.13) leads to: (11.14) Introducing Eqs.(11.12) and (11.14) into Eq. (11.11), the corresponding radial equilibrium equation is:

(11.15) Equations (11.11) and (11.15) represent the radial equilibrium condition. The points of principal interest in this derivation are associated with the treatment of the body forces, Fm and Fn. Note that the complete body force field contains a third component Fu which is orthogonal to the other two mutually orthogonal components Fm and Fn and can be directly calculated from Eq. (11.3). The remainder of this derivation is aimed at obtaining expressions for Fm and Fn which relate to geometric properties of the blading. Combining Eqs.(11.1) and (11.7) gives: (11.16)

or

1

Wennerstrom [6] calls this quantity rothalpy which was introduced by Wu [1], who termed the sum of the static enthalpy and the kinetic energy of the relative flow as the relative total enthalpy (see also Lakshminarayana [2]). In a rotating frame of reference, the expression changes along a streamline that varies in radial direction. In contrast, the relative total enthalpy given by Eq. (11.13) derived in Chapter 3 (Eq. 3.104) remains constant and is generally valid within a rotating frame of reference, where the streamlines may change in radial direction (see also Vavra [3]).

296

11 Radial Equilibrium

(11.17)

Equation (11.3) and the thermodynamic definition of work (applied to a turbomachine) may be combined to give

(11.18)

a differential form of the Euler equation of turbomachinery. Combining Equations (11.17) and (11.18), noting from the velocity triangle that (11.19) and introducing Equation (11.3) again leads to (11.20) Equation (11.20) is sufficient to solve Eqs. (11.11) or (11.15) for some bladeless cases but, in order to solve the equations within a bladed region, more information is required concerning the direction and nature of the force components. For this purpose, it is convenient to resort to vector analysis to relate the various force components and coordinate directions. Fig. 11.2b illustrates the vectors described in the following discussion. The variables m, u, n are the principal orthogonal axes of a right handed, streamline coordinate system. In this system, Fs is the body force vector acting to oppose motion of the fluid; i.e. the body force which produces an irreversible increase in entropy. It lies in the m-u plane and at the angle ȕ (relative flow angle) from the meridional direction. The vector L is coincident with the l-direction and lies in the n-m plane at the angle (ij-Ȗ) from the n-direction. The vector B is tangent to the mean blade surface and lies in the l-u plane at an angle J from the l-direction. Planes such as the m-u plane and l-u plane are understood as planes tangent to the curved surface, described by the respective coordinates at the point in question. The equation of the unit vector in Fs-direction is: (11.21) and the equation of the unit vector B is (11.22)

11 Radial Equilibrium

297

The force due to a pressure difference across a blade acts in a direction normal to both vectors Fs and B, which together define the local plane of the mean blade surface at a point. Therefore, the pressure force vector can be defined by the cross product of Fs and B. Since Fs and B are usually not perpendicular but at an angle (ʌ/2 - (ij - Ȗ)) to one another, the product of these vectors differs from a unit vector by cos(ij - Ȗ). Therefore to obtain a unit vector for the force due to pressure across a blade and because Fp must be in the positive direction of rotation, this is written

(11.23) From Fig. 11.2b, note that Fm is composed of the m-components of Fs and Fp. Consequently, from Eqs. (11.21) and (11.23) we obtain: (11.24) Similarly, Fu and Fn are composed respectively of the u and n components of Fs and Fp (11.25) (11.26) Since the orthogonal components of the force field have been defined in terms of components acting along and perpendicular to the relative flow direction, it is possible to solve for these two components. By inserting Eqs. (11.24) and (11.25) into Eq. (11.20) and simplifying, one obtains: (11.27) Subsequently, Eqs. (11.25) and (11.27) can be combined to give (11.28)

If Eqs. (11.27) and (11.28) are now inserted into Eqs. (11.24) and (11.26), the body forces appropriate to the intrinsic coordinate system are defined in terms of

298

11 Radial Equilibrium

parameters readily calculated. The most useful form of the radial equilibrium equation is the form in relative system, Equation (11.15), since it becomes identical to Eq. (11.11) in the absolute system when the angular velocity of the coordinates is set equal to zero. Combining Eq. (11.24), (11.26), (11.27) and (11.28) into Eq. (11.15) puts the radial equilibrium equation in the form:

(11.29) The force Fu can be obtained from Eq. (11.3) or from

(11.30) applicable to the relative system. If working in a relative system with specified flow angle, it is convenient to employ Eq. (11.19) to eliminate Wu. The result of this substitution, included in a combined form of Eq. (11.29) and (11.30), is

(11.31)

The form of the radial equilibrium equation most appropriate for a particular calculation depends on a number of factors. The most important of these are the nature of the numerical scheme for analysis and the choice of parameters to be specified. This derivation was guided by the intent to apply the results to a streamline curvature type of computing procedure wherein the meridional velocity was the

11 Radial Equilibrium

299

primary variable for which solutions were sought. In this type of procedure, the radial equilibrium equation is solved along each computing station for an assumed set of streamlines. The streamlines are then refined and this process repeated until a satisfactory degree of convergence is attained. This calculation method is used successfully in the turbomachinery industry for design of advanced compressors and turbines. Before discussing applications in bladed regions of a turbomachine, consider applications involving bladefree spaces. The body force due to a pressure difference across a blade was defined by the vector Fp in Eq. (11.23). In a bladefree space, this component is zero. The remaining body force, Fs, associated with the entropy increase, was defined by Eq. (11.26). Hence, Eqs. (11.24), (11.25) and (11.26) describing the three orthogonal body forces of the intrinsic coordinate system reduces to

(11.32)

(11.33) (11.34) In addition, all terms involving the blade lean angle, J, drop out of the radial equilibrium equation. Equation (11.29) then reduces to:

(11.35) It is interesting to note that, according to Eq. (11.32) and (11.33) in a swirling flow, an entropy rise in the direction of flow always leads to body force terms both in — and u-directions. This is obvious for the vaneless diffuser and a centrifugal machine where most of the entropy rise occurs from wall friction. It is much less obvious for an axial turbomachine of high aspect ratio where annulus wall friction may be negligibly small in relation to losses due to wake mixing. However, it appears that a change in angular momentum occurs in both cases if the entropy is assumed to be increasing in the flow direction. Considering bladed regions, the choice of parameters specified initially generally falls into one of two categories. The first involves cases where total

300

11 Radial Equilibrium

temperature or circumferential velocity component is specified. Practically, these variables are for all practical purposes interchangeable, being simply related through Eq. (11.18). The most convenient solution for these cases is usually obtained in the stationary frame of reference using Eq. (11.29) with Ȧ = 0, Wu = Vu, and Hr = H. During each calculation pass, the only variables in Eq. (11.29) are usually Vm and ȕ, since all the other parameters distributed along the l-direction are either input data or upgraded between passes. The second important category of calculation within the bladed region involves specified relative flow angles. For this purpose, it is most expedient to work in a relative frame of reference, specifying Ȧ within the rotors and zero elsewhere. The most convenient radial equilibrium equation is Eq. (11.31). In this instance, the only variable in Eq. (11.31) during a calculation pass is Vm and all other variations in l-direction are either input data or upgraded only between calculation passes.

11.2 Application of Streamline Curvature Method Utilizing the equations derived in section 11.1, Hearsey [7] developed a computational tool that can be used for design and analysis of turbomachines. This section summarizes the computational aspects of the streamline curvature technique by Hearsey [7]. As discussed in the previous section, the streamline curvature method offers a flexible method of determining a Euler solution of the axisymmetric flow through a turbomachine. The computational grid comprises the streamlines, as seen in a meridional view of the flow path, and quasi-normals that are strategically located in the flow, Figs. 11.1. Several stations are generally placed in the inlet duct upstream of the turbomachine proper, and several more are generally placed downstream. Within the turbomachine, the minimum number of quasi-normals, or "stations ", is simply one between each adjacent pair of blade-rows, which would then represent both outlet conditions from the previous row and inlet conditions to the next. A better choice is one at each edge of each blade-row. For some calculations, additional stations are added within the blade-rows. This is a relatively coarse grid (compared with that used for CFD computations), making for calculations that typically take just seconds to complete (when run on a typical year-2003 PC). By virtue of being an axisymmetric, Euler solution, the basic method is inherently incapable of predicting entropy rises, that is pressure losses and efficiency. This is handled by invoking a cascade performance prediction scheme to estimate losses and, in some situations, discharge flow angles from blading. Complicated boundary layers form on the flow path inner and outer walls of turbomachinery, causing "blockage" and participating in secondary flows. Especially for multi-stage axial compressors, a good estimate of blockage is required in order to correctly predict stage matching. The equation system that is broadly used caters to two situations: when the variation of absolute tangential velocity component along a computing station is known, and when the relative flow angle distribution is known. The former generally arises in “design” calculations, and the latter in "analysis" or performance prediction,

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301

cases. Thus, the streamline curvature procedure may be used for design calculations, off-design performance predictions and test-data analysis. This versatility, combined with the speed of solution, makes the method a work-horse tool. Many designs can be accomplished using only the streamline curvature method, and while some CFD computations are now offering design capability, even then a good candidate design is generally required as a starting point. Although the Euler equations are the basis of the method, properly accounting for the change in entropy that is super-imposed from station to station, that is, in the streamwise direction, requires some care. This was highlighted in Horlock [8] in which several previously-published equation systems were examined and found to be missing what is usually a small term. For the case where tangential velocities are known, the appropriate equation is Eq. (11.29). When relative flow angles are known, Eq. (11.31) is used. In conjunction with one of these equations, the continuity equation and the energy equation are required. Fluid properties are also required; these are best computed in a discreet subprogram to allow for easy replacement. The continuity equation takes the form:

(11.36)

The rate of change of flow with meridional velocity will also be required. This is given by:

(11.37)

when tangential velocity component is specified, or:

(11.38)

when relative flow angle is given. The loss coefficient ȗ included in Eq. (11.38) is the total pressure loss coefficient normalized by exit dynamic pressure. If not used, it is set zero in Eq. (11.38). The Euler equation of turbomachinery relates total enthalpy change to angular momentum change: (11.39)

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11.2.1 Step-by-Step Solution Procedure Given these equations, a step-by-step procedure to obtain a solution is as follows: 1. An initial estimate of the streamline pattern is made. An obvious choice is to divide the flow path into equal areas at each station. Slightly more uniform (final) streamline spacing may be achieved by dividing the inlet (or some other) station into equal increments and then using the resulting area fractions to guide the remaining estimates. 2. Initial estimates of the streamwise gradients that occur in Eqs .(11.29) and (11.31) are made. These are all second-order terms and zero is the usual first estimate. 3. The streamline slopes and curvatures are computed at all mesh points (although, alternatively, this can be done station-by-station as the calculation proceeds). Curvatures are normally set to zero for the first and last stations. 4. At the first station, the user is required to specify the variation of total enthalpy, entropy and angular momentum or flow angle along the station. Enthalpy and entropy are typically implied by total pressure and temperature in the users input data. The fluids package can then be used to obtain enthalpy and entropy on each streamline. 5. A first estimate of the meridional velocity on one streamline is made from the inlet flow and the first station area. It is recommended that the mid-streamline value be defined. High accuracy is not required (this will not be required after the first pass). 6. Depending upon whether angular momentum or flow angle was given in the input data, Eq. (11.29) or (11.31) is integrated from the mid-streamline to the inner wall, and from the mid-streamline to the outer wall. This yields a meridional velocity distribution that is consistent with the momentum equation, the assumed streamline characteristics and the estimated mid-line velocity estimate. Although this is not likely to occur at Station 1, extreme values can lead to negative velocities, or values greater than the speed of sound. Provisions must be made for such occurrence. 7. Equations (11.36) and (11.37) or (11.38) are integrated across all streamlines to yield the achieved flow and its rate of change with meridional velocity. 8. The sign of the flow gradient indicates on which branch of the continuity equation the current velocity profile lies, a positive value indicating the subsonic

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303

branch, and a negative value, the supersonic branch. If tangential velocity is specified, only the subsonic branch is valid for a streamline curvature solution, but if flow angle is specified, the user must specify which branch is required; both are potentially valid. (The criterion for a valid solution is actually that the meridional velocities be less than the speed of sound. Thus, if zero or small flow angles are specified, the supersonic solution will not be valid). If the profile lies on the correct branch and the achieved flow is within the desired tolerance of the specified flow, control passes to Step 9. Otherwise, a new estimate of the midline meridional velocity is required. If the profile lies on the wrong branch, a semi-arbitrary change in the mid-line meridional velocity estimate is made and control returns to Step 6 (subject to some limit on the permitted number of iterations). Otherwise, the mid-line meridional velocity is re-estimated using:

(11.40)

and again, control returns to Step 6 (subject to some limit on the permitted number of iterations). Several potential difficulties exist when applying Eq. (11.40). The gradient becomes ever smaller as the junction between the two branches of the continuity equation is approached. Thus, very large changes in Vm-mid may be calculated (although usually not at Station 1, but this same procedure will be used at stations). Some logic should therefore be employed to ensure that the result of applying Eq. (11.39) is plausible. 9. The process moves on to the next station. Although Station 2 will usually be in an inlet duct, a general description applicable to all stations after the first will be given. 10. If the station follows a blade-free space, total enthalpy, entropy and angular momentum are convected along streamlines from the previous station (these may have to be computed following a solution at the previous station if Eq. (11.29) or (11.31) in rotating coordinates was used). Equation (11.29) will be applied whenever absolute angular momentum is given or implied by the user input data. Equation (11.39) will usually be used to determine total enthalpy from angular momentum or vice versa. Entropy is determined from the efficiency or pressure loss however expressed. In some cases, for example when a total pressure loss coefficient relative to the exit dynamic pressure is given, entropy can only be estimated because the loss depends upon the as-yet unknown velocity distribution. Then, an initial estimate of meridional velocities on all streamlines is required; this can be made by assuming the values determined at the previous station. Note that this is done on the first pass through all stations only. There are numerous possible combinations of data. Total temperature, total enthalpy, total

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temperature or enthalpy change from the previous station, total pressure or pressure ratio from the previous station are some of the possibilities. In conjunction with these, various efficiencies or loss coefficients may be prescribed. For test data analysis, and following a rotor, total temperature and total pressure will be the inputs. A significant amount of programming logic is required to navigate the choices. Equation (11.31) will be applied when relative flow angle is given or implied. A design calculation might specify flow angle rather than angular momentum at a stator discharge station, where test data will normally be total pressure and flow angle. (If total temperature is also measured, there is a choice to be made between convecting the temperatures from the leading edge station or abandoning them and imposing the newly measured values. Some care would then be required with the overall thermodynamics of the calculation). Relative flow angle will be implied by any off-design performance calculation when blade geometry is the input data. This will require the estimation of losses and discharge flow angles by a cascade performance prediction scheme. Depending upon the scheme selected, the losses and/or flow angles may be dependent upon the as-yet unknown meridional velocities, leading only to estimates. Equation (11.29) or (11.31) is integrated, as was done in Step 6. If any of the data required for Eq. (11.29) or (11.31) could only be estimated, an iterative loop should be performed to obtain a meridional velocity distribution that is consistent with the performance modeling, although once a few overall passes have been performed, any discrepancy should become very small. 11. The continuity equation is applied as in Steps 7 and 8, with control passing back to Step 10 until the specified flow is achieved (or all permitted iterations have been performed). The stream function on each streamline will be required later. 12. Control passes repeatedly back to Step 9 until the last station has been processed. 13. The overall convergence of the calculation is checked, unless this point has been reached at the completion of the first pass. The following two criteria should be applied: (a) The stream functions along each streamline should be constant, and (b) The meridional velocities computed at each mesh point should not change from one pass to the next. If convergence has been achieved, the desired output data may be generated (tolerances are discussed bellow). If convergence has not been achieved, and subject to a maximum permitted number of passes, control is passed to Step 14. 14. The estimated streamline pattern for the entire machine may now be updated, the intent being to have the stream functions established at Station 1 maintained

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305

throughout. A simple interpolation of the streamline coordinates at each computing station using the Station 1 stream functions, provides the new, raw streamline coordinates. In order to ensure a stable, convergent procedure, a relaxation factor must be applied to the coordinate changes that the interpolation provides. This is discussed further below. 15. All streamwise gradients are updated. 16. Control passes back to Step 3, to start another pass. 17. The relaxation factor required when relocating the streamlines is: (11.41)

where

is the ratio of the station length to the meridional distance to the

closest adjacent station, and

if tangential velocity component is given,

if relative flow angle is given. Occasionally, the constant

shown as 6 in Eq.(11.41) needs to be set to 4. Typically, this occurs when many stations are placed relatively close together, for example, when multiple stations are placed within blade-rows. Although not supported by theory, it seems reasonable to limit the Mach numbers to maximum values of perhaps 0.7. These equations are based on Wilkinson [9]. Nested iterative calculations are involved in the solution procedure, and appropriate tolerances to judge convergence of each loop are needed. A master tolerance should be defined by the input data, and this can be used for the outermost loop, that is, the streamline location and meridional velocity convergence check. For normal engineering purposes, a tolerance of 1 or 2 parts in 104 is adequate and easily obtained. If the results are to be used in a gradient-based optimization scheme, it may be necessary to reduce the master tolerance to 1 part in 107 or less. Tolerances should be reduced by a factor of 5 or 10 for each successive nesting level. Several additional calculations can be integrated in the above procedure. Bleed flows can be modeled by arranging that, at the first station, the desired bleed flows be contained within the outermost streamtubes. One or more streamlines are then removed at appropriate stations along the flow path. Overall pressure ratio can be specified for off-design calculations by estimating the corresponding flow and then refining the estimate as the solution proceeds. Logic is required to handle the various failure situations that may occur, for example, choking of the machine when the estimated flow is too large. Loss models may be incorporated so that when a converged solution is achieved, it includes losses consistent with the derived flow both for design and off-design calculations. During design calculations, blade geometry may also be simultaneously generated.

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A major shortcoming of the basic streamline curvature method lies in its use of the Euler equations: no transfer of momentum or heat occurs in the transverse direction. This is in contrast to both real flows and CFD solutions of the NavierStokes equations. The result is that, realistic span-wise variations of losses cannot be specified through multi-stage machines because extreme profiles of properties typically develop. This can be largely overcome by superimposing a "mixing" calculation upon the basic streamline curvature calculation. While there is evidence that some of the transverse effects, such as radial wake movement in axial compressors, are not random (Adkins and Smith [10]), a simple turbulent mixing calculation, as suggested in Gallimore [11], seems to capture most important effects. With this addition to the flow-model, off-design calculations can be performed that include realistic span-wise variations of losses which result in realistic profiles of properties. A momentum equation of somewhat different form to Eq. (11.29) and (11.31) is shown in Smith [12]. Here, the gradient of meridional velocity in the meridional direction is eliminated through use of the continuity equation and streamline characteristics. This is aesthetically pleasing (at least) as it removes a streamwise gradient from those that must be updated from pass to pass. Further, for cases with few stations, such as when there are stations at blade-row edges only, intuitively it seems that the local streamwise velocity gradient might be better estimated from the streamline pattern than from velocities at points beyond a blade-row. In a case with many stations within all blade-rows, the differences should be very small. As presented, Smith's equation is less useful because it is framed in the radial rather than an arbitrary direction.

11.2 Compressor Examples In this section, some representative examples of the use of a streamline curvature program are shown. A design is pursued that aims to meet the targets defined by the NASA UEET project for a four-stage aero-engine HP-compressor with an overall pressure ratio of 12:1. The first calculations use the compressor preliminary design (CPD) procedure that is included in the program. In this mode of operation, streamline slope and curvature are assumed zero, and the blade forces acting in the station-wise direction are ignored. Radial stations are placed at each blade-row edge so that a greatly simplified form of the momentum equation applies. These simplifications are made in order to stabilize and speed the solution. An optimizer package is used to derive the design of maximum efficiency that meets various constraints that the designer specifies. Figure 11.3 shows the resulting variation of design-point efficiency with first rotor tip speed. The optimum corrected tip of 589 m/sec results from the overambitious project goals. For the purposes of this rather academic exercise, a tip speed of 548 m/sec was selected as a possible compromise between mechanical and aerodynamic considerations. Figure 11.4 is the corresponding flow-path.

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307

84

K%

82 80 78 76

500

525

550 U tip (m/s)

575

600

Fig. 11.3: Efficiency versus tip speed for NASA -UEET four stage compressor with pressure ratio 12:1

Flowpath of the four-stage HP-compressor with prssure ratio 12:1, first iteration Fig. 11.4: Preliminary flow path of NASA-UEFT for stage compressor.

Fig. 11.5: Second iteration flow path of NASA UEET four-stage HP-compressor with pressure ration 12:1

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11 Radial Equilibrium

The next calculation made is a "first detail design". As for the CPD, total pressure is specified at rotor exits and tangential velocity component is specified at stator exits. Whereas for the CPD, losses are estimated based solely on flow angles and Mach numbers, in the detail, design blading, albeit generic, is derived by the design procedure. The flow model simplifications that were used for the CPD are removed. Figure 11.5 shows the computed streamline pattern. The first detail design assumed generic blading (DCA profiles), but with a first rotor tip Mach number approaching 1.5, a more sophisticated design is required. This is accomplished by adding stations within the blade-rows, and then distributing the effects of the blade-rows amongst the station-to-station intervals.

Fig. 11.6: Refined final flow path of NASA UEET four-stage HP-compressor with pressure ration 12:1

Pressure ratio

20

1.1 1.05 Z=1.0 0.95

15 10

0.9 0.8

5 0 20

0.7

30

40 Mass flow (kg/s)

50

Fig. 11.7: Performance map of NASA UEET four-stage HP-compressor

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309

Fig. 11.8: Solid model of two adjacent blades Figure 11.6 shows the streamline pattern that is computed. Given the blading specification that the calculation created, an off-design calculation can now be performed; Figure 11.7 shows the resulting performance map. Figure 11.8 shows a solid-model of two adjacent blades looking from downstream. The procedure described may be used to produce a final design, although nowadays it will generally be used to create a "first candidate" for further analysis using CFD.

11.3 Turbine Example, Compound Lean Design The streamline curvature method has been most heavily used in turbine design and analysis. The area of turbine applications covers the HP-, IP- and LP-parts of steam turbines, power generation gas turbines and aircraft jet engines. In case of IP- and LPturbine design, the radial equilibrium condition is achieved based on a prescribed variation of the absolute tangential velocity component along a computing station. The computational results allow designing highly three-dimensional blades that follow the calculated flow angles using the streamline curvature method. Once the design process is completed and the relative angle distribution is known, off-design calculation can be performed on a routine basis using the streamline curvature method. The condition for an accurate off-design performance prediction is the implementation of appropriate loss calculation models, as we discussed in Chapter 7.

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In contrast to IP-and LP-turbines, HP-turbines have relatively small aspect ratios that cause the hub and tip secondary flow zones come closer occupying a major portion of the blade height. The secondary flow zones caused by a system of hub and tip vortices induce drag forces resulting in an increase of secondary flow losses that account for almost 40-50% of the entire stage total pressure loss. The most efficient methods of reducing the secondary flow zones in HP-turbines is utilizing the compound lean blade design by varying the lean angle. In recent years, engine manufacturers extensively use leaned blades for new turbines as well as for retrofitting the existing HP-turbines. The application of compound lean design to both stator and rotor blades has been proven as an effective measure of secondary flow control. Using the streamline curvature method, optimizing the lean angle, and placing appropriate turbine profiles from hub to tip, the HP-turbine blades are designed with stage efficiencies that are more than 1.5% higher than those with cylindrical blades. The efficacy of the 3-D leaned design in suppressing the secondary flow is demonstrated in efficiency and performance studies on a three-stage HPturbine by Schobeiri and his co-workers, among others [13] and [14]. A subsequent comparative study, [15], using cylindrical blades with identical blade height, hub and tip diameter, and inlet conditions, revealed a significant efficiency improvement of more than for the rotor with 3-D compound lean blades compared to the one with cylindrical blades. Abhari and his co-workers [16] performed similar investigations on two bladings. They reported that the compound-lean blading has a clear performance advantage of 1% to 1.5% in cascade efficiency over the cylindrical blading. While a combined stator-rotor blade lean has proven to bring efficiency improvement, the stator-row only lean does not seem to have a noticeable efficiency improvement effect, as reported among others in [17]. Before discussing some recent experimental and numerical results, we present in the following the basics of blade lean nomenclature.

11.3.1 Blade Lean Geometry Figure 11.9 shows three sets of stator blades with different lean configurations: (a) exhibits conventional cylindrical blades which have been traditionally used in HPturbines, (b) shows a blade set with constant lean blades, and (c) a blade set with a symmetric lean configuration also called compound lean.The compound lean blade is characterized by a circumferential displacement of the profiles from hub to tip that results in blade surface bow. As Fig. 11.19 (top) shows, the pressure surface has a convex bow, while the suction surface bow is concave. The corresponding lean distribution at the trailing edge is shown in Fig. 11.19 (bottom). The compound lean blade may have a symmetric lean angle distribution with +J at the hub and -J at the tip. It may also have an asymmetric J-distribution with +J1 at the hub and -J2 at the tip, Fig.11.19 bottom (c) and (d). The compound mean may also be confined to the hub and tip regions only, as shown in Fig.11.19(e).

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311

(b) Constant lean

(c) Symmetric lean

H

H

Stator blades

(a)

H

(b)

(c)

h

d

H2

Hub-Tip lean

Asymmetric lean

H

y

H

Symmetric lean

Constant lean

Zero-lean

Tip

H1 (d)

(e)

Hub

Trailing edge view

Fig. 11.9: Blades with different lean geometry

11.3.2 Calculation of Compound Lean Angle Distribution The general case of an asymmetric compound lean is shown in Fig. 11.9 (d), the dimensionless circumferential displacement distribution (bow) can be expressed in terms of a second order polynomial as a function of the dimensionless height as: (11.42) Using the following boundary conditions:

(11.43)

we find the coefficients: (11.44) and Eq. (11.42) becomes:

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(11.45) The position of the maximum bow can be calculated from

(11.46)

Figure 11.10 shows the dimensionless circumferential displacement į (bow) as a function of dimensionless variable Ș for three different J1-values with J2 as a parameter.

1.0 H2

0.8

O

O

10 O 15 20O O 25

10 O 15 20O O 25

G

0.6

O

10 O 15 20O O 25

0.4 0.2 H1 = 10

O

H1 = 15

O

H1 = 20

O

H1 = 25

O

0.0

-0.25

0.0

0.25

0.0

K

0.25

Fig. 11.10: Dimensionless bow į as a function of dimensionless blade height Ș for constant lean angles J1 with J2 as parameter. Note: The abscissa is displaced by 0.25.

Keeping J1 at a time constant and varying J2 from 10o to 25o changes the bow curvature. As shown, for the bow is symmetric. For the bow increases progressively, whereas for

, it decreases. In an actual design

process, these two angles serve as design parameters in addition to the stage parameters we discussed in Chapter 5. An optimization process with the efficiency as the target can be used in conjunction with the streamline curvature method to achieve the optimum efficiency.

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11.3.3 Example: Three-Stage Turbine Design Using the streamline curvature method, multi-stage HP-, IP-, and LP turbines can be optimally designed. As an example, a three-stage HP-research turbine with 3-D compound lean blades is presented in Fig. 11.11 with the specifications in Table 1.

Fig. 11.11: A three-stage high efficiency HP-research turbine with 3-D compound lean blades with enlarged details A, TPFL Table 11.1: Turbine research facility data Item

Specification

Item

Specification

Stage number

N=3

Mass flow

ms =3.728 kg/s

Tip diameter

Dt = 685.8mm

Speed range

n =1800-2800rpm

Hub diameter

Dh =558.8mm

Inlet pressure

pin = 101.356 kPa

Blade height

Hb = 63.5 mm

Exit pressure

pex = 71.708 kPa

Blade number

Stator row 1 = 58

Stator row 2 = 52

Stator row 3 = 56

Blade number.

Rotor row 1 = 46

Rotor row 2 = 40

Rotor row 3 = 44

As seen in Fig. 11.11, the blades have an asymmetric lean with

.The

effectiveness of the compound lean in reducing the secondary flow zone is illustrated in Fig. 11.12. It shows the total pressure contour upstream of the second stator for both rotors.

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91000 90845.5 90690.9 90536.4 90381.8 90227.3 90072.7 89918.2 89763.6 89609.1 89454.5 89300

Fig. 11.12: Total pressure contours upstream of the second stator ring, rotor with compound lean, rotational speed 2600 rpm, TPFL

Figure 11.12 shows the distribution of the total pressure upstream of the second stator (downstream of the first rotor row). The secondary flow zones are expressed by decreasing the contour level in Fig. 11.12 (dark zones). The wakes generated by the preceding rotor blade cause a reduction of the total pressure. For comparison purposes, the total pressure distribution downstream of the second stator pertaining tp a second rotor with identical blade counts and boundary conditions but with cylindrical blades is shown in Fig. 11.13. As seen, major total pressure reduction at the hub exists for the cylindrical blade. The comparison of these two blade types shows that a substantial reduction of total pressure loss is achieved by the compound lean design of stator and the rotor blades. The dimensionless loss coefficient in Fig. 11.14, shows that the typical large region of secondary flow that is present in cylindrical blades has been diminished by the compound lean design.

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91000 90863.6 90727.3 90590.9 90454.5 90318.2 90181.8 90045.5 89909.1 89772.7 89636.4 89500

Relative total pressure loss coefficient ]/]0.5

Fig. 11.13: Total pressure contours upstream of the second stator ring, rotor with compound lean, rotational speed 2600 rpm, TPFL

30

Rpm = 2600 25

20

]Rotor/]Rotor(0.5) ]Stator/]Stator(0.5)

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Immersion ratio (r-rhub)/(rtip-rhub)

Fig. 11.14: Radial distribution of the relative total pressure loss coefficients for stator and rotor row for a TPFL-rotor with compound lean stator and rotor blades

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11.4 Special Cases The radial equilibrium calculation method derived in section 11.3 was applied to design compressor stages using numerical procedures. The method is also applicable to turbine stages as we saw in the previous section. For few special cases, analytical solutions can be found. For this purpose, simplifications are necessary to obtain analytical solutions.

11.4.1 Free Vortex Flow The flow under consideration is assumed to fulfill the conditions: isentropic: /s = 0, isoenergetic: /H = 0, and constant meridional velocity, dVm = 0. With these conditions and the assumption that the streamlines have no curvature, (cylindrical stream surfaces), Eq. (11.11) reduces for a bladeless channel, where Fm = Fn = o, and (ij - Ȗ) = 0, to:

(11.47) Eq.(11.47) is the radial equilibrium equation for a free vortex flow, called Beltrami free vortex flow. Figure 11.15 shows the radial pressure distribution within an annular channel for different circumferential Mach number.

3

Free- Vortex Flow

Mui = 1.5

2.5

p/pi

1.25 2 1.0 1.5 1

0.75 0.50 0

0.25

0.5 (R-Ri)/(Ro-Ri)

0.75

1

Fig. 11.15: Radial pressure distribution of a free-vortex flow with the hub circumferential Mach number Mu as parameter

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317

11.4.2 Forced Vortex Flow This type of flow is assumed to satisfy the following conditions: isentropic, /s = 0 isoenergetic, /H= 0, and circumferential velocity is proportional to the radius: Vu  r or Vu = K r, where K is a constant. With these conditions and the assumption that the streamlines have no curvature, (cylindrical stream surface), Eq. (11.11) reduces for a bladeless channel, where Fm = Fn = o, and (ij - Ȗ) = 0, to: (11.48) Integration of Eq. (11.48) gives: (11.49) The constant K is: (11.50) Using the angle definition in Fig. 11.3 where ȕ is replaced by Į and Vui = VmitanĮi and introducing Eq.(11.50) into (11.49) gives: (11.51) Figure 11.16 shows the radial distribution of meridional velocity ratio.

3

Forced- Vortex Flow Mui = 1.5

2.5

p/pi

1.25 2 1.0

0.75

0.50

1.5 1

0

0.25

0.5 (R-Ri)/(Ro-Ri)

0.75

1

Fig. 11.16: Pressure distribution of a forced-vortex flow with the hub circumferential Mach number Mu as parameter

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11.4.3 Flow with Constant Flow Angle The flow under consideration is assumed to fulfill the conditions: isentropic, /s = 0 isoenergetic, /H = 0, and constant inlet flow angle, dĮ1 = 0. With these conditions and the assumption that the streamlines have no curvature, (cylindrical stream surface), Eq. (11.11) then reduces for a bladeless channel, where Fm = Fn = o, and (ij - Ȗ) = 0, to: (11.52) Since (11.53) introducing Eq.(11.52) into (11.53) leads to: (11.54) Integrating Eq.(11.54) gives: (11.55) Figure 11.17 depicts the axial velocity ratio as a function of the radius ratio with Į1 as parameter.

1 D=20o 30o

0.8 Va/Vai

40o 50

o

60o

0.6

70o D=90o

0.4

0

0.25

0.5 (R-Ri) /(Ro-Ri)

0.75

1

Fig. 11.17: Axial velocity ratio as a function of dimensionless blade height with the inlet angle as parameter

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319

References 1. Wu, C.-H.: A general Theory of Three-Dimensional Flow in Subsonic and Supersonic Turbomachines of Axial-, Radial, and mixed-Flow Types. NACA technical Note 2604, Washington, D.C. (January 1952) 2. Lakshminarayana, B.: Fluid Dynamics and Heat Transfer of Turbomachinery. John Wiley & Sons, Chichester (1996) 3. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960) 4. Novak, R.A., Hearsey, R.M.: A Nearly Three Dimensional Intra blade Computing System for Turbomachinery, Part I & II. Journal of Fluid Engineering, 99, 154–166 5. Wilkinson: D.H, Calculation of Blade-to-Blade Flow in Turbomachine by Streamline Curvature,” British ARC R&M 3704 (1972) 6. Wennerstrom, A.J.: On the Treatment of body Forces in the radial Equilibrium Equation of Turbomachinery. Traupel-Festschrift. Juris-Verlag, Zürich (1974) 7. Hearsey, R.M.: Computer Program HT 0300 version 2.0. Hearsey technology. Bellevue, Washington (2003) 8. Horlock, J.H.: On Entropy Production in Adiabatic Flow in Turbomachines. ASME Journal of Basic Engineering, pp. 587-593 (1971) 9. Wilkinson, D.H.: Stability, Convergence and Accuracy of 2-Dimensional Streamline Curvature Methods Using Quasi-Orthogonals. Proceedings of the Institution of Mechanical Engineers 184, 1979–1970 (1969) 10. Adkins, G.G., Smith, L.H.: Spanwise Mixing in Axial-Flow Turbomachines. ASME Journal of Engineering for Gas Turbines and Power 104, 97–110 (1982) 11. Gallimore, S.J.: Spanwise mixing in Multistage Axial Flow Compressors: Part II – Throughflow Calculations Including Mixing. ASME Journal of Turbomachinery 108, 10–16 (1986) 12. Smith Jr., L.H.: The Radial-Equilibrium Equation of Turbomachinery. ASME Journal of Engineering for Power, pp. 1-12 (January 1966) 13. Schobeiri, M.T., Gilarranz, J., Johansen, E.: Final Report on: Efficiency, Performance, and Interstage Flow Field Measurement of Siemens-Westinghouse HPTurbine Blade Series 9600 and 5600 (September 1999) 14. Schobeiri, M.T., Gillaranz, J.L., Johansen, E.S.: Aerodynamic and Performance Studies of a Three Stage High Pressure Research Turbine with 3-D Blades, Design Point and Off-Design Experimental Investigations. In: Proceedings of ASME Turbo Expo, 2000-GT-484 (2000) 15. Schobeiri, M.T., Suryanaryanan, A., Jerman, C., Neuenschwander, T.: A Comparative Aerodynamic and performance study of a three-stage high pressure turbine with 3-D bowed blades and cylindrical blades. In: Proceedings of ASME Turbo Expo, Power of Land Air and Sea, June 14-17, Vienna, Austria, paper GT200453650 (2004) 16. Treiber, M., Abhari, R.S., Sell, M.: Flow Physics and Vortex Evolution in Annular Turbine Cascades. ASME, GT-2002-30540. In.Proceedings of ASME TURBO EXPO, June 3-6, Amsterdam, The Netherlands (2002) 17. Rosic, B., Xu, L.: Blade Lean and Shroud Leakage Flows in Low Aspect Ratio Turbines. In: Proceedings of ASME Turbo Expo 2008: Power for Land, Sea and Air, June 9-13, Berlin, Germany, GT2008-50565 (2008)

Part III: Turbomachinery Dynamic Performance

12 Nonlinear Dynamic Simulation of Turbomachinery Components and Systems

The following chapters deal with the nonlinear transient simulation of turbomachinery systems. Power generation steam and gas turbine engines, combined cycle systems, aero gas turbine engines ranging from single spool engines to multi-spool high pressure core engines with an afterburner for supersonic flights, rocket propulsion systems and compression systems for transport of natural gas with a network of pipeline systems are a few examples of systems that heavily involve turbomachinery components. Considering a power generation gas turbine engine as a turbomachinery system that is designed for steady state operation, its behavior during routine startups, shot downs, and operational load changes significantly deviates from the steady state design point. Aero gas turbine engines have to cover a relatively broad operational envelope that includes takeoff, low and high altitude operation conditions, as well as landing. During these operations, the components are in a continuous dynamic interaction with each other, where the aero-thermodynamic as well as the mechanical load conditions undergo temporal changes. As an example, the acceleration/ deceleration process causes a dynamic mismatch between the turbine and compressor power resulting in temporal change of the shaft speed. In the above cases, the turbomachinery systems are subjected to the operating modes that are specific to the system operation. Besides these foreseeable events, there are unforeseeable operation scenarios that are not accounted for when designing the system. System failures, such as, blade loss during a routine operation, loss of cooling mass flow through the cooled turbine blades, adverse operation conditions that force the compressor component to surge, and failure of the control system, are a few examples of adverse operational conditions. In all of these operations, the system experiences adverse changes in total fluid and the thermodynamic process leading to greater aerodynamic, thermal and mechanical stress conditions. The trend in the development of gas turbine technology during the past decades shows a continuous increase in efficiency, performance, and specific load capacities. This trend is inherently associated with increased aerodynamic, thermal, and mechanical stresses. Under this circumstance, each component operates in the vicinity of its aerodynamic, thermal and mechanical stress limits. Adverse operational conditions that cause a component to operate beyond its limits can cause structural damages as a result of increased aerodynamic, thermal, and structural stresses. To prevent this, the total response of the system, including aerodynamic, thermal, and mechanical responses must be known in the stage of design and development of new turbomachinery systems.

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This chapter describes the physical basis for the non-linear dynamic simulation of gas turbine components and systems. A brief explanation of the numerical method for solution is followed by detailed dynamic simulation of several components described in the following chapters.

12.1 Theoretical Background Dynamic behavior of turbomachinery components and systems can generally be described by conservation laws of fluid mechanics and thermodynamics ([1], [2], [3], [4]). The fluid dynamic process that takes place within an inertial system is, as a rule, unsteady. The steady state, a special case, always originates from an unsteady condition during which the temporal changes in the process parameters have largely come to a standstill. The following accounts this fact by formulation of the fluid and thermodynamic conservation laws in a 4-dimensional temporal-spatial coordinate system. For this purpose, the conservation laws we discussed in Chapter 3 must be rearranged such that temporal changes of thermo-fluid dynamic quantities are expressed in terms of spatial changes. A summary of relevant equations are found in [5]. For an unsteady flow, the conservation of mass derived in Chapter 3, Eq. (3.4), is called upon:

(12.1) Equation (12.1), (12.54) expresses the temporal change of the density in terms of spatial change of the specific mass flow. Neglecting the gravitational force, the Cauchy equation of motion (3.22) was derived as:

(12.2)

We rearrange Eq. (12.2) to include the density in the temporal and spatial derivatives by implementing the continuity equation (3.4)

(12.3)

where the stress tensor Ȇ can be decomposed into the pressure and shear stress tensor as follows:

12 Nonlinear Dynamic Simulation

325

(12.4) with Ip as the normal stress and T as the shear stress tensor. Inserting Eq. (12.4) into (12.3) results in

(12.5)

Equation (12.5) directly relates the temporal changes of the specific mass flow to the spatial changes of the velocity, pressure, and shear stresses. For the complete description of the unsteady flow process, the total energy equation that includes mechanical and thermal energy balance is needed. Referring to Eq. (3.58) and neglecting the gravitational contribution, we find for mechanical energy that:

(12.6) Equation (12.6) exhibits the mechanical energy balance in differential form. The first term on the right-hand side represents the mechanical energy contribution due to the pressure gradient. The second term is the contribution of the shear stress work. The third term represents the production of the irreversible mechanical energy due to the shear stress. It dissipates as heat and increases the internal energy of the system. Before rearranging Eq. (12.6) and performing the material differentiation, we revert to the thermal energy balance, Eq. (3.66),

(12.7)

The combination of the mechanical and thermal energy balances, Eq. (12.6) and (12.7), results in:

(12.8)

Since all components of a turbomachinery system are considered open systems, it is appropriate to use enthalpy h rather than internal energy u. The state properties h and u can be expressed as a function of other state properties such as T, v, p etc.

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12 Nonlinear Dynamic Simulation

(12.9)

(12.10)

where T and h are the absolute static temperature and enthalpy. With the definitions:

(12.11)

and the application of the first law, the following relation between cp and cv is established:

(12.12)

For the open cycle gas turbines and also jet engines with moderate pressure ratios, the working fluids air and combustion gases behave practically like ideal gases whose internal energy is only a function of temperature and not volume. This circumstance considerably simplifies the interconnection of different thermodynamic properties if the equation of state of ideal gases is considered: (12.13) Introducing the Gibbs's enthalpy function: (12.14) the differentiation gives: (12.15) with

12 Nonlinear Dynamic Simulation

327

(12.16) As a consequence, for ideal gases, the result of this operation leads to:

(12.17)

Thus, the state properties, u and h, as well as the specific heat capacities cp, and cv and their ratio ț are solely functions of temperature. This is also valid for combustion gases with approximately ideal behavior. For combustion gases, there is a parametric dependency of the above stated properties and the fuel-air ratio. After this preparation, Eq. (12.7) can be written in terms of h and p:

(12.18) with Eq. (12.15) and considering the continuity equation (12.1), (12.54), the substantial change of the static pressure is:

(12.19) Introducing Eq. (12.19) into (12.18):

(12.20) The combination of thermal energy equation (12.20) and mechanical energy equation (12.6) leads to:

(12.21)

with as the total enthalpy and the kinetic energy. Equation (12.21) can also be obtained by introducing the relationship between the pressure and enthalpy into Eq.(12.21). The total enthalpy can be expressed in terms of total temperature:

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12 Nonlinear Dynamic Simulation

(12.22)

Finally, the total energy equation in terms of total pressure can be established by inserting Eqs. (12.14) and (12.16) into Eq. (12.8):

(12.23)

where is the total pressure. Equations (12.21), (12.22) and (12.23) express the same physical principles, namely the law of conservation of energy in two different forms. In physical terms, they are fully equivalent and mathematically convertible to each other. As shown in the following chapters, one or the other of these equations is called upon in conjunction with the other laws to deal with various dynamic problems. For example, in dealing with an unsteady exchange of energy and impulse, it is useful to apply the differential equation for total temperature. If the principal goal of a problem is the determination of the unsteady changes in pressure, the total pressure differential equation should be used. A summary of the working equations is given in Table 12.1. Table 12.1: Summary of thermo-fluid dynamic equations Equations In terms of substantial derivatives

Eq. No.

Continuity

(3.4) (12.1)

Motion

(3.22) (12.2)

Mechanical Energy

(3.58) (12.6)

Equation of thermal energy in terms of u

(3.66) (12.7)

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329

Table 12.1 (continued) Equations in terms of substantial and local derivatives

,

Eq. No.

Equation of thermal energy in terms of h for ideal gas

(12.18)

Equation of total enthalpy

(3.73) (12.21)

Equation of thermal energy in terms of cv and T for ideal gas

(3.66) (12.7)

Equation of thermal energy in terms of cp and T for ideal gas

(12.7)

Equation of continuity

(3.4) (12.1)

Equation of motion in terms of total stress tensor

(3.22) (12.2)

Equation of motion, stress tensor decomposed

(12.5)

Equation of mechanical energy including ȡ

(3.58) (12.6)

Equation of thermal energy in terms of u for ideal gas

(12.7) Rearranged

Equation of thermal energy in terms of h for ideal gas

(12.7) Rearranged

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12 Nonlinear Dynamic Simulation

Table 12.1 (continued) Equations in terms of local derivatives

Eq. No.

Equation of thermal energy in terms of cv and T

(12.7) Rearranged

Equation of thermal energy in terms of static h for ideal gases

(12.7) Rearranged

Energy equation in terms of total enthalpy

(12.21)

Energy equation in terms of total temperature

(12.22)

Energy equation in terms of total pressure

(12.23)

12.2 Preparation for Numerical Treatment The thermo-fluid dynamic equations discussed above constitute the theoretical basis describing the dynamic process that takes place within a turbomachinery component during a transient operation. A four-dimensional time space treatment that involves the Navier solution is, at least for the time being, out of reach. For simulation of dynamic behavior of a turbomachine that consists of many components, it is not primarily important to calculate the unsteady three-dimensional flow processes in great detail. However, it is critical to accurately predict the response of each

12 Nonlinear Dynamic Simulation

331

individual component as a result of dynamic operation conditions. A one-dimensional time dependent calculation procedure provides sufficiently accurate results. In the following, the conservation equations are presented in index notation. For the onedimensional time dependent treatment, the basic equations are prepared first by setting the index in Eqs. (12.24-12.27).

12.3 One-Dimensional Approximation The thermo-fluid dynamic equations discussed above constitute the theoretical basis describing the dynamic process that takes place within a turbomachinery component during a transient operation. A four-dimensional time space treatment that involves the Navier solution is, at least for the time being, out of reach. For simulation of the dynamic behavior of a turbomachine that consists of many components, it is not primarily important to calculate the unsteady three-dimensional flow processes in great detail. However, it is critical to accurately predict the response of each individual component as a result of dynamic operation conditions. A one-dimensional time dependent calculation procedure provides sufficiently accurate results. For this purpose, firstly, the basic equations are prepared for the one-dimensional treatment.

12.3.1 Time Dependent Equation of Continuity In Cartesian coordinate system the continuity equation (12.1), (12.54) is:

(12.24)

Equation (12.24), after setting

, becomes:

(12.25)

with as the length in streamwise direction and the cross sectional area of the component under investigation. Equation (12.25) expresses the fact that the temporal change of the density is determined from the spatial change of the specific mass flow within a component. The partial differential Eq. (12.25) can be approximated as an ordinary differential equation by means of conversion into a difference equation. The ordinary differential equation can then be solved numerically with the prescribed initial and boundary conditions. For this purpose, the flow field

332

12 Nonlinear Dynamic Simulation

1

2

3

Si

i

k

'x

i+1

n

n-1

Si+1

x

Fig. 12.1: Discretization of an arbitrary flow path with variable cross section S = S(x)

is equidistantly divided into a number of discrete zones with prescribed length, , inlet and exit cross sections and as Fig. 12.1 shows. The quantities pertaining to the inlet and exit cross sections represent averages over the inlet and exit heights. For the annular nozzles and diffusers the quantities are thought of as averages taken in circumferential as well as radial direction. Using the nomenclature in Fig. 12.1, Eq. (12.25) is approximated as: (12.26) with and as the mass flows at stations and with the corresponding cross sections. For a constant cross section, Eq.(12.26) reduces to: (12.27) with as the volume of the element k enclosed between the surfaces i and i+1 . The index k refers to the position at , Fig. 12.1.

12 Nonlinear Dynamic Simulation

333

12.3.2 Time Dependent Equation of Motion The index notation of the momentum equation (12.5) is: (12.28) In the divergence of the shear stress tensor in Eq. (12.28), represents the shear force acting on the surface of the component. For a one-dimensional flow, the only non-zero term is . It can be related to the wall shear stress

which is a function of the friction coefficient cf. (12.29)

In the near of the wall, the change of the shear stress can be approximated as the difference between the wall shear stress and the shear stress at the edge of the boundary layer, which can be set as

(12.30)

The distance in can be replaced by a characteristic length such as the hydraulic diameter . Expressing the wall shear stress in Eq. (12.30) by the skin friction coefficient

(12.31)

and inserting Eq. (12.31) into the one-dimensional version of Eq. (12.28), we obtain

(12.32)

Equation (12.32) relates the temporal change of the mass flow to the spatial change of the velocity, pressure and shear stress momentums. As we will see in the following sections, mass flow transients can be accurately determined using Eq. (12.32). Using the nomenclature from Fig. 12.1, we approximate Eq. (12.32) as:

334

12 Nonlinear Dynamic Simulation

(12.33)

For a constant cross section, Eq. (12.33) is modified as:

(12.34)

12.3.3 Time Dependent Equation of Total Energy The energy equation in (12.21), in terms of total enthalpy, is written in index notation

(12.35)

expressing the total enthalpy, Eqs. (12.35), in terms of total temperature results in:

(12.36)

For calculating the total pressure, the equation of total energy is written in terms of total pressure already derived as Eq. (12.23), which is presented for the Cartesian coordinate system as:

12 Nonlinear Dynamic Simulation

335

(12.37)

Before treating the energy equation, the shear stress work needs to be evaluated:

(12.38)

For a two-dimensional flow, Eq. (12.38) gives

(12.39)

Assuming a one-dimensional flow with

, the contribution of the shear stress

work Eq. (12.39) is reduced to

(12.40)

The differences in

at the inlet and exit of the component under simulation stem

from velocity deformation at the inlet and exit. Its contribution, however, compared to the enthalpy terms in the energy equation, is negligibly small. Thus, the onedimensional approximation of total energy equation (12.35) in terms of total enthalpy reads:

(12.41)

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12 Nonlinear Dynamic Simulation

For a steady state case without changes of specific mass

, Eq. (12.41) leads to:

(12.42)

Assuming a constant cross section and mass flow, Eq. (12.42) gives

(12.43)

Integrating Eq. (12.43) in a streamwise direction results in:

(12.44)

for Eq. (12.44) to be compatible with the energy equation, a modified version of Eq. (4.75) is presented.

(12.45)

Equating (12.44) and (12.45) in the absence of a specific shaft power, the following relation between the heat flux vector and the heat added or rejected from the element must hold:

(12.46)

From (12.46), it immediately follows that

(12.47)

12 Nonlinear Dynamic Simulation

337

where is the thermal energy flow added to or rejected from the component. In the presence of shaft power, the specific heat in Eq. (12.47) may be replaced by the sum of the specific heat and specific shaft power:

(12.48)

Equation (12.48) in differential form in terms of

and L is

(12.49)

With Eq. (12.49), we find:

(12.50)

Using the nomenclature in Fig. 12.1, Eq. (12.50) is written as:

(12.51)

In terms of total temperature, Eq. (12.51) is rearranged as:

338

12 Nonlinear Dynamic Simulation

(12.52)

In terms of total pressure, Eq. (12.37) is:

(12.53)

which is approximated as:

(12.54)

with:

(12.55)

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339

12.4 Numerical Treatment The above partial differential equations can be reduced to a system of ordinary differential equations by a one-dimensional approximation. The simulation of a complete gas turbine system is accomplished by combining individual components that have been modeled mathematically. The result is a system of ordinary differential equations that can be dealt with numerically. For weak transients, Runge-Kutta or Predictor-Corrector procedures may be used for the solution. When strong transient processes are simulated, the time constants of the differential equation system can differ significantly so that difficulties must be expected with stability and convergence with the integration methods. An implicit method avoids this problem. The system of ordinary differential equations generated in a mathematical simulation can be represented by:

(12.56) with X as the state vector sought. If the state vector X is known at the time t, it can be approximated as follows for the time t+dt by the trapezoidal rule: (12.57) Because the vector X and the function G are known at the time t, i.e., Xt and Gt are known, Eq. (12.57) can be expressed as: (12.58)

As a rule, the function F is non-linear. It can be used to determine Xt+dt by iteration when Xt is known. The iteration process is concluded for the time t+dt when the convergence criterion

(12.59)

is fulfilled. If the maximum number of iterations, k = kmax, is reached without fulfilling the convergence criterion, the time interval ǻt is halved, and the process of iteration is repeated until the criterion of convergence is met. This integration process, based on the implicit one-step method described by Liniger and Willoughby [6] is reliable for the solution of stiff differential equations. The computer time required

340

12 Nonlinear Dynamic Simulation

depends, first, on the number of components in the system and, second, on the nature of the transient processes. If the transients are very strong, the computer time can be 10 times greater than the real time because of the halving of the time interval. For weak transients, this ratio is less than 1.

References 1. Schobeiri, M.T.: Aero-Thermodynamics of Unsteady Flows in Gas Turbine Systems. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-51 (1985) 2. Schobeiri, T.: COTRAN, the Computer Code for Simulation of Unsteady Behavior of Gas Turbines. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-53 (1985b) 3. Schobeiri, T.: A General Computational Method for Simulation and Prediction of Transient Behavior of Gas Turbines. ASME-86-GT-180 (1986) 4. Schobeiri, M.T., Abouelkheir, M., Lippke, C.: GETRAN: A Generic, Modularly Structured Computer Code for Simulation of Dynamic Behavior of Aero-and Power Generation Gas Turbine Engines; an honor paper. ASME Transactions, Journal of Gas Turbine and Power 1, 483–494 (1994) 5. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. John Wiley and Sons, Chichester (1960) 6. Liniger, W., Willoughby, R.: Efficient integration methods for stiff systems of ordinary differential equations. SIAM. Numerical Analysis 7(1) (1970)

13 Generic Modeling of Turbomachinery Components and Systems

A turbomachinery system such as a power generation gas turbine engine, a thrust generation aero- engine, rocket propulsion, or a small turbocharger, consist of several sub-systems that we call components ([1],[2],[3],[4]). Each component is an autonomous entity with a defined function within the system. Inlet nozzles, exit diffusers, combustion chambers, compressors, and turbines are a few component examples. A component may consist of several sub-components. A turbine or a compressor stage exhibits such a sub-component. The numerical models of components are called modules. The gas turbine engine shown in Fig. 13.1 composed of an inlet nozzle, a multistage compressor component with cooling air extraction and bypass systems. The cooling air extraction provides cooling mass flow for the shaft, the high pressure turbine, and the mixing air for reducing the combustion chamber exit temperature. The bypass system prevents the compressor from getting into a rotating stall and surge during the startup and shutdown procedures (see Chapter 16). To reduce the compressor exit flow velocity and thus the kinetic energy, the remaining compressor Combustion chamber with fuel injection Bypass valve

Multi-stage compressor

Compressor inlet nozzle

Multi-stage turbine

To generator

Diffuser

Fig. 13.1: Single-spool power generation gas turbine engine, BBC-GT-9 with major components

342

13 Generic Modeling of Turbomachinery Components and Systems

mass flow passes through a diffuser before entering the combustion chamber. In the following combustion chamber component, fuel is added to establish the required turbine inlet temperature. The following multistage turbine component placed on the same shaft as the compressor, drives the compressor and the generator. The configuration of the compressor and the turbine components placed on the same shaft is called a spool. As seen in Fig. 13.1, the spool shaft is coupled with the generator shaft, which converts the net turbine power into electrical power. Figure 13.2 illustrates a decomposition of a twin-spool engine into its major components. It consists of (1) a high pressure-spool, the so-called gas generator that encompasses the multistage high pressure compressor (HP-compressor), combustion chamber component, and the high pressure turbine (HP-turbine) component, (2) a low pressure-spool shaft connects the low pressure compressor with the low pressure turbine component, (3) the combustion chamber, (4) an inlet diffuser and exit nozzle. The two spools are connected aerodynamically. The mass flow, with high kinetic energy exiting from the last stage of the HP-turbine stage, impinges on the first stator of the LP-turbine, and is further expanded within the LP-component and the following thrust nozzle, that provides the required kinetic energy for thrust generation. Besides the major components shown in Fig. 13.1 and 13.2, there are several other components such as pipes that serve the transport of mass flow from compressor to turbine for cooling purposes, valves with the operation ramp defined by the control system, control systems, lubrication systems, bearings, electric motors for power supply, etc. LP-compressor stage

Combustion chamber m

LP-turbine stage

4

F

mS

2 m

mP

mS1 mSi

5

mM mA mF

A

Inlet

3

Exit

m1

7

m2

m3

6

JT-2SZO

HP-compressor stage

Cooled HP-turbine stage

m = Air mass flow A m = Fuel mass flow

m = Primary air mass flow

HP-turbine stage

P

m = Secondary air mass flow

Fig. 13.2: A twin-spool aero-gas turbine with its major components

13 Generic Modeling of Turbomachinery Components and Systems

343

13.1 Generic Component, Modular Configuration These components shown in Fig. 13.1 and 13.2 are common to a small or large gas turbine engine regardless if it is a power generation engine, a thrust generation, or an engine that serves as a turbocharger. They can be modeled as modules to assemble a complete system ranging from a single spool power generation gas turbine to the most complicated of rocket propulsion systems. The components can be categorized into three groups according to their functions. The representatives of each of these groups are described below.

13.1.1 Plenum as Coupling Module Before describing in detail the individual component modeling in Chapters 14 to 18, we first mathematically describe the module plenum that connects the components in a turbomachine. The plenum is the coupling module between two or more successive components, Fig. 13.3. Its primary function is to couple the dynamic information of entering and exiting components. The volume of the plenum is the sum of half of all volumes of the components that enter and exit the plenum. The inlet components transfer information about mass flow , total pressure P , total temperature , fuel/air ratio, and water/air ratio w to the plenum. After entering the plenum, a mixing process takes place, where the aforementioned quantities reach their equilibrium values. These values are the same for all outlet components exiting the plenum. Each component represented by a module is uniquely defined by its name, inlet and outlet plena.

Inlet Plena

IN

n

mn , Pn , Tn , f n , wn

i

m

Out-Components Mixing Plenum

mn , P , T

,

f

,w

P,T

,

f

,

w

OUTi

m 1, P , T

,

f

,w

OUT 1

OUT n

MP IN

IN

1

i

,P,

i

T i

,

f

i

,

w

i

m 1, P1 , T1 , f1 , w1

P, T, V f w

m

i

Outlet Plena

In-Components

Fig. 13.3: Plenum as the coupling module between components

The enthalpy, temperature and pressure transients are obtained from Eq. (12.21) and (12.23) neglecting, however, the contribution of the kinetic energy and time change in mass flow relative to other terms. The enthalpy equation (12.21)

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13 Generic Modeling of Turbomachinery Components and Systems

(13.1) applying the chain rule of differentiation to Eq. (13.1), we arrive at: (13.2) Equation (13.2) can be expressed in terms of total temperature by replacing the total enthalpy with as follows: (13.3) The total pressure equation (12.23) applied to the plenum, where no heat is added and the terms that include temporal and spatial changes of the kinetic energy as well as the friction compared to the enthalpy terms are neglected is written as:

(13.4) The index notation of Eq. (13.3) is:

(13.5) For a one-dimensional time dependent flow the approximation of Eq. (13.5) leads to:

(13.6) Implementing the mass flow in Eq. (13.6) results in

(13.7) Approximating the differentials by differences, Eq. (13.7) reads: (13.8)

13 Generic Modeling of Turbomachinery Components and Systems

345

In Eq. (13.8) the quantities with subscript in and out refer to inlet and outlet, respectively. In case that the plenum has m inlet and n exit components, Eq. (13.8) is be written as:

(13.9)

In Eq. (13.9) the quantities

,

and

refer to the plenum total

temperature, total pressure and the volume, respectively. As Fig. 13.3 shows, the mass flow pertaining to the components that enter the plenum may contain certain amount of combustion products and water vapor specified by the fuel/air ratio f and water /air ratio w, respectively. This information is already included in the corresponding specific heats and the gas constants of the individual components. Once these mass flows are mixed in the plenum, the resulting f and w of the exiting mass flows are different from entering ones. This is accounted for by the exiting specific heat . Similar procedure is applied to obtain the plena pressure using Eq.(13.4) with the index notation:

(13.10)

Resulting in:

(13.11)

With Eqs. (13.9) and(13.11) the temperature and pressure information of the plenum as the connecting module is completely defined. As seen from Eq. (13.11), the plenum pressure include the plenum temperature. The implicit method of integration described in Section 12.4 utilizes the initial values for the quantities in Eqs. (13.9) and (13.11) and perform the iteration until the convergence criterium has been satisfied.

13.1.2 Group1 Modules: Inlet, Exhaust, Pipe Group 1 includes those components in which no transfer of thermal energy with the surrounding environment takes place. Their function consists, among other things, of

346

13 Generic Modeling of Turbomachinery Components and Systems

transporting the mass, accelerating the mass flow through the nozzle, and reducing the kinetic energy through a diffuser. Figure 13.4 exhibits the physical as well as the modular representation of the component’s nozzle, pipe, and diffuser. Each module is surrounded by an inlet and an outlet plenum. These plena serve as the coupling element between two or more modules. Detailed physical and mathematical modeling of these modules are presented in Chapter 14.

Fig. 13.4: Component and modular representation of inlet nozzle N, connecting pipe P, and Diffuser D

13.1.3 Group 2: Heat Exchangers, Combustion Chamber, After- Burners Group 2 includes those components within which the processes of thermal energy exchange or heat generation take place. Heat exchangers are encountered in different forms such as recuperators, regenerators, inter-coolers, and after-coolers. Combustion chambers and afterburners share the same function. They convert the chemical energy of the fuel into thermal energy. As an example, Fig. 13.5 shows the component and modular representation of a recuperator and an inter-cooler. Recuperators are used to improve the thermal efficiency of small power generation gas turbines. While hot gas from the exhaust system enters the low pressure side (hot side) of the recuperator, compressor air enters the high pressure side (cold side). By passing over a number of contact surfaces, the hot side of the recuperator transfers thermal energy to the compressor air that passes through the recuperator cold side. After exiting the recuperator air side, preheated air enters the following combustion chamber.

13 Generic Modeling of Turbomachinery Components and Systems

347

Fig. 13.5: Component and modular representations of a recuperator and an intercooler with the inlet and exit plena

Fig. 13.6: Component and modular representation of a typical combustion chamber The component combustion chamber is exhibited in Fig. 13.6. It generally consists of a primary combustion zone, a secondary zone, and a mixing zone. The primary zone, surrounded by n rows of ceramic segments, separates the primary combustion zone from the secondary zone and protects the combustion chamber casing from being exposed to high temperature radiation. The actual process of combustion occurs in the primary zone. The mixing in of the secondary air passing through mixing nozzles and holes reduces the gas temperature in the mixing zone to an acceptable level for the gas turbine that follows. The rows of segments in the combustion zone are subjected to a severe thermal loading due to direct flame radiation. Film and convective cooling on both the air and the gas sides cools these segments. The air mass flows required to cool these hot segments flow through finned cooling channels, thereby contributing to the convection cooling of the segments on the air side. The cooling air mass flow exiting from the jth segment row, effects a film cooling process on the gas side within the boundary layer in the next row of segments. At the end of that process, the cooling air mass flow is mixed with the primary air mass flow, thus, reducing the exit temperature.

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13 Generic Modeling of Turbomachinery Components and Systems

Afterburner physical component AF

I m

CG

m

+ m

CG

AF

m

= Combustion gas mass flow from turbine

m

= Afterburner fuel mass flow

CG AF

O

m

Modular representation of aftreburner

Fig. 13.7: Component and modular representation of a typical afterburner

The afterburner (AF) component is applied to supersonic aero-engines that require supersonic exit velocities. Figure 13.7 exhibits a typical afterburner component with its modular representation. After leaving the low pressure turbine component, fuel is added to the mass flow inside the afterburner component causing an increase in temperature. The combustion gas is accelerated from subsonic to supersonic in a Laval nozzle that follows the afterburner. From combustion process point of view the principle function of the afterburner component and its modular simulation is very similar to the one discussed for combustion chambers.

13.1.4 Group 3: Adiabatic Compressor and Turbine Components This group includes components within which an exchange of mechanical energy (shaft power) with the surroundings takes place. The representatives of this group are compressors and turbines. Figure 13.8 (a) and (b) show a compressor and a turbine stage with the velocity diagrams and the modular representations. The modules represent one stage within a multi-stage environment. The stages are decomposed into a stator and a rotor row that are connected by the corresponding plena. The row-byrow adiabatic compression and expansion processes are described by the conservation laws in conjunction with the known stage characteristics as discussed in Chapter 5. These components are comprehensively treated in Chapters 16 and 17.

13 Generic Modeling of Turbomachinery Components and Systems

Velocity diagram m2

Compressor module 1

1

2

3

W3

W2

V

Compressor stage representation

349

3 U2

2

V2

2

D

3

U

U3

Rotor

E D2

Stator

V3

m3

W3

V

V1

W2

V1

E

3

Stator Rotor

Fig. 13.8a: Representation of an adiabatic compressor stage, module. The stage is decomposed into two rows that are connected via plenum No. 2.

Turbine stage representation

Vm3

Velocity diagram

2

Rotor

Vm2

U3

Stator Rotor

U2

V2

V1

W

3

V

U2

V2

2

W2

D

U

Stator

VU

3

V3

3

3

1

2

1

W3

3

Turbine module representation

tuandco

Fig. 13.8b: Representation of an adiabatic turbine stage module

3

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13 Generic Modeling of Turbomachinery Components and Systems

13.1.5 Group 4: Diabatic Turbine and Compressor Components Unlike the adiabatic turbine and compressor components discussed in Section 13.1.4, this group includes turbine and compressor components within which not only the exchange of mechanical energy (in terms of shaft power) with the surroundings takes place, but also thermal energy exchange with the working fluid occurs. This group encompasses cooled turbines and compressors as well as those that are not cooled. Figure 13.9 shows a cooled turbine stage with the corresponding modular representation. It is considered a special module that is usually utilized in the first three rows of the high pressure part of gas turbine engines. Its modular representation shows the decomposition of the stage into its stator and rotor row that are separated by the corresponding plenum 2.

1

I1

2

3 3

m3

m1

Rotor row

Stator row

I2

mCR

2

1

m CS

I1

I2

COOLTU3

Fig. 13.9: Schematic of a cooled turbine stage and its modular representation

As Fig. 13.9 shows, a fraction of the air mass flow, , extracted from the HPcompressor is diverted into the first stator row and enters the cooling channels that might have several pin-fins and rib turbulators. Heat is transferred from the blade surface to the cooling air which may exit through several film cooling holes, slots at the trailing edge, and other holes. After exiting the blade surface, the cooling mass flow joins the turbine main mass flow. The same procedure may repeat in the subsequent rotor and stator rows as shown in Fig. 13.9. The module representation in Fig. 13.9 reflects the cooling and mixing procedure in the turbine stage. The main mass flow, with certain fuel/air and humidity/air ratios enters the upstream station 1 that corresponds to plenum (1) and expands through the stator row. The cooling mass flow from plenum (I1), with certain water/air ratios but zero fuel/air ratios, passes though the cooling channel and mixes with the mainstream combustion gas in plenum (2). The mixing process changes the fuel/air and humidity/air ratios, which is precisely calculated. Similar heat transfer and mixing processes occur within the rotor row with rotor cooling mass flow that enters the plenum at (I2). The non-linear dynamic behavior of the cooled turbine stage represented by the above module, is described by three differential equations for the gas side, three differential equations

13 Generic Modeling of Turbomachinery Components and Systems

351

for the cooling air side, and one differential equation for the turbine material heat conduction as the closure condition. The same number of differential equations are generated for the rotor row. A diabatic expansion is not necessarily limited to the cooled turbine component. Uncooled blades are also subjected to diabatic processes whenever there is a temperature difference between the working medium and the turbine blade material. In steady state operations, which is specific to power generation gas turbine systems, there is a temperature equilibrium between the combustion gas and the blade material. Changing the engine load condition disrupts this equilibrium leading to a temperature difference between the turbine mass flow and the blade material. In the case of load addition, opening the fuel valve causes an instantaneous gas temperature increase. Since the blade metal temperature response significantly lags behind the response of the gas temperature, heat is transferred from the gas to the turbine material. Reducing the load by closing the fuel valve causes a portion of the thermal energy stored in the turbine blade material to be transferred to the working medium, causing a reduction of the turbine blade metal temperature. In case of aero-gas turbine engines, where acceleration and deceleration is the routine operational procedure, the turbine operates under diabatic conditions. A similar diabatic process is encountered in uncooled compressor components. However, the temperature differences are significantly lower. p

3

Inter-cooler mass flow

4

CC om pre ssio n

2

2

IP-

I

HP -C

1

O

Co mp res s

Inter-cooler Compressor mass flow

p

4 ion

h

p

3

p 1

s IP-Compressor

HP-Compressor

Fig. 13.10: Modular configuration of an intermediate pressure compressor, intercooler and a high pressure compressor, h-s-diagram of inter-cooling process The same cooling and simulation principle can be applied to compressor stages. However, it is a common practice for compressor trains consisting of several compressor components with a high pressure ratio to intercool the mass flow. The process of intercooling serves to substantially reduce the temperature of the working medium. Figure 13.10 shows the modular configuration of an intermediate pressure compressor (IP-C), an intercooler (IC), and a high pressure compressor (HP-C). The h-s diagram shows the two-stage compression process with intercooling. After the compression in IP-C, the working medium is intercooled to the initial temperature and is compressed in HP-C to arrive at the final pressure. Usually, the intercooling process is applied to industrial processes, where compression ratios above are required. For higher compression ratios, it is necessary to apply

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13 Generic Modeling of Turbomachinery Components and Systems

more than one intercooler. In this case the compression-cooling process is repeated. It should be noted that compressing to high pressure ratios without intercooling results in a substantial efficiency drop. Intercooling is also applied to Compressed Air Energy Storage plants (CAES, see Chapter 18), where the working medium air is compressed to pressure levels close to 70 bar and is stored in underground caverns. The high pressure air allows operating the CAES-gas turbine to operate during the period of high electric power consumption demand.

13.1.6 Group 5: Control System, Valves, Shaft, Sensors This group of modules consists of shafts, sensors, control systems, and fuel and load schedule generators, shown in Fig. 13.11.

Fig. 13.11: Schematics of the modules: control system, valve, shaft, and sensor

The control system is a major module that controls the entire engine. Input information may contain rotor speed transferred by the speed sensors, turbine inlet temperature, pressure, and compressor inlet and exit pressures and temperatures. This information may trigger closing or opening of the fuel valves and several bypass valves, and can perform other control functions, such as the adjustment of compressor stator blades to perform active surge protection.

13.2 System Configuration, Nonlinear Dynamic Simulation From the conceptual dynamic simulation point of view, the previous survey has led to the practical conclusion that any arbitrary aircraft or power generation gas turbine engine and its derivatives, regardless of configuration, i.e., number of spools and components, can be generically simulated by arranging the modules according to the

13 Generic Modeling of Turbomachinery Components and Systems

353

engine of interest. The nonlinear dynamic simulation method discussed in this and the next following chapters, is based on this modular configuration concept, and is a generic, modularly structured computational procedure that simulates the transient behavior of individual components, gas turbine engines, and their derivatives. The modules are identified by their names, spool numbers, and the inlet and outlet plena.

Control System, Rotor Dynamics

Simulation Schematic of the Power Genaration Gas Turbine Engine BBC-GT-9 , Module Addressing

BV2 FV

BV 1 1

N1

3

2

4

1

CC

5

NS 1

S C

1

C

2

C

6

7

8

10

9

11

D1

G P

2

3

T

1

T

2

T

3

T

4

P

1

BVi Ci CC Di FT

= Bypass valve = i th compressor stage group = Combustion Chamber = Exit diffuser = Fuel tank

GT9-TST

F Vi G N NS Si Ti

= Fuel valve = Generator Load = Inlet nozzle = Speed sensor = Shaft th i turbine stage

= Signal flow = Air flow = Gas flow = Fuel flow

Fig. 13.12: Modular configuration of the gas turbine presented in Fig.13.1

This information is vital for automatically generating the system of differential equations that represent individual modules. Modules are then combined into a complete system that correspond to the system configuration. Each module is physically described by the conservation laws of thermo-fluid mechanics, which result in a system of nonlinear partial differential or algebraic equations. Since an engine consists of a number of components, its modular arrangement leads to a system containing a number of sets of equations. The above concept can be systematically applied to any aircraft or power generation gas turbine engine.The modular concept allows the choice of generating a wide variety of simulation cases from single spool, power generation engines to twin, and multi spool thrust or combined thrust-power generation engines. The following three examples explain the modular design and simulation capability. To simulate the gas turbine engine shown in Fig. 13.1, first a modular representation is constructed for the purpose of the module addressing and plena assignments. Figure 13.12 represents the schematic simulation scheme for this engine. The first plenum numbered 1, is open to the atmosphere, and is the entrance to the engine. The entrance to the compressor is represented by a inlet nozzle N1. The compression system is divided into three compressors, the low pressure compressor stage group, C1, which is assigned plenum 2 as the inlet plenum and plenum 3 as the exit plenum, the intermediate pressure stage group, C2, and the high pressure stage group, C3. Upon exiting the first compressor,

354

13 Generic Modeling of Turbomachinery Components and Systems

a portion of the flow is removed via the second pipe of the engine system, P2, and is assigned plenum 3 as its inlet plenum, and plenum 7 as its exit plenum. The temperature of the flow at this point is relatively low and the second pipe mass flow serves for the turbine blade internal cooling. A portion of the flow is removed upon exiting the high pressure compressor, C3, via the second pipe, P2, and is transferred to the turbine inlet without passing through the combustion chamber. Fuel is injected into the combustion chamber at a rate which is given in the input data. The combustion gas then exits the combustion chamber, is mixed with the exiting flow from the third pipe, P2, for the purpose of temperature control, and enters the turbine which resides between plena 6 and 10. The exit diffuser, D1, leads to plenum 11 which is also open to the atmosphere. The control system input variables are the rotational speed of the shaft and the turbine inlet temperature. Output from the control system is the opening and closing of the fuel valve to control fuel flow into the combustion chamber (Fig.13.12). Each component corresponds to a simulation module consisting of a main routine and several subroutines, which determine the complete thermo-fluid dynamic profile of the component during the transient. Figure 13.13 schematically exhibits a twin spool aircraft engine with its derivatives. The base engine (middle) is modified to construct a ducted bypass engine, where the exit jets are not mixed (top left) and an unducted bypass engine (bottom left). The same base engine also serves to design a power generation and a turboprop engine. While in the case of the power generation example a third shaft has been added, in the turboprop example, the low pressure turbine drives the propeller. Figure 13.13 presents an appropriate example of a generic, modularly structured, component Twin -Spool Fan Engine Ducted Bypass, Non-Mixed Exhaust

JT-2SDER M.T.Schobeiri

Twin-Spool Fan Engine Unducted Bypass Mass Flow

Twin-Spool Engine with a Free Power Shaft

Twin-Spool Core Engine

Twin-spool Turboprop Engine

Fig. 13.13: A twin spool base engine (middle) with different derivatives

13 Generic Modeling of Turbomachinery Components and Systems

355

and system modeling. The schematic of the base engine is shown in Fig. 13.14. Two shafts are defined as S1 and S2. The flow passes through the engine in the following manner: first its velocity is reduced and its pressure is increased in the inlet diffuser (D1) which is between plena 1 and 2. The low pressure compressor group (C11 C15) will then further increase the flow pressure. Next, the high pressure compressor group (C21 - C25), which resides on the second shaft, S2, further raises the pressure. Fuel is injected into the combustion chamber (CC1), which is located between plena 13 and 14, thus, raising its temperature. High pressure and temperature gas is expanded through the HP-turbine group (T21 - T23) which is resided on the second shaft (S2) that drives the HP-compressor. Due to high thermal loading on the blades, the Hp-turbine requires cooling. The combustion gas further expands through the LPturbine group (T11 - T13) that drives the LP-compressor before it is exhausted into the atmosphere through the exit nozzle (N1), which is between plena 21 and 22.

FT

BV FV1

BV1

Control System

8

9

10

11

12

13

CC1

14

15

16

T21

T22

T23

18

19

20

17

S2 C21

1

3

2

C23

C22

C24

5

4

C25

6

7

21

22

S C

BV = Bypass valve C = j compressor stage of i spool CC = Combustion Chamber D = Diffuser FT = Fuel tank

C12

C13

F Vi Ni NS Si Ti j

C14

= = = =

C15

Fuel valve Nozzle Speed sensor Shaft th j turbine stage of ithspool

T

T12

T13

= Signal flow = Air flow = Gas flow = Fuel flow

JT-2S2S1

Fig. 13.14: Modular configuration of the twin spool core engine of Fig.13.13 As shown in Fig. 13.14, signal flow from each module to the control system is represented by dashed lines, air flow is represented by single lines, gas flow is represented by the double lines, and fuel flow is represented by thick single lines. An example of a modular composition of a helicopter gas turbine engine charged with a wave rotor is shown in Fig. 13.15. Compared to a single spool engine, Fig.13.15 reveals two new features: (1) a wave rotor super charger component is installed between the compressor and the combustion chamber that further increases the compression ratio leading to higher thermal efficiency, and (2) a propeller is driven by the spool shaft, whose rotational speed is reduced by a gear transmission.

356

13 Generic Modeling of Turbomachinery Components and Systems

7

3 10

6 P1

V1 2

Control System, Rotor Dynamics

4

Wave Rotor P3

3

P2

Propeller

6

8

Fuel tannk

S3 FV

Gear N

N

S

1

11

D

S

SCWAV1-SCH

C1

S1

T1

1

BVi = Bypass valve Ci = i t h compressor stage CC = Combustion Chamber D i = Diffuser FT = Fuel tank

T1

T2

T2

9

F V = Fuel valve L = Generator Loasd NS = Speed sensor S i = Shaft th Ti i turbine stage

2

S2

N

12

= Signal flow = Air flow = Gas flow = Fuel flow

Fig. 13.15: Modular configuration of a helicopter gas turbine engine supercharged by a wave rotor

13.3 Configuration of Systems of Non-linear Partial Differential Equations Each component of the above modular system configuration is completely described by a set of partial differential equations and algebraic equations. These modules are connected via plena that are described by two differential equations. Thus, the modular system configuration consists of a set of differential equation systems. The solution of these equations depend on the initial and boundary conditions that trigger the nonlinear dynamic event. This and other related issues will be comprehensively discussed in the following chapters.

References 1. Schobeiri, M.T., Abouelkheir, M., Lippke, C.: GETRAN: A Generic, Modularly Structured Computer Code for Simulation of Dynamic Behavior of Aero-and Power Generation Gas Turbine Engines. ASME Transactions, Journal of Gas Turbine and Power 1, 483–494 (1994) 2. Schobeiri, T.: A General Computational Method for Simulation and Prediction of Transient Behavior of Gas Turbines. ASME-86-GT-180 (1986) 3. Schobeiri, M.T.: Aero-Thermodynamics of Unsteady Flows in Gas Turbine Systems. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-51 (1985) 4. Schobeiri, M.T.: COTRAN, the Computer Code for Simulation of Unsteady Behavior of Gas Turbines. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-53-Technical Report (1985b)

14 Modeling of Inlet, Exhaust, and Pipe Systems

14.1 Unified Modular Treatment This chapter deals with the numerical modeling of the components pertaining to group 1 discussed in section 13.1.1 The components pertaining to this category are the connecting pipes, inlet and exhaust systems, as shown in Fig. 14.1.

Fig. 14.1: Component and modular representation of inlet nozzle N, connecting pipe P, and Diffuser D The function of this group consists, among other things, of the transportation of mass flow, and of converting the kinetic energy into potential energy and vice versa. Their geometry differs only in the sign of the gradient of the cross section in streamwise direction . For , flow is accelerated for subsonic and decelerated for supersonic Mach numbers. On the other hand, for , the flow is decelerated for subsonic and accelerated for supersonic Mach numbers.

14.2 Physical and Mathematical Modeling of Modules For modeling ducts with the varying cross sections, we apply the conservation laws derived in Chapter 12 ([1], [2], [3], [4]). The temporal change of the density at position k, shown in Fig. 14.2, is determined from Eq. (12.26) as is given below:

358

I

14 Modeling of Inlet, Exhaust, and Pipe Systems

s i-1

si

s i+1

x Diffuser: dS/dx > 0

O

I

k-1

k

i+1

x Pipe: dS/dx = 0

Fig. 14.2: Modeling the components diffuser

O

I

s i-1

si

s i+1

O

x Nozzle: dS/dx < 0

, pipe

and nozzle

(14.1) The mass flows and at stations the momentum equation (12.33):

and

are determined in conjunction with

(14.2)

The energy equation in terms of total pressure Eq. (12.54) in the absence of heat addition is modified as follows:

(14.3)

For a constant cross-section, the equation of continuity (14.1) and motion (14.2) are written respectively as:

14 Modeling of Inlet, Exhaust, and Pipe Systems

359

(14.4)

(14.5)

Similarly, the equation of energy in terms of total pressure is simplified to:

(14.6)

Equations (14.4), (14.5), and (14.6) describe the transient process of a compressible flow within a tube with a constant cross section. For an incompressible flow Eq. (14.5) can be reduced to a simple differential equation:

(14.7)

The friction coefficient can be determined from the known steady-state condition, where the temporal changes of the mass flow are set equal to zero. The indices 1 and n refer to the first and last station of the component. The solution of Eqs. (14.1) (14.3) and (14.4) - (14.7) can be found using the implicit integration method discussed later. However, considerable calculation speed is reached if the component under investigation is subdivided into several subsections that are connected to each other via plena. In this case, for each discrete section, the mass flow can be considered as spatially independent, which leads to further simplification of the above equations.

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14 Modeling of Inlet, Exhaust, and Pipe Systems

14.3 Example: Dynamic Behavior of a Shock Tube Simulation of a high frequency compression-expansion process within a shock tube is an appropriate example to demonstrate the nonlinear dynamic behavior of the above components. The shock-expansion process within shock tubes has been the subject of classical gas dynamics for many decades ([5], [6], [7],[8]). With the introduction of fast response surface mounted sensors, shock tubes have gained practical relevance for calibrating the high frequency response pressure and temperature probes. In classical gas dynamics, shock-expansion process are treated using the method of characteristics. Results of studies presented in [9] show substantial disagreement between calculations using method of characteristics and experiments. In this and the subsequent chapters, we simulate the dynamic behavior of each individual component using the simulation code GETRAN [2]. In GETRAN, the system of non-linear differential equations is solved using the implicit solution method described in Chapter 12. The shock tube under investigation has a length of L=1m and a constant diameter D = 0.5m. The tube is divided into two equal length compartments separated by a membrane. The left compartment has a pressure of , while the right one has a pressure of . Both compartments are under the same temperature of . The working medium is dry air, whose thermodynamic properties, specific heat capacities, absolute viscosity, and other substance quantities change during the process and are calculated using a gas table integrated in GETRAN. The pressure ratio of 2 to 1 is greater than the critical pressure ratio and allows a shock propagation with the speed of sound. Two equivalent schemes can be used to predict the compression-expansion process through the tube. These are shown in Fig. 14.3. In schematic (a), each half of the tube is subdivided into 10 equal pieces. The corresponding coupling plane 1 though 11, and thus, the left half of the tube are under pressure of 100 bar, while the right half with the plenums 12 through 21are under the pressure of 50 bar. L /2

L /2

p = 100 bar , T = 400 K L L

p = 50 bar , T = 400 K R R Valve simulating the membrane

Membrane 1

P1

2

P2

3

P3

4

P10 11

12 P12 13 P13 14 P19 19 P20 21

Valve ramp unraptured

raptured

t=0

Fig. 14.3: Simulation schematics for a shock tube: the physical tube (top), and the simulation schematics (a) and (b)

14 Modeling of Inlet, Exhaust, and Pipe Systems

361

The membrane is modeled by a throttle system with a ramp that indicates the cross sectional area shown underneath the throttle. The sudden rupture of the membrane is modeled by a sudden jump of the ramp. The schematic (b) offers a simpler alternative. Here, as in case (a), the tube is subdivided into 20 pieces that are connected via plenums 1 to 21.

14.3.1 Shock Tube Dynamic Behavior Pressure Transients: The process of expansion and compression is initiated by suddenly rupturing the membrane. At time t = 0, the membrane is ruptured which causes strong pressure, temperatures, and thus, mass flow transients. Since the dynamic process is primarily determined by pressure, temperature, and mass flow transients, only a few representative results are discussed as shown in Figs. 14.4 through 14.9. Figure 14.4 shows the pressure transients within the left sections 1 to 9. As curve 9 shows, the section of the tube that is close to the membrane reacts with a steep expansion wave. On the other hand, the pressure within the pipe section ahead of the shock, Fig. 14.5, curve 11, increases as the shock passes through the section. Oscillatory behavior is noted as the shock strength diminishes. The pipe sections that are farther away from the membrane, represented by curves 7, 5, 3, and 1 on the left and curves 13, 15, 17, and 19 on the right section, follow the pressure transient with certain time lags. Once the wave fronts have reached the end wall of the tube, they are reflected as compression waves. The aperiodic compression-expansion process is associated with a propagation speed which corresponds to the speed of sound. The

110 100

p(bar)

90 9

80

7

3 5 1

Left section

70 60 50 40

0

0.002

0.004

t(s)

0.006

0.008

Fig. 14.4: Pressure transients within the shock tube. Left section includes all tube sections initially under high pressure of 100 bar, while the right section include those initially at 50 bar.

362

14 Modeling of Inlet, Exhaust, and Pipe Systems

110

Right section

100

p(bar)

90 80 70 11 13 17 15 19

60 50 40

0

0.002

0.004

t(s)

0.006

0.008

Fig. 14.5: Pressure transients within the shock tube. Right section includes all tube sections initially under high pressure of 50 bar, while the left section includes those initially at 100 bar. expansion and compression waves cause the air, which was initially at rest, to perform an aperiodic oscillatory motion. Since the viscosity and the surface roughness effects are accounted for by introducing a friction coefficient, the transient process is of dissipative nature. This pressure rise is followed by a damped oscillating wave that hits the opposite wall and reflects back with an initially increased pressure followed by a damped oscillation Temperature Transients: Figure 14.6 shows the temperature transients within the left sections 1 to 9. As curve 9 shows, the section of the tube that is close to the membrane reacts with a steep temperature decrease. The pipe sections that are farther away from the membrane, represented by curves 7, 5, 3, and 1 on the left and curves 13, 15, 17, and 19 on the right section, follow the temperature transient with certain time lags. Once the shock waves have reached the end wall of the tube, they are reflected as compression waves where the temperature experience a continuous increase. Slightly different temperature transient behavior of the right sections are revealed in Fig. 14.7. Compared to the temperature transients of the left sections, the right sections temperature transients seem to be inconsistent. However, a closer look at the pressure transients explains the physics underlying the temperature transients. For this purpose we consider the pressure transient curve 11, in Fig. 14.5. The location of this pressure transient is in the vicinity of the membrane’s right side with the pressure of 50 bar. Sudden rapture of the membrane simulated by a sudden ramp (Fig. 14.2) has caused a steep pressure rise from 50 bar to slightly above 80 bar.

14 Modeling of Inlet, Exhaust, and Pipe Systems

363

420 400 380 T(K)

Left section

9 7 5 31

360 340 320 300

0

0.002

0.004

t(s)

0.006

0.008

Fig. 14.6: Temperature transients within the left sections of the tube. Left and right sections includes all tube sections initially under temperature of 400 K.

500 480

17

460 T(K)

440

19

Right section

15

420 13

400 380

11

360 340

0

0.002

0.004

t(s)

0.006

0.008

Fig. 14.7: Temperature transients within the right sections of the tube. Left and right sections includes all tube sections initially under temperature of 400 K. The pattern of the pressure transients is reflected in temperature distribution, where the pressure rise causes a temperature increase and vice versa. The temperature transients at downstream locations 12 to 20 follow the same trend.

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14 Modeling of Inlet, Exhaust, and Pipe Systems

Mass Flow Transients: Figures 14.8 and 14.9 show the mass flow transients within the left and the right section of the tube. The steep negative pressure gradient causes the mass contained within the tube to perform aperiodic oscillatory motions. During the expansion process, curve 1, mass flows in the positive x-direction. It continues to stay positive as long as the pressure in individual sections are above their minimum. This means that the shock front has not reached the right wall yet. Once the shock

300 9 7

3 1

5

0

x

m(kg/s)

150

-150 -300

0

0.002

0.004 t(s)

0.006

0.008

Fig. 14.8: Mass flow transients within left section of shock tube

300

13 11 15

0

17 19

x

m(kg/s)

150

-150 -300

0

0.002

0.004 t(s)

0.006

0.008

Fig. 14.9: Mass flow transients within the shock tube. The right section includes all tube sections initially under high pressure of 50 bar, while the left section includes those initially at 100 bar.

14 Modeling of Inlet, Exhaust, and Pipe Systems

365

front hits the right wall, it is reflected initiating a compression process that causes the mass flows in the negative x-direction. Figures 14.8 through 14.9 clearly show the dissipative nature of the compression and expansion process that results in diminishing the wave amplitudes and damping the frequency. The degree of damping depends on the magnitude of the friction coefficient cf that includes the Re-number and surface roughness effects. For a sufficiently long computational time, the oscillations of pressure, temperature, and mass flow will decay. For cf = 0, the aperiodic oscillating motion persists with no decay.

References 1. Schobeiri, T.: A General Computational Method for Simulation and Prediction of Transient Behavior of Gas Turbines. ASME-86-GT-180 (1986) 2. Schobeiri, M.T., Abouelkheir, M., Lippke, C.: GETRAN: A Generic, Modularly Structured Computer Code for Simulation of Dynamic Behavior of Aero-and Power Generation Gas Turbine Engines, an honor paper. ASME Transactions, Journal of Gas Turbine and Power 1, 483–494 (1994) 3. Schobeiri, M.T., Attia, M., Lippke, C.: Nonlinear Dynamic Simulation of Single and Multi-spool Core Engines, Part I: Theoretical Method. AIAA, Journal of Propulsion and Power 10(6), 855–862 (1994) 4. Schobeiri, M.T., Attia, M., Lippke, C.: Nonlinear Dynamic Simulation of Single and Multi-spool Core Engines, Part II: Modeling and Simulation Cases. AIAA Journal of Propulsion and Power 10(6), 863–867 (1994) 5. Prandtl, L., Oswatisch, K., Wiegarhd, K.: Führer durch die Strömunglehre, 8th edn. Vieweg Verlag, Braunschweig (1984) 6. Shapiro, A.H.: The Dynamics and Thermodynamics of Compressible Fluid Flow, 1954th edn., vol. I. Ronald Press Company, New York (1954) 7. Spurk, J.: Fluid mechanics. Springer, New York (1997) 8. Becker, E.: Gasdynamik. Teubner Studienbücher Mechanik, Leitfaden der angewandten Mathematik und Mechanik. Teubner, Stuttgart (1969) 9. Kentfield, J.A.C.: Nonsteady, One-Dimensional, Internal, Compressible Flows, Theory and Applications. Oxford University Press, Oxford (1993)

15

Modeling of Recuperators, Combustion Chambers, Afterburners

This category of components includes recuperators, preheaters, regenerators, intercoolers, and aftercoolers. Within these components the process of heat exchange occurs between the high and low temperature sides. The working principle of these components is the same ([1], [2], [3]). However, different working media are involved in the heat transfer process. More recently recuperators are applied to small and medium size gas turbine engines to improve their thermal efficiency. The exhaust thermal energy is used to warm up the compressor exit air before it enters the combustion chamber. A typical recuperator consists of a low pressure hot side flow path, a high pressure cold side flow path, and the wall that separates the two flow paths. A variety of design concepts are used to maximize the heat exchange between the hot and the cold side by improving the heat transfer coefficients. A cold side flow path may consist of a number of tubes with turbulators, fin pins, and other features that enhance the heat transfer coefficient. Based on the individual recuperator design concept, hot gas impinges on the tube surface in cross flow or counter flow directions. The working media entering and exiting the recuperator is generally combustion gas that exits the diffuser (hot side) and air that exits the compressor (cold side). Intercooler is applied for intercooling the compressor mass flow before it enters the next stage of compression. In contrast to recuperators, the working medium on the intercooler hot side is air that exits the intermediate pressure compressor before

Fig. 15.1: Schematics of a recuperator and an intercooler and their modular representation, recuperator heat exchange between low pressure hot combustion gas from exhaust and high pressure cold air from compressor

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

entering the high pressure compressor. On the low pressure side, the working medium is water that removes the heat from the compressed air before entering the high pressure compressor for further compression. Aftercooler is used to aftercool the high pressure working medium for storage purposes in order to reduce the specific volume, thereby increasing the amount of stored air mass. This procedure is used, among other things, in compressed air storage facilities. Figure 15.1 shows schematics of a recuperator and an intercooler and their modular representation.

Recuperator LP-In

Qh

I

T(1)

T(1)

T(1)

O

Q HP-In

h

w

c

LP-Out

mh T(2)

T(2)

T(2)

h

w

c

T(i-1)

T(i-1)

T(i-1)

h

w

c

T(i)

T(i)

T(i)

h

w

c

T(i+1)

T(n-1)

T(n)

T(i+1)

T(n-1)

T(n)

T(i+1)

T(n-1)

T(n)

h

w

c

h

w

c

O

h

w

I

c

mc

c

HP-out

X

1

i-1

i

i+1

n

LP=Low pressure side HP=High pressure side c=cold side h=hot side Fig. 15.2: A schematic segment of a heat exchanger with heat transfer, temperature distribution inlet and outlet plena

15.1 Modeling Recuperators Within this component, the heat exchange occurs between the high temperature low pressure combustion gas, and low temperature high pressure air. A schematic representation of a counter flow heat exchanger is given in Fig. 15.2. As shown, the heat exchange zone between the inlet, and exit plena on the hot, and cold sides is divided into (n-1) sections. The flow direction as well as the direction of the heat flow is shown in Fig.15.2. Since both the cold and hot gas streams have low subsonic velocities, they can be considered as incompressible, resulting in . For the sake of unifying

15 Modeling of Recuperators, Combustion Chambers, Afterburners

369

treatment, we adopt the positive sign for heat transfer whenever heat is added to the wall. Furthermore, no cross section changes occur, so that . These specifications lead to a significant simplification of Eq. (12.33) and (12.52) that described the mass flow and temperature transients on the hot side.

15.1.1 Recuperator Hot Side Transients Starting from the equation of motion, the above simplifications lead to the following relations that describe the mass flow and temperature transients on the hot side. The mass flow dynamics is obtained by rearranging Eq. (12.34):

(15.1)

The temperature transients are calculated by rearranging Eq. (12.52) as follows:

(15.2)

The quantities in Eq. (15.2) with the subscript k refer to the averages as in Eq. (12.55).

15.1.2 Recuperator Cold Side Transients The dynamic behavior of the recuperator cold side is determined by the equations of motion and energy, adopting the same simplification as for the flow on the hot side. The mass flow transient is calculated by rearranging Eq. (12.34) for the cold side as:

(15.3)

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

The temperature transients are calculated by rearranging from Eq. (12.52) as:

(15.4) The subscripts h and c in Eqs. (15.1), (15.2) , (15.3), and (15.4) refer to the hot and cold sides, respectively. The subscripts i, i+1 refer to computational stations in Fig. 15.2. The quantities with the subscript k are averaged between i and i+1. Equations (15.1), (15.2), (15.3), and (15.4) describe the mass flow transients as well as the temperature transients within volume elements of the width . The thermal energy flow portions and assume positive (heat addition) or negative signs (heat rejection) based on the direction of the heat flow that pertains to the individual volume elements. They are calculated using the heat transfer coefficient and the temperature difference: (15.5) In Eq. (15.5), temperature and

are the averaged heat transfer coefficients,

the mean

the contact surfaces on the cold and hot sides, respectively.

The mean temperatures are: (15.6) The subscripts S and  in Eq. (15.6) refer to the surface temperature and the temperature outside the boundary layer, respectively.

15.1.3 Coupling Condition Hot, Cold Side The coupling condition between the cold and hot side is provided by the material temperature differential equation: (15.7) where the and assume negative values if the heat is rejected. For the special steady state case, Eq. (15.7) is reduced to: (15.8)

15 Modeling of Recuperators, Combustion Chambers, Afterburners

371

15.1.4 Recuperator Heat Transfer Coefficient The heat transfer coefficients on both sides are determined by using Nusselt based correlations: (15.9) Kays and London [4] give a summary of empirical correlations for heat exchangers with different geometries. A more detailed explanation of the heat transfer process within different exchangers is found in Hausen [5]. The Re-number in Eq. (15.9), which must be individually determined for each side, is determined using the individual mass flow and hydraulic diameter (15.10) With Eq. (15.9) and (15.10), the Statnon number can be calculated that determines the heat transfer coefficient: (15.11) From Eq. (15.11), the individual heat transfer coefficient

is calculated as: (15.12)

with as the specific heat at constant pressure and the cross section. To account for the thermal resistance on the cold and hot side of the recuperator, the surface temperatures in Eqs. (15.5) and (15.6) must be replaced by the wall mean temperature. This is done by introducing the combined thermal resistance into Eq. (15.5)

(15.13)

In most of the recuperators’ high pressure sides, the working medium passes through a number of tubes. It is appropriate to approximate the thermal resistance and by an equivalent cylindrical pipe resistance. For determining the material temperature transients, Eq. (15.13) is inserted into (15.7). With the above equations and the following boundary conditions (15.14)

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

(15.14) the transient behavior of the recuperator is completely described.

15.2 Modeling Combustion Chambers A generic representation of three different types of combustion chambers used in aero- and power generation gas turbine engines is given in Fig. 15.3. Key design features of gas turbine combustion chambers are given in [6]. While the annular (a) and tubular type (b) combustion chambers are encountered in both aircraft and power generation gas turbines, the single cylindrical silo type (c) is utilized only in power generation gas turbine engines. All three types, regardless of their size and application fields, share the following features: (1) Primary combustion zone, (2) secondary air zone for protecting the combustion chamber casing against excessive flame radiation temperature, (3) mixing zone, where the hot gas and the rest of the secondary air are mixed, and (4) the fuel/air inlet nozzle. Since these features are common to the above, and almost all other types of combustion chambers, a generic module can be designed that entails the above features. Figure 15.4 exhibits such a generic module. Figure 15.4 is a modular representation of a gas turbine component.

Fig. 15.3: Three different types of combustion chambers utilized in aircraft and power generation gas turbines. (a) Annular type, (b) tubular type, (c) heavy duty, single cylindrical power generation (BBC)

15 Modeling of Recuperators, Combustion Chambers, Afterburners

373

It consists of a primary combustion zone, or primary zone, surrounded by n rows of segments, the secondary air zone, and the mixing zone. The actual combustion process occurs in the primary zone. The secondary air zone separates the hot primary combustion zone from the combustion chamber casing. The rows of segments in the combustion zone are subjected to a severe thermal loading due to direct flame radiation. Film and/or convection cooling on both the air and the gas sides cools these segments. The air required to cool the hot segments flows through finned cooling channels, thereby contributing to the convection cooling of the segments on the air side. The cooling air flow exiting from the jth segment row affects the film cooling process on the gas side within the boundary layer in the next row of segments. At the end of that process, the cooling air mass flow is mixed completely with the primary air mass flow. The mass flow relationships prevailing in the primary, the cooling, and the mixing zones are substituted into the energy equation, already formulated. Their effect is significant, particularly in the case of energy balance, because they determine temperature distribution in the individual combustion chamber stations. For that reason, we first determine the mass flow relationships and then deal with the energy balance.

Fig. 15.4: Generic modular representation of a combustion chamber

15.2.1 Mass Flow Transients The combustion chamber mass flow is divided into primary mass flow, secondary (cooling) mass flow, and mixing mass flow, as shown in Fig. 15.4.

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

(15.15) In Eq. (15.15), , and denote the primary, secondary, and mixing mass flow ratios, respectively. If the primary zone consists of n rows of segments, the cooling mass flow for the row of segments is: (15.16) Considering the fuel/air ratio, segments is:

, the mass flow within the

row of the

(15.17)

To determine the transient behavior of a combustion chamber that has already been designed, it is necessary to start from the given ratios , , , and . For a new design, it is possible to vary these ratios until the desired solution is attained that fulfils the required material temperature conditions dictated by the local heat transfer. The mass flow in the combustion chamber is obtained as the solution of the modified version of Eq. (12.34):

(15.18)

In Eq. (15.18), the subscripts I and O stand for the inlet and outlet plena, respectively. The volume of the combustion chamber is replaced here with an equivalent volume, which is the product of a constant cross section S and the characteristic length . Thus, the pressures and temperatures in Eq. (15.18) represent the inlet and outlet quantities, which must be known at the design point.

15.2.2 Temperature Transients To determine the temperature transients for gas within the primary combustion zone and air within the secondary zone, we start from Eq. (12.52). Considering air, fuel, and eventually, water as the main combustion constituents, Eq. (12.52) needs to be rearranged as follows:

15 Modeling of Recuperators, Combustion Chambers, Afterburners

375

(15.19)

with as the gas side heat flow. The index i refers to the computation station in question. The mixing components are identified with the sequential index k, the upper summation limit for which K represents the number of constituents involved in a mixing and combustion process. The mixing components at the inlet station are the cooling air from secondary zone, the combustion air, and the fuel. For the cooling zone, Eq. (12.52) yields: (15.20) with as the air side heat flow. The temperature distribution within the segment material can be determined as follows using the heat conductance equation: (15.21) with and as the heat flows supplied and carried from the combustion chamber segment, respectively. For the primary zone surrounded by a row of segments, is made up of the fuel heat, , the flame radiation heat, , and the convection heat, , on the gas (or hot) side: (15.22) The increased temperature level produced in the segments from the direct flame radiation is reduced to an acceptable level by an intensive film cooling on the gas side and convective heat removal on the air side. The segments are therefore subjected to the following thermal loading on the gas side (or hot side): (15.23) with standing for the amount of heat carried off by the film cooling. The heat flow carried off on the air side (cold side), , consists of a convection and a

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

radiation component. The latter is due to the difference in temperature between the cold air enclosure liner and the hot surface of the fins. As a result we find: (15.24) For the film cooling: (15.25) with as the average heat transfer coefficient and the average film temperature. The convective heat removal on the air side is obtained from: (15.26) The calculation of the radiation heat flows section.

and

is explained in the following

15.2.3 Combustion Chamber Heat Transfer Heat transfer processes in combustion chambers involve several mechanisms. To distinguish these mechanisms, we consider a section of the heavy duty combustion chamber shown in Fig. 15.3 with the corresponding generic module Fig. 15.4. Different heat transfer mechanisms are shown in Fig. 15.5. Considering the combustion zone on the hot (or gas side) of a segment that separates the primary combustion zone from the secondary flow zone, as shown in Fig. 15. 5, the following heat transfer types are distinguished: (1) is added by

Fig. 15.5: Details of heat transfer radiation, conduction, and convection heat transfer along a generic segment of a gas turbine combustion chamber

15 Modeling of Recuperators, Combustion Chambers, Afterburners

377

the flame radiation (2) , is added to the segments by convection, (3) is removed from the gas side by film jets, (4) is removed from the segment on the air side (cold side) by convection, and (5) the conduction through the segment wall couples the heat transfer portions on both sides of the segment. Convective and conductive heat transfer types are discussed in Section 15.1. For the jet film cooling heat transfer, correlations by Juhasz and Marek [7] can be used. On the primary combustion side, the flame radiation is, by far, the major contributor to the radiation heat transfer. There is also radiation on the secondary air side between the segment material, enclosed secondary air, and the casing. However, its contribution compared to the flame radiation is negligibly small. The basis for the radiative heat transfer is the Planck’s [8] spectral distribution of radiation for an ideal radiating body, called black body given as:

(15.27)

In Eq. (15.27),

is the spectral emissive power of a black body as a function

of wave length and temperature of the emitting surface, is the wave length in microns (or meters), T the absolute temperature of the body, and the subscript b stands for black ideal body. The constants are and . Equation (15.27) is derived for an emitting surface in a vacuum, where the index of refraction is unity. This index is defined as the ratio of velocity of light in a vacuum to the velocity of light in a non-vacuum environment. The spectral distribution described by Eq. (15.27) is plotted in Fig. 15.6 with body temperature as a parameter. Equation (15.27) describes the radiation power for each given wave length with c as the speed of light and Ȟ wave frequency. For combustion chamber application, it is necessary to find the total energy emitted by the body. This can be found by integrating Eq. (15.27): (15.28)

with ı as the Stefan-Bolzmann constant. Equation (15.28) can be applied to two or more bodies that are in radiative interaction. Consider the simplest case of two parallel black surfaces that are at different temperatures. The net rate of heat transfer between these two surfaces is: (15.29) For radiative heat transfer, the flame shape may be approximated as a cylinder.

15 Modeling of Recuperators, Combustion Chambers, Afterburners 3

Emmisive Power EO,b(T) (W/m )

378

T=3000K 8E+06

2500K 2000K 1500K

4E+06

1000K 0

0

4

8

Wave length O (P)

12

Fig. 15.6: Spectral emissive power as a function of wave length with body temperature as parameter described by Planck’s Eq. (15.27)

15.3 Example: Startup and Shutdown of a Combustion Chamber-Preheater System The example deals with the startup and shutdown process of a generic system consisting of a combustion chamber, a pipe and a preheater as shown in Fig. 15.7.

Fig. 15.7: A system consisting of a combustion chamber, a preheater and a pipe

15 Modeling of Recuperators, Combustion Chambers, Afterburners

379

These components are parts of a Compresses Air Energy Storage (CAES) plant discussed in Chapter 18, Section 18.5.1.1. The combustion chamber is connected to plenum (1), which operates at a constant pressure and temperature. Constant air flow enters the combustion chamber which is connected to plenum (2). A major portion of the combustion gas passes through a pipe and exits the system at plenum (6). A minor portion of the combustion gas enters the hot side of the preheater and leaves it at plenum (3). The cold side of the preheater continuously receives cold air at constant pressure and room temperature from plenum (4) and exits the cold side preheater at plenum (5). The fuel valve is programmed to provide the combustion chamber with the fuel schedule . The fuel schedule shows the startup, normal operation and the combustion chamber shutdown, Fig. 15.8.

1.2

Normal operation

x x

n

0

do w

0.4

ut Sh

Startup

m/mD

0.8

0

100

200

t(s)

300

400

Fig. 15.8: Combustion chamber fuel schedule: Startup, design operation, shutdown, calculated by GETRAN

Similar to the cases discussed previously, each component of the above system represented by the corresponding module is described by a system of differential equations. Prescribing the following boundary conditions:

(15.30) the systems of the corresponding differential equations can be solved using an appropriate solver for stiff differential equations [1], [2].

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

Combustion Chamber Startup: Using the boundary conditions (15.30), we first simulate a cold start of the combustion chamber followed by its design point operation and the subsequent shutdown. This process is controlled by the fuel schedule shown in Fig. 15.8 with and as the actual and the design point fuel mass flow, respectively. Starting from a cold combustion chamber and preheater, cold air flows through the entire system for about 10 seconds. At , the fuel valve starts to open adding fuel continuously until the design fuel mass flow has been reached. During the fuel addition, the temperature of the entire system starts to rise. The combustion chamber gas temperature, Fig. 15.9, reacts to this event with a corresponding increase.

3000

Combustion chamber gas temperature

T(K)

2000 1: Core temperature

1000 2: Exit temperature 0

100

200

t(s)

300

400

Fig. 15.9: Combustion chamber gas temperature, startup, design operation, shutdown, calculated by GETRAN Curve 1 exhibits the temperature of the core flow within the primary combustion zone. Mixing with the secondary air significantly reduces the gas temperature to the exit temperature level of slightly above 1100 K, curve 2. Preheater Temperature Transient: The combustion gas enters the hot side of the preheater and increases its temperature as seen in Fig. 15.10. Curves 1 represents the gas temperature at the rear portion of the preheater. Heat is transferred from the hot combustion gas to the cold side air of the preheater by means of convection and conduction through the preheater metal raising its temperature. The metal temperature transient is shown in Fig. 15.10, curve 2. The heat transferred from the preheater metal to its air side causes the cold side air temperature to rise as seen in Fig. 15.10, curve 3.

15 Modeling of Recuperators, Combustion Chambers, Afterburners

381

800 1: Hot side gas temperature

T(K)

700 600

2: Metal temperature

500 3: Cold side exit air temperature 400 300

0

50

100

150

200

250

300

350

400

450

t(s) Fig. 15.10: Gas, metal and air temperature transients at the exit portion of the preheater Accurate prediction of the design and off-design dynamic behavior of the components discussed in this chapter requires the implementation of appropriate heat transfer coefficients. For design purposes, useful correlations can be found, among others in [4] and [5]. The heat transfer coefficient (Į) utilized for dynamic calculations in conjunction with the skin friction coefficient are expected to deliver the desired temperature and pressure distributions within these components. Is this not the case, Į can be calculated using steady state condition, where the temporal changes of plena temperature in Eq. (15.19) and (15.20) are set equal to zero. As a results, the amount of the heat transferred and thus the heat transfer coefficient are accurately calculated. This allows a correction of the heat transfer coefficient initially used for dynamic calculation.

15.4 Modeling of Afterburners This component, from a modeling point of view, exhibits a simplified version of the combustion chamber discussed in the previous section. This component, and its modular representation is shown in Fig. 15.11. The modeling of this component follows the same procedure as the combustion chamber. However, because of the absence of secondary flow, the mass flow ratios , , and have the following values: = 1, = 0, and = 0.

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15 Modeling of Recuperators, Combustion Chambers, Afterburners

Fig. 15.11: Schematic of an afterburner module

References 1. Schobeiri, T.: A General Computational Method for Simulation and Prediction of Transient Behavior of Gas Turbines. ASME-86-GT-180 (1986) 2. Schobeiri, M.T., Abouelkheir, M., Lippke, C.: GETRAN: A Generic, Modularly Structured Computer Code for Simulation of Dynamic Behavior of Aero-and Power Generation Gas Turbine Engines. ASME Transactions, Journal of Gas Turbine and Power 1, 483–494 (1994) 3. Schobeiri, M.T., Attia, M., Lippke, C.: Nonlinear Dynamic Simulation of Single and Multi-spool Core Engines, Part I, II: Theoretical Method, Simulation Cases. AIAA, Journal of Propulsion and Power 10(6), 855–867 (1994) 4. Kays, W.M., London, A.L.: Compact Heat Exchangers, 3rd edn. McGraw-Hill, New York (1984) 5. Hansen, H.: Wärmeübertragung im Gegenstrom, Gleichstrom und Kreuzstrom, 2nd edn. Springer, New York (1976) 6. Lefebvre, A.H.: Gas Turbine Combustion. Hemisphere Publishing Corporation (1983) 7. Marek, C.J., Huhasz, A.J.: Simulataneous Film and Convection Cooling of a Plate inserted in the Exhaust Stream of a Gas Turbine Combustor, NASA TND7689 (1974) 8. Planck, M.: Theory of Heat Radiation. Dover Publication, Inc, New York (1959)

16 Modeling the Compressor Component, Design and Off-Design

As mentioned in Chapter 1, the function of a compressor is to increase the total pressure of the working fluid. According to the conservation law of energy, this total pressure increase requires external energy input, which must be added to the system in the form of mechanical energy. The compressor rotor blades exert forces on the working medium thereby increasing its total pressure. Based on efficiency and performance requirements, three types of compressor designs are applied. These are axial flow compressors, radial or centrifugal compressors, and mixed flow compressors. Axial flow compressors are characterized by a negligible change of the radius along the streamline in the axial direction. As a result, comparison of the contribution of the circumferential kinetic energy difference to the pressure buildup is marginal. In contrast, the above difference is substantial for a radial compressor stage, where it significantly contributes to increasing the total pressure as discussed in Chapter 5. During the compression process, the fluid particles are subjected to a positive pressure gradient environment that may cause the boundary layer along the compressor blade surface to separate. To avoid separation, the flow deflection across each stage and thus the stage pressure ratio is kept within certain limits discussed in the following section. Compared to an axial stage, much higher relative stage pressure ratios at relatively moderate flow deflections can be achieved by centrifugal compressors. However, geometry, mass flow, efficiency, and material constraints place limits on utilizing radial compressors. Radial compressors designed for high stage pressure ratios and mass flows comparable to those of axial compressors require substantially larger exit diameters. This can be considered an acceptable solution for industrial applications, but is not a practical solution for implementing into gas turbine engines. In addition, for gas turbine applications, high compressor efficiencies are required to achieve acceptable thermal efficiencies. While the die efficiencies of advanced axial compressors have already exceeded 91.5% range, those of advanced centrifugal compressors are still below 90%. Power generation gas turbine engines of 10 MW and above as well as medium and large aircraft engines use axial compressor design. Small gas turbines, turbochargers for small and large Diesel engines have radial impellers that generate pressure ratios above 5. Compact engines for turboprop applications may have a combination of both. In this case a relatively high efficiency multi-stage axial compressor is followed by a lower efficiency centrifugal compressor to achieve the required engine pressure ratio at smaller stage numbers.

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16 Modeling the Compressor Component, Design and Off-Design

Further stage pressure buildup is achieved by increasing the inlet relative Mach number . In case of subsonic axial flow compressors with , the compression process is primarily established by diffusion and flow deflection. However, in the case of supersonic relative Mach number that occupies the entire compressor blade height from hub to tip, the formation of oblique shock waves followed by normal shocks as discussed in Chapter 4 substantially contributes to a major pressure increase. However, the increase of stage pressure ratio as a result of compression shocks is associated with additional shock losses that reduce the stage efficiency. To achieve a higher stage pressure ratio at an acceptable loss level, the compressor stage can be designed as transonic compressor stage. In this case, the relative Mach number at the hub is subsonic and at the tip supersonic, with transonic Mach range in between. Transonic compressor stage design is applied to the first compressor stage with a relatively low aspect ratio of high performance gas turbine engines. In this chapter, we first investigate several loss mechanisms and correlations that are specific to compressor component. Using these correlations, first the basic concept for a row-by-row adiabatic calculation method is presented that accurately predicts the design and off-design behavior of single and multi-stage compressors. With the aid of this method, efficiency and performance maps are easily generated. The chapter is then enhanced to calculate the diabatic compression process where the blade rows exchange thermal energy with the working medium and vice versa. The above methods provide three different options for dynamically simulating the compressor component. The first option is to utilize the steady state compressor performance maps associated with dynamic coupling. The second option considers the row-by-row adiabatic calculation. Finally, the third option uses the diabatic compression process. Examples are presented.

16.1 Compressor Losses In Chapter 7 we attempted to provide the basic physics of different loss mechanisms for accelerating and decelerating cascades from a unifying point of view. This section deals with particular loss mechanisms encountered in compressors. Because of its significance to aircraft gas turbine engines, manufacturers and research centers have been focusing their attention on developing compressor components with higher efficiency. To predict the compressor stage efficiency accurately, compressor designers often use loss correlations that reflect different loss mechanisms within the compressor stage flow field. In the early fifties, Lieblein and his co-workers [1], [2], [3], [4] and [5] conducted fundamental research in compressor cascade and stage aerodynamics. Their research work, NASA-Report SP-36 [6], is a guideline for compressor designers. Miller and Hartmann [7], Miller et al. [8], and Schwenk et al. [9] initiated their fundamental research on transonic compressors where they primarily investigated the shock losses. Gostelow et al. [10], Gostelow [11], Seylor

16 Modeling the Compressor Component, Design and Off-Design

385

and Smith [12], Seylor and Gostelow [13], Gostelow and Krabacher [14], Krabacher and Gostelow [15] and [16] focused their experimental research on single-stage high Mach number compressor stages. Their comprehensive experimental research includes the performance evaluation of several rotors. Monsarrat et al. [17] performed similar investigations on single-stage high Mach number compressor stages. Koch and Smith [18] presented a method for calculating the design point efficiency potential of a multi-stage compressor. Schobeiri [19] investigated the individual loss mechanisms that occur in an advanced compressor stage. He developed a new shock loss model, introduced a modified diffusion factor and re-evaluated the relevant experimental data published by NASA. Recent investigations by König et al. [20] investigate the loss and deviation angles for transonic blading. The total pressure losses encountered in an advanced compressor stage are: (1a) The blade primary losses generated by the wall shear stress, which is proportional to the local velocity deformation. Since the blade mid-section is not affected by the secondary flow that originates from the hub and casing, the primary losses are dominant. (1b) The trailing edge mixing losses are due to the thickness of the trailing edge and the boundary layer thicknesses on the suction and pressure surfaces that causes wake defect, mixing, and thus additional entropy increase. From an experimental point of view, these two losses are not separable since the total pressure measurements occur at a certain distance downstream of the trailing edge plane and inherently include the wake total pressure defect. The combination of these two losses is frequently called profile loss. (2) Shock losses are encountered in compressor stages with high transonic to supersonic inlet relative flow conditions. Based on an angle incidence and the shock position, these losses may generate considerable entropy increase that result in significant deterioration of the stage efficiency. The shock losses are approximately of the same order of magnitude as the profile losses. (3) Secondary losses due to the end wall boundary layer development and blade tip clearances. (4) Secondary flow losses are also present for compressor blades with shrouds. A comprehensive treatment of compressor losses is found in Schobeiri [19], [21], [22], [23] and Schobeiri and Attia [24]. This chapter focuses on three issues: (1) A new modified diffusion factor that describes the blade loading for the rectilinear and annular cascades, as well as for the entire compressor stage. This new diffusion factor, which includes the compressibility effects, allows the loss parameters to be systematically correlated with the diffusion factor. (2) A new shock loss model is presented that overcomes the weakness of the existing loss models described by Levine [25], Balzer [26] and Swan [27]. (3) The existing published data are re-evaluated and detailed correlations are presented.

16.1.1 Profile Losses Lieblein and Roudebusch [3]derived the expression for profile loss coefficient as a function of cascade geometry, flow angles, and the boundary layer parameters shown in Eq. (16.1):

386

16 Modeling the Compressor Component, Design and Off-Design

(16.1)

with the form factor H32 = f(H12). Among the boundary layer parameters in Eq. (16.1), the momentum thickness į2 is of primary importance. It gives a direct relationship between the separation point and the free-stream velocity gradient (or pressure gradient) shown in the following von Kàrmàn integral equation for incompressible flow:

(16.2)

with IJw as the wall shear stress and H12 the form parameter. For a highly loaded compressor blade, flow separation may occur, where the velocity profile starts to separate at the profile inflection point. Consequently, the wall shear stress at that point vanishes and Eq. (16.2) reduces to:

(16.3)

Equation (16.3) shows a direct relation between the blade velocity gradient and the momentum thickness. As an appropriate measure for the velocity gradient, Lieblein et al. [1] introduced the equivalent diffusion factor:

(16.4)

with Vmax as the maximum velocity on the suction surface shown in Fig.16.1. This velocity ratio, which changes by changing the flow deflection given by the velocity triangle, properly reflects the blade loading situation. However, it requires the knowledge of the maximum velocity at different flow deflections, which is not always given. From a compressor designer point of view, relating the blade loading to the actual velocity triangle is most appropriate.

16 Modeling the Compressor Component, Design and Off-Design

387

S V1

VS

b

V

max

1

E1

C

2

E2

V2

Fig. 16.1: Schematic representation of a compressor cascade flow

16.1.2 Diffusion Factor Using Fig. 16.1, we introduce the dimensionless parameters , approximate the dimensionless velocity distribution Ș by a polynomial

and

(16.5) The Taylor expansion in the near of ȟmax results in:

(16.6)

with ǻȟ = ȟ - ȟmax. Neglecting the higher order terms, Eq. (16.6) can reduce to:

(16.7)

388

16 Modeling the Compressor Component, Design and Off-Design

The velocity slope is found from Eq. (16.5): (16.8)

Incorporating Eq. (16.8) into Eq. (16.7) results in: (16.9) with the constants: and . The circumferential component of the force acting on the blade in Fig.16.1 is calculated by integrating the pressure distribution along the blade suction and pressure surfaces using the Bernoulli equation:

(16.10) Since only the contribution of the suction surface is considered for estimating the diffusion factor, the second term in the integrand may be set equal to zero. Incorporating Eq. (16.9) into Eq. (16.10) results in:

(16.11)

The force component T can also be calculated using the momentum equation in the circumferential direction: (16.12) with s and ȡ as the blade spacing and the flow density, Vm and Vu as the meridional and circumferential velocity components. Inserting the velocity components defined in Fig.16.1 into Eq. (16.12), it becomes:

(16.13) Equating Eq. (16.11) and (16.13) results in the following relationship: (16.14)

16 Modeling the Compressor Component, Design and Off-Design

389

with the coefficients Di as:

(16.15)

The solution of Eq. (16.14) after neglecting the higher order terms in D3 yields: (16.16) Equation (16.16) exhibits a special case of Eq. (16.14) and gives an explicit relationship between the maximum velocity Vmax and the cascade circulation function in the bracket. Using the NACA-65(A10) series and the circular arc blade C.4, for the optimum flow condition that causes a minimum loss denoted by (*), the constants in Eq. (16.16) are experimentally determined by Lieblein [5] as K1=0.6 and K2= 1.12: (16.17) Equation (16.14) is generally valid for any arbitrary inlet flow angle including the off-design incidence angles, Fig.16.2 . However, the compressor designer prefers to E *1

V1

(a)

V2

E1

V1

+i

(b)

E *1

+i

E1

V2

Fig. 16.2: Positive incidence angle: (a) definition by Lieblein [4], (b) based on the unified velocity diagram angle definition in this book

390

16 Modeling the Compressor Component, Design and Off-Design

relate the off-design to the design point . For this purpose, Lieblein [4] introduced an empirical correlation for for positive incidence angles as defined in Fig.16.2:

(16.18)

where a = 0.0117 for the NACA 65(A10) and a = 0.0070 for the C.4 circular arc blades. Equation (16.18) accurately estimates the maximum velocity ratio for a positive incidence. However, it cannot be used for negative incidence angles because of the rational exponent of the argument . To eliminate this deficiency, we replace Eq. (16.18) with the following polynomial: (16.19)

and neglect the terms with n > 2, Eq. (16.19) becomes: (16.20)

1.9

1.7

E-E *=8o 1.6

4

C l0 = 0

6o

12 16

8

1.7

4o 2o

1.5

Vmax /V1

Vmax /V1

0o

1.4

1.5

1.3 1.3 1.2

1.1

0

0.2

G

0.4

0.6

1.1 0

10

20

30

40

E -E *

Fig. 16.3: Velocity ratio (left) as a function of the circulation function G with the incidence angle i = ȕ - ȕ * as parameter, and (right) as a function of incidence angle i = ȕ - ȕ * with lift coefficient Cl0 as parameter, experiments represented by symbols taken from Lieblein

16 Modeling the Compressor Component, Design and Off-Design

391

Re-evaluating the experimental results obtained by Lieblein [4] leads to a1 = 0.746 and a2 = 6.5. Equation (16.20) enables the calculation of the velocity ratio and thus the diffusion factor for any off-design operation range. Fig.16.3 shows the results of Eq. (16.20), where the velocity ratio is plotted against the circulation function G with the incidence angle i = ȕ1 - ȕ*1 as a parameter. Compared with the Lieblein's correlation Eq. (16.18), the new correlation Eq. (16.20) yields more accurate results. This statement is also true for Fig.16.3, where the maximum velocity ratio is plotted versus the incidence angle with the lift coefficient Cl0 as parameter.

16.1.3 Generalized Maximum Velocity Ratio for Cascade, Stage Lieblein’s correlations for the maximum velocity ratio and their experimental verifications were based exclusively on the two-dimensional incompressible cascade flow situation. Significant effects such as compressibility and three-dimensionality were not considered. Furthermore, changes of axial velocity component and streamline curvature, which are always present in a modern compressor, were ignored. Finally, the effect of rotational motion on circulation was disregarded. This section includes the above-mentioned effects in the maximum velocity ratio by employing a generalized circulation concept that leads to a modified diffusion factor. Starting from the Kutta-Joukowsky’s lift equation (lift force/unit span) with ȡ, V as the free stream density, velocity, and ī as the circulation: (16.21) The circulation for an annular compressor cascade is shown in Fig.16.4. Using the definition in Fig.16.4, the circulation is expressed as: (16.22)

1

1 2

r1

Stator row

r2

2

2

r1

Rotor row

r2

s1

1 1

2 3

s

2

4

Row circulation

CAS-Radcir

Fig. 16.4: Unified station definition to circulate the compressor row circulation

392

16 Modeling the Compressor Component, Design and Off-Design

where Vu1 and Vu2 represent the circumferential velocity components at the inlet and exit and ī23 = -ī41. For a rectilinear cascade with constant height or a stator cascade with constant radius cylindrical streamlines, the spacings are equal at the inlet and exit, s1 = s2 = s. For a stator cascade with conical streamlines as discussed in Chapter 6, different spacings at the inlet and exit are present that relate to each other by the radius of the streamline curvature, and . As shown in Fig.16.4, the stator and the rotor row have identical station numbers as opposed to the ones defined in Chapter 6. Also for both rows the generic angle ȕ is used. As we see below, this convention allows developing unified equations for diffusion factor that can be applied to stator row as well as to rotor row by inserting the absolute and relative flow angles defined in Chapter 6. Using the following velocity ratios:

(16.23)

and defining the specific circulation function Ȗ, we obtain the following relations for the linear cascade ȖC, stator ȖS and rotor ȖR:

(16.24)

(16.25)

Replacing in Eq.(16.22) the circumferential components of the absolute velocities by the relative one and the rotational velocities at the same streamline, we have:

(16.26)

Equation (16.26) exhibits a generalized relation for the specific circulation function. As seen, to calculate the circulation, the absolute velocity components Vu1 and Vu2 are utilized and refer to the absolute circulation rather than the relative one. Special cases such as the linear cascade, stator with cylindrical streamlines, and stator with axisymmetric streamlines can follow immediately by setting:

16 Modeling the Compressor Component, Design and Off-Design

case 1: case 2: case 3: case 4:

u = 0, u = 0, u g 0, u g 0,

ij = , ij = , ij g , ij g ,

ȝ = 1, ȝ g 1, ȝ = 1, ȝ g 1,

393

Ȟ = 1(linear cascade, cylindrical stator) Ȟg 1(axisymmetric stator) Ȟ = 1(cylindrical rotor) Ȟ g 1(axisymmetric rotor)

Using the most general case (16.26), we obtain the circulation function for the rotor as:

(16.27)

With Eqs. (16.24)-(16.27), the maximum velocity ratio at the optimum point for linear cascade (C), stator (S), and rotor (R) are calculated from:

(16.28)

Correspondingly, we obtain the off-design maximum velocity ratio by using Eq. (16.20):

(16.29)

In Eqs. (16.28) and (16.29), the individual quantities denoted by the subscripts C, S, and R pertain to cascade, stator and rotor, respectively. Thus, the Lieblein’s equivalent diffusion factor defined in Eq. (16.4) becomes:

(16.30)

16.1.4 Compressibility Effect To consider the effect of compressibility on the maximum velocity ratio and thus on the diffusion factor, we modify the specific circulation function for the simplest case, namely a linear cascade, by introducing the inlet density ȡ1:

394

16 Modeling the Compressor Component, Design and Off-Design

(16.31) The second subscript c refers to compressible flow. The freestream density ȡ can be expressed in terms of the density at the inlet and a finite increase ȡ = ȡ1 + ǻȡ. Outside the boundary layer we assume a potential flow that is not influenced by small perturbations. With this assumption, the Euler equation combined with the speed of sound may be applied: (16.32) with C as the speed of sound. For small changes, the flow quantities can be related to the quantities at the inlet:

(16.33) We introduce the above relations into Eq. (16.32) and approximate the differentials by differences and neglect the higher order terms. After some rearranging we obtain the density changes by:

(16.34)

Introducing Eq. (16.34) into the relation

results in:

(16.35)

Implementing Eq. (16.35) into Eq. (16.31) obtains the specific circulation functions for linear cascade and stators with cylindrical streamlines:

(16.36) The expression in the above bracket reflects the Mach number effect on the specific circulation function. Accounting for the compressibility effects and using the same procedure applied above, the generalized circulation function for the rotor is obtained by:

16 Modeling the Compressor Component, Design and Off-Design

395

(16.37)

Equation (16.37) allows calculating the specific circulation function for cases 1 to 4. As seen from Eqs. (16.36) and (16.37) and comparing them with Eqs. (16.24) and (16.26), the specific circulation of compressible and incompressible flows are related to each other by the density ratio and thus the Mach number. Considering the simplest case, namely the linear cascade described by Eq. (16.36), because of the compression process with V2 < V1, the bracket representing the compressibility effect is always greater than unity. With Eq. (16.36) and the conditions for case 1 to case 4, the circulation functions for cascade, stator and rotor considering the compressibility effect are: (16.38)

Using Eq. (16.38) for optimum conditions, the velocity ratio for compressible flow is obtained from:

(16.39)

Introducing Eq. (16.39) into the relationship for the equivalent diffusion factor for the rotor as a generalized case, reads:

(16.40)

The angles used in the above equations correspond to those defined in Fig.16.1 and Fig.16.5. Equations (16.39) and (16.40) show the direct relationship between the maximum velocity ratio and the specific circulation function. Equation (16.40)

396

16 Modeling the Compressor Component, Design and Off-Design

inherently includes the compressibility effect and the actual and optimum flow angles. An alternative diffusion factor, which was first proposed by Smith [28], is: (16.41)

This relation has been accepted and widely used by compressor aerodynamicists. It includes the effect of the rotation in its third term.

Fig. 16.5: Velocity triangle, angle definition Using the angle definition in Fig.16.5 and the dimensionless parameters previously defined, the re-arrangement of Eq. (16.41) results in:

(16.42)

The crucial part of this equation is the expression in the bracket, which is identical to the specific circulation function in Eqs.(16.26). Introducing the compressibility effect results in a modified version of Eq. (16.42):

(16.43)

The diffusion factors presented above are used to establish correlations for the compressor total pressure losses. The theoretical background and the above

16 Modeling the Compressor Component, Design and Off-Design

397

discussion showed a direct correlation between the profile losses and the boundary layer quantities, particularly the boundary layer momentum thickness. Investigations by NACA, summarized in NASA SP-36 [6] and briefly reviewed in this chapter, showed that measuring the total pressure losses can experimentally determine the momentum thickness. Further investigations presented in [10], [11], [14], [29], [30] and [31] deal with the spanwise distribution of the total pressure and the total pressure loss coefficient. For the aerodynamic design of a single-stage compressor, Monsarrat et al. [31] presented correlations between the profile loss parameter and the diffusion factor using the experimental data by Sulam et al. [32]. The loss correlations by Monsarrat et al. [30] are frequently used as a guideline for designing compressor stages with the profile similar to that described in [30]. Although the experimental data revealed certain systematic tendencies, no attempt was made to develop a correlation to describe the loss situation in a systematic manner. These facts gave impetus to consider the above experimental data in the present analysis.

16.1.5 Shock Losses Several studies have discussed experimental and theoretical shock loss investigations. As indicated previously, Miller and Hartmann [7], Miller et al. [8], and Schwenk et al. [9] initiated their fundamental research on transonic compressors where they investigated shock losses. Schwenk et al. [9] considered a normal shock in the entrance region of the cascade using a Prandtl-Meyer expansion. Levine [33], Balzer [34], and Swan [35] made efforts to calculate shock losses by estimating the shock position. Their proposed methods, particularly, Levine's [33] and Swan's [35] found their application in compressor design. Similar to Schwenk et al. [9], the methods by Levine, Balzer and Swan include the assumption of a normal shock. While Levine and Swan considered the acceleration on the suction surface by using the continuity and Prandtl-Meyer expansion, Balzer disregarded the expansion completely and used the continuity requirement. The deficiencies in the existing methods can be summarized as: a) They cannot accurately calculate the shock position, which is a prerequisite for accurately predicting the shock losses. b) The Mach number calculated by the Prandtl-Meyer expansion on the suction surface does not represent the shock Mach number along the channel width. Swan partially corrected this deficiency by building an average Mach number. c) The description of the physical process is not complete: the Prandtl-Meyer expansion combined with the continuity requirement is not sufficient to describe the physics. The above deficiencies gave impetus to generate the following new shock loss model, [36]. For the development of this model we assume the passage shock as an oblique shock, whose position changes according to the operating point and may include normal shock as a special case. Furthermore, we assume that the blading has a sufficiently sharp leading edge, where the shocks are attached at least at the design point with no detached bow waves expected. Fig.16.6 shows the shock situation with the inlet flow angle ȕ1 , the metal angle ȕt (camber angle), and the incidence angle i. To determine the shock position we use the continuity equation, the Prandtl-Meyer expansion, and the momentum equation. For the control volume in Fig.16.6, the continuity requirement for a uniform flow is:

398

16 Modeling the Compressor Component, Design and Off-Design

V1 E

J G

-i

E1

n1

D

Es C

Et

nW

B Vs

E2

R

Ss

ns

+i

4

S

V2 Fig. 16.6: Shock position and angle definition, n1 , ns, nw are normal unit vectors at the inlet, shock location and the wall, from Schobeiri [21]

(16.44) with h1 and hS as the height of the stream tube at the inlet and at the shock position. Using the gas dynamics relationships, Eq. (16.44) is written as:

(16.45)

with the geometric relation: (16.46) The incidence and expansion angle are coupled by: (16.47)

16 Modeling the Compressor Component, Design and Off-Design

399

where Ȟ is determined from the Prandtl-Meyer expansion law:

(16.48)

The momentum equation in circumferential direction is given by:

(16.49) with as the inlet surface, the shock surface characterized by AB, the portion of the blade suction surface denoted by BC and the variable angle between the normal unit vector and the circumferential direction (Fig.16.6). As Fig.16.6 shows, because of the cascade periodicity, the pressure at the point A is identical to the pressure at point C. Furthermore, point B on the suction surface represents the common end point for both distances, AB and CB. This means that the pressure distributions along AB and CB have exactly the same beginning and ending values, but may have different distributions between the points AB and CB. Assuming that the pressure integrals along the shock front AB and the blade contour portion CB are approximately equal, their projections in circumferential direction may cancel each other simplifying Eq.(16.49) as:

(16.50)

Finally, we arrive at a geometric closure condition that uses the mean stream line, which is assumed to be identical to the mean camber line of the blade with the radius R as shown in Fig.16.7. From Fig.16.7, it follows immediately that (16.51) and (16.52) The shock angle is determined from:

400

16 Modeling the Compressor Component, Design and Off-Design

J M

n ea

r St

m ea

e lin

Et

Ss R

4 S Fig. 16.7: Introduction of mean streamline with curvature radius

(16.53) Considering the above procedure, the continuity equation yields:

(16.54)

With Eqs. (16.46), (16.47), (16.50), and (16.54), we have a system of four equations that easily calculates the four unknowns, namely: į, ȕs, Ȗ, and Ms. The shock loss is:

(16.55)

where Pb and Pa represent the total pressure before and after the shock. For ȕt = 30o and the incidence angle i = 0o, the above equation system is used to calculate the shock Mach number, the expansion angle ș, the shock position Ȗ, the total pressure ratio, and the shock losses.Fig.16.8 shows the shock Mach number as a function of spacing ratio S/R with the inlet Mach number M1 as the parameter. This figure shows that increasing the spacing ratio cause the shock Mach number to continuously

16 Modeling the Compressor Component, Design and Off-Design

401

increase and approach an asymptotic value. These results are similar to those presented by Levine [37]. However, the values are slightly different because of the simplifying assumptions by Levine. Keeping the inlet Mach number constant, the increase of spacing ratio leads to higher expansion angle ș as shown in Fig.16.9. However, increasing the inlet Mach number at a constant spacing ratio S/R leads to a smaller expansion angle.

2 1.8 M1=1.6

1.6 MS

1.5 1.4

1.4

1.3 1.2

1.2 1

0

0.2

0.4 S/R 0.6

0.8

1

Fig. 16.8: Shock Mach number as a function of the spacing ratio S/R with the inlet Mach number M1 as parameter with ȕt = 30o, and the incidence angle i = 0o

15 M1=1.2 1.3

12

1.4 1.3

9 4

1.5 1.6

6 3 0

0

0.2

0.4

S/R

0.6

0.8

1

Fig. 16.9: Expansion angle Ĭ as a function of S/R with inlet Mach number M1 as parameter with ȕ2 = 30o and incidence angle i = 0(

402

16 Modeling the Compressor Component, Design and Off-Design

180

160 J

MS=1.6 1.5

140

1.4 1.3

120

1.2

0.2

0.4

S/R 0.6

0.8

1

Fig. 16.10: Shock angle Ȗ as a function of S/R with shock Mach number as parameter The same tendency can be read from the charts by Levine. For an inlet Mach number M1 = 1.2 and S/R = 0.5, the Levine’s method gives an expansion angle ș = 8o, while the method presented in here calculates ș = 9.7o. Fig.16.10 shows the shock angle Ȗ as a function of spacing ratio S/R. This figure shows the significant effect of the inlet Mach number on the shock position. Once the shock angle is calculated at the design point, it may undergo changes during an off-design operation. The off-design operation may place a limitation on the shock angle range as Fig.16.11 illustrates.

J

E

a

Attached normal shock limit line

E1b

M 1b

E1a

M 1a

Za

Jb

Off-design operation regime with dettached bow waves a

b 2

p c

J lim

E1d

M 1d

E1c

M 1c

Surge limit

1

Ze

a

Jd

Jc

c

b p

J

d

e c Attached normal shock

d Dettached bow waves

. m Compressor performance map

Fig. 16.11: Effect of shock angle Ȗ change described in Fig.16.8 on compressor performance at 4 different operation conditions (a) through (d), map

16 Modeling the Compressor Component, Design and Off-Design

403

Beginning with a design point speed line characterized by , Fig.16.11a, the operating point (a) is given by the inlet Mach number M1 with a uniquely allocated inlet flow angle ȕ1 that satisfies the unique shock angle criterion. Increasing the back pressure from the design point back pressure to a higher level (b) causes the shock angle to increase ( ). Thus the end point of the passage moves toward the cascade entrance. By further increasing the back pressure from (b) to (c), a normal shock is established, which is still attached. The corresponding shock angle Ȗ can be set equal to . This particular shock angle that we call unique shock angle corresponds to an inlet flow angle which is frequently referred to as the unique incidence angle. Increasing the back pressure and thus decreasing the mass flow beyond this point, causes the shock to detach from the leading edge, as shown in Fig.16.11d. Reducing the rotational speed changes the incidence and may lead to further moving the shock from the leading edge as shown in Fig.16.11d. These operating points are plotted schematically in a compressor performance map shown in Fig.16.11(right). To establish the loss correlations, the existing available experimental data were re-evaluated, particularly those in Gostelow et al. [10] and Krabacher and Gostelow [15] and [16] which used four single-stage compressors with multi-circular-arc profiles. A detailed description of the compressor facility and the stages are found in their reports. The data analysis used the following information: (1) the total pressure losses as a function of diffusion factor in the spanwise direction, (2) inlet, exit, and incidence angles, (3) Mach numbers, (4) velocities, and (5) geometry. To consider the compressibility effect discussed previously, the modified diffusion factor Dm was obtained using the information from Gostelow’s report mentioned previously. Figures 16.12 and 16.13 show the results. Starting from the tip with an immersion ratio of , Fig.16.12 (top) shows the total loss parameter as a function of the modified diffusion factor. As shown, the highest total pressure loss that consists of shock loss, profile loss, and secondary flow loss is encountered at the tip region, with shock and secondary flow losses as the predominant losses. Subtracting the shock loss coefficients from the total loss coefficients provides loss coefficients that contain the primary and the secondary loss effects. The resulting loss coefficients are plotted in Fig.16.13 (bottom) and are approximately 30% smaller than the total pressure loss coefficient. Figure 16.13 (top) exhibits the shock losses as a function of modified diffusion factor with the immersion ratio as a parameter. Highest shock losses occur at the tip region, with the relative Mach number above unity and decrease toward the hub. The fact that the sum of primary and secondary flow losses exhibit a minimum range of , where the secondary flow effects diminish, enables the compressor designer to estimate the secondary flow losses. This is done by subtracting the losses at from the loss distribution at different spanwise locations. As a result, these losses include the effect of the secondary flows associated with wall boundary layer development and clearance vortices. Figure 16.13 (bottom) shows the distribution of the secondary loss parameter as a function of the modified diffusion factor. It exhibits a linear dependency of the secondary loss parameter as a function of the modified diffusion factor with the immersion ratio as

404

16 Modeling the Compressor Component, Design and Off-Design

Fig. 16.12: Total loss parameter (top) and profile loss parameter(bottom) as a function of modified diffusion factor

16 Modeling the Compressor Component, Design and Off-Design

405

Fig. 16.13: Shock loss (top) and difference in profile loss parameter with respect to the minimum profile loss parameter (bottom) as a function of the modified diffusion factor with the immersion ratio as parameter

406

16 Modeling the Compressor Component, Design and Off-Design

a parameter. Since the diffusion factor is directly related to the lift force and thus to the lift coefficient (CLc/s) as a linear function, one may conclude that the secondary flow losses are linearly proportional to (Clc/s). This is in agreement with the measurements by Grieb et al. [38] and in contrast to the correlation proposed earlier by Carter [39] that includes the term (CLc/s)2 and is adopted by many other researchers. A similar dependency for HR = 50, 70 and 90%, where the smallest losses are encountered at Hr = 60 - 70. Moving further toward the hub at HR = 90%, the total pressure losses experience a continuous increase attributed to higher friction losses and the secondary flow caused by the secondary vortices at the hub. A comparison of the loss parameters plotted in Fig.16.13 (top) shows that the total pressure losses in the tip region (Hr = 10%) are much higher than those in the hub region (Hr = 90%). The higher losses at the tip are due to an increased secondary flow effect that is attributed to the tip clearance vortices.

16.1.6 Correlations for Boundary Layer Momentum Thickness As shown, the profile loss coefficient and the boundary layer momentum thickness are interrelated by: (16.56) The momentum thickness in Eq. (16.56) is the projection of the suction surface and pressure surface momentum thicknesses given by: (16.57) where the subscripts S and P refer to the suction and pressure surface, respectively, and the function F is given by:

(16.58)

with H12, H32 as the displacement and energy form factors, į2 as the boundary layer momentum thickness, ı as the solidity, and c as the blade chord. In the literature (see also Hirsch [40] and Swan [41]), the function F is frequently approximated as a constant with the value F = 2. For a realistic velocity distribution, Schobeiri [19] showed that the value of F may differ from 2. To arrive at a better estimation for F, the boundary layer velocity profile is approximated by several simple functions such as a linear function, a power law, a sine function, and an exponential function. A close examination of the results and their comparison with the experiments showed

16 Modeling the Compressor Component, Design and Off-Design

407

that the velocity approximation by a power function yields better results. However, the exponential approximation would be more appropriate for those profiles that are close to separation. Using the law function approximation, we arrive at: (16.59) Introducing Eq. (16.59) into Eq. (16.58) and the results into Eq. (16.56) leads to:

(16.60)

With Eq. (16.60) and (16.58), the momentum thickness is determined from:

(16.61) Using the profile losses as previously discussed, the correlation for the momentum thickness as a function of modified equivalent diffusion factor are plotted in Fig.16.13 with the immersion ratio as a parameter. The highest value for the momentum thickness is encountered in the vicinity of the tip that includes the viscosity effects as well as the secondary flow effects. Similar to the profile losses, the momentum thickness continuously decreases by moving toward the blade mid-section up to Hr = 0.6. It assumes a minimum at Hr = 0.7. At this radius, the secondary flow effect apparently diminishes completely so that the momentum thickness corresponds to the one generated by the blade surface friction only. For immersion ratios greater than Hr = 0.7, the momentum thickness starts increasing again, which indicates the strong effect of the secondary flow.

16.1.7 Influence of Different Parameters on Profile Losses The correlations presented above are based on experimental results performed on typical high performance compressors with specific flow characteristics and blade geometries similar to those discussed previously. These correlations may be applied to other compressors with similar geometries but different flow conditions by considering the effect of the following individual parameters. 16.1.7.1 Mach Number Effect: Estimating the Mach number effect requires calculating the critical Mach number. When the Mach number reaches unity locally in a compressor cascade, the corresponding inlet Mach number is said to have reached

408

16 Modeling the Compressor Component, Design and Off-Design

its critical value. Jansen and Moffat [42] made the assumptions that below the critical Mach number, the total pressure losses and the turning angle are essentially constant. The pressure losses increase rapidly beyond this value. Using the gas dynamics relations, Jansen and Moffat [42] determined the local critical Mach number by the following implicit relation:

(16.62)

To estimate the critical Mach number directly, Davis [43] suggested the following explicit relation:

(16.63) As seen in Part I, the velocity ratio Vmax/V1 is directly related to the circulation function and thus the diffusion factor. With the critical Mach number from Eq. (16.7) or (16.8), the profile loss coefficient can be corrected as:

(16.64) with A = 1.8-2.0 (see Moffat [42] and Davis [43]). For DCA-profiles, Dettmering and Grahl [44] found that Eq. (16.9) underestimates the correction and suggested the following modified approximation:

(16.65)

16.1.7.2 Reynolds Number Effect: This effect is only at lower Reynolds number ranges of practical significance. For high performance compressors, the Reynolds number is high enough so that its changes do not affect the profile losses. The following profile loss correction is suggested for Reynolds number ranges Re :

(16.66)

16 Modeling the Compressor Component, Design and Off-Design

409

16.1.7.3 Blade Thickness Effect: To consider the effect of the thickness ratio t/c, the boundary layer momentum thickness may be corrected using the correlation by Fottner [45].

(16.67)

16.2 Compressor Design and Off-Design Performance The prerequisite for simulating the dynamic behavior of a compressor is a detailed knowledge of the design point properties as well as the overall performance characteristics. Three levels of simulation are discussed. The first level deals with steady state performance maps. It provides global information about the design and off-design efficiency and performance behavior of a compressor. It does not contain any detailed information about the individual stage parameters such as stage flow, load coefficients, degree of reaction, absolute and relative flow angles. However, this information is necessary to construct the compressor performance map. The map can be generated either by a stage-by-stage or by a row-by-row compression calculation procedure, which treats the compression procedure fully adiabatic, neglecting the heat transfer to/from the blade material. The following row-by-row adiabatic compression calculation method, which provides the basis for the second level simulation will be followed by the diabatic compression process that is necessary for the third level comprehensive dynamic simulation.

16.2.1 Stage-by-Stage and Row-by-Row Adiabatic Compression Process The stage-by-stage and row-by-row methods for calculating the compression process within a multi-stage compressor, Fig.16.14, presented in this chapter are equally applicable to axial and radial turbines and compressors, as discussed in Chapter 6. These methods are based on a one-dimensional calculation of the compression or expansion process using a set of dimensionless stage or row characteristics. These characteristics, along with loss correlations discussed in Section 16.1 describe the design and off-design behavior of the compressor or turbine component under consideration. In the following we discuss and evaluate both methods. The equation derived for the stage-by-stage calculation method follows the nomenclature in Fig.16.14. 16.2.1.1 Stage-by-Stage Calculation of Compression Process: The performance behavior of the compressor stage is completely described by the stage characteristics. The meridional view, blade configuration, compression diagram and velocity diagram of a typical compressor are shown in Fig.16.14. For stage-by-stage analysis, we resort to the following dimensionless variables introduced in Chapter 5 in conjunction with

410

16 Modeling the Compressor Component, Design and Off-Design

(a) Multistage compressor

(c) Compression diagram

h

2

V3

2

P3 p

H3 h3

'h''

l

m

'h''d

3

P2 2

2

V2

p

h2

2 d

'h' 'h'

p

s

COMUSTAG 1

h1

(b) Row arrangement S1

R1

S2

V1

S R2

W

3

W2

S3

U

U

W4

R3

(d) Velocity diagram

U W

V4

'h'

7

2

V2

2

6

'hs

5

4

2

3

2

2

V1

1

H2

'h''s

P1

W

5

W V6

7

m3

E3

D3

E

2

D

2

2

6

m2

2

3

2

3

3

Fig. 16.14: Multi-stage compressor (a) flow path, (b) row-by-row arrangement, (c) compression diagram, (d) velocity diagram the stage-by-stage arrangement, Fig.16.14a and b, compression diagram, Fig.16.14c and velocity diagram in Fig.16.14d.

(16.68) As shown in Chapter 5, these dimensionless variables were incorporated into the conservation equations of mass, momentum, moment of momentum, and energy leading to the following relations (5.50) to (5.53): (16.69)

(16.70)

(16.71)

16 Modeling the Compressor Component, Design and Off-Design

411

(16.72) Using the above set of stage characteristics in conjunction with the stage loss coefficients discussed in Chapter 7, the compression process is accurately calculated. This process, however, requires few assumptions and several iterations. The four equations (16.69) to (16.72) contain nine unknowns. The following quantities are assumed to be known: (1) the compressor mass flow, (2) the compressor pressure ratio, (3) the type of blades, their exit flow angles and for each individual stage, and their configuration in terms of degree of reaction. Further, we may assume the absolute inlet flow angle of the first stage and the exit flow angle of the last stage to be 90o. With these assumptions, all nine stage characteristics are estimated in the first iteration step, which usually does not deliver the required compression pressure ratio and exit enthalpy. The process of iteration is continued until the enthalpy and pressure convergence have been reached. This stage-by-stage compression calculation method delivers accurate results, however, it requires some experience in guessing the initial values for unknown characteristics. As an alternative, the row-by-row calculation is discussed in more detail in the following section. It is computationally more efficient and does not require a lot of iterations.

16.2.1.2 Row-by-Row Adiabatic Compression: As discussed in Chapter 5, the compression process within stator and rotor row, shown in Fig.16.15 , is summarized as:

(16.73)

The quantities in Eqs. (16.73), the decomposition of a compressor stage in its stator and rotor rows, and the compression diagram are presented in Fig.16.15 . In the above and the following, all quantities with superscripts “/ ” and “// “ refer to stator and rotor rows, respectively. For the stator row, the compression diagram shows the details of an adiabatic compression, where the absolute total enthalpy remains constant. The following rotor row with the corresponding compression diagram exhibits the details of energy balance in relative frame of reference with constant relative total enthalpy.

412

16 Modeling the Compressor Component, Design and Off-Design

Compressor stator row W

Compressor rotor row

Compressor stage

2

V1 U W

H3

h

P3r

P2r

H1 H2

V

P2

H2r= H 3r

p

h

=

W3 - U3 2

2

V P2

S

1

2

'h

H= H

' hs

P2

h2

p 1 S

V2

2

p2

h2

'h

h1 Cas-C-SR-N

2

'h ' hs

' hs p1

P1

2

h 3s

h3

2 ' hs

' hd

h2

'h

V

2

2

p

' hd

p3

2

( W 22- U 2 )

2

2

V2

2

2

m

P1

2

2

V2

h

P3

h 3

h1 S

Fig. 16.15: Stator, rotor, stage compression diagrams. P: total pressure, p: static pressure, “/”, “//” mark stator and rotor, respectively

The compression diagram for the entire stage exhibits a composite picture of energy balance within stator and rotor row. Introducing the efficiency definition for the stator and rotor row, respectively:

(16.74)

Incorporating Eq. (16.74) into Eqs. (16.73), the isentropic enthalpy difference for stator and rotor row, respectively, is expressed by:

(16.75)

The expression for the dimensionless isentropic enthalpy difference is obtained by dividing Eqs. (16.75) by the circumferential kinetic energy at the exit of the stage. The dimensionless isentropic enthalpy difference is written in terms of the stage parameters of Eq. (16.68) as:

16 Modeling the Compressor Component, Design and Off-Design

413

(16.76)

Further analysis of Eqs. (16.76) shows that knowledge of the entire stage parameter is necessary to determine the row isentropic enthalpy difference, which is similar to the stage-by-stage procedure discussed in the previous section, and requires many iterations. To avoid the iterative procedure, we subdivide the stage specific polytropic mechanical energy l, as well as the isentropic mechanical energy ls, into two virtual contributions and that we allocate to stator and rotor row as shown in Fig.16.15. This step does not imply that the stator row is performing shaft power, which would violate the energy balance. It merely provides a smooth transition from stator row to rotor row when calculating the compression process. With this re-arrangement, we arrive at the contributions allocated to the stator and rotor rows as:

(16.77)

Similarly we find:

(16.78)

As seen, the sum of stator and rotor contributions leads to the stage specific mechanical energy. Using the polytropic specific mechanical energy expressions given in Eqs. (16.78) and dividing them by the circumferential kinetic energy at the exit of the individual row, the dimensionless row polytropic load coefficients for stator and rotor are:

(16.79)

where, ,and .Combining Eqs. (16.79) with the efficiency definition for the stator and rotor rows, Eq. (16.74), the following expressions for the efficiency are obtained:

414

16 Modeling the Compressor Component, Design and Off-Design

(16.80)

The isentropic row load coefficient ȥ is defined as the dimensionless isentropic specific mechanical energy for the row. Implementing the efficiency expressions from Eqs. (16.80), the isentropic row load coefficients for the stator and rotor rows are obtained:

(16.81)

(16.82)

All the information necessary to pursue the compression process on the h-s diagram is now available to predict the compressor behavior with sufficient accuracy and reliability. Given the pressure and temperature at the inlet, the remaining thermodynamic properties are calculated using the property tables of the working medium that can be implemented as a subroutine within the calculation procedure. The row flow coefficients are calculated from the continuity equation:

(16.83) Given the exit stator and rotor blade angle (Į2, ȕ3) as input data necessary to describe the geometry and the above flow coefficient, we determine the flow angles (Į3, ȕ2) for the stator and rotor rows, respectively:

(16.84)

The incidence and deviation angles are calculated according to the procedure described in Chapter 9. Upon determining all the angles involved in the velocity

16 Modeling the Compressor Component, Design and Off-Design

415

diagram, the velocities and their components are completely known. The velocity triangle at the exit of the row is calculated and the flow behavior is completely described by the velocity diagram. The flow coefficient, ij' and ij", and the flow angles ( Į3, ȕ3, Į2, ȕ2,) are necessary tools to determine the polytropic and isentropic enthalpy differences between the inlet and exit of the row. The amount of mechanical energy consumed by the flow to increase its total pressure is represented by the polytropic load coefficients, Ȝ', Ȝ", and the isentropic load coefficients ȥ', ȥ" as presented above. Finally, from the energy balance relationships, the complete compression process for the stage is determined by the following set of equations:

(16.85)

(16.86)

the above procedure can be easily repeated for all the stages of the compressor in question. However, it should be pointed out that the above analysis is completely dependent on an accurate and reliable method for determining the off-design efficiency. This is the subject of the next section. 16.2.1.3 Off-Design Efficiency Calculation: The off-design efficiency calculation is based on the analysis of the diffusion factor. The modified diffusion factor, Eq. (16.45), includes compressibility effects and, therefore, is capable of handling highly compressible flow such as in transonic compressors. To apply the diffusion factor definition to stators and rotors on an individual basis, separate expressions were derived in terms of known stage and row quantities. The expression for the stator row given is:

(16.87)

and for the rotor row:

416

16 Modeling the Compressor Component, Design and Off-Design

(16.88)

The parameters ȝ, Ȟ, and ı in Eqs. (16.87) and (16.88) pertain to the individual row under consideration. Using the above modified diffusion factor, the total loss parameter can be taken as a function of modified diffusion factor with the immersion ratio defined in Fig.16.16 as a parameter. 0.09 0.08

1: Hr = 0.0 2: Hr = 0.1 3: Hr = 0.2 4: Hr = 0.3, 1.0 5: Hr = 0.4, 0.9 6: Hr = 0.5, 0.8 7: Hr = 0.6, 0.7

]tsin(Din,Ein)/2V

0.07 0.06

1 2

0.05

3

0.04

4 5 7 6

0.03 0.02 0.01 0

0

0.1

0.2

0.3

Dm

0.4

0.5

0.6

0.7

Fig. 16.16: Total loss parameter as a function of modified diffusion factor with immersion ratio as parameter from Schobeiri [21] As shown, a set of curves representing the blade total loss parameter ȗt sin(Įin, ȕin )/2ı is plotted, with Įin, ȕin as the inlet flow angle of stator or rotor row and ı as the blade solidity ratio. These curves may be presented in the form of polynomials to be implemented into a calculation procedure. The highest loss is encountered close to the tip at an immersion ratio of , where secondary flow and tip clearance losses predominate the loss picture. These losses decrease as the immersion ratio approaches the blade mid height. However, for this particular compressor blade, the lowest loss is located around . Approaching the hub, the secondary losses dominate the losses leading to higher total loss parameter. It should be pointed out that, the loss parameter presented in Fig.16.16 represents only the loss situation for a particular compressor, whose efficiency and performance are calculated and presented in this section. As indicated previously, the generation of compressor

16 Modeling the Compressor Component, Design and Off-Design

417

efficiency performance maps requires knowledge of the individual losses of the stator and rotor rows. The loss parameter curves of Fig.16.16 are not symmetric and, therefore, must be applied to the blade in a manner compatible with their distribution over the blade spanwise direction. For this purpose, we apply energy balance to the blade, which results in: (16.89) where the index j refers to the blade spanwise positions from tip to hub. The efficiency is defined in terms of the row loss coefficient Z, introduced in Chapter 7, which is a function of ȗt and may be written for the stator and rotor, respectively: (16.90) Using Eq. (16.90), the row efficiency can be expressed as: (16.91) Eq. (16.91), coupled with the total loss definition, can be reduced to: (16.92) Equation (16.92) is an integral representation of the loss distribution in the spanwise direction. An alternative calculation method for calculating the efficiency is using the entropy change as a result of a row-by-row compression process. The design and offdesign efficiency is then determined as an outcome of the loss calculation which reduces the number of iterations involved. Applying Eq. (4.103) to the stator row, the entropy change is directly related results in: (16.93) With the given initial entropy and total pressure , the row exit entropy is calculated by using the row loss coefficient. For the stator row it is expressed by:

(16.94)

418

16 Modeling the Compressor Component, Design and Off-Design

Substituting Eq. (16.94) into (16.93) after some rearrangement, we obtain the entropy at the exit of the row: (16.95) With the exit entropy and the corresponding enthalpy, the stator row exit thermodynamic condition is fully determined. The following rotor row is treated similarly considering the relative frame of reference. The entropy increase is related to the change of the relative total pressure which is expressed in the following relation: (16.96) Similar to the stator, with the given initial entropy and relative total pressure , the row exit entropy is calculated by using the rotor row loss coefficient. For the rotor row it is expressed by:

(16.97)

Substituting Eq. (16.90) into (16.89), we obtain (16.98) With the exit entropy and the corresponding enthalpy, the rotor row exit thermodynamic condition is fully determined. The row-by-row calculation is continued until the compression process for the entire compressor is completed.

16.3 Generation of Steady State Performance Map Using the row-by-row calculation method described above, the compression process for design and off-design operation conditions can be accurately calculated. As a result, the calculation procedure provides efficiency and performance maps for singlestage axial, radial, and multi-stage axial compressors. The prerequisite for an accurate efficiency and performance prediction is the knowledge of the compressor blade’s loss behavior. Given the geometry of a nine-stage subsonic axial compressor, steady state calculations are performed by varying the rotational speed and the compressor mass flow. Figures 16.17 and Fig.16.18 exhibit the performance and efficiency behavior of a 3-stage subsonic compressor at its design and off-design operating

16 Modeling the Compressor Component, Design and Off-Design

419

points. The rotational speed is non-dimensionalized by the design angular velocity. Figure 16.17 shows the compressor pressure ratio versus the non-dimensionalized mass flow ratio. Starting with the design speed ratio of , we first reduce the mass flow. This in turn changes the stage velocity diagrams, and thus, the incidence angle of blades resulting in higher flow deflection, greater stage load coefficient, and

3 Z/ZD=1.1

2

1.0

lim

it

0.8

S

e u rg

0.9

pZ/pD

2.5

0.6

e Chok

0.5

a

1 0.2

0.7

1.5

0.4

0.6

li m it

0.8 x

1

1.2

x

m/mD Fig. 16.17: Compressor pressure ratio as a function of relative mass flow with relative angular velocity as parameter

1 Z/ZD=1.1

1.0

0.9

0.8

0.7

0.6

K

0.50

0.8

0.6

0.4

0.2

0.4

0.6

0.8 x

1

1.2

x

m/mD Fig. 16.18: Efficiency as a function of relative mass flow with relative angular velocity as parameter

420

16 Modeling the Compressor Component, Design and Off-Design

therefore, higher pressure ratio. The mass flow reduction may be caused by specific operation conditions such as closing a valve of a compression system or increasing the compressor back pressure of a gas turbine system. For the case that mass flow at a constant speed is further reduced, a subsequent increase in inlet flow incidence will cause a partial or total flow separation resulting in a compressor surge. On the other hand, if at a constant rotational speed the compressor back pressure is reduced, the incidence angle assumes negative values causing the axial velocity component to increase. Further reducing the back pressure results in higher velocity that may approach the speed of sound. In this case, the compressor operates in a state of choke. Thus, the operational envelope of the compressor is bounded by a surge limit and a choke limit plotted in Fig.16.17 and Fig.16.19.

3

pZ/pD

2.5 2

Su

rg

im el

it (

Z/ZD=1.1

i t) -l i m

Dm 2

D

1

1.5 Choke

1

0.4

0.6

0.8 x

limit

1

1.2

x

m/mD Fig. 16.19: Construction of surge and choke limits, point D refers to design point, point 2 corresponds to the maximum diffusion factor, 1-2 precautionary surge margin set by compressor designer.

16.3.1 Inception of Rotating Stall Details of the performance map is shown in Fig.16.20. After passing through the design point D, by further decreasing the compressor mass flow rate, the pressure ratio first reaches a maximum followed by a continuous decrease. Further decreasing the mass flow may trigger a sequence of events, sketched in Fig.16.20. First, a boundary layer separation inception may occur at a few blades, as sketched in Fig.16.20(a). While the channel of these blades are occupied by low energetic stall cells, the rest of the blades operate at normal flow condition, however, at a different incidence angle, as shown in Fig.16.20(b). Since the stall cells have blocked a portion of the cross sectional area, a redistribution of the mass flow in circumferential direction occurs, leading to a redistribution of the incidence angles. Based on the separation extent, the cells may partially occupy the tip and hub region, as well as the entire blade channels from hub to tip, as shown in c).

16 Modeling the Compressor Component, Design and Off-Design +i

+i

421

+i

a) Inception of stall cells, boundary layer separation. due to an increase of positive incinece angle

b) Manifestation of stall cell, deflection of velocity direction caused by stall cells (3a)Partial span stall (3b)Full span stall (3c) Surge

c) Growth of stall cells to surge Fig. 16.20: Inception of rotating stall and surge If these stall cells are located within rotor blades, they rotate with the corresponding frequency causing the compressor to operate in rotating stall mode. This operating mode is characterized by temporal fluctuations of the mass flow. Further reducing the mass flow may lead to a complete breakdown of the compressor operation that is called compressor surge. In this case, the compressor mass flow periodically oscillates between positive and negative. In conjunction with the combustion chamber, the compressor surge may suck hot gas and eject it in a counter flow direction. To prevent the compressor from getting into surge operation, the compressor designer places a surge limit on the performance map. This limit may be experimentally determined or empirically estimated by setting certain flow deflection limits, diffusion factor limits, or even boundary layer separation criterion. For the performance map presented in Fig.16.19, we set the diffusion factor to as the limiting criterion for the beginning of a rotating stall. This is shown in Fig. 16.20

422

16 Modeling the Compressor Component, Design and Off-Design

as the surge limit that intersects the pressure curve at point 2. The compressor aerodynamicist wishes to have certain margins to this limit, and may construct a second surge limit. This is illustrated in Fig.16.19, where the diffusion factor limit was set at , point 1.

16.3.2 Degeneration of Rotating Stall into Surge The surge process is graphically explained in Fig.16.21, where an axial compressor is connected to an air storage tank. The compressor is driven by an electric motor that operates at a constant frequency. At the time t = 0, where the tank is assumed to be completely empty, the compressor starts pressurizing the tank Fig.16.21 (A). The process of pressurization continues until the tank pressure is equal to the compressor exit design pressure. Further compressor operation causes the blade incidence angles to become greater than the design incidence angle resulting in an adverse increase of the compressor stage flow deflection. As a consequence, the compressor exit pressure and thus the storage pressure will raise, causing first a rotating stall followed by a stable surge with a reversed flow direction as indicated in Fig.16.21(B) if operation continues.

(A)

(B)

Storage tank Pressurizing at p < p Surge

Storage tank depressurizing at p > p Surge

Fig. 16.21: Explaining the phenomenon of compressor surge

The details of the surge process is qualitatively shown in Fig.16.22(a) with the pressure performance curves, the surge and the chock limits. Starting from an arbitrary point D bellow the design point, the compressor continuously sucks air from the environment pumps it into the storage and raises the storage pressure approaching the design point pressure DP. During this process the blades boundary layer is fully attached Fig.16.22(b) and the compressor operates in normal compression mode.

16 Modeling the Compressor Component, Design and Off-Design

423

Passing through the design point and approaching the surge limit at point A, a few blades may experience rotating stall with partial separation on the blade suction surface, Fig.16.22(c). As the compressor operation continues the flow deflections increase, the number of the blades that operate in rotating stall mode increases and the storage pressure exceeds the compressor surge limit. At this point a flow reversal takes place and the mass flow jumps from positive to negative causing the tank to depressurize. Since the compressor blades are subject to a reversed flow, they function as an energy dissipator that might has a characteristic qualitatively shown as the curve BC Fig.16.22(a). Once the tank’s pressure is substantially lowered below the compressor design point DP, for instance D, the compressor resumes its normal operation. If the compression process continues at the same rotational speed, the surge process will repeat itself leading to a hysteresis pattern. This non-linear dynamic operation causes the blades to be subjected to severe periodic forces that may result in a complete de-blading of the compressor rows if immediate actions are not taken. 4.0

3.6

ZZD= 1.2

(d)

1.1

Design surge limit

B

A 1.0

Compressor back flow characteristic

3.2 3.0

+i

0.9

U

(c) u 2

0.8

3

2.8

B: Stable surge

1

C

2.6

D

0.7

4

Pressure ratio

S

3.4

(a)

DP

3.8

A: Rotating stall

0.6

2.4

U

0.5

2.2

Choke Limit

2.0 -0.2

0.0

0.2

0.4

0.6

0.8

Mass flow ratio

1.0

. . m mD

(b)

DP:Design point 1.2

Fig. 16.22: Dynamic operation of a compressor

16.4

Compressor Modeling Levels

Three levels of compressor modeling are presented in this section. The first level utilizes the steady performance map. The second level uses the row-by-row adiabatic compression process, and the third level employs a diabatic row-by-row calculation method. The first level exhibits the global performance behavior of a compression system under dynamic operation condition. The second level delivers a detailed

424

16 Modeling the Compressor Component, Design and Off-Design

calculation of the adiabatic compression process under any dynamic operation condition . In addition to the information provided by the second level model, the third level provides detailed information about the blade temperature and the effect of the heat transfer on an operation envelope that includes the performance, surge, and choke limits.

16.4.1 Module Level 1: Using Performance Maps The efficiency and performance maps generated in section16.2.3 can be used for a first level compressor simulation. This simulation level obviously does not provide details regarding the dynamic events within compressor stages. However, it is capable of globally reflecting the state of the compressor during a dynamic event. The global compressor module is modeled mathematically by a set of algebraic equations. It receives the dynamic information from the inlet plenum, performs off-design calculations, and transfers the results to the exit plenum. This arrangement allows a quasi dynamic simulation of the compressor component. The set of algebraic equations determines the off-design values of the mass flow rate , total temperature To), power consumption , efficiency and the volume flow rate as functions of the efficiency, pressure ratio, inlet temperature, and angular velocity (Ȧ), as follows:

(16.99)

To account for the effect of temperature on the results obtained from the performance map, we introduce a non-dimensional relative volume flow rate,

(16.100)

a relative rotational speed, (16.101)

16 Modeling the Compressor Component, Design and Off-Design

425

and a relative pressure ratio

(16.102)

where, the subscript R and D refer to relative and design point values, respectively. By introducing the above non-dimensionalized parameters into Eqs. (16.99), we obtain two functional relations for the efficiency and the volume flow rate:

(16.103) (16.104)

As seen from Eqs. (16.103) and (16.104), the number of the parameters has reduced by one, resulting in simpler relationships that can be expressed in terms of twodimensional polynomials as functions of , and as given below: (16.105) (16.106) with

and

. The matrices

, and

represent

the coefficients of the two dimensional polynomial describing the behavior of the volume flow rate and efficiency, respectively, as functions of the given pressure ratio and angular velocity. Considering the non-dimensionalized Eqs. (16.104)-(16.106), the first element of the matrices A and C are determined by the map as and rate

. Expanding Eqs. (16.105) and (16.106) we find the relative volume flow as:

(16.107)

and the efficiency

426

16 Modeling the Compressor Component, Design and Off-Design

(16.108)

The elements are (16.109) Using Eqs.(16.107) to (16.109), the off-design mass flow, (16.110) the total exit temperature (16.111) and the power consumption (16.112) are calculated with as the volume flow rate, the inlet stagnation temperature, the exit stagnation temperature, the average specific heat at constant pressure, and as the compressor efficiency. 16.4.1.1 Quasi-dynamic Modeling Using Performance Maps: The compressor components are modeled quasi-dynamically using a steady state performance maps. The compressors are operating between two plena as Fig. 16.23 reveals. For a compressor with up to three-stages, a single performance map is sufficient to calculate the global transient behavior of the compressor. However, if the compressor I Pi Ti

O

Global Compressor

1 Po P i To T i

3

2

Map 1

Map 2

Po To

Performance map GLOBAL-COMP

Three-Stage compressor One performance map

Six-Stage compressor Two performance maps

Fig. 16.23: Modeling the dynamic performance of a multi-stage compressor using (a) one performance map, (b) two or more maps

16 Modeling the Compressor Component, Design and Off-Design

427

has more than three stages, two or more performance maps may be necessary to satisfactorily predict the transient behavior of the compressor. The steady performance map is obviously not able to handle transient events. However, if it is placed between two plena that continuously feed the steady map with unsteady data, it may deliver reasonable dynamic results. The inlet plenum transfers the time dependent pressure and temperature to the global map which calculates the compressor performance and transfers the information to the outlet plenum. For this purpose, the following system of differential and algebraic equation are used: (1) The inlet and outlet plena are described by Eq. (13.9) and (13.11). For the sake of completeness, these equations are listed below:

(16.113)

(16.114)

with Eqs. (16.113) and (16.114) and the performance map described by algebraic equations (16.107), (16.108), (16.110), (16.111) and (16.112), we are now able to quasi-dynamically model the compressor component. 16.4.1.2 Simulation Example: As an example, the global dynamic performance of a three-stage compressor is quantitatively investigated. The compressor, Fig.16.24, running at a constant rotational speed is connected to a large air storage facility similar to the one presented in Fig.16.21. It continuously pumps air into the storage. The system, schematically shown in Fig.16.24, consists of an electric motor drive, a three-stage compressor with the performance maps shown in Fig.16.17 and Fig.16.18

Pipe 1

Electric Motor

2 Global Compressor Performance map

Fig. 16.24: Simulation schematics using a compressor performance map

3

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16 Modeling the Compressor Component, Design and Off-Design

and a pipe that connects the compressor exit with the storage. Air is sucked from the environment at a constant pressure and temperature which is modeled by plenum 1. The storage is represented by plenum 3. Starting from an absolute storage pressure of one atmosphere, air is pumped raising the pressure. Approaching the surge limit at point A, Fig.16.25, the compressor operation experiences an instantaneous breakdown that causes the mass flow to reverse the direction, point B.

3

im

it

Z/ZD=1.0

A

D

pZ/pD

Su

rg

el

B

2.5

2

E

C 1.5

1

Characteristic curve for compressor back flow

-0.2

0

0.2

0.4

0.6 x

0.8

1

x

m/mD Fig. 16.25: Compression, surge, and recovery plotted in performance map

At this point the compressor functions as an energy dissipater with the characteristic curve BC shown in Fig.16.25. The air storage system starts rapidly to depressurize to arrive at point C, which is below the compressor design point D. This depressurization causes the compressor to return to its normal compression mode. The mass flow jumps to the right side, passes through the design point D, reaches the surge limit and repeats the surge cycle, while the electric motor continues to drive the compressor. It should be pointed out that the characteristic curve for the compressor back flow is not a part of the performance map that can be accurately predicted. It is merely an approximation that assumes the compressor channel as an annular pipe filled with blades that are exposed to a reverse flow. The blades function as obstacles that dissipate the mechanical energy of the rotating shaft without being able to convert it into potential energy. Figure 16.26 reflects the details of the surge cycle that includes discharging from A to B and filling from B to E.

16 Modeling the Compressor Component, Design and Off-Design

1.00

429

E

A

0.40

x

x

m/m

ref

0.60

Filling

Discharging

0.80

0.20 0.00 B

-0.20

250

500

750

1000

Time (s) Fig. 16.26: Quantitative details of surge cycle

16.4.2 Module Level 2: Row-by-Row Adiabatic Calculation Procedure This compressor module provides detailed information about the compressor behavior during dynamic events. Two methods can be used for describing this module. The first method utilizes the time dependent conservation laws for continuity, momentum, and energy Eqs. (14. 26), (14.33), and (14.51) derived in Chapter 14. For adiabatic stage compression calculation, Eq. (14.51) is reduced to:

(16.115)

The stage power

in Eq. (16.115) is directly related to the specific stage mechanical

energy with . For a row-by-row dynamic calculation, Eq. (16.115) may be decomposed using the row-parameters discussed in section 16.2.1.2. This method allows a detailed, dynamic calculation of row properties. The second method is based on the row-by-row adiabatic calculation procedure outlined in section 16.2.1.2 under utilization of plena Eqs.(16.113) and (16.114) for dynamic coupling. Three alternative coupling configurations for a two-stage compressor are shown in Fig.16.27. In configuration (a) each row has an inlet and an exit plenum, whereas in (b) each stage is enclosed by an exit and an inlet plenum. In (c) the entire

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16 Modeling the Compressor Component, Design and Off-Design

compressor is placed between an inlet and an exit plenum. This configuration provides satisfactory results for small size compressors such as those of a helicopter gas turbine as shown in Fig 16.27. To accurately account for the volume dynamics of a multi-stage compressor, it is more appropriate to decompose the compressor into several stage groups, where each group may consist of two to three stages that have configuration (b).

Fig. 16.27: Row-by-row plena configuration, (a) each row enclosed by two plana, (b) each stage enclosed by two plena, (c) the entire compressor enclosed by two plena

Fig. 16.28: Decomposition of a six stage compressor into two three-stage groups. Each group is enclosed by two plena. 16.4.2.1 Active Surge Prevention by Adjusting the Stator Blades: The row-by-row calculation procedure not only provides a detailed information about the compression process of individual rows and the entire compressor, but it also enables an active control of compressor instability and surge by adjusting the stagger angles of the stator rows. It is probably the most effective active aerodynamic control mechanism. The configuration of a multi-stage compressor with adjustable stator blade rows is shown schematically in Fig.16.29. As shown in Fig.16.29(a) and Fig.16.29(b), the stagger angle of each individual stator row can be changed according to a Ȗscheme controlled by a multi-variable control system with the row exit pressure as one of its input variables. The effect of the adjustment on the stage velocity diagram is shown in Fig.16.30. Figure 16.30(a) exhibits the velocity diagram pertaining to the stator row in its design stagger position under an adverse operation condition which is associated with an increase in compressor pressure ratio. This pressure ratio, however, is established by a deflection angle ș that may cause a boundary layer separation on stator and rotor blades thus, leading to an inception of rotating stall and surge condition. To prevent this, the stagger angle Ȗ is adjusted resulting in a reduced ș as Fig.16.30(b) shows.

16 Modeling the Compressor Component, Design and Off-Design

431

Fig. 16.29: (a) Multistage compressor with adjustable stator blades, (b) Stagger angle adjustment

E

D E2

3

a

D

2

3

T

V3

E

U3

3

V2 U2

D

W3

W2 b

D

E 3

2

2

T V2

V3

U2

W3

W2

U3

Fig. 16.30: (a) Velocity diagram with increased flow deflection ș as a result of an adverse dynamic operation, (b) adjusted stator blades causing a reduction in flow deflection ș, thus preventing rotating stall and surge

16.4.2.2 Simulation Example: Surge and Its Prevention: This example demonstrates the surge prevention capability of the row-by-row calculation method by simulating the dynamic behavior of an actual single spool gas turbine engine, Fig. 16.30, under an adverse transient operation condition. To force the compressor into an unstable regime, we reconfigured the engine by splitting the shaft and decomposing the gas turbine into a gas generator and a power generator part, as shown in Fig.16.31. The gas generator unit incorporates a multistage compressor that we decompose into a low pressure, an intermediate pressure, and a high pressure compressor, as discussed earlier. The compressor is followed by a combustion chamber, a three-stage turbine that drives the compressor, and a twostage power turbine connected with a generator. The LP, IP, and HP-compressors are modeled using the row-by-row method presented in this paper. The simulation schematic of the engine is shown in Fig.16.32, where the individual components are placed between the plena. It also shows the interaction of individual components with the control system. As seen in Fig.16.32, the speed controller is connected to the power shaft, thus sensing the rotational speed and the temperature of the power shaft and its time derivative to control the fuel mass flow.

432

16 Modeling the Compressor Component, Design and Off-Design

Fig. 16.31: A split shaft gas turbine engine for simulating rotating stall and surge

Fig. 16.32: Simulation schematic of gas turbine shown in Fig. 16.30

Case I: Simulation of Compressor Surge: Starting from a steady state operating point, the dynamic behavior of the above engine is simulated for a transient operation which is controlled by a prescribed generator power schedule that acts on the power shaft as shown in Fig.16.33. The simulation includes a sudden generator load drop from 100% to 25% that lasts about 7 seconds and is followed by a sudden load addition to 100% and a subsequent incremental load decrease to about 25%. The sudden load drop causes the shaft rotational speed to increase. This increase in rotational speed causes the controller to trigger a rapid throttling of the fuel mass flow. The throttling process lasts until a constant idling speed of the power shaft is attained. After about seven seconds, full load is suddenly added and then reduced

16 Modeling the Compressor Component, Design and Off-Design

433

slowly such that after the completion of load addition, the gas turbine is supplying 25% of its rated load, as shown in Fig.16.33. The rotor reacts to this sudden addition of load with a sharp decrease in rotational speed. That, in turn, causes a quick opening of the fuel valve.

40 Power turbine response

Power (MW)

30

Generator power schedule

20 10 0

-10 Gas generator power response

-20

0

2.5

5

7.5

10

12.5

15

Time (s) Fig. 16.33: Generator power schedule, response of power turbine and gas generator

During this process, the power generation capability of the gas generation turbine deteriorates significantly causing a major power imbalance between the turbine and compressor components. This imbalance results in a continuous decrease of the compressor rotor speed, causing the compressor to operate partially in rotating stall and surge regimes. As shown in Fig.16.34, reducing the rotor speed below 90% forces the LP-compressor stage group into rotating stall and a short duration surge process with reversal in the mass flow direction. Since the magnitude of the reversed mass flow is relatively small and of very short duration, a total engine mass flow reversal does not occur. The IP-compressor stage group exhibits similar instability behaviors, where the compressor mass flow reversal occurs at slightly lower frequency and almost the same amplitude. The HP-stage group displays a distinctively different behavior. While the mass flow experiences fluctuation at a similar frequency, the amplitude remains always positive. These fluctuations apparently are not caused by the HP-stage group itself and are propagated downstream from the LP- and IP-parts respectively. This behavior is fully consistent with the continuity requirement that leads to an integrally positive mass flow rate because of the high frequency and short duration mass flow reversal in LP- and IP-compressors.

Compressor Mass Flow (kg/s)

434

16 Modeling the Compressor Component, Design and Off-Design

160 120

LP-Compressor

80 40 0 -40

0

2.5

5

7.5

10

12.5

15

Time (s) Fig. 16.34: LP-compressor operating in surge mode

Case II: Surge Prevention by Stator Stagger Angle Adjustment: To prevent the compressor instability and surge described in Case I, the stagger angles of the LP- and IP-compressor stage groups were dynamically adjusted. Similar to Case I, the engine was forced into an adverse off-design operation condition with the same load schedule as shown in Fig.16.33. Starting from the steady state point, in accordance with the load schedule displayed in Fig.16.35, a generator loss of load was simulated first. The power generation shaft responds to this event with a rapid increase in rotational speed, Fig.16.36, which triggers closing of the fuel valve. As a consequence, without active control, the power of the gas generation turbine would not be sufficient to cover the power consumption of the compressor. As observed in Case I this imbalance of power has led to a decrease in rotational speed of the gas generation shaft, which forced the first two compressor stage groups into an unstable regime. In order to avoid the power imbalance of the gas generation shaft as in Case I, the stagger angle pertaining to the stator blades of the LP- and IP- stage groups are continuously adjusted according to a prescribed dynamics schedule. This procedure starts immediately at the time where the loss of load occurs and lasts until the prescribed Ȗ- values at t = 4s are reached. These values are kept constant for the rest of the simulation. As shown, it is sufficient to reduce the stagger angle of the LP- and IP-compressors, while the stagger angle of the HP-compressor remained unchanged. This intervention causes a substantial shift in the surge limit preventing all three compressor stage groups from entering into the instability regime. As Fig.16.37 shows, the compressor mass flow does not experience any fluctuations. The stable operation of the compressors is reflected in Fig.16.35, where the load schedule and the response of the power generator and gas generator turbines are displayed.

16 Modeling the Compressor Component, Design and Off-Design

435

Fig. 16.35: Power response of gas generator and power turbine shaft following a dynamic stator stagger angle adjustment

Fig. 16.36: Response of the gas generator and power turbine shaft. following a dynamic stagger angle adjustment

436

16 Modeling the Compressor Component, Design and Off-Design

In contrast to the unadjusted Case I shown in Fig.16.33, no power fluctuations are encountered. The rotational speed behavior of the gas generator shaft shown in Fig.16.36 is substantially different from the one shown in Case I. The positive power difference of the gas generation spool plotted in Fig.16.35 is the result of a decreased compressor load caused by stator angle adjustment. It prevents the rotational speed of the gas generation spool from decreasing and moves the compressor operation into a more stable operation regime.

Fig. 16.37: Mass flow transient after stator blade adjustment

16.4.3 Module Level 3: Row-by-Row Diabatic Compression During startup, shutdown, load change, or any other transient operation, there is a temperature difference between the compressor blade, hub, casing, discs, and the compressor working medium. This temperature difference causes a heat transfer from the working medium to the blade material and vice versa. In case of high pressure compressor trains utilized in the industry, the working medium exiting the IPcompressor passes through an inter-cooler, thus substantially reducing its temperature before entering the HP-compressor. Moreover, the trend in the development of future generation of high performance gas turbine engines point toward higher pressure ratios that inherently result in higher temperature requiring cooling of rear stages. In these cases, the compression process is no longer adiabatic, it is termed diabatic compression. Figure 16.38 displays a diabatic compressor stage with the corresponding instantaneous velocity diagram during a transient event, where heat from a working medium at a higher temperature is transferred to stator and rotor blade material, thus, increasing the blade temperature. The total heat added to the stage is the sum of the heat transferred to the stator and to the rotor. Heat transfers from a working medium to the compressor structure and

16 Modeling the Compressor Component, Design and Off-Design /

q=q+q

h 3

//

Q

H3

2

2

P

V3

2

Q

//

q

1

/

437

p l

m

h3

Rotor

Stator

1

2

W2

hs 2

V1

2

W3

V1

p2

h2

hs

H= H

2

P2

V2

P1

h

U

S

2

Q

STAGE-CO-HEAT

p1

Q

h1

R

Fig. 16.38: Diabatic compression process, h-s-diagram thus, blading, hub and casing occurs when the compressor is required to increase the higher pressure ratio. Heat rejection from the blading occurs when the compressor is relaxed. The process of heat transfer described in this section automatically captures both directions of heat transfer. 16.4.3.1 Description of Diabatic Compressor Module: This compressor module provides a detailed information about the compressor behavior during a transient operation that causes heat transferred to/from the blade during the operation. Similar to the adiabatic case discussed in Section 16.4.2, two methods can be used for describing this module. The first method utilizes the time dependent conservation laws for continuity, momentum, and energy Eqs. (12. 26), (12.33), and (12.51) derived in Chapter 12. For diabatic stage compression calculation, Eq. (12.51) is:

(16.116)

The stage power in Eq. (16.116) is directly related to the specific stage mechanical energy with . The heat transferred to the blades is with q as the specific heat (kJ/kg) added/rejected to/from the stage. Based on the sign

438

16 Modeling the Compressor Component, Design and Off-Design

convention in Chapter 5, for compressors the sign the power input automatically assumes a positive sign. However the sign of may be positive or negative based on transient operation discussed below. For steady state case with a constant mass flow, Eq. (16.116) immediately yields the conservation of energy . Thus, the equation of energy is rearranged as:

(16.117)

Equation. (16.117) together with equations of continuity and momentum, and the additional information about the heat transfer coefficient, describes the dynamic behavior of the diabatic compressor stage. For a row-by-row analysis it can be decomposed into two equations that describe the individual stator and rotor rows. For stator row , we find:

(16.118)

For the rotor row, because

and

, we find:

(16.119)

with lm as the stage specific mechanical energy. To completely describe the diabatic compression process with stator and rotor rows, heat transfer equations as well as blade material temperature equations must be added to the set of equations (16.118),

16 Modeling the Compressor Component, Design and Off-Design

439

(16.119), Eqs. (12. 26), and (12.33). The terms in the brackets in Equations (16.118) and (16.119) are of second order, therefore, may be neglected. As a result we obtain for stator:

(16.120)

and for the rotor row

(16.121)

with as net volume of the row space occupied by the working medium. The heat added to or rejected from the working medium in Equations (16.120) and (16.121) is transferred to the stator and rotor blade material. The heat transfer aspect is treated in the subsequent section. 16.4.3.2 Heat Transfer Closure Equations: Utilizing the stator blades as an example, details of heat transfer mechanism are shown in Fig. 16.32. As shown, during the cold start of power generation gas turbine engines or the acceleration phase of aircraft engines the temperature of the working medium quickly reaches its polytropic compression temperature, while the blade material temperature lags behind, Fig. 16.32(a). The temperature difference causes a heat flow from the working medium (air) into the blade material. During the deceleration and shutdown process, Fig. 16.32(b), heat is transferred from the blade material to the working medium. For the sake of completeness, we may assume that the compressor blades are cooled internally. For gas turbine engines in general, this assumption is, at this stage of compressor development, merely a hypothetical one. Its realization requires a high pressure ratio far above the one that is present in the current gas turbine engines. For the future generation of high performance gas turbine with sequential combustion such the one discussed in Chapter 18, for the sake of Carnitization of the gas turbine process, pressure ratios above 40 are necessary that may require compressor blade cooling. In this case, the compressor working medium transfers certain amounts of thermal energy flow or heat flow to the compressor blades. To maintain a desired blade temperature, the cooling mass flow through the

440

16 Modeling the Compressor Component, Design and Off-Design

V1

Q

V1

/

Stator W2

Q

/

W2

//

(a) Cold start Compression transient

Q

U

W3

Rotor

(b) Decompression transient

U

Q

//

W3

Fig. 16.39: (a) Heat transferred from compressor working medium to compressor blades, (b) heat transferred from blade material to the compressor working medium blade internal channels have to remove the heat flow from the blade. The transferred thermal energies flows on hot and cold sides are:

(16.122) In Eq. (16.122) and are the averaged heat transfer coefficients for the blade external hot side and internal cooling channels. Furthermore, , and are the mean temperatures of coolant flow, compressor working medium, wall temperature and are the contact surfaces on blade hot and cold side, respectively. The coupling condition between the cold and the hot side is provided by the material temperature differential equation: (16.123)

The heat flow in Eq. (16.123) may assume positive or negative values, whereas is always negative, since heat is rejected. For the case of no cooling is zero. In case of a dynamic temperature equilibrium, where the blade temperature change approaches zero, Eq. (16.123) is reduced to: (16.124) To calculate and for inserting into Eq. (16.123), we use the Nusselt number correlation as a result of dimensional analysis:

16 Modeling the Compressor Component, Design and Off-Design

441

(16.125) with as the averaged heat transfer coefficient, the blade chord length and k the thermal conductivity. Major parameters affecting the compressor or turbine blade heat transfer are Reynold’s number Re, Prandtl number Pr, Strouhal number Str, turbulence intensity Ti, surface roughness R, and acceleration ratio Ar : (16.126) The dimensionless parameters in Eq. (16.126) contain a group of flow quantities and geometric parameters that are relevant for heat transfer calculations. Besides the Reand Pr- number, the turbulence intensity which is the magnitude of the fluctuation velocity divided by the blade mean velocity V is a major parameter in heat transfer enhancement. Likewise, the Strouhal number that contains the wake passing frequency, the blade chord and the blade mean velocity plays a similar role. Despite tremendous amounts of publications that deal with the heat transfer issues in turbomachinery, no correlation can be found that incorporates all the above parameters. Therefore, we resort to simple correlations such as the one given below: (16.127) The coefficient as well as the exponents m and n in Eq. (16.127) depend upon the type of flow, the surface roughness, and the heat transfer direction. For turbulent flow, empirical correlations by Dittus-Boelter were found to be adequate.

References 1. Lieblein, S., Schwenk, F., Broderick, R.L.: Diffusions factor for estimating losses and limiting blade loadings in axial flow compressor blade elements, NACA RM E53D01 June (1953) 2. Lieblein, S.: Review of high performance axial flow compressor blade element theory, NACA RME 53L22 April (1954) 3. Lieblein, S., Roudebush, W.H.: Theoretical loss relations for low speed two dimensional cascade flow NACA Technical Note 3662 March (1956) 4. Lieblein, S.: Analysis of experimental low-speed loss and stall characteristics of two-dimensional compressor blade cascades, NACA RM E57A28 March (1957) 5. Lieblein, S.: Loss and stall analysis of compressor cascades, ASME Journal of Basic Engineering (Sept. 1959) 6. NASA SP-36 NASA Report (1965) 7. Miller, G.R., Hartmann, M.J.: Experimental shock configuration and shock losses in a transonic compressor rotor at design point NACA RM E58A14b (June 1958)

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16 Modeling the Compressor Component, Design and Off-Design

8. Miller, G.R., Lewis, G.W., Hartman, M.J.: Shock losses in transonic compressor blade rows ASME Journal for Engineering and Power, pp. 235-241 (July 1961) 9. Schwenk, F.C., Lewis, G.W., Hartmann, M.J.: A preliminary analysis of the magnitude of shock losses in transonic compressors NACA RM #57A30 March (1957) 10. Gostelow, J.P., Krabacher, K.W., Smith, L.H.: Performance comparisons of the high Mach number compressor rotor blading NASA Washington, NASA CR1256 (1968) 11. Gostelow, J.P.: Design performance evaluation of four transonic compressor rotors, ASMEJournal for Engineering and Power (January 1971) 12. Seylor, D.R., Smith, L.H.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part I, Design of rotor blading. NASA CR54581, GE R66fpd321P (1967) 13. Seylor, D.R., Gostelow, J.P.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part II, Performance of rotor 1B. NASA CR54582, GE R67fpd236 (1967) 14. Gostelow, J.P., Krabacher, K.W.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part III, Performance of rotor 2E. NASA CR-54583 (1967) 15. Krabacher, K.W., Gostelow, J.P.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part IV, Performance of Rotor 2D. NASA CR-54584 (1967) 16. Krabacher, K.W., Gostelow, J.P.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part V, Performance of Rotor 2B. NASA CR-54585 (1967) 17. Monsarrat, N.T., Keenan, M.J., Tramm, P.C.: Design report, Single stage evaluation of high Mach number compressor stages, NASA CR-72562 PWA3546 (July 1969) 18. Koch, C.C., Smith, L.H.: Loss sources and magnitudes in axial-flow compressors. ASME Journal of Engineering and Power 98(3), 411–424 (1976) 19. Schobeiri, M.T.: Verlustkorrelationen für transsonische Kompressoren. BBCStudie, TN-78/20 (1987) 20. König, W.M., Hennecke, D.K., Fottner, L.: Improved Blade Profile Loss and Deviation Angle Models for Advanced Transonic Compressor Bladings: Part I-A Model for Subsonic Flow, ASME Paper, No. 94-GT-335 21. Schobeiri, M.T.: A New Shock Loss Model for Transonic and Supersonic Axial Compressors With Curved Blades. AIAA, Journal of Propulsion and Power 14(4), 470–478 (1998) 22. Schobeiri, M.T.: Advanced Compressor Loss Correlations, Part I: Theoretical Aspects. International Journal of Rotating Machinery 3, 163–177 (1997) 23. Schobeiri, M.T.: Advanced Compressor Loss Correlations, Part II: Experimental Verifications. International Journal of Rotating Machinery 3, 179–187 (1997) 24. Schobeiri, M.T., Attia, M.: Active Aerodynamic Control of Multi-stage Axial Compressor Instability and Surge by Dynamically Adjusting the Stator Blades. AIAA-Journal of Propulsion and Power 19(2), 312–317 (2003) 25. Levine, P.: Two-dimensional inlet conditions for a supersonic compressor with curved blades. Journal of Applied Mechanics 24(2) (1957)

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443

26. Balzer, R.L.: A method for predicting compressor cascade total pressure losses when the inletrelative Mach number is greater than unity, ASME Paper 70-GT-57 27. Swan, W.C.: A practical method of predicting transonic compressor performance, ASME Journalfor Engineering and Power, vol. 83, pp. 322-330 (July 1961) 28. Smith, L.H.: Private communication with the author and the GE-Design Information Memorandum 1954: A Note on The NACA Diffusion Factor (1995) 29. Seylor, D.R., Smith, L.H.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part I, Design of rotor blading. NASA CR54581, GE R66fpd321P (1967) 30. Seylor, D.R., Gostelow, J.P.: Single stage experimental evaluation of high Mach number compressor rotor blading, Part II, Performance of rotor 1B. NASA CR54582, GE R67fpd236 (1967) 31. Monsarrat, N.T., Keenan, M.J., Tramm, P.C.: Design report, Single stage evaluation of high Mach number compressor stages, NASA CR-72562 PWA3546 (July 1969) 32. Sulam, D.H., Keenan, M.J., Flynn, J.T.: Single stage evaluation of highly loaded high Mach number compressor stages. II Data and performance of a multicircular arc rotor. NASA CR-72694 PWA (1970) 33. Levine, Ph., Two-dimensional inlet conditions for a supersonic compressor with curved blades,Journal of Applied Mechanics, Vol. 24, No. 2 (June 1957) 34. Balzer, R.L.: A method for predicting compressor cascade total pressure losses when the inlet relative Mach number is greater than unity, ASME Paper 70GT-57 35. Swan, W.C.: A practical method of predicting transonic compressor performance. ASME Journal for Engineering and Power 83, 322–330 (1961) 36. Schobeiri, M.T.: A New Shock Loss Model for Transonic and Supersonic Axial Compressors With Curved Blades. AIAA, Journal of Propulsion and Power 14(4), 470–478 (1998) 37. Levine, P.: Two-dimensional inlet conditions for a supersonic compressor with curved blades. Journal of Applied Mechanics 24(2) (1957) 38. Grieb, H., Schill, G., Gumucio, R.: A semi-empirical method for the determination of multistage axial compressor efficiency. ASME-Paper 75-GT-11 (1975) 39. Carter, A.D.S.: Three-Dimensional flow theories for axial compressors and turbines. Proceedings of the Institution of Mechanical Engineers 159, 255 (1948) 40. Hirsch, C.: Axial compressor performance prediction, survey of deviation and loss correlations AGARD PEP Working Group 12 (1978) 41. Swan, W.C.: A practical method of predicting transonic compressor performance, ASME Journal for Engineering and Power, Vol. 83, pp. 322-330 (July 1961) 42. Jansen, W., Moffat, W.C.: The off-design analysis of axial flow compressors ASME, Journal of Eng for Power, pp. 453–462 (1967) 43. Davis, W.R.: A computer program for the analysis and design of turbomachinery, Carleton University Report No. ME/A (1971) 44. Dettmering, W., Grahl, K.: Machzahleinfluß auf Verdichtercharakteristik. ZFW 19 (1971) 45. Fottner, L.: Answer to questionnaire on compressor loss and deviation angle correlations, AGARD-PEP, Working Group 12 (1979)

17 Turbine Aerodynamic Design and Off-Design Performance

As briefly discussed in Chapter 13, within a turbine component, an exchange of mechanical energy (shaft work) with the surroundings takes place. In contrast to compressors, the total energy of the working medium is partially converted into shaft work, thus supplying necessary power to drive a variety of components. In base load power generation area the primary function of a turbine unit is to drive generators supplying the electricity. Considering a large steam turbine shown in Fig. 17.1, the power generation is accomplished by a series of multi-stage turbines that consists of a high pressure part (HP), an intermediate pressure part (IP), and a low pressure part (LP).

Fig. 17.1: A large Brown Boveri steam turbine with an integrated HP, IP-part and two LP-units. Steam exiting from IP-part is divided into two equal portions to feed the two LP-units.

While all three turbine parts are coupled serially driving the generator shaft, the LPpart may consist of two LP-units, each of which is fed by 50% of steam mass flow that exits the IP-turbine. The HP-turbine generally has a blade height to mean diameter ratio of and an aspect ratio of . Figure 17.2 shows a Brown Boveri HP-turbine unit with a control stage followed by a multi-stage HP-part. The relatively short blade height allows two-dimensional blade designs with a constant stagger angle from hub to tip.

446

17 Turbine Aerodynamic Design and Off-Design Performance

Fig. 17.2: HP-turbine unit of a large steam turbine, Brown Boveri

Figure 17.3 exhibits a three-stage research turbine rotor with cylindrical blades (constant profile shape and stagger angle) from hub to tip. The blades are designed to have a minimum profile loss at the mid-section. The cylindrical configuration causes the flow angle to change from hub to tip resulting in higher profile losses and consequently lower efficiencies. The low aspect ratio also causes relatively high secondary flow losses at the hub and tip sections that originate from the secondary flow, as extensively discussed in Chapter 7. Stacking different profiles from hub to tip improves the efficiency only marginally, but it does not counteract the secondary flow cause and effect. As a result, in the past, most of HP-turbines used a cylindrical blade design, accepting higher losses. In recent years, however, the implementation of computational fluid dynamics (CFD) in the turbine design process has made it possible to design HP-units with efficiencies close to 94% by using three-dimensional (3-D) convex bowed blades (also called compound lean), where the pressure surface is bowed in a convex shape as shown in [1]. Before discussing the effect of 3-D blade design on efficiency and performance of turbine components, we will present the necessary aero-thermodynamic tool for a 1-D turbine design, which is a prerequisite for any 3-D design process. Regarding the turbine component modeling for dynamic simulation, similar to the compressor component we discussed in Chapter 16, we present three simulation levels. First, we treat the fundamental aspects of 1-D adiabatic design, followed by a section that deals with generation of global turbine performance characteristics. Three dynamic simulation levels using simple performance characteristics, row-by-row adiabatic, and row-by-row diabatic dynamic performance simulation, concludes the chapter.

17 Turbine Aerodynamic Design and Off-Design Performance

447

Fig. 17.3: The TPFL-turbine rotor with cylindrical blades

17.1 Stage-by-Stage and Row-by-Row Adiabatic Design and Off-Design Performance The unifying treatment of the design calculation procedure outlined in Chapter 16 for compressors is now extended to the turbine component. Stage-by-stage and row-byrow methods for calculating the expansion process within a multi-stage turbine component are presented. The performance behavior of a single- or multi-stage turbine component is completely described by the individual stage characteristics in conjunction with the total pressure losses that we extensively discussed in Chapter 7. The meridional view, the blade row configuration, the expansion and the velocity diagram of a typical multi-stage turbine, are shown in Fig. 17.4.Similar to Chapter 16, using the variables in Fig. 17.4(c) and (d), the following dimensionless parameters are defined:

(17.1)

448

17 Turbine Aerodynamic Design and Off-Design Performance

Fig. 17.4: Meridional view, blade row configuration, expansion and velocity diagram of a typical multi-stage turbine

The dimensionless parameters from Eq. (17.1) are used to treat the following stageby-stage as well as row-by-row design and off-design analysis.

17.1.1 Stage-by-Stage Calculation of Expansion Process Using the characteristics defined in (17.1) in conjunction with the stage efficiency discussed in Chapter 7, the expansion is accurately calculated from (see also Chapters 5 and 16):

(17.2)

17 Turbine Aerodynamic Design and Off-Design Performance

449

As we saw in Chapters 5 and 16, the four equations in (17.2) contain nine unknowns. The following quantities are assumed to be known: (1) the turbine mass flow, (2) the turbine pressure ratio, (3) the type of blades, and (4) their exit metal angles, Į2m and ȕ3m, for each individual stage and their configuration in terms of degree of reaction. Further, we may assume the absolute inlet flow angle of the first stage and the exit flow angle of the last stage to be 90o. With these assumptions, all nine stage characteristics are estimated in the first iteration step, which usually does not deliver the required turbine power at the given conditions (pressure ratio, mass flow, inlet or exit enthalpy). The process of iteration is continued until the enthalpy and mass flow convergence has been reached. This stage-by-stage expansion calculation method delivers accurate results, however, it requires some experience in guessing the initial values for unknown characteristics. In the following row-by-row calculation, we will discuss in more detail how it is computationally more efficient and does not require a lot of iterations.

17.1.2 Row-by-Row Adiabatic Expansion As discussed in Chapter 5, the expansion process within stator and rotor row shown in Fig. 17.5 is summarized as:

(17.3)

The quantities in Eqs. (17.3), the decomposition of the turbine stage in its stator and rotor rows, and the expansion diagram are presented in Fig. 17.5. The quantities with superscripts “/ ” and “// “ in Fig. 17.5, refer to stator and rotor rows, respectively. For the stator row, the expansion diagram shows the details of an adiabatic expansion where the absolute total enthalpy remains constant. The following rotor row with the corresponding expansion diagram exhibits the details of energy balance in a relative frame of reference with constant relative total enthalpy. The expansion diagram for the entire stage exhibits a composite picture of energy balance within the stator and rotor row. We introduce the following efficiency definition for the stator and rotor row:

450

17 Turbine Aerodynamic Design and Off-Design Performance

Fig. 17.5: Energy transfer in turbine stator row, turbine rotor row, and turbine stage

(17.4)

For the sake of completeness, the rotor efficiency definition in Eq. (17.4) includes the difference of circumferential kinetic energies in denominator and numerator. However, this difference may be neglected without causing any noticeable error. The choice of efficiency definition varies throughout the literature (see Traupel [2], Vavra [3], NASA [4]). Engine aerodynamicists and researchers frequently prefer to use their own efficiency definition that may serve their design purpose. Regardless how the efficiency is defined, it must be kept consistently throughout the calculation procedure. We chose definition (17.4) which is consistent with the one defined in Chapter 16. Incorporating Eq. (17.4) into Eqs. (17.3), the isentropic enthalpy difference for stator and rotor row, respectively, is expressed by:

17 Turbine Aerodynamic Design and Off-Design Performance

451

(17.5)

Introducing the dimensionless isentropic enthalpy differences by dividing Eqs. (17.5) by the circumferential kinetic energy at the exit of the turbine stage and incorporating the stage parameters Eqs. (17.2), the dimensionless row parameters are developed as follows:

(17.6)

(17.7)

Equations (17.6) and (17.7) represent the dimensionless isentropic enthalpy differences for the expansion through stator and rotor row, respectively. As the above equations indicate, the calculation of each row necessitates the knowledge of entire stage parameters which requires few iterations. Similar to the compression process outlined in Chapter 16, to substantially reduce the number of iterations, we subdivide the stage specific polytropic mechanical energy lm, as well as the isentropic mechanical energy lms, into two virtual contributions and that we allocate to stator and rotor row. This step does not imply that the stator row is performing shaft power which would violate the energy balance. It merely provides a smooth transition from stator row to rotor row when calculating the compression and expansion processes. With this re-arrangement, we arrive at the contributions allocated to the stator and rotor rows as:

(17.8)

Similarly we find the isentropic parts:

(17.9)

452

17 Turbine Aerodynamic Design and Off-Design Performance

As seen, the sum of stator and rotor contributions leads to the stage specific mechanical energy. Using the polytropic specific mechanical energy expressions given in Eqs. (17.8) and dividing them by the circumferential kinetic energy at the exit of the individual row, the dimensionless row polytropic load coefficients for stator and rotor are:

(17.10)

with stator and rotor flow coefficient , and . Combining Eqs. (17.10) with the efficiency definition for the stator and rotor rows, Eq.(17.4) , the following expressions for the efficiency are obtained:

(17.11)

The isentropic row load coefficient ȥ is defined as the dimensionless isentropic specific mechanical energy for the row. Implementing the efficiency expressions from Eqs. (17.11), the isentropic row load coefficients for the stator and rotor rows are obtained: (17.12)

(17.13)

Similar to compressor component treated in Chapter 16, all the information necessary to pursue the expansion process on the h-s diagram is now available to predict the turbine performance behavior at the design point. Likewise, the off-design operation can be predicted accurately providing that a reliable off-design efficiency calculation method exists. Such a method will be presented in the next section.

17 Turbine Aerodynamic Design and Off-Design Performance

453

Given the pressure and temperature at the inlet, the density can be calculated from the equation of state, as well as the enthalpy and entropy of the gas at the inlet. Utilizing the gas tables, the first point on the h-s diagram can be determined easily. The row flow coefficients ( ) and ( ) are calculated using the continuity equation: (17.14) Given the stator and rotor exit metal angles Į2m, ȕ3m as the input data necessary to describe the geometry, the flow angles Į2, ȕ3 are calculated using deviation angles outlined in Chapter 9. The remaining flow angles ȕ2, Į3 are determined by using the flow coefficient from Eq. (17.14):

(17.15)

From the velocity diagram of Fig. 17.4, we introduce the relationship: (17.16) Implementing Eq. (17.16) into(17.15) and introducing the stator flow coefficient we obtain the expression:

,

(17.17)

Similarly for the rotor: (17.18)

Upon the determination of all the angles involved in the velocity diagram, the velocities and their components are fully described. The flow coefficients, and , and the flow angles Į2, ȕ2, Į3, ȕ3 are necessary tools to determine the polytropic and isentropic enthalpy differences between the inlet and exit of the row. The amount of work produced by the flow is fully represented by the load coefficients Ȝ and ȥ, as presented in the previous sections. Finally, from the energy balance relationships presented above, the complete expansion process for the turbine stage is determined as follows:

454

17 Turbine Aerodynamic Design and Off-Design Performance

(17.19)

The above procedure can easily be repeated for all the rows and stages of the turbine under consideration.

17.1.3

Off-Design Efficiency Calculation

Accurate prediction of the turbine cascade off-design efficiency is essential for calculating performance behavior of a turbine stage and component. For multi-stage axial flow turbines, the primary loss coefficient is the major contributor to the overall stage efficiency. Numerous correlations are given in the literature for off-design efficiency calculation procedure, where the off-design primary loss coefficient is related to its design point value (see Kroon and Tobias [5], Kochendorfer and Nettles [6] ). In a comprehensive and systematic study, Bammert and Zehner [7] and Zehner [8] investigated the off-design behavior of single- and multi-stage turbines with different turbine blade geometry. Based on the experimental results, Zehner [7] established the following relationship for the off-design primary loss coefficient:

(17.20)

with

(17.21)

where represent the stator/rotor inlet flow angle at the off-design point and at the design point respectively. The coefficients a, b are functions of the cascade characteristic described by Zehner [7]. Once the off-design primary loss coefficient has been calculated, the row efficiency is calculated by considering the other individual losses such as secondary and clearance losses (see Chapter 7, Schobeiri [9], Zehner [7]):

17 Turbine Aerodynamic Design and Off-Design Performance

455

(17.22) with as the individual loss coefficient such as primary, trailing edge, secondary, and shock losses. Equations (17.20) and (17.21) explicitly relate the primary loss coefficient of the off-design operation to the design point. A similar relation does not exist for the rest of the losses. This is partially due to the complexity of the secondary flow phenomena. An alternative off-design efficiency correlation presented in [10] directly relates the off-design efficiency to the design efficiency as follows: (17.23) where, the exponent c is expressed for the stator, and rotor as:

(17.24) The coefficients may, among other things, depend on the Renumber, Ma-number, and blade geometry. For a particular set of turbine blade profiles, N-8000, developed by Brown Boveri & Cie, the following coefficients were found:

(17.25)

where, the angles are in degrees and the angle directions follow the convention in Fig. 17.4. Equations (17.23), (17.24), and (17.25) satisfactorily relate the off-design efficiency of typical turbine blades to the design point efficiency. With the procedure described above, it is possible to accurately predict the off-design efficiency of a turbine stage provided that the given off-design mass flow permits a normal turbine operation. However, if the mass flow has been reduced to such an extent that the turbine stage is no longer able to produce mechanical energy, the efficiency has to be redefined in such a way that it reflects the dissipative nature of the energy conversion. To evaluate the degree of dissipation, the following loss coefficients and efficiencies are introduced: (17.26) (17.27)

456

17 Turbine Aerodynamic Design and Off-Design Performance

with and from Fig. 17.4c. For the extreme off-design cases, the relationships between the stage parameters based on the normal turbine stage operation as previously described, are no longer valid. In the following section, a method is presented that predicts accurately the off-design behavior of a turbine stage for extreme low mass flows.

17.1.4

Behavior under Extreme Low Mass Flows

During the shutdown process of a steam turbine or a compressed air storage gas turbine, mass flow and turbine power continuously decrease. By approaching a certain mass flow at which the turbine ceases to produce power, the rear stages of a multi-stage turbine are not able to produce mechanical energy and act as an energy dissipater, Fig 17.6. As a result, the kinetic energy of the rotating shaft dissipates into heat, resulting in excessive temperature and entropy rise.

Stage power L

Theoretical solution

Normal operation

Ventilation range 0.0

Real solution

LV0 x

mmin

x

mT

x

mL=0

x

m

Fig. 17.6: Turbine performance behavior under low mass flow conditions

For the turbine starts to ventilate (wind milling) and is no longer able to remove the heat. In order to prevent the turbine blades and structure from damages caused by excessive temperature development, cooling mass flow must be injected into the flow path to remove the heat. Figure 17.6 shows the detail of the ventilation process. The ventilation process cannot be captured by the method used for predicting the design and normal off-design behavior discussed above. In the following section we present a method discussed in [12] and [13] that accurately predicts the dissipation process.

17 Turbine Aerodynamic Design and Off-Design Performance

457

Consider a turbine stage operating under normal condition. The stage power determined by a second order polynomial in is: (17.28)

The stage flow coefficient flow rate as follows:

in Eq. (17.28) is expressed in terms of the stage mass

(17.29)

where,

is the mass flow parameter defined as:

(17.30) with

as the mean radius and the blade height at the exit of the stage.

Substituting Eqs. (17.29) and (17.30) into (17.28), the power expression is then rewritten:

(17.31) Equation (17.31) accurately represents the stage power behavior under normal operating conditions. However, for , it leads to the erroneous theoretical solution shown in Fig. 17.6 dashed line. For ventilation condition, a different expression for the stage power is written that accurately describes the ventilation branch as a second order polynomial: (17.32)

The constant is found immediately by setting in Eq. (17.32) . This results in as seen from Fig. 17.6. For the calculation of the ventilation power at zero mass flow, , Traupel [2] suggests: (17.33)

458

17 Turbine Aerodynamic Design and Off-Design Performance

where the constant C is obtained from the experiment. Since this constant changes with the blade geometry, it seems appropriate to replace it by a blade geometry functional with G as a geometry parameter: (17.34)

Zehner [7] proposed for G the following relationship that accounts for the blade geometry:

(17.35) with s as the blade spacing, ș the total flow deflection, Ȗ the stagger angle and f the maximum camber height shown in Fig. 17.7.

Stator row

Rotor row

f

f

T

J

T

J

s J = Stagger angle

s f = Maximum camaber height

T = Blade deflection s = Cascade spacing

Fig. 17.7: Blade nomenclature for defining the geometry parameter Ȗ For the functional

we set: (17.36)

with CG = 0.97121. To determine the constant C2 in Eq.(17.32), we assume that the maximum power consumption occurs at zero mass flow rate. This assumption leads to the conclusion that C2 = 0.The constant C1 in Eq.(17.32) is calculated from the requirement that must follow the continuous course of L given by Eq. (17.31) with the same value, and slope at a common tangent point at shown in Fig. 17.6. Equating (17.31) and (17.32) and their respective derivatives at the common tangent point given below:

17 Turbine Aerodynamic Design and Off-Design Performance

459

(17.37) As a result, Eqs. (17.37) determine the mass flow at the tangent point

:

(17.38)

and:

(17.39)

Finally, an expression for the stage power during the ventilation is written as:

(17.40)

Equation (17.40) describes the performance behavior of a turbine stage during the ventilation operation. It can be decomposed into portions that are allocated to stator and rotor row, as discussed previously.

17.1.5

Example: Steady Design and Off-Design Behavior of a MultiStage Turbine

The row-by-row expansion calculation discussed above provides information relevant for performance and efficiency prediction during design and off-design operation. It also supplies essential information related to reliability aspects of the design that includes thermal and mechanical stress analysis of blade rows. This analysis requires accurate information about the temperature and pressure distributions along the expansion path. Considering these aspects, the method discussed above is applied to a multi-stage turbine, where its design and off-design performance is calculated and the results are compared with the measurement. As an appropriate application example, a seven-stage research turbine is chosen, whose geometry, design, and offdesign performance behavior were extensively studied and well documented in [7], [8], and [14]. For this turbine, three different off-design points are calculated and compared with the measurement. The corresponding mass flow and speed ratios are

460

17 Turbine Aerodynamic Design and Off-Design Performance

1.20 0.13 0.20

T/T1

1.00

0.30 0.38 0.50 0.63 0.83

0.80 x mD=11.0 kg/s

P = 1.0

0.60 0.40

0

2

4

6

8 10 Stations

12

14

16

Fig. 17.8: Temperature distribution in a seven stage turbine for design and off-design mass flow rates, experiments denoted by y from [7]

1.2 0.13

1 p/p1

0.20

0.8 0.30 0.38

0.6

0.50 x

0.4 0.2

0.65

x

P = m/mD=1.0

0

2

4

6

8 10 Stations

0.83

12

14

16

Fig. 17.9: Pressure distribution in a seven stage turbine for design and off-design mass flow rates, experiments denoted by y from [7]

17 Turbine Aerodynamic Design and Off-Design Performance

461

given as ȝ = = 0.13, 0.38, 0.83 and . Figure 17.8 exhibits the dimensionless temperature distribution along the expansion path at different stations with mass flow ratio as the parameter. As shown, temperature increases occur at the rear stages for mass flow ratios less than 0.4, indicating the beginning of a dissipation process associated with entropy and temperature increases. Decreasing the mass flow ratio results in strong dissipation with corresponding temperature rise. A similar tendency is displayed by the pressure distribution in Fig. 17.9. As shown, the reduction of the turbine mass flow ratio to 0.13 leads to a considerable distortion of the stage velocity diagram and flow deflection that is associated with dissipation of mechanical energy resulting in pressure increase along the flow path. Figure 17.8 also includes complementary calculations for ȝ = = 0.20, 0.30, 0.50, 1.00.

17.2 Off-Design Calculation Using Global Turbine Characteristics Method This method introduced by Stodola [11] and Traupel [2] replaces the turbine component by a nozzle that can be described by a set of algebraic equations. Horlock [12], Pfeil [13], and Schobeiri [14] derived differential equations for predicting the off-deign behavior of multi-stage turbines. These methods require the knowledge of efficiency behavior of the turbine during off-design operation. The method suggested by Stodola [11] exhibits a simple method for calculating the mass flow. However, it does not provide any details relative to the expansion process. The off-design mass flow is determined from the following expression:

(17.41)

where ʌ is the turbine pressure ratio defined as polytropic exponent given by the expression:

, and n as the

(17.42) with as the isentropic exponent. Accurate determination of the off-design performance behavior of a multi-stage turbine using the global characteristics method requires accurate calculation of the efficiency. This can be done utilizing the above row-by-row calculation procedure, where the dimensionless velocity parameter Ȟ, as defined below, is varied: (17.43) In Eq. (17.43), U is the circumferential velocity and is the isentropic enthalpy difference of the turbine component. Once the efficiency versus Ȟ is obtained, the

462

17 Turbine Aerodynamic Design and Off-Design Performance

relative efficiency can be plotted versus the relative velocity parameter , where the superscript “*” refers to the design point. Figure 17.10 qualitatively shows the relative efficiency versus relative velocity parameter. The circumferential velocity U in Eq.(17.43) may be calculated using an average mean radius for the entire turbine. The characteristic curve of depends on the type of turbine blade geometry and the stage characteristics. Once it is calculated for a particular turbine, it can be expressed as a polynomial such that: (17.44) The coefficients are determined by a least square fit or the data calculated from the steady state row-by-row procedure. The turbine exit temperature may then be determined by the expression: (17.45) and finally, the net power production is calculated by: (17.46) where is the total temperature difference expressed by is the mechanical efficiency, and accounts for bearing friction losses.

, and

1.2

K/ K*

0.9 0.6 0.3 0.0

0

0.5

1

1.5

Q/Q* Fig. 17.10: Relative efficiency as a function of relative velocity parameter

2

17 Turbine Aerodynamic Design and Off-Design Performance

463

17.3 Modeling the Turbine Module for Dynamic Performance Simulation Three levels of turbine modeling are presented in this section. The first level utilizes the steady performance characteristics discussed in Section 17.2. It exhibits the global performance behavior of a turbine component under dynamic operation conditions. The second level uses the row-by-row adiabatic expansion process and delivers a detailed calculation of the adiabatic expansion process under any dynamic operation condition. In addition to the information provided by the second level model, the third level provides detailed information about the blade temperature under the effect of the different cooling procedures.

17.3.1

Module Level 1: Using Turbine Performance Characteristics

The efficiency and performance characteristic generated in Section 17.2 can be used for a first level turbine simulation. This simulation level obviously does not provide details regarding the aerodynamic events within turbine stages. However, it is capable of globally estimating the thermodynamic state of the turbine component during a dynamic event. In contrast to the compressor module, the global turbine module does not have a performance map that defines surge limits for each individual rotational speed. This is due to the accelerating nature of the flow through the turbine expansion path. Although local flow separation may occur on turbine blade surfaces, particularly on the suction surface of low pressure turbine blades, a total flow separation and reversal under normal operating conditions will never occur. However, strong flow separation may occur at extremely low mass flows far below the normal operating range. The global turbine module is modeled mathematically by a set of algebraic equations. It receives dynamic information from the inlet plenum, performs off-design calculations, and transfers the results to the exit plenum. This arrangement allows a quasi dynamic simulation of the turbine component. The set of algebraic equations (17.41) to (17.46)determine the off-design values of the efficiency, the mass flow rate, , the total temperature, To, and the power consumption.

P T

KK

KK

IP

0.9 0.6 0.3 0.0

OP 0.5

1.0

1.5

2.0

QQ

P T

1 P T

1.2

KK

1.2

1.2

0.9 0.6 0.3 0.0

0.5

1.0

1.5

QQ

Charcteristic 1

Global Turbine

2.0

2 P T

0.9 0.6 0.3 0.0

0.5

1.0

1.5

2.0

QQ

Characteristic 2

Turbine-Global

Fig. 17.11: Global simulation of a multi-stage turbine using one turbine characteristic (left) and two characteristics (right)

3 P T

464

17 Turbine Aerodynamic Design and Off-Design Performance

Figure 17.11 shows a turbine component using a steady state performance characteristic enclosed within two plena. For a turbine with up to three stages, a single performance characteristic is sufficient to calculate the global transient behavior of the turbine. However, if the turbine has more than three stages, two or more performance characteristics may be necessary to satisfactorily predict the transient behavior of the turbine. The steady performance characteristic is obviously not able to handle transient events. However, if it is placed between two plena that continuously feed the steady map with unsteady data, it may deliver reasonable dynamic results. The inlet plenum transfers the time dependent pressure and temperature to the global map which calculates the turbine performance and transfers the information to the outlet plenum. For this purpose, the following system of differential equations and the algebraic equations (17.41) to (17.46) are used. The inlet and outlet plena are described by Eq. (13.9) and (13.11). For the sake of completeness, these equations are listed below:

(17.47)

(17.48)

with Eqs.(17.47), (17.48), and the performance characteristic described by algebraic equations (17.41) to (17.46), we are now able to quasi-dynamically model the turbine component.

17.3.2

Module Level 2: Row-by-Row Adiabatic Expansion Calculation

This turbine module provides detailed information about the compressor behavior during dynamic events. Two methods can be used for describing this module. The first method utilizes the time dependent conservation laws for continuity, momentum, and energy Eqs. (12.26), (12.33), and (12.51) derived in Chapter 12. For adiabatic stage expansion calculation, Eq. (12.51) is reduced to:

(17.49)

17 Turbine Aerodynamic Design and Off-Design Performance

465

The stage power L in Eq. (17.49) is directly related to the specific stage mechanical energy with . Similar to the compressor module discussed in Chapter 16 for row- by-row dynamic calculation of a turbine component, Eq. (17.49) may be decomposed using the row parameters discussed in Section 17.1. This method allows detailed dynamic calculation of the row properties. The second method is based on the row-by-row adiabatic calculation procedure outlined in Section 17.1 under utilization of plena Eqs. (17.47) and (17.48) for dynamic coupling. Three alternative coupling configurations for a two-stage turbine are shown in Fig. 17.12. In Configuration (a), each row has an inlet and exit plenum, whereas in (b) each stage is bounded by an exit and inlet plenum. In (c) the entire turbine is placed between an inlet and exit plenum. This configuration provides satisfactory results for gas turbine engines where stage number does not exceed 4. It also can be applied to high pressure steam turbine units, where the volume dynamic is not substantial. However, to accurately account for the volume dynamics of an intermediate or low pressure turbine unit, it is more appropriate to decompose the turbine into several stage groups, where each group may consist of two to three stages that have Configuration (b).

Fig. 17.12: Row-plena configuration, (a) each row is placed between two plane, (b) each stage is placed between two plena, (c) the entire turbine is placed between two plena.

17.3.3

Module Level 3: Row-by-Row Diabatic Expansion

During startup, shutdown, load change, or other transient operation of gas turbine engines, the turbine component experiences adverse temperature changes between the turbine blade material, rotor hub, casing, discs, and the combustion gas. This temperature change causes a heat transfer from the working medium to the blade material and vice versa. In addition, the first three rows of gas turbine engines require cooling. In these cases, the expansion process is no longer adiabatic. Figure 17.13 displays a cooled (diabatic) turbine stage, where heat is removed from the stator and rotor blade material by the stator and rotor cooling mass flows and . Gas turbine manufacturers may use internal, external, or the combination of both cooling schemes. During the past three decades, numerous papers have been published that discuss different cooling aspects. Since a detail discussion of the

466

17 Turbine Aerodynamic Design and Off-Design Performance 1

Q out Q in

4

2

3

Q in

3

m3

m1

Rotor row

Stator row

5

m CR

2

1

m CS

4

5

COOLTU3

Fig. 17.13: Cooled module component (left) and simulation schematic (right)

cooling issues is beyond the scope of this book, we refer to Han et al. [15], who reviewed the relevant papers and provided a comprehensive summary of the relevant turbine heat transfer results. In conjunction with the modeling, we generically use the major cooling techniques. A high efficiency gas turbine engine with advanced cooling technology requires a total cooling mass flow of extracted from the compressor air to cool the front turbine rows. This amount is distributed among the rows that require cooling. Air is extracted from different compressor locations and is injected into the stator and rotor blades. The cooling mass flow at the compressor extraction point has a slightly higher pressure than the pressure at the injection points of the turbine. The mass flow extraction from the compressor causes a decrease of the engine thermal efficiency. However, the thermal efficiency gained by higher turbine inlet temperature offsets the loss caused by the extraction. In case of internally cooled blades, the cooling air mass flow and passes through different channels that have internal turbulators and pin fins for enhancing the heat transfer as schematically shown in Fig. 17.13. The cooling mass flow and may eject at the blade trailing edge or at the blade tip to join the primary mass flow. The trailing edge ejection loss and its optimization were discussed in Chapter 7. As seen, the optimization of the ejection slot geometry minimized the ejection losses. In case of an external cooling that includes, among others, shower head, full coverage film cooling, and transpiration cooling, the cooling mass flow passes though different blade cooling plena and ejects though a number of holes that are arranged on the blade surface with certain spacings and angles. The purpose of external cooling is to build a thin “cool” buffer layer distributed on the blade surface to prevent hot gas from directly transferring heat to the blade material. Because of discrete spacings of holes, a uniform distribution of cooling air is not possible. This leads to a mixing of the cooling mass flow and the primary hot gas. To circumvent this deficiency, a transpiration cooling technique can be used. The cooling passes through uniformly porous material that builds a uniformly distributed protective cool air shield on the blade surface. Although transpiration cooling is the most efficient cooling technique, it has not reached the state of maturity to be implemented into gas turbines.

17 Turbine Aerodynamic Design and Off-Design Performance

467

For turbine blades without internal or external cooling, heat exchange between the blade material and the primary hot gas occurs during off-design operations. Heat transfer from hot gas to turbine blade and structure occurs when the turbine component undergoes a transient power increase by adding fuel in the combustion chamber. On the other hand, heat is rejected from blades, when a transient operation triggers throttling of fuel to reduce the turbine power. As shown in an h-s-diagram in Fig. 17.14, the total heat added to the stage is the sum of the heat transferred to the stator and to the rotor . These partial heat contributions may originate from blade cooling, Fig. 17.14(a), or from transient operation of uncooled blades.

Fig. 17.14: Cooled turbine component with expansion diagram

17.3.3.1 Description of Diabatic Turbine Module, First Method: This turbine module provides detailed information about the turbine behavior during dynamic operations and the heat transferred to/from the blade during a transient event. Similar to the adiabatic turbine case discussed in Section 17.3.2, two methods can be used for describing this module. The first method utilizes the time dependent conservation laws for continuity, momentum, and energy, Eqs. (12. 26), (12.33), and (12.51), derived in Chapter 12. For diabatic turbine stage calculation, Eq. (12.51) is re-written:

468

17 Turbine Aerodynamic Design and Off-Design Performance

(17.50)

The stage power in Eq.(17.50) is directly related to the specific stage mechanical energy with . The heat added to (+) or rejected from(-) the stage is with as the specific heat (heat per unit mass kJ/kg) transferred to/from the stage. For a steady state case with a constant mass flow, Eq.(17.50) immediately yields the conservation of energy . Thus, the equation of energy is rearranged as:

(17.51)

Equation (17.51) together with equations of continuity, momentum, and additional information about the heat transfer coefficient, describes the dynamic behavior of the diabatic turbine stage. For a row-by-row analysis it can be decomposed into two equations that describe the individual stator and rotor rows. For stator row, because , and , we find:

(17.52)

For the rotor row, because

and

, we find:

17 Turbine Aerodynamic Design and Off-Design Performance

469

(17.53)

with lm as the stage mechanical energy. To completely describe the diabatic expansion process within stator and rotor rows, heat transfer equations as well as blade material temperature equations must be added to the set of Eqs. (17.52) , (17.53), (12. 26), and (12.33). The terms in the brackets in Eqs. (17.52) and (17.53) are of second order significance, therefore may be neglected. As a result we obtain for stator:

(17.54)

and for the rotor row,

(17.55)

with as net volume of the row space occupied by the working medium. The heat added to or rejected from the working medium in Equations (17.54) and (17.55) is transferred to the stator and rotor blade material. The heat transfer aspect is treated in the subsequent section. 17.3.3.2 Description of Diabatic Turbine Module, Second Method: The second method is an enhanced diabatic calculation procedure that is based on the row-by-row calculation method outlined in Section 17.1. In this case, the thermal energy supplied to the system is included in the governing equations. The new equation set is combined with the plena Eqs. (17.47) and (17.48) for dynamic coupling. The amount

470

17 Turbine Aerodynamic Design and Off-Design Performance

of heat supplied to the stage is expressed in stage mechanical energy balance, Eq.(5.8): (17.56) with q as the specific heat (heat per unit mass kJ/kg) added to or rejected from the stage. As in Eq. (17.8), to substantially reduce the number of iterations, we decompose the stage specific polytripic mechanical energy lm, as well as the specific isentropic mechanical energy lms, into two virtual contributions and that we allocate to stator and rotor row. This decomposition results in a modified version of Eqs.(17.8) and (17.9). Thus, we arrive at the thermal and mechanical energy contributions allocated to the stator and rotor rows as:

(17.57)

Similarly we find the isentropic parts:

(17.58)

Similar to the polytropic specific mechanical energy expressions given in Eqs. (17.8), we divide Eqs. (17.57) and (17.58) by the circumferential kinetic energy at the exit of the individual row. The dimensionless row polytropic load coefficients for stator and rotor with heat transfer are:

(17.59)

with the dimensionless parameters

and

. The rotor row

parameters in Eq. (17.59) have the same definition as those in Section 17.1.2. For the row-by-row diabatic calculation, the row load coefficients in Eqs. (17.8) must be replaced by (17.59).

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17.3.3.3 Heat Transfer Closure Equations: The heat transfer mechanism for cooled and uncooled blades are shown schematically in Fig. 15.15.

Qh

Qc

Qh

Cooled blades

Qh

Qh

unooled blades

Fig. 17.15: Heat flow is transferred from turbine working medium to turbine blades , the amount of is removed by the cooling medium.

Considering cooled turbine blades, the hot gas transfers certain amounts of thermal energy flow or heat flow to the stator , and rotor . To maintain a desired blade temperature, the cooling mass flow through the blade internal channels has to remove the heat flow from the blade. The transferred thermal energies on hot and cold sides are:

(17.60)

In Eq. (17.60), are the averaged heat transfer coefficients for cold side (internal cooling channels) and the hot side (external flow path through the blade row), , and are the mean temperatures of coolant flow, row working medium, row wall temperature, and AC, Ah are the contact surfaces on the cold and hot side, respectively. The coupling condition between the cold and the hot side is provided by the material temperature differential equation:

(17.61)

the heat flow in Eq. (17.61) may assume positive or negative values, whereas is always negative, since heat is rejected. For the case of no cooling, is zero. In case of a dynamic temperature equilibrium where the blade temperature change approaches zero, Eq. (17.61) is reduced to:

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17 Turbine Aerodynamic Design and Off-Design Performance

(17.62) To calculate and for inserting into Eq. (17.61), we use the Nusselt number correlation, Eq.(16.127), as discussed in Chapter 16. For blades without cooling, generic Nusselt number correlations from Chapter 7 can be used. For determination of heat transfer coefficients for particular cooling schemes, Han et al. [15] offers a comprehensive source.

References 1. Schobeiri, M.T., Gilarranz, J.L., Johansen, E.S.: Aerodynamic and Performance Studies of a Three-Stage High Pressure Research Turbine with 3-DBlades, Design Point and Off-Design Experimental Investigations. ASME paper: 2000GT-484 (2000) 2. Traupel, W.: Thermische Turbomaschinen, 3rd edn. Springer, Heidelberg (1977) 3. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960) 4. NASA SP-290, Turbine Design and Application, vol. 2 (1975) 5. Kroon, R.P., Tobiasz, H.J.: Off-Design Performance of Multistage Turbines. Trans. ASME, Journal of Eng. Power 93, 21–27 (1971) 6. Kochendorfer, F.D., Nettles, J.C.: An Analytical Method Estimating Turbine Performance. NACA Report 930 (1948) 7. Bammert, K., Zehner, P.: Measurement of the Four-Quadrant Characteristics on a Multistage Turbine. Trans. ASME, Journal of Eng. Power 102(2) (1980) 8. Zehner, P.: Vier-Quadranten Charakteristiken mehrstufiger axialer Turbinen. VDI-Forsch.-Bericht. VDI-2, Reihe 6, Nr. 75 (1980) 9. Schobeiri, T.: Thermo-Fluid Dynamic Design Study of Single and Double Inflow Radial and Single-Stage Axial Steam Turbines for Open-Cycle Ocean Thermal Energy Conversion, Net Power Producing Experiment Facility. ASME Transaction, Journal of Energy Resources 112, 41–50 (1990) 10. Schobeiri, M.T., Abouelkheir, M.: A Row-by-Row Off-Design Performance Calculation Method for Turbines. AIAA, Journal of propulsion and Power 8(4), 823–826 (1992) 11. Stodola, A.: Dampf- und Gasturbinen, 6th edn. Springer, Berlin (1924) 12. Horlock, J.H.: Axial Flow Turbines. Robert E. Krieber Publishing Company (1973) 13. Pfeil, H.: Zur Frage des Betriebesverhaltens von Turbinen. VDI-Zeitschrift, Forschung im Ingenieurwesen 41(2), 33–36 (1975) 14. Schobeiri, M.T.: Eine einfache Näherungsmethode zur Berechnung des Betriebsverhaltens von Turbinen. VDI-Zeitschrift, Forschung im Ingenieurwesen 53(1), 33–36 (1987) 15. Han, J.-C., Duta, S., Ekkad, S.: Gas Turbine Heat Transfer Technology. Tylor and Francis (2000)

18 Gas Turbine Engines, Design and Dynamic Performance

A gas turbine engine is a system that consists of several turbomachinery components and auxiliary subsystems. Air enters the compressor component which is driven by a turbine component and is placed on the same shaft. Air exits the compressor at a higher pressure and enters the combustion chamber, where the chemical energy of the fuel is converted into thermal energy producing combustion gas at a temperature that corresponds to the turbine inlet design temperature. The combustion gas expands in the following turbine component, where its total energy is partially converted into shaft work and exit kinetic energy. For power generation gas turbines, the shaft work is the major portion of the above energy forms. It covers the total work required by the compressor component, the bearing frictions, several auxiliary subsystems, and the generator. In aircraft gas turbines, a major portion of the total energy goes toward generation of high exit kinetic energy that is essential for thrust generation. Gas turbines are designed for particular applications that determine their design configurations. For power generation purposes, the gas turbine usually has a single spool. A spool combines a compressor and a turbine that are connected together via

Fig. 18.1: A single-spool power generation gas turbine, BBC-GT9

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18 Gas Turbine Engines, Design and Dynamic Performance

Fig. 18.2: A twin-spool Pratt & Whitney high bypass ratio aircraft engine with multistage compressors and turbines

Fig. 18.3: Schematic of a twin-spool core engine with its derivatives

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475

a shaft. Fig. 18.1 exhibits a single-spool power generation gas turbine, where a 14stage compressor shares the same shaft with a 3-stage turbine. While in power generation gas turbine design the power/weight ratio does not play an important role, the thrust/weight ratio is a primary parameter in designing an aircraft gas turbine. High performance aircraft gas turbine engines generally have twin-spool or multi-spool arrangements. The spools are usually rotating at different angular velocities and are connected with each other aerodynamically via air or combustion gas. Fig. 18.2 exhibits a typical high performance twin spool aircraft gas turbine with a ducted front fan as the main thrust generator. Gas turbine engines with power capacities less than 20 MW might have a split shaft configuration that consists of a gas generation spool and a power shaft. While the turbine of the gas generation spool provides the necessary shaft work to drive the compressor, the power shaft produces the net power. In addition to the above design configurations, a variety of engine derivatives can be constructed using a core engine as shown in Fig. 18.3.

18.1 Gas Turbine Steady Design Operation, Process Starting with the single-spool power generation gas turbine that consists of a multistage compressor, a combustion chamber, and a turbine, the h-s diagram is shown in Fig. 18.4(a). The compression process from 1 to 2 is accomplished by the compressor with a polytropic efficiency that can be accurately calculated using the row-by-row or stage-by-stage methods discussed in Chapters 16. The combustion process from 2 to 3 is associated with certain total pressure loss coefficient thus, it is not

Fig. 18.4: H-s-diagram of a single spool power generation gas turbine (a) and a twinspool aircraft engine (b)

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18 Gas Turbine Engines, Design and Dynamic Performance

considered isobaric. The expansion process from 3 to 4 causes an entropy increase that is determined by the turbine efficiency. Figure 18.4(b) shows the h-s diagram for a twin spool aircraft engine. In contrast to the single spool engine, the compression process is accomplished by two compressors that are operating at two different angular velocities. Air enters the low pressure compressor (LP-compressor) driven by the LP-turbine and is compressed from 1 to 2. Further compression from 2 to 3 occurs in the high pressure compressor (HP-compressor) driven by the HP-turbine. After addition of fuel in the combustion chamber, fist expansion occurs in the HPturbine, whose power exactly matches the sum of HP-compressor power and the power required to compensate bearing frictions. Second expansion in LP-turbine matches the power by the LP-compressor, bearing friction, and the auxiliary subsystems. In off-design operation, there is always a dynamic mismatch between the turbine and compressor power that changes the spool rotational speed. This dynamic mismatch is brought to an equilibrium by taking appropriate control measures, as we discuss in the following sections. As mentioned briefly, small gas turbines may have a split shaft configuration as shown in Fig. 18.5. The single spool gas generator unit provides the power turbine with combustion gas that has the required pressure and temperature to produce the power. As seen in Fig. 18.5(a), the specific turbine enthalpy difference is , leaving the rest enthalpy difference for power generation. Figure 18.5(b) shows the h-s diagram of a high efficiency power generation gas turbine. The schematic cross section of this gas turbine is shown in Fig. 18.9. It consists of a multi-stage compressor, C, a combustion chamber, CC1, providing a lean combustion gas that expands in a single-stage reheat turbine, RT. The exhaust gas from RT enters the second combustion chamber, CC2, where the remaining fuel

Fig. 18.5: H-s-diagram of a single spool power generation gas turbine with a power shaft (a) and a single spool power generation gas turbine with a reheat stage and two combustion chambers (b)

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is added to insure a stoichiometric combustion. It expands in the multi-stage turbine that produces the major portion of power. As seen in the following sections, the implementation of the reheat process substantially increases the thermal efficiency of gas turbines. The underlying thermodynamic principles of this concept is the reheat process, which has been very well known in steam turbine design for more than a century. However, in gas turbine design, adding a second combustion chamber to a conventionally designed gas turbine seemed to be associated with unforeseeable problems. Based on design experiences with Compressed Air Energy Storage (CAES) gas turbines with two combustion chambers, Brown Boveri designed and successfully manufactured the first series of power generation gas turbines with a reheat stage and two combustion chambers.

18.1.1 Gas Turbine Process Accurate prediction of thermal efficiency of a gas turbine engine requires the knowledge of the compressor, combustion chamber, and turbine efficiencies as well as the bearing losses and the losses in auxiliary systems. Furthermore, detailed knowledge of the amount of mass flows with their extraction and injection pressures for cooling the turbine blades and the rotor discs are necessary. In addition, a detailed gas table that accounts for thermodynamic properties of humid air as well as the properties of the combustion gas must be implemented into the calculation procedure. Assuming air and combustion gas as calorically perfect gases results in significant errors. Figure 18.6 exhibits a schematic diagram that shows in detail the extraction of different cooling mass flows and their injection locations. Mass flow through P1 extracted from plenum 3 cools the rotor and does not participate in power generation; mass flows through P2 and P3 cool the second and first turbine stages and remain in the system; and finally, mass flow through P4

Fig. 18.6: Schematic of a single spool gas turbine illustrating the mass flow extraction from compressor for different cooling purposes

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18 Gas Turbine Engines, Design and Dynamic Performance

reduces the combustion chamber exit temperature before it enters the turbine. At stations 6, 7, 8, and 11 humid air is mixed with combustion gas resulting in a local change of water/air and fuel/air ratios, therefore changing the entire thermodynamic properties including the special gas constant R. In the absence of the above information, reasonable assumptions relative to component efficiencies can be made to qualitatively determine the thermal efficiency and its tendency with regard to parameter variation. In the following section, a simple thermal efficiency calculation procedure is derived that is appropriate for varying different parameters and qualitatively determining their impacts on thermal efficiency.

Fig. 18.7: Simple sketch of a gas turbine with recuperator

The gas turbine with its corresponding process is sketched in Fig. 18.7. It consists of a compressor, a recuperator, a combustion chamber and a turbine. Exhaust gas from the turbine is diverted into the recuperator heating up the compressed air before entering the combustion chamber. The individual processes are compression, expansion, fuel addition and combustion, and heat exchange in recuperator. The compressor and turbine enthalpy differences are calculated from:

(18.1)

We introduce the following definitions for the recuperator air- and gas-side (RA, RG), as well as combustion chamber (CC) pressure loss coefficients:

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479

(18.2)

The thermal efficiency is defined as:

(18.3) The specific net power is calculated from:

(18.4)

with the fuel air ratio . Replacing the specific turbine power enthalpy difference from Eq. (18.1), we find:

by the

(18.5)

Equation (18.5) expresses the isentropic enthalpy difference in terms of a product of averaged specific heat at constant pressure and the isentropic temperature difference. The specific heat in Eq.(18.5) exhibits an averaged value between the two given temperatures:

(18.6) The temperature ratio in Eq. (18.5) can be related to the pressure ratio as follows:

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18 Gas Turbine Engines, Design and Dynamic Performance

(18.7)

with Eq. (18.7), Eq. (18.5) becomes:

(18.8)

Because of the pressure losses across the combustion chamber, the turbine and compressor pressure ratios are not the same (ʌT g ʌc). Implementing the pressure losses of the combustion chamber and recuperator air side, we find:

(18.9)

We set the fraction on the right hand side of Eq. (18.9): (18.10) and arrive at: (18.11) For parameter variation, following values may be used: , . Following exactly the same procedure defined by Eqs. (18.4) through (18.11), we find the compressor specific work as: (18.12) Inserting Eqs. (18.8) and (18.12) into Eq. (18.4), we arrive at:

(18.13)

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481

The turbine inlet temperature T3, the environmental temperature T1, and thus their ratio T3 /T1 is considered as a known parameter. This parameter can also be used for parametric studies. Therefore it is desirable to express the ratio T5 /T1 in terms of T3 /T1.We find this ratio by utilizing the recuperator effectiveness ȘR: (18.14) From compressor and turbine energy balance, Eq. (18.1) we find

(18.15)

Equation (18.15) in dimensionless form yields:

(18.16)

Introducing the temperature ratio

, the temperature ratio

Eq. (18.16)

becomes:

(18.17)

To determine the temperature ratio

, we re-arrange Eq. (18.14)

(18.18)

Using Eqs. (18.16) and (18.17), Eq. (18.18) is re-arranged as (18.19)

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Introduce Eqs. (18.19) and the definition

into Eq. (18.13), the thermal

efficiency equation for a gas turbine with a recuperator is written as:

(18.20)

From Eq. (18.20) special cases are obtained. Setting gives the thermal efficiency of a gas turbine without recuperator. The ideal case of Brayton cycle is obtained by setting all loss coefficients equal to zero, all efficiencies equal to unity, and Equation (18.20) properly reflects the effects of individual parameters on the thermal efficiency and can be used for preliminary parameter studies. As an example, Fig. 18.8 shows the effect of pressure ratio, the turbine inlet temperature, and the component efficiency on thermal efficiency for two different cases. As Fig. 18.8 shows, for each turbine inlet temperature, there is one optimum pressure ratio. For temperature ratios up to ș = 3.5 pronounced efficiency maxima are visible within a limited ʌ-range. Approaching higher inlet temperature, however, this range widens significantly.

0.45

0.45 T=4.5 T=4.5 4.0

4.0 0.30 3.5

0.15

3.0

Case 2: KT=90% KC=80%

3.0

Case 1: KT=90% KC=90% 5

10

S

T=4.5 4.0

Kth

Kth

3.5

0.15

0.00

T=4.5 4.0 0.30

15

20

0.00

5

3.0

10

3.5

3.0

S

15

3.5

20

Fig. 18.8: Thermal efficiency as function of pressure ratio with turbine inlet temperature ratio as parameter for (a) gas turbine with recuperator, dashed curves and (b) without recuperator, solid curves. , for Case 1 and Case 2. In Case 2 the turbine efficiency is lowered from 90% to 80%. For a gas turbine without recuperator, the thermal (efficiency solid curves in Fig. 18.8) shows that for ș = 4.0, increasing the pressure ratio above 15 does not bring any noticeable efficiency increase. However, it requires the compressor to have one or two more stages. The temperature ratio ș = 4.0 corresponds to a turbine inlet temperature of at a compressor inlet temperature of .

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Figure 18.8, dashed curves indicate that tangibly higher thermal efficiencies at a substantially lower pressure ratio can be achieved by utilizing recuperators. This is particularly advantageous for small gas turbines (so called “micro-turbines”) with power ranging from 50 to 200 kW. The required low maximum pressure ratio can easily be achieved by a single-stage centrifugal compressor. Comparing Case1 and Case 2 in Fig. 18.8 shows that thermal efficiency reduces if low efficiency components are applied.

18.1.2 Improvement of Gas Turbine Thermal Efficiency The above parameter study indicates that for a conventional gas turbine with a nearoptimum pressure ratio with or without the recuperator, the turbine inlet temperature is the parameter that determines the level of thermal efficiency. For small size gas turbines recuperator is an inherent component of the gas turbine. For large power generation gas turbines, however, it is not a practical option. Using a recuperator in a large gas turbine requires significantly lower pressure ratio that results in a large volume recuperator and turbine. As a result in order to improve the thermal efficiency of conventional gas turbines, increasing the turbine inlet temperature seems to be the only option left. Considering this fact, in the last three decades, gas turbine manufacturer have been concentrating their efforts to introduce more sophisticated cooling technologies that is essential for increasing the turbine inlet temperature of conventional gas turbines.

Fig. 18.9: A schematic cross section of GT-24 gas turbine engine with a single stage reheat turbine and a second combustion chamber

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To substantially improve the thermal efficiency without a significant increase in turbine inlet temperature, a well known reheat principle as a classical method for thermal efficiency augmentation is applied. Although this standard efficiency improvement method is routinely applied in steam turbine power generation, it did not find its way into the aircraft and the power generation gas turbine design. The reason was the inherent problem of integrating a second combustion chamber into a conventionally designed gas turbine engine. This issue raised a number of unforeseeable design integrity and operational reliability concerns. ABB (formerly Brown Boveri & Cie) was the first to develop a gas turbine engine with one reheat stage turbine followed by a second combustion chamber and a multi-stage turbine, Fig. 18.9.

Fig. 18.10: Comparison of a conventional baseline gas turbine process (a) with the GT-24 process (b) from [1]

A comparative study by Schobeiri [1] emulates, among others, two conceptually different power generation gas turbine designs utilizing components, whose detailed aerodynamic performance characteristics were known. The first one is a conventional gas turbine, whereas the second one has a reheat turbine stage and a second combustion chamber resembling GT-24 [2]. Starting from a given environmental condition (pressure, temperature) and a consolidated turbine inlet temperature T3BL = 1200 oC for both engines, Fig 18.10a, the thermal efficiency is determined by the compressor pressure ratio and the compressor and turbine polytropic efficiencies and is plotted in Figure 18.11a.As curve 1 shows, for the given pressure ratio, which is not identical with the optimum pressure ratio, an efficiency of is calculated. Substantial efficiency improvement is achieved by introducing a singlestage reheat principal as applied to GT-24. Details of the process are sketched in Fig. 18.10b with the baseline process as the reference process. The vertically crosshatched area in Fig. 18.10b translates into the efficiency improvement, which in the

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485

case of GT-24 resulted in efficiency improvement of 5.5% above the baseline efficiency. A detailed dynamic engine simulation of the GT-24 gas turbine engine with GETRAN© [3] verified a thermal efficiency of Șth = 40.5% plotted in Fig. 18.11a, curve 2. This tremendous efficiency improvement was achieved despite the facts that (a) the compressor pressure ratio is much higher than the optimal one for baseline engine and (b) the introduction of a second combustion chamber inherently causes additional total pressure losses. Further calculation showed that introducing a third combustion chamber would result only in a marginal improvement of 1 to 1.5% thermal efficiency, which does not justify the necessary R&D efforts to integrate a third combustion chamber. The specific work comparison is plotted in Fig. 18.11(b), which shows a significant increase in specific work. 450

0.45

(b)

(a) 400

0.4

Kth

w (KJ/kg)

(2) GT-24/26

0.35

350 300

0.3 0.25 5

(2) GT-24/26

(1) Baseline GT

(1) Baseline GT

10

15

20

25

S

30

35

40

45

250

5

10

15

20

25

S

30

35

40

45

Fig. 18.11: Comparison of efficiency and specific work between a conventional baseline gas turbine and GT-24, (a) efficiency, (b) specific work Additional efficiency improvement requires a major technology change. As Schobeiri [2] showed, major improvement can be achieved by using the UHEGT-technology (ultra high efficiency gas turbine technology). This technology eliminates the combustion chambers altogether and places the combustion process inside the stator and rotor blade passages (see section 18.6).

18.2 Non-linear Gas Turbine Dynamic Simulation The continuous improvement of efficiency and performance of aircraft and power generation gas turbine systems during the past decades has led to engine designs that are subject to extreme load conditions. Despite the enormous progress in the development of materials, at the design point, the engine components operate near their aerodynamic, thermal, and mechanical stress limits. Under these circumstances, any adverse dynamic operation causes excessive aerodynamic, thermal, and subsequent mechanical stresses that may affect the engine safety and reliability, and, thus, the operability of the engine if adequate precautionary actions are not taken. Considering these facts, an accurate prediction of the above stresses and their cause is critical at the early stages of design and development of the engine and its components. This section focuses on simulation of dynamic behavior of gas turbine

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18 Gas Turbine Engines, Design and Dynamic Performance

engines and their components. The simulation spectrum encompasses single- and multi-spool gas turbine engines, turbofan engines, and power generation gas turbine engines. The simulation concept is based on a generic modularly structured system configurations. In the last six chapters, the gas turbine components were represented by individual modules described mathematically by systems of differential equations. Based on these and other necessary modules, a generic concept is presented that provides the reader with necessary tools for developing computer codes for simulation of arbitrary engine and plant configurations ranging from single-spool thrust generation to multi-spool thrust/power generation engines under adverse dynamic operating conditions. It can easily be extended to rocket engines, combined cycles, co-generation cycles and steam power plants. A multi-level system simulation treats different degrees of complexity ranging from global adiabatic simulation to detailed diabatic one. The dynamic behavior of the subject engine is calculated by solving a number of systems of partial differential equations which describe the unsteady behavior of the individual components. Accurate prediction of the dynamic behavior of the engine and the identification of critical parameters by the method enables the engine designer to take appropriate steps using sophisticated control systems. The method may also be used to proof the design concept of the new generation of high performance engines. The modular structure of the concept enables the user to independently develop new components and integrate them into the simulation code. As representative examples, four different case studies are presented that deal with dynamic simulation of a compressed air energy storage gas turbine, different transient cases with single- and multi-spool thrust and power generation engines were simulated. The transient cases range from operating with a prescribed fuel schedule, to extreme load changes and generator shut down.

18.2.1 State of Dynamic Simulation, Background Dynamic behavior of aircraft gas turbine engines was investigated earlier by several researchers using the component performance map representation for simulating engines. Seldner et al. [4] and Koenig and Fishbach [5] utilized overall component performance maps in their simulation program GENENG, which performs purely steady-state computations. In order to account for system dynamics, Seller and Daniele [6] extended the above code by introducing simplified dynamic equations. In a report about a hybrid simulation of single- and twin-spool turbofan engines, Szuch[7] described the representation of engine components by overall performance maps. To estimate gas turbine starting characteristics, Agrawal and Yunis [8] generated a set of steady component characteristics, where the turbine and compressor components are represented by overall steady performance maps. The engine representation by performance maps, as briefly addressed above and comprehensively discussed by Schobeiri [3], exhibits a useful tool for approximating engine behavior within the operation range defined by the component maps. However, the detailed information that is crucial for engine development and design cannot be provided at this simulation level. Furthermore, the above representation is not viable for providing

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the control system designer with the necessary input parameters, such as those describing the aero-thermodynamic and structural conditions of the compressor and turbine blade rows. Consequently, the response of the real system to the intervention of the controller cannot be verified. These and other parameters are required inputs to the controller for triggering precautionary actions such as active surge control, achieved by adjusting variable compressor stator blades. In order to address the above issues, Schobeiri [9], [10], [11], [12], [13], [14] developed the modularly structured computer code COTRAN for simulating the nonlinear dynamic behavior of single-shaft power generation gas turbine engines. To account for the heat exchange between the material and the working fluid during a transient event, diabatic processes are employed in COTRAN for combustion chamber and recuperator components. The dynamic expansion process through the turbine component is accomplished by a row-by-row calculation using the stage characteristics. COTRAN reflects real engine configurations and components, and is routinely used at the early stages of design and development of new gas turbine engines. Dynamic simulations of different single-shaft engines performed with COTRAN were reported by Schobeiri and Haselbacher [15]. Although COTRAN represents an advanced, nonlinear dynamic code, its simulation capability is limited to single-shaft power generation gas turbine engines and, thus, cannot be used for simulating multi-spool aircraft engines. Considering this circumstance, Schobeiri and his co-workers [3], [16], [17] developed a new computational method with the corresponding, generic, modularly structured computer code GETRAN for simulating the nonlinear dynamic behavior of single- and multi-spool high pressure core engines, turbofan engines, and power generation gas turbine engines. The code is capable of simulating aircraft engines having up to five spools with variable geometry, with or without additional power generation shafts.

18.3 Engine Components, Modular Concept, Module Identification A schematic component arrangement and modeling of a twin-spool core engine is shown in Fig.18.12. The corresponding modules are implemented into the engine modular configuration schematic Fig.18.13. Figures 18.14 and 18.15 display the lists of components with their corresponding modular representations and symbols that are described by the method presented in Chapters 14 through 19. They exhibit the basic components essential for generically configuring any possible aero- and power generation gas turbine engines. These modules are connected with each other with a plenum, which is a coupling component between two or more successive components. As briefly explained in Chapters 14, the primary function of the plenum is to couple the dynamic information of entering and exiting components such as mass flow, total pressure, total temperature, fuel/air ratio, and water/air ratio. After entering the plenum a mixing process takes place, where the aforementioned quantities reach their equilibrium values. These values are the same for all outlet components.

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Fig. 18.12: Schematic of a twin spool core engine, component decomposition

Fig. 18.13: Modular configuration of engine exhibited in Fig.18.12

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489

A survey of power and thrust generation gas turbine engines has led to the practical conclusion that any arbitrary aircraft or power generation gas turbine engine and its derivatives, regardless of configuration, i.e., number of spools and components, can be generically simulated by arranging the components according to the engine configuration of interest. The present nonlinear dynamic method is based on this generic, modularly structured concept that simulates the transient behavior of existing and new engines and their derivatives. The modules are identified by their names, shaft number, and inlet and outlet plena. This information is vital for automatically generating the system of differential equations representing individual modules. Modules are then combined into a complete system which corresponds to the engine configuration. Each module is physically described by the conservation laws of thermo-fluid mechanics which result in a system of nonlinear, partial differential or algebraic equations. Since an engine consists of a number of components, its modular arrangement leads to a system containing a number of sets of equations. The above concept can be systematically applied to any aircraft or power generation gas turbine engine. The general application of the modular concept is illustrated in Figs. 18.12 and 18.13. The twin-spool engine shown in Fig. 18.12 exemplifies the modular extension of the single-spool base engine. It consists of two spools with the shaft S1 and S2, on which the low and high pressure components such as compressors and turbines are assembled. The two shafts are coupled by the working media air and combustion gas. They rotate with different speeds which are transferred to the control system by the sensors Ns1 and Ns2. Air enters the inlet diffuser D1, which is connected with the multi-stage compressor assembled on S1, and is decomposed in several compressor stages C1i. The first index, 1, refers to the spool number and the second index, i, marks the number of the compressor stage. After compression in the S1 compressor stage group, the air enters the second compressor (HP- compressor) assembled on the S2 shaft that consists of stages C21-C25. In the combustion chamber (CC1) high temperature combustion gas is produced by adding the fuel from the tank FT. The gas expands in the high pressure turbine that consists of stages T21 - T23. By exiting from the last stage of HP-turbine, the combustion gas enters the low pressure turbine consisting of stages T11 - T13 and is expanded through the exit nozzle. Two bypass valves, BV1 and BV2, are connected with the compressor stator blades for surge prevention. The fuel valve, FV1, is placed between the fuel tank, FT, and the combustion chamber, CC1. The pipes, Pi, serve for cooling air transport from the compressor to cooled turbines. The compressor stage pressures, the turbine inlet temperature, and the rotor speed are the input signals to the control system, which controls the valve cross sections and the fuel mass flow.

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Gas Turbine Generic Components, Modules, and Symbols

Fig. 18.14: Components, modules, and their symbols: Plenum, Control system CS, Shaft S, with moment of inertia I and the rotational velocity Ȧ, Speed sensor N, Valve with an arbitrary ramp for closing and opening the cross section S, Adjusting mechanism AM for stator blade adjustment, Subsonic nozzle N, Subsonic diffuser D, Supersonic Diffuser SSD, Supersonic nozzle SSN, Recuperator R, Combustion Chamber CC, and Afterburner AB

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Fig. 18.15: Adiabatic turbine stage with the module T, adiabatic compressor stage with the module C, cooled turbine stage with the module CT

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Fig. 18.16: Schematic of a 3-spool high performance core engine, component decomposition

Fig. 18.17: Modular configuration of the 3-spool engine shown in Fig. 18.16. The three compressors and turbines are connected aerodynamically. The plena addressing, the spool number, and the component number uniquely identify the set of differential equations that describe the module.

18 Gas Turbine Engines, Design and Dynamic Performance

493

Figure 18.16 shows a more complex example of a three-spool supersonic engine with its modular decomposition. Figure 18.17 exhibits a systematic modular configuration of Fig. 18.16 that is represented by a large system of differential and algebraic equations.

18.4 Levels of Gas Turbine Engine Simulations, Cross Coupling Accuracy of gas turbine dynamic simulation is determined by the level of component modeling. It increases by increasing the level of simulation complexity. Four levels of simulation are introduced: Zeroth Simulation Level: Is applied to simple cases such as those in [4] through [7], utilizing a fixed system configuration with steady state component characteristics that are described by algebraic equations, simplified differential equations, and lookup tables and maps. Furthermore, there is no dynamic coupling between the components. Since this simulation level does not account for engine dynamics, it will not be discussed further. First Simulation Level: This level uses component global performance map only for turbines and compressors. The maps are generated using the row-by-row adiabatic calculation method detailed in Chapters 18 and 19. The other components such as recuperators, coolers, combustion chambers, pipes, nozzles, and diffusers are simulated according to methods discussed in Chapters 14 through 17. Primary air, secondary combustion gas, and metal temperature of the combustion chamber are calculated. All modules are coupled with plena insuring a dynamic information transfer to all modules involved. Modules are described by algebraic and differential equations. Second Simulation Level: This level utilizes adiabatic row-by-row or stage-by-stage calculation for compressor and turbine modules. For combustion chamber, primary air, secondary combustion gas, and metal temperature are calculated. Dynamic calculations are performed throughout the simulation, where the modules are coupled by plena. Each module is described by differential and algebraic equations. Third Simulation Level: This level uses diabatic row-by-row calculation for compressor and turbine modules. This level delivers a very detailed diabatic information about the compressor and turbine component dynamic behavior. It utilizes cooled turbine and compressor stages and simultaneously calculates the blade temperatures. For combustion chamber, primary air, secondary combustion gas, and metal temperature are calculated. Dynamic calculations are performed throughout the simulation, whereas the modules are coupled by plena. Each module is described by differential and algebraic equations. The details of information delivered by this level and degree of complexity is demonstrated by the following example. The first two stages of a four-stage turbine component of high performance gas turbine engine must

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18 Gas Turbine Engines, Design and Dynamic Performance

be cooled. For the first four turbine rows we use the diabatic expansion process that requires three differential equations for describing the primary flow, three differential equations for describing the cooling flow, and one differential equation for describing the blade temperature. This leads from two cooled turbine stages to 28 differential equations. The generic structure allows to cross-couple level 1 to 3. For example, we wish to simulate a gas turbine engine with a global compressor performance map, but need to obtain detailed information about turbine blade temperature, which is necessary to calculate the relative expansion between the blades and the casing, then we may use the diabatic calculation method. In this case, we cross-couple the first and third level simulation.

18.5 Non-linear Dynamic Simulation Case Studies Three different case studies dealing with three completely different gas turbine systems are presented. Table 18.1 shows the matrix of the cases where the engine types and transient-type simulations are listed. These studies demonstrate the capability of the generic structured method we discussed in Chapters 15 through 18 to dynamically simulate complex systems with high accuracy.

Table 18.1: Simulation Case Studies TESTS

GAS TURBINE TYPE

TRANSIENT TYPE

CASE 1

CAES: Compressed air energy storage power generation gas turbine engine, zero-spool, single shaft, two turbines, two combustion chambers

Generator trip, emergency shut down.

CASE 2

Single-spool, single-shaft, power generation gas turbine engine, BBC-GT9

Adverse load changes.

CASE 3

Three-Spool, four-shaft, thrust and power generation core engine

Operation with fuel schedule.

The case studies presented in this chapter are related to the real world engine simulation and are intended to provide the reader with an insight into the non-linear engine dynamic simulation. The selected cases ranging from zero-spool, single shaft power generation to three-spool four shaft thrust and power generation gas turbine engines provide detailed information about the engine behavior during design and off-

18 Gas Turbine Engines, Design and Dynamic Performance

495

design dynamic operation. For each engine configuration the simulation provides aero-thermodynamic details of each individual component and its interaction with the other system components. Since the presentation of the complete simulation results of the three cases listed in Table 1 would exceed the frame of this chapter, only a few selected plots will be displayed and discussed for each case.

18.5.1 Case Study 1: Compressed Air Energy Storage Gas Turbine The subject of this case study is a zero-spool, single-shaft compressed air energy storage (CAES) gas turbine[18], which is utilized to efficiently cover the peak electric energy demand during the day. Continuous increase of fuel costs has motivated the power generation industry to invest in technologies that result in fuel saving. Successful introduction of combined cycle gas turbines (CCGT) has drastically improved the thermal efficiency of steam power plants that is equivalent to a significant fuel saving. Further saving is achieved by using the excess electrical energy available during the period of low electric energy demand (6-8 hours during the night) to compress air into a large storage. During the peak demand, the compressed air is injected into the combustion chambers and is mixed with the fuel. After the ignition process is completed, the high pressure, high temperature gas expands in the turbine generating electric energy for about 2 to 4 hours. In contrast to a CCGT, the period of operation of a CAES plant is restricted to a few hours per day resulting in a daily startup followed by a shutdown procedure. This relatively high frequency of startups and shutdowns may cause structural damages resulting in reduced life time if the startup and shutdown procedures are not performed properly. The condition for a safe startup procedure is outlined in this study that helps the engine and control system designer to integrate into their design procedure. The CAES gas turbine system, Fig. 18.18, with the simulation schematic shown in Fig. 18.19 features a large volume plenum (8) for storing the compressed air, a high-pressure combustion chamber (HPCC), a high-pressure turbine (HPT), a low-pressure combustion chamber (LPCC), a low-pressure turbine (LPT2), a cold-air pre-heater with a low pressure and high pressure side (LPP and HPP-side) and a generator (G). During the steady-state turbine operation, cold air from the air-storage facility, plenum 8, passes through the shutdown valve (V1) to the inlet plenum (1), where it is divided into combustion and cooling-air flows. The addition of fuel in HPCC causes the combustion air to be heated up to the combustion chamber's exit temperature. Immediately upstream of HPT, the combustor mass flow mixed with a portion of the cooling-air flow, which has already been preheated in HPP. As a result, the gas temperature of the turbine mass flow lies below the combustion chamber's exit temperature. After expansion in HPT, the combustion chamber (LPCC) mass flow is mixed in LPT inlet plenum (4) with the rest of the preheated cooling-air flow and the sealing-air flow. After expansion in LPT, the gas gives off some of its heat in LPP before leaving the gas turbine system.

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18 Gas Turbine Engines, Design and Dynamic Performance

Fig. 18.18: BBC-CAES-Huntdorf gas turbine engine [18]

Fig. 18.19: Simulation schematic of the CAES shown in Fig. 18.18

18 Gas Turbine Engines, Design and Dynamic Performance

497

Figure 18.19 shows how the various components are interconnected. Plenum 8, the air storage facility, is connected via two identical pipes (P6) to two shutdown valves (V1). During steady-state operation, the blow-off valve (V2) remains closed, being opened in the event of a disturbance likely to cause rapid shutdown. In such an event, the valve blows off some of the gas, thereby limiting the maximum rotor speed. For the sake of clarity, the pre-heater (P) has been separated into its air and gas sides, designated HPP and LPP, respectively. 18.5.1.1 Simulation of Emergency Shutdown: Starting from a steady operating point, a generator trip with rapid shutdown was simulated assuming a failure of the control system. This circumstance necessitates an intervention by the hydraulic emergency system. This incident simulates an extreme transient process within some of the components, as explained briefly. After the generator trip, the rotor is strongly accelerated because of full turbine power acting on it, Fig. 18.20(a). The hydraulic emergency system intervenes only when the speed corresponding to the hydraulic emergency overspeed trip is reached. This intervention involves

1.20

250

(a)

(b)

Closing shurdown valves V1

200

1.16

150 m(kg/s)

Z/Zo

1.12

x

1.08

50

1.04

0

1.00

-50 0

0.5

1

1.5 t(s)

2

2.5

Opening bypass valves V2

100

3

0

0.5

1

1.5 t(s)

2

2.5

3

Fig. 18.20: Relative angular velocity (a) and mass flows (b) as function of time. The Inlet-shutdown valves V1 remain open until the trip speed at t= 0.35 s has been reached. The same procedure is true for opening the blow-off valves V2. Closing the shutdown valves follow the ramp shown in Fig. 18.20(b).

5E+06

1200

plenum 8

Plena temperature ( K)

4E+06

o

Plena Pressure (pa)

9

3E+06 2E+06 1E+06

1,2,7,10 3, 4

800

1

2

t(s)

3

4

5

2 5

600

3 7

400 9, 8

5, 6

0

Pl: 4

1000

200 0

1, 10

1

2

3 t(s)

4

5

6

Fig. 18.21:Plena pressure and temperature as functions of time. Shutdown process causes rapid depressurization in high pressure plena1, 2, 7, and 10.

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18 Gas Turbine Engines, Design and Dynamic Performance

closing the fuel valves, FV1 and FV2, and air valves, V1, after which the system no longer receives any energy from outside, Fig. 18.20(b). It also involves opening the bypass/blow-off valve V2 that allows the high pressure air contained in both large volume combustion chambers as well as in the HP-side of pre-heater to discharge. The closing process of the inlet and shutdown valves and the opening of the bypass valves are shown in Fig. 18.20(b). This process results in a steady drop in plena pressures and temperatures. As Fig. 18.21 shows, the pressure drop in the highpressure section is initially steeper than in the low-pressure section. This means that the enthalpy difference of the high-pressure turbine is reduced more rapidly than that of the low-pressure turbine. Immediately after the blow-off valve is opened, an abrupt pressure drop takes place in plenum 10, which is connected to plenum 1 via pipe P5. Thereafter, dynamic pressure equalization takes place between the two plena. This drop in pressure and temperature causes a corresponding drop in the shaft power and the mass flow throughout the engine.

700 Turbine exit temperature (oK)

LPT

o

Turbine Inlet temperature ( K)

1200 1000 800 600

HPT

400 0

1

2

t(s)

3

4

600 500

HPT 400

200

5

LPT

300 0

1

2

t(s)

3

4

5

Fig. 18.22: Turbine inlet and exit temperature as functions of time. Note changes of the exit temperature at t=3.4 s 300 250

400

Shaft power (MW)

Turbine Mass Flow (kg/sec)

500

LPT

300 200

HPT

100 0

200

Total gas turbine power Ptotl = PT1 + PT2

150 100 50 0

0

1

2

t(s)

3

4

5

-50 0

0.5

1

1.5

t(s)

2

2.5

3

Fig. 18.23: Turbine mass flow and shaft power as function of time

Figure 22.22 shows the resulting drop in turbine inlet and exit temperature. The continuous decrease in turbine mass flow causes a strong dissipation of shaft power resulting in the excessive increase of turbine exit temperature. In order to avoid thermal damages to blades, a small stream of cold air is injected into the turbine flow path that causes a reduction in temperature gradient. This is shown in Fig. 18.22 for the exit temperature at t = 3.4 s.

18 Gas Turbine Engines, Design and Dynamic Performance

499

Dynamic behavior of the rotor speed is generally determined by the turbine power acting on the rotor. How the rotor behaves in response to a generator trip depends, in particular, on how long the full turbine power is available, a process monitored by the control and safety monitoring system. When the control system functions normally, trip is signaled without delay to the shutdown valve. Failure of the control system causes the hydraulic emergency system to intervene. The intervention begins only when the speed corresponding to the hydraulic emergency overspeed trip is reached. During this process, and also the subsequent valve dead time, the rotor receives the full turbine power. The closing phase is characterized by a steady reduction in energy input from outside, which finally becomes zero. The total energy of the gases still contained in the system is converted by the two turbines into mechanical energy, causing the rotor speed to increase steadily, Fig. 18.20. When the instantaneous turbine power is just capable of balancing the friction and ventilation losses, the rotor speed reaches its maximum, after which it begins to decrease. Reducing the turbine mass flow, Fig. 18.23 (left), below the minimum value discussed in Chapter 19, causes the shaft power to dissipate completely as heat resulting in negative values as Fig. 18.23 (right) shows. From this point on, the rotational speed starts to decrease. The figure depicts the mass flow through both turbines as representatives for the entire engine as well as the total shaft power.

18.5.2 Case Study 2: Power Generation Gas Turbine Engine The subject of this case study is the dynamic simulation of a BBC-GT9 gas turbine which is a single-spool single shaft power generation gas turbine engine. It is utilized as a stand-alone power generator or in conjunction with combined cycle power generation. The engine shown in Fig. 18.24 consists mainly of three compressor stage groups, a combustion chamber, a turbine, a control system, and a generator. The simulation schematic of this engine is presented in Fig. 18.25. The rotor speed and turbine inlet temperature are the input parameters for the controller, its output parameters are the fuel mass flow (fuel valve opening), and the mass flows of the bypass valves (bypass valve opening). The dynamic behavior of BBCGT9 was experimentally determined for transient tests with extreme changes in its load. Its transient data was accurately documented by Schobeiri [3]. Starting from a given network load schedule, the dynamic behavior of the gas turbine is predicted and the results are presented.

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18 Gas Turbine Engines, Design and Dynamic Performance

Fig. 18.24: A single spool power generation gas turbine, BBC-GT9

The engine under consideration consists mainly of three compressor stage groups, a combustion chamber, a turbine, a control system, and a generator. The simulation schematic of this engine is shown in Fig. 18.25. For dynamic simulation, the first, second and third stage groups are simulated in row-by-row fashion. A similar row-byrow calculation procedure is applied to the turbine component. The rotor speed and the turbine inlet temperature are the input parameters for the controller, its output parameters are the fuel mass flow (fuel valve opening), and the mass flows of the bypass valves (bypass valve opening).

Fig. 18.25: Simulation schematic of BBC-GT9 shown in Fig. 18.24

18 Gas Turbine Engines, Design and Dynamic Performance

501

Simulation of an Adverse Dynamic Operation: Starting from steady state, in accordance with the load schedule indicated in Fig. 18.26 (left), curve 1, after one second, a generator loss of load is simulated that lasts for six seconds. The rotor at first reacts with a corresponding increase in rotational speed, Fig. 18.26 (right), which results in a rapid closing of the fuel valve, Fig. 18.31 (right). The rotational speed is then brought to an idling point and held approximately constant. The process of control intervention lasts until a constant idling speed is attained. After that, there is an addition of load in sudden increases, such that the gas turbine is supplying approximately 25% of its rated load, Fig. 18.26. The rotor first reacts to this addition of load with a sharp decrease in rotational speed, as exhibited in Fig. 18.26, causing a quick opening of the fuel valve, Fig. 18.31. After completion of the transient process, the steady off-design state is reached.

1.06

30 1: Generator power schedule

1.04

Z/Zo

Power (MW)

20 10

-10

1.02

2: Engine power response

0

0

2.5

5

7.5 Time (s)

10

12.5

1

15

0

2.5

5

7.5 Time (s)

10

12.5

15

Fig. 18.26: Left, generator load schedule curve 1, sequence of events: Steady operation from 0 to 1 second, sudden loss of load, idle operation, sudden addition of load, continuous decrease of load to 25%. Curve 2: engine power response. Right, relative shaft speed as a function of time.

Plena Pressure and Temperature Transients: The above adverse dynamic operation has triggered temporal changes of flow quantities within individual components. Figure 18.27 shows how the plena pressure and temperature change with time.

1400

800

Plenum Temperature (K)

Plena Pressure (kpa)

1000

1200

5

6

400 4

200

3 Plena: 1, 2, 7, 8

0

0

2.5

5

7.5 Time (s)

10

12.5

Plenum No.6

1000

600

15

800 7

600 400 200 0

5

1

3

2

2.5

5

4

7.5 Time (s)

10

12.5

15

Fig. 18.27: Plena pressure and temperature as functions of time. Individual plena are labeled.

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18 Gas Turbine Engines, Design and Dynamic Performance

Decrease of turbine power and increase of the shaft speed, Fig. 18.26, has caused the HP-compressor exit pressure in plenum 5 to decrease. Temperature at combustion chamber exit, plenum 6, and turbine exit, plenum 7, follow the course of fuel injection shown in Fig. 18.31(right). The plena temperature upstream of the combustion chamber are not affected.

160

Comb. Ch. Mass Flow (kg/s)

Compressor Mass Flows (kg/s)

Compressor and Combustion Chamber Mass Flow Transients: Figure 18.28 exhibits the mass flow transients through LP-, IP-, and HP-compressors. While IPand HP-stage groups have the same mass flow, the LP-part has a greater mass flow. The difference of 1 kg/s is due to the cooling mass flow extraction. As briefly mentioned, the increase in shaft speed and the simultaneous decrease in compressor power consumption leading to compressor pressure drop has caused an increase in compressor mass flow during the process of loss of load that lasts up to t = 6 sec.

LP-Compressor stage group

155 150 IP, HP-compressors

145 140

0

2.5

5

7.5 Time (s)

10

12.5

15

145 140 135 130 125 120

0

2.5

5

7.5 Time (s)

10

12.5

15

Fig. 18.28: Compressor and combustion chamber mass flows as functions of time

Fig. 18.29:Combustion chamber module, stations and segments, = total secondary air,

= individual secondary air

= primary air,

18 Gas Turbine Engines, Design and Dynamic Performance

503

The sudden load addition reduces the compressor mass flow. The combustion chamber mass flow shows a similar course with a substantial difference. A substantial portion of compressor mass flow is extracted for combustion chamber exit temperature mixing cooling. Combustion Chamber Gas and Metal Temperature Transients: The combustion chamber component used in this simulation has three segments that separate the primary combustion zone from the secondary cooling air zone. Its module is shown in Fig. 18.29. Figure 18.31 exhibits the combustion chamber gas and metal temperatures as functions of time. Compressed air enters the combustion chamber at station 1, Figs. 18. 29, and 18.31. Fuel is added and the segment cooling occurs according to the procedure described in Chapter 17. The secondary mass flow portions serve as cooling jets and are mixed with the combustion gas, thus reducing the gas temperature. Before exiting, the combustion gas is mixed with the mixing air stream , further reducing the temperature. Figure 18.31 (right) shows the mean segment temperatures. In accordance with the measurements on this gas turbine, the flame length extends from station 1 to 3, which makes the segment number 2 the hottest one. We assumed that all secondary cooling channels are open.

1300 Metal Temperature (K)

Gas Temperature (K)

3000 2500 2000 2

1500 4

3

5

1000

Segment 2

1200 Segment 3

1100 1000 Segment 1

900 800

Station 1

500

0

2.5

5

7.5 Time (s)

10

12.5

15

700

0

2.5

5

7.5 Time (s)

10

12.5

15

Fig. 18.30: Combustion chamber gas and metal temperature at different positions as functions of time

Turbine and Fuel Mass Flow Transients: Figure 18.31 (left) exhibits the turbine mass flow transient, which is dictated by the compressor dynamic operation. Difference between the turbine and the compressor mass flow is the injected fuel mass flow. The particular course of fuel mass flow shown in Fig. 18.31 (right) is due to the intervention of the control system. An increase in rotational speed causes the controller to close the fuel valve. Subsequent addition of generator load results in a steep drop of rotational speed which causes an opening of the fuel valve.

18 Gas Turbine Engines, Design and Dynamic Performance

160

3.0

158

2.5

Fuel mass flow (kg/s)

Turbine Mass Flows (kg/s)

504

156 154 152 150

0

2.5

5

7.5 Time (s)

10

12.5

15

Response to shaft speed behavior

2.0 1.5 1.0 0.5 0.0

0

2.5

5

7.5 Time (s)

10

12.5

15

Fig. 18.31: Turbine and fuel mass flow as functions of time. The fuel mass flow is controlled by the shaft rotational speed.

18.5.3 Case Study 3: Simulation of a Multi-spool Gas Turbine Engine The subject of this study is the non-linear dynamic simulation of a gas turbine engine with a higher degree of complexity than the previous cases. For this purpose a threespool thrust generating gas turbine engine is designed that incorporates advanced components. The three-spool four shaft high performance gas turbine engine consists of a low pressure spool that incorporates the LP-compressor and turbine connected via shaft S1. The intermediate pressure spool integrates the IP-compressor and turbine connected via shaft S2. The high pressure spool carries the HP-compressor and HPturbine on shaft S3. To increase the level of engine complexity, a fourth shaft, S4, with

Fig. 18.32:Simulation schematic of three-spool four shaft high performance gas turbine engine. Spool 1 incorporates LP-compressor and LP-turbine connected via shaft S1; Spool 2 incorporates IP-compressor and IP-turbine connected via shaft S2; Spool 3 incorporates HP-compressor and HP-turbine connected via shaft S3.

18 Gas Turbine Engines, Design and Dynamic Performance

505

the power generating turbine, T4, was attached to the exit of the three-spool gas generating unit as shown in Fig. 18.32. The transient operation is controlled by a given fuel schedule. The component nomenclature for this configuration is the same as for the previous cases. The simulation schematics shown in Fig. 18.32 represents the modular configuration of the gas turbine. Fuel Schedule, Rotor Response: The dynamic behavior of the above engine is simulated for an adverse acceleration-deceleration procedure. The transient operation is controlled by an open loop fuel schedule shown in Fig. 18.33 (left). The threespools and the fourth shaft run independently at different rotational speed, Fig. 18.33 (right). 1.05

3.5

Power shaft S 4

1.02 S1

1.00

Design point

S2

A cc

n atio eler

eler atio n

3.0

Z/Zo

Dec

Fuel mass flow (kg/s)

Off-design point

2.5

S3

0.97 2.0

0

10

20 30 Time (s)

40

50

0.95

0

10

20 30 Time (s)

40

50

Fig. 18.33: Fuel schedule (left) starts with an off-design mass flow followed by a cyclic acceleration-deceleration procedure. Rotational speed of the three spools and the power shaft.(right).

The fuel schedule generated fully arbitrarily simulates an acceleration-deceleration procedure with emphasis on deceleration. We start with the steady-state operation and reduce the fuel mass flow to for about two seconds. During this short period of time, the engine operates in a dynamic state which is followed by a cyclic acceleration-deceleration event with the ramps given in Fig. 18.33. The dynamic operation triggers a sequence of transient events within individual components that are discussed in the following sections. Rotor Speed Behavior: The transient behavior of the three spools as well as the power shaft is determined by the net power acting on the corresponding rotor. For each individual spool, the cyclic acceleration-deceleration event has caused a dynamic mismatch between the required compressor power consumption and the turbine power generation as shown in Fig. 18.34. While LP- and IP-spools 2 and 3 decelerate under the influence of negative net power, the HP-spool 3 reacts faster to the acceleration. Since the fuel schedule places special weight upon deceleration, the rotational speeds of all three spools have a decelerating tendency as shown in Fig. 18.34.

506

18 Gas Turbine Engines, Design and Dynamic Performance

30

1.05

20

1.02 Power generation shaft

S1

Z/Zo

Power (MW)

Power shaft S 4

1.00

10

S2 S3

Net poer spool 1, 2, and 3

0.97

0 -10

0

10

20 30 Time (s)

40

0.95

50

0

10

20 30 Time (s)

40

50

Fig. 18.34: Net power acting on the three spools causing a dynamic mismatch; power generated by the fourth shaft (left). Relative rotor speed of three spools and the fourth shaft. Pressure and Temperature Transients within Plena: The change in fuel mass flow triggers a chain of transient events within the plena as shown in Fig. 18.35. 100

1400

80

Plena Temperature (K)

Plena Pressure (bar)

Plenum No. 5 6

60 40

7 8

4

20

3

0

10

20

9

40

8

1000

7

9

800

10

600

4

400

3 Plena: 1, 2, 11

1, 2, 10, 11

30 Time (s)

6

1200

50

200 0

10

20 30 Time (s)

40

50

Fig. 18.35: Plena pressure (left) and temperature (right) as functions of time Plena pressure 5 and 6 which corresponds to the exit pressure of the HP-compressor and the combustion chamber, are strongly affected by the cyclic fuel change, whereas the other plena that correspond to the inlet and exit plena of the remaining components experiences moderate changes. The plena temperature distributions downstream of the combustion chamber shown in Fig. 18.35 (right) reflect the course of the fuel schedule. Combustion Chamber Gas and Metal Temperature Transients: Figure 18.36 exhibits the combustion chamber gas and metal temperatures as functions of time. The combustion chamber component used in this simulation has three segments that separate the primary combustion zone from the secondary cooling air zone. Its module is shown in Fig. 18.32. Compressed air enters the combustion chamber at station 1, Figs. 18. 36 (left). Fuel is added and the segment’s cooling occurs according to the procedure described in Chapter 17. The secondary mass flow portions serve as cooling jets and are mixed with the combustion gas, thus reducing the gas temperature. Before exiting, the combustion gas is mixed with the mixing air stream further reducing the temperature. Figure 18.36 (right) shows the mean segment

18 Gas Turbine Engines, Design and Dynamic Performance

507

temperatures. The flame length extends from station 1 to 3, which makes the segment number 2 the hottest one. As seen, the gas temperature at station 2 follows the sharp changes in the fuel schedule. By convecting downstream, these sharp changes are smoothed out. Wall temperatures shown in Fig. 18.36 (right) exhibit similar tendencies. 1400 Metal Temperature (K)

Gas Temperature (K)

2500 2

2000

3

1500 4

1000 Station 1

500

0

10

20 30 Time (s)

40

2

1200 1100 1000 900

50

3

1300

Segment 1

0

10

20 30 Time (s)

40

50

Fig. 18.36: Combustion chamber gas and metal temperature as functions of time

155.0

155.0 Trubine mass flow (kg/s)

Compressor mass flow (kg/s)

Compressor and Turbine Mass Flow Transients: Figure 18.37 (left) exhibits the compressor mass flow transients, which are dictated by the compressor dynamic operation. The difference in compressor mass flow is due to the mass flow extraction for cooling purposes. Turbine mass flows are illustrated in Fig. 18.37 (right). Except for a minor time lag, they show identical distributions. The difference in turbine and compressor mass flow is due to the addition of fuel.

LP-& IP-Compressors

152.5 150.0 147.5 HP-Compressor

145.0

0

10

20 30 Time (s)

40

50

LP-, IP-, HP-, power turbine

152.5 150.0 147.5 145.0

10

20

30 Time (s)

40

50

Fig. 18.37: Compressor and turbine mass flow as functions of time

18.6 A Byproduct of Dynamic Simulation: Detailed Efficiency Calculation One of the interesting aspects of a dynamic simulation is the capability to dynamically calculate the gas turbine thermal efficiency during steady state and dynamic operations. Such calculations are performed to compare the thermal efficiencies of four gas turbines with different design methodologies. The calculations are performed with the nonlinear dynamic code GETRAN. The first gas turbine dynamically simulated for efficiency calculation is a conventional single-shaft, single-combustion

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18 Gas Turbine Engines, Design and Dynamic Performance

chamber power generation gas turbine. The second one is the ABB, GT 24/26. The third is an ultra-high efficiency gas turbine (UHEGT) with a pre-combustor, a reheat turbine stage and an integrated stator internal combustion as illustrated in Fig. 18.38. For the fourth gas turbine, the combustion process is placed entirely within the stator rows, thus eliminating the combustion chambers all together, Fig. 18.39. The dynamic efficiency calculation results are presented in Fig. 18.40. To accurately determine the thermal efficiency and specific work of the gas turbines, calculations are performed with GETRAN© and the results are presented in Fig. 18.40. To compare the degree of efficiency improvement, the thermal efficiency and specific work of a baseline GT, GT-24, and the three UHEGT gas turbines are included in the figures. For a UHEGT with three-stator combustion, denoted by the curve UHEGT0C3S, a thermal efficiency above 45% is calculated. This exhibits an increase of at least 5% above the gas turbine engine GT-24, which is close to 40.5% as Fig. 18.40 (left) shows. Increasing the number of stator internal combustion to 4, curve labeled with UHEGT-0C4S, raises the efficiency above 48%. This is an enormous increase compared to any existing gas turbine engine. In the course of this calculation, the UHEGT technology is applied to a gas turbine engine with a pre-combustion chamber, such as the first one in GT-24, Fig.18.48. Using this combustion chamber with two-stator combustion, the curve labeled with UHEGT-1C2S shows an efficiency of 44%. This is particularly interesting for upgrading the existing gas turbines with UHEGT technology. Figure 18.38 (right) reveals the specific work comparison for the gas turbines discussed above. Compared to GT-24, UHEGT technology has about 20% higher specific work, making these engines very suitable for aircraft, stand-alone, as well as for combined cycle power generation applications.

Fig. 18.38: A derivative of the Ultra-High Efficiency Gas Turbine Engine with a multi-stage compressor, a conventional combustion chamber PC, a single-stage reheat turbine RT, a three-stage turbine with an integrated stator internal combustion, B1, B2: compressor bypass blow-off, Fl1, Fl2: Fuel lines to stator. The combustion process takes place inside the pre-combustor and stator flow path, [1].

18 Gas Turbine Engines, Design and Dynamic Performance

509

Fig. 18.39: An Ultra-High Efficiency Gas Turbine Engine with a multi-stage compressor, and a three-stage turbine with an integrated stator internal combustion, B1, B2: compressor bypass blow-off, Fl1, Fl2: Fuel lines to stator. The combustion process takes place inside the stator flow path, Schobeiri [1]. 600

0.5

UHEGT-0C4S

UHEGT-0C4S

UHEGT-0C3S

UHEGT-0C3S UHEGT-1C2S

Kth

w (KJ/kg)

GT-24

0.4

0.3

GT-24

400

UHEGT-0C3S

200

Baseline GT

Baseline GT

0.2

0

10

20

S

30

40

50

0

10

20

S

30

40

50

Fig. 18.40: Thermal efficiency (left) and specific work (right) as a function of compressor pressure ratio for the reference baseline gas turbine engine, ABB-GT-24 with two-combustion chamber, the UHEGT-1C2S with one conventional combustion chamber and two UHEGT-stator combustion, UHEGT-03S with three-stator combustion, and UHEGT-0C4S with four-stator combustion. Turbine inlet temperature = 1,2000C, calculation with GETRAN [3]. It is interesting to note that this efficiency increase can be established at a compressor pressure ratio of , which can be achieved easily by existing compressor design technology with high polytropic efficiency. In performing the GETRAN© calculation, compressor and turbine efficiencies are calculated on a row-by-row basis. This automatically accounts for an increase of secondary flow losses based on aspect ratio decrease. Thus, in a compressor case, efficiency decrease with pressure ratio increase is inherently accounted for.

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18 Gas Turbine Engines, Design and Dynamic Performance

18.7 Summary Part III, Further Development This chapter concludes Part III of the book that started with Chapter 14. The last six chapters dealt with providing the basic non-linear dynamic equations essential for describing the aero-thermodynamic process within individual turbomachinery components from a generic point of view. The four-dimensional time-space equations were simplified and modules were developed based on two-dimensional time-space coordinates. These modules were used to configure simulation schematics of complex gas turbine engines. Four levels of gas turbine engine simulations with cross coupling were presented. Three representative case studies were presented that aimed at better understanding the concept and the necessity for non-linear dynamic simulation. Improvement of Thermal Efficiency: To introduce the reader to the problematic of gas turbine operation and process, Chapter 18 started with a brief presentation of a simple method for calculating the thermal efficiency of gas turbines, also a few methods for improving the gas turbine process and thermal efficiency were discussed. The parameter study using Eq. (18.20) illustrated the impact of the turbine inlet temperature (TIT) on thermal efficiency. More accurate results are obtained using the dynamic method applied to the corresponding simulation schematics that may incorporate many components and subsystems that are relevant for gas turbine operation. Considering the current advanced state-of-the-art turbine cooling technology and the technologies utilized to protect the blades from excessive thermal stresses by using thermal barrier coatings, there is not much room left for increasing the TIT. Likewise, the development of turbine and compressor components has reached a high level of polytropic efficiency (above 94%) that additional marginal improvement would not significantly improve the thermal efficiency. However, substantially higher thermal efficiencies at a consolidated TIT level can be obtained by taking advantage of the basic thermodynamic principles as demonstrated by the performance evaluation of GT-24/26. Further efficiency improvement can be achieved by utilizing the UHEGT concept for future generation of gas turbine engines. This concept is based on sound thermodynamic principles and offers a substantial improvement, however it requires substantial R&D efforts. Necessity for Dynamic Simulation: The case studies showed the capability of a generic modularly structured simulation tool based on equations presented beginning in Chapter 14 through 18. From today’s computational point of view, simulating a complex gas turbine power plant or an aircraft engine consisting of various components and subsystems that are described by a large number of systems of differential and algebraic equations, does not exhibit any computational obstacle. It is well known that problems with new gas turbine types arise during dynamic operation. Many of these problems can be eliminated in the early stage of design and development by dynamically simulating possible adverse scenarios that may lead to component and system failures. Modeling Enhancement: As explained, the modular concept is based on a twodimensional time space approximation. The models described in Part 3 can be

18 Gas Turbine Engines, Design and Dynamic Performance

511

extended to fully four-dimensional time space models. Using the unsteady NavierStokes equations requires considerable computational effort and time as we briefly discussed. A time dependent version of the streamline curvature method may deliver sufficiently accurate results for evaluating a dynamic simulation.

References 1. Schobeiri, M.T.: The Ultra-High Efficiency Gas Turbine Engine with Stator Internal Combustion, UHEGT. U.S. Patent Pending, 1389-TEES-99 (1999) 2. Schobeiri, M.T., Attia, S.: Advances in Nonlinear Dynamic Engine Simulation Technology. ASME 96-GT-392, presented at the International Gas Turbine and Aero-Engine Congress and Exposition, Birmingham, UK- June 10- 13 (1996) 3. Schobeiri, M.T., Abouelkheir, M., Lippke, C.: GETRAN: A Generic, Modularly Structured Computer Code for Simulation of Dynamic Behavior of Aero-and Power Generation Gas Turbine Engines, an honor paper. ASME Transactions, Journal of Gas Turbine and Power 1, 483–494 (1994) 4. Koenig, R.W., Fishbach, L.H.: GENENG- A Program for Calculating Design and Off-Design Performance for Turbojet and Turbofan Engines. NASA TN D-6552 (1972) 5. Seldner, K., Mihailowe, J.R., Blaha, R.J.: Generalized Simulation Technique for Turbojet Engine System Analysis. NASA TN D-6610 (1972) 6. Seller, J., Daniele, C.J.: DYGEN – A Program for Calculating Steady-State and Transient Performance of Turbojet and Turbofan Engines, NASA TND-7901 (1975) 7. Szuch, J.R.: HYDES- A Generalized Hybrid Computer Program for Studying Turbojet or Turbofan Engine Dynamics. NASA TM X-3014 (1974) 8. Agrawal, R.K., Yunis, M.: A Generalized Mathematical Model to Estimate Gas Turbine Starting Characteristics. Journal of Eng. Power 104, 194–201 (1982) 9. Schobeiri, M.T.: Aero-Thermodynamics of Unsteady Flows in Gas Turbine Systems. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-51 (1985) 10. Schobeiri, T.: COTRAN, the Computer Code for Simulation of Unsteady Behavior of Gas Turbines. Brown Boveri Company, Gas Turbine Division Baden Switzerland, BBC-TCG-53 (1985) 11. Schobeiri, T.: Digital Computer Simulation of the Dynamic Response of Gas Turbines. VDI-Annual Journal of Turbomachinery 1985, 381–400 (1985) 12. Schobeiri, T.: A General Computational Method for Simulation and Prediction of Transient Behavior of Gas Turbines. ASME-86-GT-180 (1986) 13. Schobeiri, T.: Digital Computer Simulation of the Dynamic Operating Behavior of Gas Turbines. Journal Brown Boveri Review 3, 87 (1987) 14. Schobeiri, T.: Digital Computer Simulation of the Dynamic Operating Behavior of Gas Turbines. Journal Brown Boveri Review 3, 87 (1987) 15. Schobeiri, H., Haselbacher, H.: Transient Analysis of Gas Turbine Power Plants Using the Huntorf Compressed Air Storage Plant as an Example. ASME-85-GT197 (1985c)

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16. Schobeiri, M.T., Attia, M., Lippke, C.: Nonlinear Dynamic Simulation of Single and Multi-spool Core Engines, Part I: Theoretical Method. AIAA, Journal of Propulsion and Power 10(6), 855–862 (1994) 17. Schobeiri, M.T., Attia, M., Lippke, C.: Nonlinear Dynamic Simulation of Single and Multi-spool Core Engines, Part II: Modeling and Simulation Cases. AIAA Journal of Propulsion and Power 10(6), 863–867 (1994) 18. Schobeiri, M.T.: Dynamisches Verhalten der Luftspeichergasturbine Huntorf bei einem Lastabwurf mit Schnellabschaltung. Brown Boveri, Technical Report, TA58 (1982)

Part IV: Turbomachinery CFD-Essentials

19 Basic Physics of Laminar-Turbulent Transition

19.1 Transition Basics: Stability of Laminar Flow The phenomena of stability of laminar flows, transition, and turbulence were systematically studied first by O. Reynolds [1] in the eighties of the eighteenth century. H. Schlichting [2] and [3] and in his classical textbook Boundary Layer Theory [4] gives an excellent treatment of these complex flow phenomena and critically reviews the contributions up to 1979, where the seventh and last edition of his book appeared. In this chapter, we first treat the fundamental issues pertaining to the subject matter followed by original contributions recently made in the area of steady and unsteady boundary layer transition. The flow in a turbine or compressor component is characterized by a threedimensional, highly unsteady motion with random fluctuations due to the existing freestream turbulence and the interactions between the stator and rotor rows. Considering the flows within the blade boundary layer, based on the blade geometry and pressure gradient, three distinctly different flow patterns can be identified: 1) laminar flow (or non-turbulent flow) characterized by the absence of stochastic motions, 2) turbulent flow, where flow pattern is determined by a fully stochastic motion of fluid particles, and 3) transitional flow characterized by intermittently switching from laminar to turbulent at the same spatial position. Of the three patterns, that is predominant in turbomchinery components is the transitional flow pattern. The Navier-Stokes equations presented in Chapter 3 generally describe the unsteady flow through a turbomachinery component. Using a direct numerical simulation (DNS) approach delivers the most accurate results. However, the application of DNS, for the time being, is restricted to simple flows at low Reynolds numbers. For calculating the complex turbomachinery flow field within a reasonable time frame, Reynolds averaged Navier-Stokes (RANS) equations with appropriate turbulence models are used. A detailed discussion of this topic is found in Chapter 20.

19.2 Laminar-Turbulent Transition, Fundamentals The complex process of transition is affected by several parameters. The most significant ones are the Reynolds number, pressure gradient, surface roughness, and the external disturbance such as the inherent freestream turbulence intensity and the interaction of the wakes stemming from one blade row with the next following blade row. To understand the fundamentals of the laminar turbulent transition, we try first to reduce the number of parameters affecting the

516

19 Basic Physics of Laminar-Turbulent Transition

transition process. This is done effectively by investigating the transition within the boundary layer along a flat plate with a smooth surface at zero pressure gradient. This is particularly important for the development of the boundary layer and its onset which is primarily responsible for the inception and magnitude of the drag forces that exert on any surface exposed to a flow field. Figure (19.1) schematically explains the transition process that takes place within the boundary layer along a flat plate at zero pressure gradient. 1

x

3

2

3

Laminar flow region 11

0.5V

0.88V Turbulent spot

x 2

xV

0

xV=0 Turbulent boundary layer

Laminar boundary layer x1

Fig. 19.1: Transition process along a flat plate at zero-pressure gradient sketched by Schubauer and Klebanoff [5] Starting from the leading edge, the viscous flow along the plate generates two distinctly different flow regimes. Close to the wall, where the viscosity effect is predominant, a thin boundary layer is developed, within which the velocity grows from zero at the wall (no-slip condition) to a definite magnitude at the edge of the boundary layer (the boundary layer and its theory is discussed in Chapter 21). Inside this thin viscous layer the flow initially constitutes a stable laminar boundary layer flow that starts from the leading edge and extends over a certain range ±. By further passing over the plate surface, the first indications of the laminar flow instability appear in form of infinitesimally unstable two-dimensional disturbance waves that are referred to as Tollmien-Schlichting waves ². Further downstream, discrete turbulent spots with highly vortical core appear intermittently and randomly ³. Inside these wedge-like spots the flow is predominantly turbulent with , whereas outside the spots it is laminar. According to the experiments by Schubauer and Klebanoff [5], the leading edge of a turbulent spot moves with a velocity of , whereas its trailing edge moves with a lower velocity of . As a consequence, the spot continuously undergoing deformation decomposes and builds new sets of turbulence spots with increasingly random fluctuations characteristic of a turbulent flow. Schubauer and Klebanoff [5] also noted the existence of a calmed region trails behind the turbulent spot. This region was named calmed because the flow is not receptive to disturbances.

19 Basic Physics of Laminar-Turbulent Transition

517

Analytical investigations by McCormick [6] indicate that artificially created turbulent spot does not persist if the Reynolds number satisfies the condition which results from linear stability theory. Schlichting [4] summarized the transition process as follows: Å A stable laminar flow is established that starts from the leading edge and extends to the point of inception of the unstable two-dimensional Tollmien-Schlichting waves. ² Onset of the unstable two-dimensional Tollmien-Schlichting waves. ³ Development of unstable, three-dimensional waves and the formation of cascades of vortices. ´ Bursts of turbulence in places with high vorticity. µ Intermittent formation of turbulent spots with high vortical core at intense fluctuation. ¶ Coalescence of turbulent spots into a fully developed turbulent boundary layer. White [7] presented a simplifying sketch, Fig. (19.2), of transition process of a disturbance free flow along a smooth flat plate at zero pressure gradient by assembling the essential elements of transition measured by Schubauer and Klebanoff [5]. The process of flow transition from laminar to turbulent in the sequence discussed above takes place at low level of freestream turbulence intensity of 0.1% or less. In 1

2

3

4

5

6

U

x3 x1

G (x1)

x1 U Laminar

Transition Re

Cr

Turbulent Re

Tu

Fig. 19.2: Sketch of transition process in the boundary layer along a flat plate at zero pressure gradient, a composite picture of features in [5] after White [7]

518

19 Basic Physics of Laminar-Turbulent Transition

this case, the presence of Tollmien-Schlichting waves are clearly present leading to a process of natural transition. In many engineering applications, particularly in turbomachinery flows, the mainstream is periodic unsteady associated with highly turbulent fluctuations. In these cases, the boundary layer transition mainly occurs bypassing the amplification of Tollmien-Schlichting waves. This type of transition is called bypass transition, [8].

19.3 Physics of an Intermittent Flow To better understand the physics of an intermittently laminar-turbulent flow, we consider a flat plate, Fig. 19.3, with a smooth surface placed within a wind tunnel Constant current input

Sensing wire: (D= 0.25- 5 P, L = 1.5-2mm)

x2

i x1

x2 x3

Single hot wire

P

V Laminar

Transition

Turbulent

x1P

(a)

't1

turbulent

't2

turbulent

'ti

turbulent

'tN

turbulent

V non-turbulent

non-turbulent

ti i=1

t ts T

i=N

(b) Intermittent velocity distrubution measured at point P(x1 , x2 )

Fig. 19.3: Measurement of an intermittently laminar-turbulent flow, (a) positioning a hot wire sensor within the transitional region of the boundary layer, (b) high frequency velocity signals acquired at point P

19 Basic Physics of Laminar-Turbulent Transition

with statistically steady inlet flow velocity

519

and a low turbulence fluctuation velo-

city . It should be noted that, in contrast to the theoretical assumption made for a fully laminar flow, the real flow in wind tunnel or cascade test facilities always contains certain degree of turbulence fluctuations superimposed on the main flow velocity . This is expressed in terms of turbulence intensity defined as . Thus, from a practical point of view, it is more appropriate to use the term non-turbulent flow rather than laminar one. Downstream of the laminar region, we place a miniature hot wire sensor at an arbitrary point P within the boundary layer to measure the velocity (see Chapter 21 for details). The position of the sensor relative to the plate is such that the axis of the sensing wire coincides with x3-axis which is perpendicular to the x1-x2-plane, Figure 19.3(a). The wire and the associated anemometer electronics provide a virtually instantaneous response to any high frequency incoming flow. Figure 19.3(b) schematically reflects the time dependent velocity of an otherwise statistically steady flow. As seen, the anemometer provides a sequence of signals that can be categorized as non-turbulent characterized by a time independent, non-turbulent pattern followed by a sequence of time dependent highly random signals that reflect turbulent flow. Since in a transitional flow regime, sequences of non-turbulent signals are followed by turbulent ones, we need to establish certain criteria that must be fulfilled before a sequence of signals can be called non-turbulent or turbulent. This is issue is treated in the following Section.

19.3.1 Intermittent Behavior of Statistically Steady Flows To identify the laminar and turbulent states, Kovasznay, et al. [9], introduced the intermittency function . The value of I is unity for a turbulent flow regime and zero otherwise: (19.1) Figure 19.4 schematically exhibits an intermittently laminar-turbulent velocity with the corresponding intermittency function for a statistically steady flow at a given position vector x and an arbitrary time t. Following Kovasznay, et al. [9], the time-averaged value of I(x,t) is the intermittency factor Ȗ, which gives the fraction of the time that a highly sensitive probe spends in turbulent flow in a sufficiently long period of time T: (19.2) which is for

equivalent to (19.3)

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19 Basic Physics of Laminar-Turbulent Transition

Experimentally, the intermittency factor Ȗ is determined from a set of N experimental data. This requires that the integral in Eq. (19.3) be replaced by Eq. (19.4): (19.4) The hatched areas in Fig. 19.4 labeled with ±indicate the portion of the velocity with random fluctuations, whereas, the blank areas point to signals lacking random fluctuations.

't1

't2

'ti

'tN

non-turbulent

turbulent

non-turbulent

turbulent

non-turbulent

turbulent

non-turbulent

turbulent

non-turbulent

V

0

1

0

1

0

1

0

1

0

1 I 0 T Fig. 19.4: Identification of non-turbulent (I = 0) and non-turbulent flow (I = 1)

19.3.2 Turbulent/Non-turbulent Decisions To make an instantaneous decision about the non-turbulent/turbulent nature of a flow it is possible to use a simple probe, such as a single hot-wire for measuring the velocity fluctuations and identifying the fine-scale structure in the turbulent flow, as shown in Fig.19.5. Since the velocity fluctuation is not sufficient for making instantaneous decisions for or against the presence of turbulence, the velocity signals need to be sensitized to increase discriminatory capabilities. The commonly used method of sensitizing is to differentiate the signals. The sensitizing process generates some zeros inside the fully turbulent fluid. These zeros influence the decision making process for the presence of turbulence or non-turbulence. The process of eliminating these zeros is to integrate the signal over a short period of time T, which produces a criterion function S(t). After short term integration, a threshold level C is applied to the criterion function to distinguish between the true turbulence and the signal noise.

19 Basic Physics of Laminar-Turbulent Transition

521

Applying the threshold level results in an indicator function consisting of zeros and ones satisfying: (19.5) The resulting random square wave, I(x,t), along with the original signal is used to condition the appropriate averages using the equations above.

V(t)

11

10

9

S(t)

0.1

0 1.5

I(t)

1 0.5 0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t(s)

Fig. 19.5: Processing the instantaneous velocity signals for intermittency calculation for a statistically steady flow along a turbine blade. V(t)= velocity signals, S(t)= Detector function, I(t)= indicator function, for non-turbulent I=0, for turbulent flow I=1, measurement TPFL. Performing the averaging process using Eqs. (19.3) or Eq. (19.4) for the statistically steady flow shown in Fig. 19.5, we find an intermittency factor . For the case this means that flow is transitional. The time averaged intermittency at any point along the surface in streamwise direction can be obtained that reflects the intermittent behavior of the flow under investigation. As an example, Fig. 19.6 exhibits the intermittency distribution along the concave surface of a curved plate at zero longitudinal pressure gradient. Using an entire set of velocity distributions along the concave surface of a curved plate at zero longitudinal pressure gradient, a detailed picture of the intermittency behavior of the boundary is presented in Fig. 19.6. It exhibits the intermittency contour within the boundary layer along the concave side

522

19 Basic Physics of Laminar-Turbulent Transition

_

J y (mm)

30 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Steady State Trs at s/s0 = 0.5

20 15 10 5 0.25

0.5

0.75 s/s0

1

Fig. 19.6: Time-averaged intermittency contour for steady flow along the concave surface S of a curved plate at zero streamwise pressure gradient, S0 is the arc length of the curved plate, measurement TPFL.

y (mm) 0.10 0.55 0.60 1.15 1.30 3.40 5.00 7.00 8.00 10.0

_

J

0.8

0.4

0

0

0.25

0.5

0.75

s/s0

1

Fig. 19.7: Time-averaged intermittency distribution along the concave surface of a curved plate at zero streamwise pressure gradient with normal distance y as a parameter, measurement TPFL

19 Basic Physics of Laminar-Turbulent Transition

523

of a curved plate under statistically steady flow condition at zero pressure gradient. Close to the surface, the intermittency starts from zero and gradually approaches its maximum value. The dark area with encloses locations with the maximum turbulent fluctuations. Moving from the surface toward the edge of the boundary layer, the intermittency factor decreases approaching the non-turbulent freestream. Figure 19.7 presents a more quantitative picture of the intermittency distribution with normal distance y as a parameter. Substantial changes of occur within a range of y = 0.0 to 1.3 mm with the maximum intermittency which means that the velocity has not reached a fully turbulent state. In fact in many engineering applications, in general, and turbomachinery aerodynamics in particular, the flow is neither fully laminar nor fully turbulent . It is transitional with . The change of in normal direction reflects the distribution of spots cross section in y- direction that decreases toward the edge of the boundary layer. The knowledge of -distribution is crucial in assessing the computational results of CFD-code, understanding the development of spot structure and the flow situation within a transitional boundary layer. For calculating the transition boundary layer characteristics, the values close to the surface are used. Figure 19.8 exhibits the -distribution along the concave side of the curved plate mentioned above at y = 0.1 mm above the surface as a function of Re-number in streamwise direction s, . Up to , the boundary layer is fully non-turbulent with = 0. This point marks the start of the transition ReS. Similarly, the end of the transition is marked with . The locations of transition start and end depend strongly on pressure gradient in streamwise direction and the inlet flow condition. The latter includes the free-stream turbulence intensity for steady inlet flow condition. For a periodic unsteady flow condition as is present in many

1.2

J

_

0.8

g

ReS=1.76X105 5 ReE=4.0X10

0.4

0 100000

200000

300000

400000 500000 Re

600000

Fig. 19.8: Intermittency as a function of Re along the concave surface of a curved plate at y = 0.1 mm from, experiment(z), solid line Eq. (19.42)

524

19 Basic Physics of Laminar-Turbulent Transition

engineering applications such as in turbomachinery fluid mechanics, periodic disturbances with specific characteristics play a key role in determining the start and end of the transition.

19.3.3 Intermittency Modeling for Flat Plate Boundary Layer The transition process was first explained by Emmons [10] through the turbulent spot production hypothesis. Adopting a sequence of assumptions, Emmon arrived at the following intermittency relation: (19.6) with ı as the turbulent spot propagation parameter, g the spot production parameter, x the streamwise distance and U the mean stream velocity. While the Emmon’s spot production hypothesis is found to be correct, Eq. (19.6) does not provide a solution compatible with the experimental results. As an alternative, Schubauer and Klebanoff [11] used the Gaussian integral curve to fit the Ȗ-distribution measured along a flat plate. Synthesizing the Emmon’s hypothesis with the Gaussian integral, Dhawan and Narasimha [12] proposed the following empirical intermittency factor for natural transition: (19.7) with , , as the streamwise location of the transition start and A as a constant. The solution of Eq. (19.7) requires the knowledge of Ȝ which contains two unknowns and the location of transition start . In [12] the constant A was set equal to 0.412. Thus, we are dealing with three unknowns, namely , and the two streamwise positions at which the intermittency factor assumes values of 0.75 and 0.25. While the transition start can be estimated, the two streamwise positions and are still unknown. Further more, the quantity A which was set equal to 0.412, may be itself a function of several parameters such as the pressure gradient and the free-stream turbulence intensity. As we discuss in the following Section, a time dependent universal unsteady transition model was presented in [13] for curved plate channel under periodic unsteady flow condition and generalized in [14] for turbomachinery aerodynamics application. The intermittency model for steady state turned out to be a special case of the unsteady model presented in [13] and [14], it reads: (19.8) with C1 = 0.95, C2 = 1.81. With the known intermittency factor, the averaged velocity distribution in a transitional region is determined from: (19.9)

19 Basic Physics of Laminar-Turbulent Transition

525

with and as the solutions of laminar and turbulent flow, respectively. As an example, we take the Blasius solution for the laminar and the Prandtl-Schlichting solution for the turbulent portion of a transitional flow for details see Chapter 21) along a flat long plate at zero streamwise pressure gradient and construct the transitional velocity distribution using Eq. (19.9). The results are plotted in Fig.19.9, where the non-dimensional velocity u/U is plotted versus the non-dimensional variable y/į with į as the boundary layer thickness. Two distinctively different curves mark the start and end of the transition denoted by and . As seen, within the two -values the velocity profile changes significantly resulting in boundary layer parameters and particularly and skin friction that are different from those pertaining to laminar or turbulent flow (see Chapter 21).

1

u/U

0.8 0.6 0.4

Turbulent solution 1.0 0.7 0 .4 0 0. _ = J

Laminar solution

Transitional: Eq. (19.8)

0.2 0 0

0.25

0.5

y/G

0.75

1

Fig. 19.9: Velocity profile in transition region using the intermittency function (19.8)

19.4 Physics of Unsteady Boundary Layer Transition The flow through a turbomchinery components is of periodic unsteady nature. Steam and gas turbine stages, jet engines, turbines, compressors and pumps are a few examples. Within these machines unsteady interaction between individual components takes place. Figure 19.10 schematically represents the unsteady flow interaction between the stationary and rotating frame of a turbine stage. A stationary probe traversing downstream of the stator at station (2) records a spatially periodic velocity distribution. Another probe placed on the rotor blade leading edge that rotates with the same frequency as the rotor shaft, registers the incoming velocity signals as temporally periodic. The effect of this periodic unsteady inlet flow condition on the boundary layer along the blade surface is qualitatively and quantitatively different from the one with steady inlet flow condition. The difference is shown in a simplified sketch of a flat plate boundary layer presented by Herbst [15], Fig .19.11.

526

19 Basic Physics of Laminar-Turbulent Transition

1 Stationary frame (satator) 2

Wake flow

Rotating frame (rotor) 3

Direcion of rotation Wake flow

Fig. 19.10: Unsteady flow interaction within a turbine stage .

(a) Steady uniform inlet velocity Flat plate under zero pressure gradient

Laminar

Transitional

GL

Turbulent

GT

GTr

(b) Periodic unsteady inlet velocity Flat plate under zero pressure gradient

GT G GL

Fig. 19.11: Comparison of boundary layer thickness developments, (a) steady state, (b) instantaneous image of the boundary layer thickness

19 Basic Physics of Laminar-Turbulent Transition

527

While the boundary layer thickness į in case of a steady inlet flow condition (a) is temporally independent, the one in case of an unsteady inlet condition (b) experiences a temporal change.

19.4.1 Experimental Simulation of the Unsteady Boundary Layer As mentioned above, the complex three-dimensional structure of the flow through a turbomchinery component, the existence of the centrifugal and Coriolis forces and the interaction of the wakes originating from the suction surface, pressure surface and the trailing edge thickness do not allow extracting meaningful data from a turbine or compressor experiment essential for establishing an unsteady transition model. Of the above mutually interacting parameters, the wake flow impingement on the blade surfaces is of primary significance. This circumstance motivated several researchers to simulate the effect of periodic unsteady wakes on the boundary layer development. Pfeil and his co-workers [15] , [16] and [17] were the first to simulate the impact of unsteady wakes on a flat plate boundary layer at zero, negative and positive pressure gradients. They utilized a squirrel cage-type wake generator that generated wakes by a set of cylindrical rods periodically impinging on a flat plate and changing the transition behavior as Fig. 19.11 showed. They developed a wake-induced transition model and showed that the boundary layer grew naturally in between the induced transition regions by wakes. In contrast to the flat plate boundary layer at zero longitudinal as well as lateral pressure gradients, the boundary layer development along a turbine or a compressor blade is associated with non-zero pressure gradients that stem from inherent curvatures on blades suction and pressure surfaces. 1 2 50

0. 0

3 70 2

920.0

6 7

4

8

5

10

9

Fig. 19.12: Test section, 1. Traversing system, 2. Rotatable outer curved wall, 3. Wake generator with rods, 4. Reduction belt, 5. electric motor, 6. Curved plate, 7.Plexiglass side wall, 8. Movable inner wall, 9. Large vernier, 10. Small vernier

528

19 Basic Physics of Laminar-Turbulent Transition

To account for the curvature effects, Schobeiri and his co-workers [18] and [19] used a similar squirrel cage-type wake generator and a curved plate as shown in Figs. 19.12 and 19.13. They acquired systematic unsteady boundary layer data for developing a new unsteady boundary layer transition model discussed in the following sections. The wake generator shown in Fig. 19.12 produce a periodic unsteady flow condition at the inlet of the test section. It consists of two parallel, rotating circular disks between which cylindrical rods are circumferentially arranged. Since the wakes that emanate from the rods are intended to emulate the wakes that emanate from the preceding blade rows, the rod diameter must be chosen such that its drag coefficient cD corresponds to the one of the preceding blades. Under this condition, the cylinder wakes and the corresponding blades have identical turbulence structures in terms of Reynolds stress components in terms of . Figure 19.13 schematically shows the trajectory of the wake generating rods, the wakes and their impingement on the curved plate. The unsteady flow produced by the wake generator is characterized by an unsteady parameter ȍ that accounts for different spacings of wake generating rods and turbine blades. This parameter is similar to the Strouhal number and is defined for the turbine cascade as ȍ = ı/ij, where ı is the ratio of the arc length of the plate so, and the spacing between the rods sR, and ij is the ratio of the inlet velocity V and the circumferential velocity of the wake generator U. To experimentally investigate the transition behavior of compressor or turbine blade cascades under periodic unsteady flow condition, Schobeiri et al. [20], [21], and [22] utilized a conceptually different type wake generator, Figs. 19.14 and 19.15. Figure 19.14 compares the wakes generated by rotating turbine blades and cylindrical rods that are moving in upward direction. As in curved plate case, to simulate the wakes with similar turbulence structure as those from a turbine blades, the diameter

Wakes Moving rods

W

Curved plate

U

V U Fig. 19.13: Wakes generated by rods that are circumferentially attached to two opposite discs of the wake generator

19 Basic Physics of Laminar-Turbulent Transition

529

of the rods must be chosen such that it generates the same drag coefficient as the blade. Two-dimensional periodic unsteady flow is simulated by the translational motion of a wake generator shown in Figs. 19.14 and 19.15 with a series of cylindrical rods attached to two parallel operating timing belts driven by an electric motor. U

U

U

V

V

U

U

W

(a) Wakes from moving rods

W

(b) Wakes from turbine blades

Fig. 19.14: Simulation of wakes: (a) Wake from a moving cylinder rod, (b) wake from turbine rotor Turbine cascade

Test section

Rod cluster Spacing S3

Rod cluster Spacing S3

Rod cluster Rod cluster Spacing S2 Spacing S1

Wake generator unit

Single wire miniature boundary layer probe Static pressure blade Blade with surface hot film instrumentation Blade with surface hot film instrumentation Wake generating rods Boundary layer traversing slots

Timing belts with rod attachments Cascade test section

Traversing slot (one) at leading edge

Inlet nozzle Electric motor

Traversing slots (three) at trailing edge

Precission rotary probe holder Boundary layer traversing system

Fig. 19.15: Unsteady turbine cascade research facility, test section, boundary layer traversing system, different rod clusters attached to the wake generator

530

19 Basic Physics of Laminar-Turbulent Transition

Measuring the boundary layer development along a turbine or a compressor blade requires traversing the boundary layer along the suction and the pressure surface in streamwise as well as in normal directions. The traversing process must be repeated, if the effect of wake frequency impingement on the boundary layer is to be investigated. This is extremely time consuming. To significantly reduce the data acquisition time, the research facility is designed to allows measurements of up to four frequency ranges in one revolution. This is done by attaching up to four clusters of rods with different spacings to the two parallel timing belts moving with translational motion (Fig. 9.15).. The periodic unsteady flow produced by the wake generator is characterized by an unsteady parameter ȍ that accounts for different spacings of wake generating rods and turbine blades. For the turbine cascade it is defined as , where ı is the cascade solidity and ij is the flow coefficient. The values of ȍ cover a broad range that are typical of turbomachines and are specified in Table 1. Table 19.1: Specifications of unsteady turbine cascade research facility at TPFL Parameters Inlet velocity

Values Uin = 15 m/s

Nozzle exit height HN =1000.mm

Parameters

Values

Blade exit metal angle Į2 = 61.8( Nozzle exit width

WN =200.0 mm

Blade inlet angle

Į1 = 0(

Blade spacing

SB = 159.31 mm

Blade height

L = 200.0 mm

Blade Re-number

Rec = 264,187

Blade chord

c = 281.8 mm

Cascade flow coefficient

ij = 2.14

Cascade solidity

ı = 1.76

Rod diameter

DR = 5.0 mm

Steady reference

sR =  mm

ȍ - parameter steady case

ȍ = 0.0

Cluster 1 rod spacing

sR = 160.0 mm ȍ- parameter for cluster 1

ȍ = 0.755

Cluster 2 rod spacing

sR= 80.0 mm

ȍ - parameter for cluster 2

ȍ = 1.51

No. of rods in cluster 1

nR = 14

No. of rods in cluster 2 nR = 21

19.4.2 Ensemble Averaging High Frequency Data Figure 19.16 schematically represents three sets of unsteady velocity data taken at three different times but during the same time interval ǻt corresponding to the sequence i = 0 to i = N. Each of these sets is termed an ensemble. Considering the velocity distribution at an arbitrary position vector x and at an ensemble j such as , we use the same procedure that we applied to the statistically steady flow to

19 Basic Physics of Laminar-Turbulent Transition

531

u(t) [m/s]

Fig. 19.16: Periodic unsteady flow velocity with corresponding distribution of I(x,t) at a particular position x 10.5 10.0 9.5 9.0 10

S(t)

8 6 4 2 0 1

I(t)

0.8 0.6 0.4 0.2 0

1

0.5

0

0.01

0.02

0.03

0.04

0.05

time [s]

0.06

0.07

0.08

0.09

0.1

Fig. 19.17: Processing of instantaneous velocity signals to calculate ensemble averaged intermittency function for a periodic unsteady flow at y = 0.1 mm, S/S0 = 0.5235 and a reduced frequency ȍ = 3.443

532

19 Basic Physics of Laminar-Turbulent Transition

identify the nature of the periodic unsteady boundary layer flow. The corresponding intermittency function at a given position vector x is shown in Fig.19.16. For a particular instant of time identified by the subscript i for all ensembles, the ensemble average of over M number of ensembles results in an ensemble averaged intermittency function . Referring to Fig. 19.16, the ensemble averaged intermittency is defined as: (19.10) Figure 19.17 shows the steps necessary to process the instantaneous velocity data to obtain the ensemble averaged intermittency. Using the same procedure discussed in Section 19.3.2, the periodic unsteady velocity shown in Fig. 19.17 is produced by moving a set of cylindrical rods with the diameter of 2 mm in front of a curved plate, Fig. 19.12, placed in the mid height of a curved channel (for details see [23], [24] and [25]). For each ensemble, the velocity derivatives are obtained leading to a time dependent intermittency function . Taking the ensemble average of as defined by Eq. (19.10) results in an ensemble averaged , shown in Fig. 19.18. It shows a contour plot of the ensemble averaged intermittency taken at y = 0.1 mm above the plate along the concave surface of the curved plate from leading to trailing edge. The contour variable is plotted for two unsteady wake passing periods. Figure 19.18 also includes the transition start for steady state

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

: = 1.75

1

B >

t/W

2.0

1.0

Steady Transition S/S0 = 0.5

: L2 0.0 0.0

L3 :

0.

3

> ke with ke 0

V1 + 'V1 V1 - 'V 1

x1 Fig. 20.8: Explaining the mechanism of Prandtl mixing length theory

20 Turbulent Flow and Modeling in Turbomachinery

velocity

and the fluctuation momentum

579

. It arrives at the layer

which has a lower velocity . According to the Prandtl hypothesis, this macroscopic momentum exchange most likely gives rise to a positive fluctuation . This results in a negative non-zero correlation particle B moving upward with the velocity a lower longitudinal velocity

. Inversely, the fluid

from the layer

, where

prevails, causes a negative fluctuation

. In both cases, the particles experience a velocity difference which can be approximated as: (20.133) by using the Taylor expansion and neglecting all higher order terms. The distance between the two layers lm is called mixing length. Since has the same order of magnitude as

, we may replace in (20.133)

by

and arrive at: (20.134)

Note that the mixing length lm is still an unknown quantity. Since, by virtue of the Prandtl hypothesis, the longitudinal fluctuation component was brought about by the impact of the lateral component

, it seems reasonable to assume that

such that with Eq. (20.134), we may find for the lateral fluctuation component

with C1 as a constant. Thus, the

component of

the Reynolds stress tensor becomes: (20.135)

Since the constant C1 as well as the mixing length lm are unknown, the constant C1 may be included in the mixing length such that we may write (20.136)

Considering Eqs. (20.91) and (20.130), the shear stress component becomes:

580

20 Turbulent Flow and Modeling in Turbomachinery

(20.137) Equation (20.137) does not take into account that the sign of the shear stress component IJ12 changes with . To correct this, we may write

(20.138) with as the eddy viscosity. This is the Prandtl mixing length hypothesis. From Eq. (20.138) we deduce that the eddy kinematic viscosity can be expressed as:

(20.139) To find an algebraic expression for the mixing length lm, several empirical correlations were suggested that are discussed by Schlichting [15] and summarized by Wilcox [16]. The mixing length lm does not have a universally valid character and changes from case to case, therefore it is not appropriate for three-dimensional flow applications. However, it is successfully applied to boundary layer flow (for details see Chapter 21) and particularly to free turbulent flows. Utilizing the two-dimensional boundary layer approximation by Prandtl, and for the sake of simplicity, we use the boundary layer nomenclature with the mean-flow component, as the significant velocity in -direction, the distance from the wall , the dimensionless velocity and the dimensionless distance from the wall . The wall friction velocity is related to the wall shear stress by the relation . Figure 20.9 exhibits the non-dimensionalized flow velocity distribution of a fully turbulent boundary layer as a function of the nondimensionalized normal wall distance . The plot with the log-scale for reveals three distinct layers: the viscous sublayer ranging from to 5, followed by a buffer layer that is tangent to the logarithmic layer at about . The buffer layer extends from to 200. The viscous sublayer is approximated by the linear wall function: (20.140) followed by the logarithmic layer which is approximated by (20.141)

20 Turbulent Flow and Modeling in Turbomachinery

25 (1) u+=y+ y+sub = 5

20

u+

15

+

+ = u (2 )

l 1/N

ny

+

581

C

(3) y+ = f(u+, N, C) = 200

10

y

+

log

5 100

101 y+

102

103

Fig. 20.9: Approximation of velocity distribution for a fully turbulent flow by its decomposition into a laminar sublayer curve (1), a logarithmic layer (2) and the buffer layer (3) extending from y+ = 5 to 200 For a fully developed turbulent flow, the constants in Eq. (20.141) are experimentally found to be and 0, Fig. 20.9,Curve 2. For transitional boundary layer flows, these constants change significantly, as detailed in a study by Müller [17]. A third layer, the outer layer, tangents the logarithmic layer described by a so-called wake function, is discussed in Chapter 10. Outside the viscous sublayer marked as the logarithmic layer, the mixing length is approximated by a simple linear function (20.142) Accounting for viscous damping, the mixing length for the viscous sublayer is modeled by introducing a damping function D into Eq. (20.142). As a result, we obtain (20.143) with the damping function D proposed by van Driest [18] as (20.144) with the constant for a boundary layer at zero-pressure gradient. As we will discuss in Chapter 11, based on experimental evaluation of a large number of velocity profiles, Kays and Moffat [19] developed an empirical correlation for that

20 Turbulent Flow and Modeling in Turbomachinery

582

accounts for different pressure gradients and boundary layer suction/blowing. For zero suction/blowing this correlation reduces to: (20.145) with pressure gradient P+in Eq. (20.145) is defined in as

. The dimensionless

(20.146) Introducing Eq. (20.145) into Eq. (20.144), the Van Driest damping function is plotted in Fig. 20.10.

1.00

03 0. 01 0.

.01 +

0. 0

0.75

-0

D

P

=-

0.0

2

0.50 0.25 0.00

0

50

100 y+

150

200

Fig. 20.10: Van Driest Damping function with p+ as a parameter Figure 20.10 exhibits the damping function D as a function of y+ with the pressure gradient P+ as a parameter. The implementation of the damping function into the mixing length accounts for the non-linear distribution of the mixing length in the lateral direction as shown in Fig. 20.11. For comparison purposes, the linear distribution is also plotted as a dashed line. For and the curves pertaining to asymptotically approach the linear distribution. Inside the viscous sublayer significant differences are clearly visible, as shown in Fig. 20.12 which is a partial enlargement of Fig. 20.11. It exhibits the mixing length distribution very close to the wall. For comparison purposes, the linear distribution is also plotted.

20 Turbulent Flow and Modeling in Turbomachinery

583

2.0

lm (mm)

1.5 p+ = 0.03

1.0

Dashed line: lm= N y

0.5 +

p = -0.02

0.0 0.0

1.0

' p+ = 0.01

2.0 y (mm)

3.0

4.0

Fig. 20.11: Mixing length in lateral wall-direction with p+ as parameter

0.075 Dashed line: lm= N y

+

p

0.050

=0 +

p

0.025 0.000

.0 3

=0

.0

Solid lines: lm= N D y

lm (mm)

0.100

+ -0.02 p =

0.05

0.10

0.15

0.20 y (mm)

0.25

0.30

Fig. 20.12: Details of the mixing length from Fig. 20.11 Considering these differences and the asymptotic approach mentioned above, it seems that Eq. (20.143) is suitable for describing the mixing length within a boundary layer for which the von Kármán constant k assumes the value k = 0.41. This value, as previously mentioned, is valid only for fully turbulent flows at zero-pressure gradient. It is not valid for transitional boundary layers, where ț changes significantly. Moreover, Eq. (20.139) implies that whenever , the eddy kinematic viscosity approaches zero, which contradicts the experimental data. In addition, the mixing length concept does not apply to boundary layer flow cases where a flow separation occurs. Furthermore, it is not suitable for three-dimensional flow

584

20 Turbulent Flow and Modeling in Turbomachinery

calculation as mentioned before. Concluding the discussion about the mixing length approach, it can be stated that the turbulence model based on the Prandtl mixing length theory, despite its shortcomings, is still used for boundary layer calculation and delivers satisfactory results, as seen in Chapter 11. For the sake of completeness, in what follows, we present a system of equations that includes the algebraic model. The system can be used to solve steady incompressible free turbulent flow, as well as boundary layer problems, where no separation occurs. Utilizing the two-dimensional boundary layer flow assumption by Prandtl, for the sake of compatibility with his convention, we use the Prandtl nomenclature with , , and as the mean flow velocities and the fluctuation components in and -direction, thus the continuity and momentum equations are reduced to (20.147) (20.148) According to the Prandtl boundary layer assumptions (see for details Chapter 21), the pressure gradient outside the boundary layer may be approximated by the Bernoulli equation, where the flow is assumed isentropic (20.149) with a known velocity distribution shear stress in Eq.(20.148) becomes

outside the boundary layer. The turbulent

(20.150) with 5m, from Eq. (20.143) in conjunction with Eqs. (20.147), (20.148) and (20.149), a solution can be obtained for the main-velocity field in terms of and .

20.3.2 Algebraic Model: Cebeci-Smith Model Another algebraic model is the Cebeci-Smith [20] which has been used primarily in external high speed aerodynamics with attached thin boundary layer. It is a two-layer algebraic zero-equation model which gives the eddy viscosity by separate expressions in each layer, as a function of the local boundary layer velocity profile. The model is not suitable for cases with large separated regions and significant curvature/rotation effects. The turbulent kinematic viscosity for the inner layer is calculated from (20.151)

20 Turbulent Flow and Modeling in Turbomachinery

585

For the outer layer kinematic viscosity is (20.152) with (20.153) , the velocity at the edge of the boundary layer, the boundary layer displacement thickness and as the Klebanoff intermittency function [21]. The mixing length in Eq. (20.151) is determined by combining Eqs. (20.143) and (20.144) (20.154) with

and

.

20.3.3 Baldwin-Lomax Algebraic Model The third algebraic model is the Baldwin-Lomax model [22]. The basic structure of this model is essentially the same as the Cebeci-Smith model with the exception of a few minor changes. Similar to Cebeci-Smith, this model is a two-layer algebraic zero-equation model which gives the eddy kinematic viscosity Ȟt as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbomachinery applications. While this model is quite robust and provides quick results, it is not capable of capturing details of the flow field. Since this model is not suitable for calculating flow situations with separation, its applicability is limited. We briefly summarize the structure of this model as follows. The kinematic viscosity for the inner layer is (20.155) with (20.156) and

as the rotation tensor. The outer layer is described by (20.157)

with the wake function Fwake

20 Turbulent Flow and Modeling in Turbomachinery

586

(20.158) and Fmax and ymax as the maximum of the function (20.159) The velocity difference

is defined as the difference of the velocity at ymax and ymin: (20.160)

with the closure coefficients listed in the Table 20.1 Table 20.1: Closure Coefficients of Eqs.(20.157) through (20.159) k

Į

A+

Ccp

CKleb

Cw

0.4

0.0168

26

1.6

0.3

1

The above zero-equation models are applied to cases of free turbulent flow such as wake flow, jet flow, and jet boundaries.

20.3.4 One-Equation Model by Prandtl A one-equation model is an enhanced version of the algebraic models we discussed in previous sections. This model utilizes one turbulent transport equation originally developed by Prandtl. Based on purely dimensional arguments, Prandtl proposed a relationship between the dissipation and the kinetic energy that reads (20.161) where the turbulence length scale 5t is set proportional to the mixing length, 5m, the boundary layer thickness į or a wake or a jet width. The velocity scale in Eq. (20.132) is set proportional to the turbulent kinetic energy as suggested independently by Kolmogorov [23] and Prandtl [24]. Thus, the expression for the turbulent viscosity becomes: (20.162) with the constant Cȝ to be determined from the experiment. The turbulent kinetic energy, k, as a transport equation is taken from Section 20.2.2 in the form of Eq. (20.111) or (20.126) where the dissipation is implemented. For simple twodimensional flows where no separation occurs, with the mean-flow component as the significant velocity in -direction, and the distance from the wall ,the following approximation by Launder and Spalding [25] may be used

20 Turbulent Flow and Modeling in Turbomachinery

587

(20.163) where ık = 1 and CD = 0.08 are coefficients determined from experiments utilizing simple flow configurations. The one-equation model provides a better assumption for the velocity scale Vt than . Similar to the algebraic model, the oneequation one is not applicable to the general three-dimensional flow cases since a general expression for the mixing length does not exist. Therefore the use of a oneequation model does not offer any improvement compared with the algebraic one. The one-equation models discussed above are based on kinetic energy equations. There are a variety of one-equation models that are based on Prandtl’s concept and discussed in [14].

20.3.5 Two-Equation Models Among the many two-equation models, three are the most established ones: (1) the standard k -J model, first introduced by Chou [26] and enhanced to its current form by Jones and Launder [27], (2) k -Ȧ model first developed by Kolmogorov and enhanced to its current version by Wilcox [14] and (3) the shear stress transport (SST) model developed by Menter [28], who combined and models by introducing a blending function with the objective to get the best out of these two models. All three models are built-in models of commercial codes that are used widely. In the following, we present these models and discuss their applicability. 20.3.5.1 Two-Equation k-J Model: The two equations utilized by this model are the transport equations of kinetic energy k and the transport equation for dissipation J. These equations are used to determine the turbulent kinematic viscosity . For fully developed high Reynolds number turbulence, the exact transport equations for k (20.126) can be used. The transport equation for J (20.129) includes triple correlations that are almost impossible to measure. Therefore, relative to J, we have to replace it with a relationship that approximately resembles the terms in (20.129).To establish such a purely empirical relationship, dimensional analysis is heavily used. Launder and Spalding [26] used the following equations for kinetic energy (20.164) and for dissipation (20.165) and the turbulent viscosity,

, can be expressed as

20 Turbulent Flow and Modeling in Turbomachinery

588

(20.166) The constants ık, ıJ, CJ1, CJ2 and Cȝ listed in the following table are calibration coefficients that are obtained from simple flow configurations such as grid turbulence. The models are applied to such flows and the coefficients are determined to make the model simulate the experimental behavior. The values of the above constants recommended by Launder and Spalding [11] are given in the following Table 20.2 Table 20.2: Closure Coefficients of Eq. (20.165) Cȝ

ık

ıJ

CJ 1

CJ 2

0.09

1

1.3

1.44

1.92

As seen, the simplified Eqs. (20.165) and (20.166) do not contain the molecular viscosity. They may be applied to free turbulence cases where the molecular viscosity is negligibly small compared to the turbulence viscosity. However, one cannot expect to obtain reasonable results by simulation of the wall turbulence using these equations. This deficiency is corrected by introducing the standard k-J model. This model uses the wall functions where the velocity at the wall is related to the wall shear stress by the logarithmic law of the wall. Jones and Launder [26] extended the original k-J model to the low Reynolds number form, which allows calculations right up to a solid wall. In the recent three decades, there have been many twoequation models, some of which Wilcox has listed in his book [14]. In general, the modified k- and J-equations, setting v = ȝ/ȡ and vt = ȝt /ȡ, are expressed as

(20.167) (20.168) The closure coefficients are listed in Table 20.2. The Reynolds stress, be expressed as

, can

(20.169) where is the eddy length scale. Using the k - J model, successful simulations of a large variety of flow situations have been reported in a number of papers that deal with internal and external aerodynamics, where no or minor separation occurs. However, no satisfactory results

20 Turbulent Flow and Modeling in Turbomachinery

589

are achieved whenever major separation is involved, indicating the lack of sensitivity to adverse pressure gradient. The model tends to significantly overpredict the shear-stress levels and thereby delays (or completely prevents) separation. This exhibits a major shortcoming, which Rodi [29] attributes to the overprediction of the turbulent length-scale in the near wall region. Menter [30] pointed to another shortcoming of the k - J model which is associated with the numerical stiffness of the equations when integrated through the viscous sublayer. 20.3.5.2 Two-Equation k-Ȧ-Model: This model replaces the J-equation with the Ȧtransport equation, first introduced by Kolmogorov. It combines the physical reasoning with dimensional analysis. Following the Kolmogorov hypotheses, two quantities, namely J and ț, seem to play a central role in his turbulence research. Therefore, it seemed appropriate to establish a transport equation in terms of a variable that is associated with the smallest eddy and includes J and ț. This might be a ratio such as or . Kolmogorov postulated the following transport equation (20.170) with ȕ and ı as the two new closure coefficients. As seen, unlike the k-equation, the right-hand-side of Eq. (20.170) does not include the production term. This equation has undergone several changes where different researchers tried to add additional terms. The most current form developed by Wilcox [31], reads (20.171) with equation as

as the specific Reynolds stress tensor. Wilcox also modified the k-

(20.172) He also introduced the kinematic turbulent viscosity (20.173)

With

as the matrix of the mean deformation tensor. Wilcox

defined the following closure coefficients and auxiliary relations

590

20 Turbulent Flow and Modeling in Turbomachinery

(20.174)

Furthermore,

(20.175)

The k-Ȧ model performs significantly better under adverse pressure gradient conditions than the k - J model. Another strong point of the model is the simplicity of its formulation in the viscous sublayer. The model does not employ damping functions, and has straightforward Dirichlet boundary conditions. This leads to significant advantages in numerical stability, Menter [28]. A major shortcoming of the k - Ȧ model is its strong dependency on freestream values. Menter investigated this problem in detail, and showed that the magnitude of the eddy-viscosity can be changed by more than 100% just by using different values for . 20.3.5.3 Two-Equation k-Ȧ-SST-Model: Considering the strength and the shortcomings of and models briefly discussed in the previous two sections, Menter [32], [33] and [34] introduced a blending function that combines the best of the two models. He modified the Wilcox k-Ȧ model to account for the transport effects of the principal turbulent shear-stress. The resulting SST-model (Sear Stress Transport model) uses a formulation in the inner parts of the boundary layer down to the wall through the viscous sublayer. Thus, the SST- k-Ȧ model can be used as a low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k-J mode in the free-stream and thereby avoids the common k-Ȧ problem that the model is too sensitive to the turbulence free-stream boundary conditions and inlet free-stream turbulence properties. For the sake of completeness, we present the Menter’s SST-model in terms of Ȧ-equation with the blending function F1

20 Turbulent Flow and Modeling in Turbomachinery

591

(20.176)

and the turbulence Kinetic Energy (20.177) The term in Eq. (20.176) and (20.177) is a production limiter and is defined in Eq. (20.183). The blending function F1 is determined from (20.178) with the argument arg1 (20.179)

(20.180)

and F1 is equal to zero away from the surface (k-model), and switches over to one inside the boundary layer (k-model). The turbulent eddy viscosity is defined as follows: (20.181) with

and F2 as a second blending function defined by: (20.182)

A production limiter in Eq. (20.183) is used in the SST model to prevent the build-up of turbulence in stagnation regions. It is defined as (20.183) with

592

20 Turbulent Flow and Modeling in Turbomachinery

(20.184) All constants are computed by a blend from the corresponding constants of the k- J and the k- Ȧ model via , etc. The constants for this model are

According to [35], the above version of the k-J and k-Ȧ equations, including constants listed above, is the most updated version. Closing Remarks: The multitude of the closure constants in the above discussed turbulence models have been calibrated using different experimental data. Since the geometry, Re-number, Mach number, pressure gradient, boundary layer transition and many more flow parameters differ from case to case, the constants may require new calibrations. The question that arises is this: can any of the models discussed above a priori predict an arbitrary flow situation? The answer is a clear no. Because all turbulence models are of purely empirical nature with closure constants that are not universal and require adjustments whenever one deals with a completely new case. As we saw, in implementing the exact equations for k and J that constitute the basis for k-J as well as k-Ȧ model, major modifications had to be performed. Actually, in the case of J-equations, the exact equation is surgically modified beyond recognition. Under this circumstance, none of the existing turbulence models can be regarded as universal. Considering this situation, however, satisfactory results can be obtained if the closure constants are calibrated for certain groups of flow situations. Following this procedure, numerous papers show quantitatively excellent results for groups of flow cases that have certain commonalities. Examples are flow cases at moderate pressure gradients and simple geometries. More complicated cases where the sign of the pressure gradient changes, flow separation and re-attachment occur and boundary layer transition plays a significant role still not adequately predicted. The models presented above are just a few among many models published in the past three decades and summarized in [14]. In selecting these models, efforts have been made to present those that have been improved over the past three decades and have a longer lasting prospect of survival before the full implementation of DNS that makes the use of turbulence models obsolete.

20.4 Grid Turbulence Calibration of closure coefficients and a proper model assessment require accurate definition of boundary conditions for experiments as well as computation. These include, among other things, information about inlet turbulence such as the turbulence intensity, length, and time scales. This information can be provided by using turbulence grids, Fig. 20.13.

20 Turbulent Flow and Modeling in Turbomachinery

593

Table 20.3: Turbulence Grids: Geometry, turbulence intensity and length scale Turbulence Grid

Grid Opening GO

Rod Thickness RT (mm)

Turbulence Intensity Tu

Length Scale ȁ(mm)

No Grid

100%

0

1.9%

41.3

TG1

77%

6.35

3.0%

32.5

TG2

55%

9.52

8.0%

30.1

TG3

18%

12.7

13.0%

23. 4

Turbulence generator grid with quadratic rod cross section

RT

RT = Rod thickness, GO = Grid Opening ratio

Fig. 20.13: Turbulence grid The grids may consist of an array of bars with cylindrical or quadratic cross sections. The thickness of the grid bars and the grid openings determine the turbulence intensity, length, and time scale of the flow downstream of the grid. Immediately downstream of the grid, a system of discrete wakes with vortex streets are generated that interact with each other. Their turbulence energy undergoes a continuous decay process leading to an almost homogeneous turbulence. The grid is positioned at a certain distance upstream of the test section in such a way that it generates homogeneous turbulence. The examples show how to achieve a defined inlet turbulence condition. Table 20.3 shows the data of three different turbulence grids for producing inlet turbulence intensities Tu = 3.0%, 8.0%, and 13.0%. The grids consist of square shaped aluminum rods with the thickness RT and opening GO. The turbulence quantities were measured at the test section inlet with a distance of 130 mm from the grid. Figure 20.14(a) shows the power spectral density of the velocity

20 Turbulent Flow and Modeling in Turbomachinery

594 -2

10

TG3: Tu = 13% -3

10

TG2: Tu = 8%

-4

PSD

10

-5

10

TG1: Tu = 3%

-6

10

(a) 10

2

3

10

Frequency

4

10

10 Grid specifications are listed in Table 2

/ (cm)

8

No grid Grid: TG1

6

Grid: TG2 Grid: TG3

4 2 0

(b) 0

5

10

Tu%

15

20

Fig. 20.14: (a) Power spectral distribution PSD as a function of frequency for three different grids described in Table 20.3. The results from (a) is used to generate the turbulence length scales as a function of turbulence intensity (b). signals from a hot wire sensor as a function of signal frequency. The length scale is calculated from [mm], Fig. 20.14 (b).

20.5 Examples of Two-Equation Models 20.5.1 Internal Flow, Sudden Expansion The following representative examples should illustrate the substantial differences between the two-equation models we presented above. The flow through a sudden

20 Turbulent Flow and Modeling in Turbomachinery

595

expansion is appropriate for comparison purposes for two reasons: (1) It has a flow separation associated with a circulation zone and (2) it is very easy to obtain experimental data from this channel. Standard k-J vs. k-Ȧ: Figures 20.15 and 20.16 show flow simulations through a channel with a sudden expansion ratio of 2/1 using k-Ȧ and standard k-J models. The purpose was to simulate the flow separation. The k-J simulation, Fig. 20.15, delivers a single large corner vortex. However, experiments show that for this type of flow generally a system of two or more vortices, are present, Fig. 20.16.

Fig. 20.15 (a) Simulation with k-J model Fig. 20.16 Simulation with k-Ȧ-model

20.5.2 Internal Flow, Turbine Cascade Flow simulation with CFD has a wide application in aerodynamic design of turbines, compressors, gas turbine inlet nozzles and exit diffusers. As an example, Fig.20.17 shows contour plots of velocity and pressure distributions in a high efficiency turbine blade using SST-turbulence model. On the convex surface (suction surface), the flow is initially accelerated at a slower rate from the leading edge and exits the channel at a higher velocity close to the trailing edge. The acceleration process is reflected in pressure contour.

Fig. 20.17: Flow simulation through a turbine cascade, TPFL-Design

596

20 Turbulent Flow and Modeling in Turbomachinery

20.5.3 External Flow, Lift-Drag Polar Diagram This example presents two test cases to predict the lift-drag polar diagram of an aircraft without and with engine integration, Figs. 20.18. Figure 20.19 shows the predicted lift-drag polar diagrams for two geometries, one the aircraft with engines and one without. The computation was performed using the SST-turbulence model and the results compared with the experiments. The lift and

Fig. 20.18: Geometry with engine integration forpredicting the polar diagram, from [32]

Fig. 20.19: Lift-Drag polar diagram for an aircraft model without engine (WB) and with engines (WVBN), from [32]

20 Turbulent Flow and Modeling in Turbomachinery

597

drag coefficients plotted in Fig. 20.19 are integral quantities that represents the lift and drag forces acting on the entire aircraft. Thus, they represent the lift and drag distribution integrated over the entire surface. The polar diagram is obtained by varying the angle of attack and measuring and computing the lift and drag forces. These forces are then non-dimensionalized with respect to a constant reference force, which is a product of a constant dynamic pressure and a characteristic area of the aircraft. Once a complete set of data for a given range of angle of attack is generated, then for each angle the lift coefficient is plotted against the drag coefficient as shown in Fig. 20.19.

20.5.4 Case Study: Flow Simulation in a Rotating Turbine The above examples were dealing with relatively simple flow cases, where a flow simulation using k-J, k-Ȧ or the combination thereof renders satisfactory results. However, simulating a complex flow situation, where different parameters are involved, leads to results, whose proper interpretations require the understanding of the model shortcomings. As an example, we discuss in detail the flow through a high pressure (HP) turbine. The parameters involved in this case are, among others, change of frame of reference from stationary to rotating frame, unsteady inlet flow condition, impingement of periodic wakes from preceding rotating blade rows on the succeeding ones, changes of pressure gradient, presence of secondary vortices and laminar-turbulent transition. The case study presented in the following deals with the numerical simulation of the aerodynamic behavior of a two-stage high pressure (HP) turbine. As discussed in previous Chapter 7, HP- turbines have relatively small aspect ratios which develop major secondary flow regions at the hub and tip. For the shrouded HP-blades, close to the hub and casing, the fluid particles move from the pressure to the suction side and generate a system of vortices. In unshrouded blades, a system of tip clearance vortices are added to the tip endwall vortices. These vortices induce drag forces that are the cause of the secondary flow losses. Focusing on the secondary flow loss mechanisms, the fluid particles within the endwall boundary layers are exposed to a pitchwise pressure gradient in the blade channel. In addition, their interaction with the main flow causes angle deviation inside and outside the blade channel, resulting in additional losses. Thus, in HP-turbines with small aspect ratios, the secondary flow loss of almost 40-50% is the major loss contributor. The degree of accuracy in simulating the complex flow physics of the unsteady, transitional and highly vortical flow of a HP-turbine is an indicator of the predictive capability of any computational method including the steady Reynolds averaged Navier Stokes (RANS) and unsteady Reynolds averaged Navier Stokes (URANS). Numerous papers published over the past three decades show that, there are still substantial differences between the experimental and the numerical results pertaining to the individual flow quantities. These differences are integrally noticeable in terms of major discrepancies in aerodynamic losses, efficiency and performance of turbomachines. As a consequence, engine manufacturers are compelled to frequently calibrate their simulation package by performing a series of experiments before issuing efficiency and performance guaranties. In the following an attempt is made at identifying the quantities, whose simulation inaccuracies are preeminently responsible for the aforementioned differences. This task requires (a) a meticulous experimental investigation of all individual thermo-fluid quantities and their interactions resulting in an integral behavior of the

20 Turbulent Flow and Modeling in Turbomachinery

598

turbomachine in terms of efficiency and performance, (b) a detailed numerical investigation using appropriate grid densities based on simulation sensitivity, and (c) steady and unsteady simulations to ensure their impact on the final numerical results. To perform the above experimental and numerical tasks, a two-stage high pressure axial turbine rotor has been designed for generating benchmark data and to compare with the numerical results. Detailed interstage radial and circumferential traversing presents a complete flow picture of the second stage. Performance measurements were carried out for design and off-design rotational speed. As mentioned earlier, for comparison with numerical simulation, the turbine was numerically modeled using ANSYS-CFX code. An extensive mesh sensitivity study was performed to achieve a grid-independent accuracy for both steady and transient analysis. In discussing the results, a close look is taken at the physics behind the models. This helps to better understand the deficiency of the existing J-equation and thus the associated Ȧ-equation and explains, why these models are not capable of a priory predicting any arbitrary flow situation and require calibration for each single flow case. Experimental Research Facility: To address the major turbine performance and flow research issues, a state-of-the-art multipurpose turbine research facility was developed and designed at Turbomachinery Performance and Flow Research Laboratory of Texas A&M University, Figs 20.20 and 20.21. Details of the facility, its data acquisition and data analysis are presented in the papers by Schobeiri and his co-workers [36], [37] and [38]. The modular structured design of the research turbine enables to incorporate up to three turbine stages. The stator and rotor blades may be of any arbitrary design, with degree of reaction ranging from zero to a higher degree of reaction. Furthermore, the blade may be of any geometric configuration such as simple cylindrical, 3-D-stacked, or 3-D with compound lean and sweep. They also may be test blades of high, intermediate, or low pressure turbine units. The facility is capable of accurately measuring the efficiency and performance of the stages and the entire turbine. In particular, the highly advanced interstage traversing system provides a detailed flow picture of the turbine middle stage. The turbine rotor pertaining to this investigation includes two stages as described below.

6 1

9

2

10

3

4a

4b

4c

7 7a

7a

8

5

1

Electric drive

4a

Honeycomb

2

Compressor

4b

Flow straightener 6

3

Calibrated Venturi 4c

Exit diffuser

5

7

7a

Flex. couplings

Turbine inlet with heater

8

Dynamometer

Torque meter

9

Traversing system

Research turbine

10 Silence chamber

Fig. 20.20: The TPFL research turbine facility with its components, the circumferential traversing system (9) is driven by another traversing system sitting on a frame and is perpendicular to this plane.

20 Turbulent Flow and Modeling in Turbomachinery

Station 5

Station 4

Station 3

Station 2

599

Station 1

Fig. 20.21: Positions of the three five hole probes for radial and circumferential traverse of the flow field downstream of the first rotor, second stator and second rotor. The probes are position in such a way that their wakes do not interfere with each other. Turbine Component, Blading: The turbine rotor specifically designed for the current benchmark data generation is a two-stage turbine shown in Fig. 20.21 with the dimensions specified in Table 20.4. To achieve a high degree of versatility, the turbine was designed with a casing that incorporates rotatable stator rings with a clocking system externally attached to the ring. Thus, the stator rings can be indexed individually and externally without disassembling the turbine. For the current investigation, the clocking system was not used. The casing also incorporates three T-rings each of which incorporates a custom designed five hole probe for circumferentially traversing the entire stage flow field from hub to tip. The T-rings seal the turbine internal flow from outside while circumTable 20.4: Specifications of the two-stage turbine Item

Specification

Item

Stage no.

N=2

Mass flow

Hub diam.

Dh = 558.8 [mm]

Speed

n = 17503000 [rpm]

Tip diam.

Dt = 685.8 [mm]

Power

Power = 2696 [kW]

Blade Height

hB = 63.5 [mm]

Pressure Ratio

Blade no.

Stator 1,2 = 66

Blade no.

Design point quantities :

= 3.84 [kg/s],

Specification = 2.563.84 [kg/sec]

= 1.161.45 Rotor 1,2 = 63

= 1.45,

nD = 3000 rpm

600

20 Turbulent Flow and Modeling in Turbomachinery

ferentially moving within three slots that are cut in the casing. They can cover a maximum traversing range of 150 degree which is the size of multiple turbine blade spacings. The turbine blades implemented in this rig were designed to operate at adverse off-design operation conditions without suffering major losses. Particular attention was paid to the stator blade design which has a symmetric compound lean, Fig. 20.22. The rotor blades, Fig.20.23, however, are fully three-dimensional and are tapered from pub to tip.

Fig. 20.22: Stator blade

Fig. 20.23: Rotor blade

To ensure that the stator blades meet the design expectations, comprehensive design and off-design operation experiments were conducted in a cascade facility prior to implementing the blades into the turbine rig. The results presented in [39] show that the profile loss coefficient remains constant over a wide incidence range of ±25o. Both rotor and stator blades are shrouded with segments that are attached to the hub and tip of stators and rotor blades, respectively. Both stator and rotor blades were designed using streamline curvature method described in Chapter 11. Interstage flow measurements were conducted at stations 3, 4 and 5, Fig. 20.21. For this purpose, three miniature five-hole probes were utilized using a non-nulling calibration method. Traversing the flow field in radial and circumferential direction was accomplished by a seven axes traversing system with step motor controller with encoder and decoder on each axis. Numerical Treatment: This section deals with the numerical simulations of the research turbine. Starting with the mesh generation, numerical simulations are conducted using Reynolds averaged Navier-Stokes (RANS) simulation first. As we will see in the following, the comparison of the numerical results with the experimental ones reveals certain discrepancies in quantities such as velocities, pressures, temperatures and particularly row loss coefficients and turbine efficiency. To find the cause of these discrepancies, before expediting the physics behind the turbulence and transition models that may cause the discrepancies, we performed a rather time consuming unsteady simulation using unsteady RANS, the URANS. The results of both simulations will be presented in the section Results and Discussions. The flow solver used in this study is the commercially available ANSYS-CFX finite volume solver [40] with the SSTturbulence model publically known and a proprietary transition model. Both simulation cases were run on the Texas A&M university supercomputing facility. Mesh Specifications: Stator and seal domains were meshed using a multi-block meshing technique to produce a structured hexahedral mesh. During the meshing process the blade surfaces were surrounded with an O-grid to create a boundary layer

20 Turbulent Flow and Modeling in Turbomachinery

601

mesh, which used 16 node points normal to the wall. Using Fig. 20.21 as the turbine geometry prior to discretization into a numerical grid, five mesh densities were created for mesh sensitivity study that requires a y+ values close to 1. The results of several sensitivity studies showed that a mesh with the density of 4.8×106 can be deemed a good compromise between resource requirements and the required results accuracy. Fig. 20.24 shows one of the numerical grids that were used for the mesh sensitivity study and the subsequent numerical analyses. Two mesh versions were created. Version 1 is used for steady-state (RANS) analysis, where the stator and rotor blade numbers are identical with the actual experimental setup listed in Table 20.4. In contrast to RANS, for URANS-simulation, it is not possible to have unequal areas across the stage interface and therefore the stator numbering was reduced from 66 to 63 to match the rotor blade numbering. This is further explained in the steadystate and transient analysis sections.

Fig. 20.24: Mesh for numerical analysis Boundary Conditions: The boundary conditions used for CFD-simulation were taken from turbine rig experiments conducted at various rotational speed, including 1800 and 3000 rpm. A list of all the boundary conditions used for the two simulation cases are shown in Table 20.5. The global machine inlet and exit points are labeled stations 1 and 6 respectively. Referring to Fig. 20.21, Station 3 is the interstage measurement station between the first rotor exit (R1) and the second stator inlet (S2). Station 4 is between the second stator exit (S2) and the second rotor inlet (R2), and finally station 5 is the measurement point at the exit of the second rotor R2.

602

20 Turbulent Flow and Modeling in Turbomachinery

Table 20.5: Boundary Conditions for three RPMs a pressure ratio of 1.38 Boundary Condition

1800 RPM

2600 RPM

3000 RPM

Reference Pressure [kPa]

101.328

101.382

101.382

Total Pressure Inlet [kPa]

100.744

100.826

100.357

Static Pressure Outlet [kPa]

70.0

69.7

69.4

Total Temperature Inlet [K]

317.2

318.6

318.3

Total Temperature Outlet [K]

293.4

291.4

290.7

Inlet Turbulence Intensity [%]

5

5

5

Simulations: In the following, a RANS and a URANS simulation procedure are presented. For the RANS simulation a specific row-to-row interface model is used which spatially averages velocity and pressure data from one row and passes this onto to the next row as boundary conditions. In the case of URANS simulation, the flow field is assumed to be unsteady and, therefore the rotor row is actually rotated when compared to the stator rows via the use of a sliding mesh interface. The sliding mesh allows for a full model of the interface between each row, without the need of any kind of averaging. This provides a full time-dependent solution and accounts for various interactions between stator and rotor rows. The main disadvantage of using a transient analysis are the greater computing resources required. RANS: Steady State Simulation: For steady state cases, each turbine row was defined as a separate domain within the solver workspace, and specific interfaces were defined for the seals' entrance and exit points for both the rotor and stator stages. Between the stator and rotor domains a "Stage" interface was used which performs a circumferential average of the fluxes through bands on the interface between the stator and rotor, this is the mixing-plane method introduced by [41]. Using this interface model each fluid zone (blade row) is solved and two adjacent frames are coupled by exchanging flow variables at the prescribed interface location. The spatial averaging at the interfaces removes any unsteady effects that would arise due to changes from one blade row to the next blade row, example of such effects include, wakes, shock waves and separated flow. This method allows for a steady-state calculations of multistage turbomachinery flows. In addition, circumferential periodic boundary conditions are placed on the upper and lower surfaces of both the stator and rotor domains. The use of the stage interface condition allows for unequal blade numbering between stator and rotor domains, in this case the stator blade number is 66 and the rotor blade number is 63 as listed in Table 20.4, which translates to a ratio of 1.047 at the stator to rotor interface which is lower than the 1.2 limit as prescribed by [41]. Since the turbine facility operates at a low inlet and exit temperatures, the turbine working medium air was assumed as an ideal gas with the constant specific heat capacities (perfect gas). Each simulation case was iterated until the global root mean square error residuals for the RANS-equations reached values below 1x10-6.

20 Turbulent Flow and Modeling in Turbomachinery

10

Residual Value

10

603

0

-1

10-2 10-3

RMS P-Equation RMS U-Equation RMS V-Equation RMS W-Equation RMS K-Equation RMS Omega-Equation

10-4 10-5 10

-6

10

-7

50 100 Number of iterations

150

Fig. 20.25: Change of residual values in function of iteration for RANS-simulation Moreover, various parameters were monitored to observe their behavior as the solver is iterated. Example of such parameters monitored include mass-averaged total pressure at stations 3, 4 and 5, overall machine mass flow, and machine isentropic efficiency. Convergence was achieved when the residuals' target value was reached, and stability was observed in the monitored variables, Fig. 20.25. For extreme off-design cases with low expansion pressure ratios close to 1.15 and low shaft speeds Ȧ/ȦD close to 60%, a 30% reduction in the false-time step was required, which in turn increased the computer wall time to achieve convergence. All cases used a second order spatial discretization for the velocity components Vi, the pressures, the energy and turbulence equations k and Ȧ. URANS Simulation: For the URANS-unsteady analysis, the stage interfaces were changed to a fully unsteady-stator-rotor interface which creates a sliding mesh interface between the stator and rotor patches. The unsteady stator-rotor interface does not allow an unequal stator and rotor blade number. Therefore, the stator numbering was changed to from 66 to 63 to create an equal interface between stator and rotor, or a 1:1 interface across the sliding mesh. A second order backward Euler temporal scheme along with a second order spatial scheme was used for the velocity components, pressure, energy equations, and turbulence equations. The time step was selected to create approximately 1000 steps for each rotor blade pass. Every 10 blade passage steps was saved. This translated to a time step of 1.25×10-6 seconds for the 3000 rpm case. After each time step a series of inner-iterations are started until convergence was achieved, a residual error fo 1×10-6 was used as a convergence criteria. The cases were run for 10 blade passage periods. The transient case required approximately 8 days to complete. Figure 20.25 shows the equation residuals versus

20 Turbulent Flow and Modeling in Turbomachinery

604

2.0E-07

Residual Value

1.5E-07

1.0E-07

RMS P-Equation RMS U-Equation RMS V-Equation RMS W-Equation RMS K-Equation RMS Omega-Equation

5.0E-08

0

2

4

6

t/T Fig. 20.26: Change of residual values as a function of time t with T as the blade passing period for URANS-simulation iteration number for the steady state case, showing case convergence whilst Fig. 20.26 shows the equation residuals versus simulation time for the transient case. Using sliding mesh URANS accounts for the unsteady flow interaction effects at the frame change locations. Simulation with URANS allows capturing the time dependent interactions between the stator and rotor rows, particularly the wake interaction with the subsequent row. This interaction, however, affects the boundary layer development, onset of the boundary transition process, separation and its reattachment as evidenced by a series of studies published by Schobeiri and his coworkers [42], [43], [44], [45], [46], [47], [48], [49], [50], and [51]. The creation of the calm region as a result of the wake passing and its impingement on the subsequent stator row affects the integral boundary layer quantities, the profile loss coefficient of individual blades and the efficiency of the entire turbine component. A RANSsimulation is based on the steady-state mixing plane approach, where the time dependent interactions are averaged out. In contrast, the URANS unsteady simulation provides a means of simulating time-dependent stator rotor interactions. Considering the unsteady results from the above studies, conducting a URANS-simulation was expected to deliver more accurate results than RANS. Turbulence Model: The turbulence model used in this study was the Shear Stress Transport model (SST),introduced by Menter [52] and [53] and implemented into CFX. This model combines and models by introducing a blending function with the objective to get the best out of these two models (for more detailed discussion of turbulence models we refer to the recent textbook by Schobeiri [54]).

20 Turbulent Flow and Modeling in Turbomachinery

605

The basic equations, assumptions and closure coefficients for the model are listed in [38] and [40]. In addition to the turbulence model a transition model was also employed. It is worth noting that the transition model built in the CFX-Package is an ANSYS-proprietary [55] not identical with the one published by Menter et al. [56].

20.5.5 Results, Discussion Interstage Flow Traversing, Total, Static Pressure Distributions: For stage 2 interstage measurements were carried at stations 3, 4 and 5, Fig. 20.21, with a rotational speed of 3000 rpm and the results were compared with RANS and URANS simulation. At the outset, it should be emphasized that the experiments were repeated at least two to three times to ensure the reproducibility of the entire data set. For numerical simulation, the boundary conditions were taken from the experiment as it was without any modification. Total and static pressure distributions are shown in Figure 20.27 as a function of the dimensionless immersion . In contrast to many papers that use dimensionless quantities, whenever experimental and numerical results are compared, all quantities, with the exception of a few, presented in this study are dimensional. This allows a direct quantitative assessment of the numerical results, when compared with the experimental ones. For the same reason, presentation of contour plots is limited to a few ones, just to show the qualitative differences. Total Pressure: Measurement, Numerical Simulations: As Fig. 20.27 top shows, the

"

measured total pressures up- and downstream of the stator (Station 3 and 4, , d) differ from each other only slightly confirming the positive effect of the circumferential lean of the stator blades on reducing the secondary flow zones at the hub and tip. The

"

comparison of the total pressure measured at the stator inlet ( ), station 3, with the steady RANS-results (SS) shows a substantial difference of more than 3 kPa. Downstream of the stator at station 4, the difference between the measured and RANSsteady results (d, SS), is more than 3.5 kPa. Considering the pressure uncertainty of 0.19%, which amounts to 35.1 Pa, the above differences are more than 100 times the measurements uncertainty. The comparison of the measured and the RANS-results downstream of the rotor row (station 5) shows similar differences. The extraction of mechanical energy from the rotor causes a reduction in total pressure as shown in Fig.

!

20.27 (station 5, ). Compared to the RANS-simulation (SS) a difference of more than 3.4 kPa is observed. The above comparison shows that we are dealing with significant quantitative differences between the experiment and RANS-simulation. Differences are also observed in course of both distributions particularly at the exit of the stator, rotor rows and close to the hub and tip. While the trend prediction outside the hub and tip regions is satisfactory, Major differences are observed in those regions as illustrated in Fig. 20.27. However, from boundary layer and secondary flow development point of view, its is vital for any CFD-simulation to predict the same trend at above regions. Considering the above, we are dealing with two simulation deficiencies: (1) a substantial quantitative underprediction of total pressures and (2) an inaccurate prediction of the total pressure trend close to the hub and tip. Being aware of the RANS-simulation deficiencies, the differences encountered above were of much higher magnitude that

4

Rotor 2

3

Stator 2

2

Rotor 1

1

Stator 1

20 Turbulent Flow and Modeling in Turbomachinery

606

5

Total Pressure [kPa]

95 90 85 80

Station 3 exp. RANS URANS Station 4 exp. RANS URANS Station 5 exp. RANS URANS

3000 rpm pr = 1.44

75 70 65 0

0.25 0.5 0.75 1 * Immersion Ratio r (r  rhub) e (rtip  rhub)

Static Pressure [kPa]

95 90 85 80 75

3000 rpm pr = 1.44

70 65 0

0.25 0.5 0.75 1 * Immersion Ratio r (r  rhub) e (rtip  rhub)

Fig. 20.27: Comparison of the measured total pressures (top) and static pressure

"

!

(bottom) at stations 3, 4 and 5 ( , d, ) with the RANS and URANS computational results, (solid, dashed curves, red, black, blue)

20 Turbulent Flow and Modeling in Turbomachinery

607

we expected. To expedite the cause of these differences, we resorted to URANS simulation, whose results are also presented in Fig. 20.27 as dashed curves. After performing the URANS calculations, the data were averaged at any single point from hub to tip in radial, circumferential and axial direction. The purpose of this procedure was to account for the stator-rotor wake interaction and to preserve integrally the impact of the wakes on the boundary layer of the subsequent blade rows. As seen, in case of the stator inlet, Station 3, the URNAS simulation has reduced the total pressure differences from 3 kPa to 2.3 kPa which exhibits a marginal improvement only. At the stator exit, station 4, which is identical with the rotor inlet station, the total pressure difference has reduced from 3.5 kPa to 2.84 kPa which is again a marginal improvement. The total pressure difference between the stator inlet and exit calculated by URANS is almost 1 kPa which results in a significant total pressure loss coefficient. The rotor row URANS-simulation presents a slightly different picture. The pressure difference has been reduced substantially from 3.4 kPa to 1.1 kPa. Although URANS has reduced the differences, particularly in the rotor exit, the remaining differences are so large that cannot be accepted as satisfactory. Furthermore, despite the improvement in rotor exit total pressure, because of differences at the stator exit (rotor inlet), the rotor total pressure loss coefficient is significantly higher than the measured one. Static Pressure: Measurement, Numerical Simulations: The static pressure as the difference between the total and dynamic pressure reveals much larger differences between the experiment and RANS simulation as shown in Fig. 20.27 bottom. Here, as in the above case, the calculated static pressures at the inlet and exit of the stator row are substantially underpredicted. Similar to the total pressure discussed above, URANS simulation brings an improvement only at the exit of the rotor, while the pressure difference at the inlet has become much larger than the total pressure difference discussed above. The increase in static pressure differences is attributed to an inaccurate calculation of the velocity distribution within the turbine blade flow path. In this context, an accurate prediction of the boundary layer development and transition is the prerequisite for accurately calculating the velocity components and thus the velocity vector that constitutes the dynamic pressure. We shall discuss this issue later in this paper. Absolute and Relative Velocities: Measurement, Numerical Simulations: The absolute velocity is the vector sum of the axial, radial and circumferential components, measured by the five hole probes locates at station 3, 4 and 5 and computed by the Reynolds averaged Navier Stokes equations. Measured absolute velocities at station 3, 4, and 5 are shown in Fig. 20.28 (top) and compared with the RANS and URANS simulation results. At the stator inlet, Station 3, the experimental results differ substantially from RANS and URANS-results, particularly at the hub and tip regions, where the secondary flow is predominant. While within the immersion range of 0.25 to 0.85, the numerical simulations follow the experimental trend, they differ substantially at the hub and tip secondary flow regions. The velocity at the stator exit, Station 4, exhibits a similar picture. As Fig. 20.28 (top) shows, substantial differences up to 20 m/s exist at the rotor exit, Station 5. Here, as in case of the stator, the velocity distribution close to the hub and tip is neither predicted quantitatively nor qualitatively by both RANS and URANS. Interestingly, while the velocity up and down-

4

Rotor 2

3

Stator 2

2

Rotor 1

1

Stator 1

20 Turbulent Flow and Modeling in Turbomachinery

608

5

Absolute Velocity [m/sec]

200 175 150

3000 rpm pr = 1.44

125

Station 3 exp. RANS URANS Station 4 exp. RANS URANS Station 5 exp. RANS URANS

100 75 50 25 0 0

0.25 0.5 0.75 1 * Immersion Ratio r (r  rhub) e (rtip  rhub)

Relative Velocity [m/sec]

200 175 150 125 100 75

3000 rpm pr = 1.44

50 25 0 0

0.25 0.5 0.75 1 * Immersion Ratio r (r  rhub) e (rtip  rhub)

Fig.20.28: Comparison of the measured absolute (top) and relative velocities (bottom) with the RANS and URANS computational results, (solid, dashed curves, red, black, blue)

20 Turbulent Flow and Modeling in Turbomachinery

609

stream of the stator are overpredicted, the one downstream of the rotor is substantially underpredicted. The relative velocity is obtained by subtracting the constant rotational velocity vector from the absolute velocity, Fig. 20.28 (bottom). Thus, the results are directly related to each other and the differences are the same. Absolute, Relative and Meridional Flow Angles: Measurement, Numerical Simulations: Before discussing the angle calculation results, it should be emphasized that the degree of accuracy of angle calculation is instrumental in determining the blade loss coefficient. The inaccuracy in angle calculation leads to positive or negative incidence leading to an inaccurate loss calculation. In this regard, the blade geometry, particularly its leading edge radius plays a crucial role. While the stator blades with a relatively large leading edge radius, Fig. 20.22, are insensitive to incidence changes, the rotor blades, Fig.20.23 has a much smaller leading edge radius making their loss coefficient more susceptible to incidence change. Figure 20.29 shows the absolute flow angle Į (top), relative angle ȕ (middle) and the meridional flow angle Ȗ with the angle convention shown in the figure. The results of RANS and URANS calculations are compared with the measurements. At station 3, upstream of stator 2, both RANS and URANS results follow the experimental trend for r* = 0.15 to 0.75 with a ǻĮ of almost 9o. Much larger differences are exhibited for station 5, downstream of the second rotor. At station 4, downstream of the second stator, the absolute flow angle Į, shows small differences because of the diagram Į-scale. However reducing the Į-range from (0o-175o) to (5o25o), reveals at the hub and tip differences of ǻĮ of more than 5o. Similar pattern, however with smaller differences with a maximum ǻȕ of almost 5o are observed for relative flow angles, Fig. 20.29 (middle). Trend wise URANS delivers a quasi similar trend as the experiment, however with a displaced position of minima and maxima close to the hub and tip. This is also true for the meridional angle Ȗ. These differences affect the calculation of the losses as discussed below. Total Pressure Loss Coefficients: The interstage traversing provides the entire flow quantities, from which the absolute and relative total pressures and thus the total pressure loss coefficient for the stator and the rotor are determined. For this purpose, first the traversed quantities are averaged in the circumferential direction. The averaged quantities must be consistent with conservation laws of thermodynamics and fluid mechanics. The significance of an appropriate averaging technique and its impact on the results has been very well known in the turbomachinery community for more than four decades and does not need to be reviewed here (see extensive review by Traupel [57]). A simple consistent averaging technique that yields consistent results at a plane under inhomogeneous flow conditions was introduced by Dzung [58]. This averaging method is considered as a standard averaging technique and is used on routine basis by many researchers, among others Emund et al. [59] and Day et al. [60]. For a given control volume, the averaging technique by Dzung yields integral quantities that are inherently consistent with the conservation laws. To obtain consistently averaged data for distributed quantities in the radial direction, the technique required a further enhancement, which was done at the TPFL by Schobeiri and his co-workers [18].Using consistently averaged quantities, we define the total pressure loss coefficient for stator and rotor as:

20 Turbulent Flow and Modeling in Turbomachinery

610

175 150 D [deg]

125 100 75 3000 rpm pr = 1.44

50 25 0

Station 3 exp. RANS URANS Station 4 exp. RANS URANS Station 5 exp. RANS URANS

Angle definition

0.25 0.5 0.75 Immersion Ratio r* (r  rhub) e (rtip  rhub)

1

175 150

Station 3 exp. RANS urans Station 4 exp. RANS URANS Station 5 exp. RANS URANS

E [deg]

125 100 75

3000 rpm pr = 1.44

50 25 0

0.25 0.5 0.75 * Immersion Ratio r (r  rhub) e (rtip  rhub)

40 30

3000 rpm pr = 1.44

J [deg]

20 10

1

Station 3 exp. RANS URANS Station 4 exp. RANS URANS Station 5 exp. RANS URANS

0 -10 -20 -30

0

0.25 0.5 0.75 Immersion Ratio r* (r  rhub) e (rtip  rhub)

1

Fig.20.29: Comparison of the measured loss coefficient (top), (middle) and the meridional angle with the RANS and URANS computational results, (solid, dashed curves, red, black, blue)

20 Turbulent Flow and Modeling in Turbomachinery

611

(20.191) with as the absolute total pressure up- and downstream of the stator, Stations 3 and 4, as the relative total pressure up- and downstream of the rotor, and as the static pressure at station 4 and 5. Figure 20.30 (top) and its enlarged portion (middle) show the stator and rotor loss coefficients. As seen, the experimental results for the stator row are below ȗ = 0.1. Close to the hub and tip, where usually increased secondary flows dictate the flow pattern, the loss coefficient is characterized by zones of reduced losses. This reduction is due to the re-configuration of the blades by introducing a symmetric circumferential lean at the hub and tip. As Fig. 20.30 (middle) shows, neither RANS nor URANS predict the stator loss coefficient accurately. Significant differences in loss coefficient with an order of magnitude for RANS and URANS show that both simulation methods cannot even approximately determine the loss coefficient. For the rotor, the results of RANSsimulation exhibits much higher differences. While RANS significantly overpredict the rotor loss coefficient, URANS overpredict the losses in within 50% of the blade height from r* = 0.25 to r* = 0.75 and from r* = 0.75 onward it does underpredic the losses. As Table 20.6 shows the averaged loss coefficient for stator row calculated by RANS is with 0.1174 almost 300% higher than the experimental one 0.042, while for the rotor row the RANS calculation with is about 8% lower than the experiment . Switching to URANS, the stator loss coefficient is reduced to 0.091, almost 200% higher than experiment, while the URANS calculation delivers . These facts show that neither RANS nor URANS can be used for accurately predicting the HP-turbine stator blade losses and thus the efficiency. For the rotor the URANS integral loss calculation delivers acceptable results. Regarding the experiments, it should be pointed out that the interstage measurements were repeated and the experimental data were reproduced to exclude any possible experimental errors. Table 20.6: Stator and rotor loss coefficients, comparison between Experimental, RANS and URANS averaged total pressure loss coefficient for case N=3000 RPM, ans the turbine pressure PR=1.44. Stator row 2

Rotor row 2

Averaged

Averaged

Experimenat

0.042

0.128

RANS

0.1174

0.1169

URANS

0.091

0.127

Investigation

20 Turbulent Flow and Modeling in Turbomachinery

612

1.6

]

1.2 0.8

Stator: exp. RANS URANS Rotor: exp. RANS URANS 3000 rpm pr = 1.44

0.4 0.0

0.25 0.5 0.75 Immersion Ratio r* (r  rhub) e (rtip  rhub)

1

0.4 ] distribution enlarged

]

0.3 0.2

Stator: exp. RANS URANS Rotor: exp. RANS URANS

0.1 0.0

0.25 0.5 0.75 * Immersion Ratio r (r  rhub) e (rtip  rhub)

1

0.4 STATOR 1 ROTOR 1 STATOR 2 ROTOR 2

]

0.3 0.2 0.1 0

0

0.25 0.5 0.75 Immersion Ratio r* (r  rhub) e (rtip  rhub)

1

Fig. 20.30: Comparison of the measured stator and rotor loss coefficients (top) and their enlargement (middle) with RANS and URANS computational results. Bottom: RANS calculations of stator 1, 2, and rotor 1, 2 loss coefficients

20 Turbulent Flow and Modeling in Turbomachinery

613

In context of the turbine rig performance determination, it is essential to have also knowledge of stage 1 aerodynamics behavior, particularly its loss behavior. This is shown in Fig. 20.30(bottom), where the stator and rotor loss coefficients are calculated using RANS simulation. As seen, stator 1 has a substantially lower loss coefficient than stator 2. This is attributed to a very uniform velocity distribution at the turbine inlet, where stator 1 is located and the turbulence intensity have a small order of magnitude. The rotor 2, however has a slightly higher loss coefficient than rotor 1. Table 20.7 summarizes the RANS based total pressure loss and total-to-static efficiencies at various machine shaft rotational speeds. This situation is reflected in performance curve, Fig. 20.31. It represents the results of the performance experiments, where the total-to-static efficiency is plotted vs, u/co. Using RANS only, at the Table 20.7: RANS based total pressure loss and total-to-static efficiencies at various machine shaft rotational speeds. Shaft Speed [RPM]

Stator 1 Rotor 1 Stator 2 Rotor 2

Stage 1

Stage 2

1800

0.0683

0.1083

0.1678

0.1488

0.73547

0.62175 0.714995

2000

0.0678

0.1046

0.1535

0.1443

0.7757

0.66587

2200

0.0674

0.1036

0.1407

0.1286

0.80754

0.71321 0.786891

2400

0.0670

0.1055

0.1297

0.121

0.83387

0.75224

0.81317

2600

0.0667

0.1102

0.1193

0.1161 0.854785 0.78387

0.83463

2800

0.0644

0.1232

0.1178

0.1141

0.86941

0.8088

0.84784

3000

0.0662

0.1473

0.1174

0.1169

0.8791

0.8273

0.8579

0.90

0.80

Kts

Experiment RANS

0.70 Performance of the two-stage turbine

0.60 0.3

0.4

0.5

0.6 0.7 u  co

0.8

0.9

1

Fig. 20.31: Performance behavior of the two-stage turbine in terms of total-to-static efficiency vs. u/co

Turbine

0.74967

614

20 Turbulent Flow and Modeling in Turbomachinery

maximum points it exhibits an efficiency difference of 2.5%. Given the number of points calculated, this difference might be slightly smaller using URANS, which would have taken significantly longer computation time. Figure 20.31 integrally summarizes the deficiencies in calculating the individual flow quantities discussed above in detail.

20.5.6 Rotating Turbine RANS- URANS-Shortcomings, Discussion Considering the experimental and numerical data presented above, in the following an attempt is made to explain the cause of the CFD-deficiencies in accurately predicting the efficiency and performance of HP-Turbines. As mentioned in the Introduction, the flow through the stages of a HP- turbine is extremely complex due to the existence of the secondary flow at the hub and tip regions, wake interaction between blade rows, strong negative pressure gradient and the transitional nature of the flow to name a few parameters. The existing turbulence and transition models are not capable of accurately predicting the flow behavior. At the outset it should be emphasized that CFD with the existing models is capable of simulating flow situations that are not as complex as the HP-turbine flow is. To explain the cause of the deficiencies in context of HP-turbine aerodynamics, we resort to the coordinate invariant Navier-Stokes equation: (20.201) The turbulent shear stress associated with the intermittency function, , plays a crucial role in affecting the solution of the Navier-Stokes equations. This is particularly significant for calculating the wall friction under intermittently laminarturbulent flow condition. As is well known, two quantities have to be accurately modeled: One is the intermittency function Ȗ, and the other is the Reynolds stress tensor with its nine components, from which six are distinct. Inaccurate modeling of these two quantities leads to a multiplicative error of their product . This error particularly affects the accuracy of the total pressure losses (through wall friction) and efficiencies. A deficient intermittency modeling affects the boundary layer development leading to under/over prediction of wall shear stress thus the velocity distribution leading to inaccurate prediction of mass flow, total pressure and thus losses and the efficiency. A deficient modeling of the Reynolds stress tensor has an equally negative impact on the flow simulation. In this context, an accurate modeling of the dissipation equation that appears in both ț-J and ț-Ȧ equations is of crucial importance. Implementing the exact equations for ț and J that constitute the basis for k-J as well as k-Ȧ model major modifications and simplification had to be performed. Actually, in the case of J-equation, the exact equation is surgically modified beyond recognition. Under this circumstance, none of the existing turbulence models can be regarded as universally applicable. Considering this situation, however, satisfactory results can be obtained if in a particular flow situation the impact of the above

20 Turbulent Flow and Modeling in Turbomachinery

615

parameters are not as adverse as in HP-turbine case. Examples of satisfactory simulations are flow cases with moderate pressure gradients, no wake interactions and simple geometries. Even in simple geometries where the sign of the pressure gradient changes abruptly, a flow separation and re-attachment may occur. In this case, an accurate prediction of boundary layer transition is crucial for an adequate numerical prediction. In more complicated cases where the frame of reference changes from stationary to rotating and vice versa with unsteady wake interaction present, caution must be exercised in interpreting the results. Despite the facts discussed above, cases have been observed, where even a RANS calculation delivers a reasonable results. In these cases, an over/under prediction of intermittency combined with under/over prediction of the Reynolds stress cancels each other’s effects leading to reasonable results.

References 1. Taylor, G.I.: Diffusion by Continuous Movements. Proc. London Math. Soc. Ser2 20, 196–211 (1921) 2. von Kármán, T.: Aeronaut. Sci. 4, 137 (1937) 3. Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975) 4. Rotta, J.C.: Turbulente Strömungen. B.C.Teubner-Verlag, Stuttgart (1972) 5. Richardson, L.F.: Weather Prediction by Numerical Process. Cambridge University Press, Cambridge (1922) 6. Kolmogorov, A.N.: Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Number. Doklady Akademia Nauk, SSSR 30, 299–303 (1941) 7. Grant, H., Stewart, H.R.W., Moilliet, A.: Turbulence Spectra from a Tidal Channel. J. Fluid Mech. 12, 241 (1962) 8. Onsager, L.: Phys. Rev. 68, 286 (1945) 9. Weizsäcker, C.F.: 1948, Zeitschrift Physik 124, 628, also proc. Roy. Soc. London 195A, 402 (1948) 10. Bradshaw, P., Perot, J.B.: A note on turbulent energy dissipation in the viscous wall region. Physics of Fluids A 5, 3305 (1993) 11. Launder, B.E., Reece, G.I., Rodi, W.: Progress in the Development of ReynoldsStress Turbulent Closure. J. of Fluid Mechanics 68, 537–566 (1975) 12. Launder, B.E., And Spalding, D.B.: The Numerical Computation of Turbulent Flows. Comp. Method in Applied mechanics and Engineering 3, 269–289 (1974) 13. Boussinesq, J.: Mé. pré. par. div. savants á l’ acad. sci. Paris, 23, 46 (1887) 14. Prandtl, L.: Über die ausgebildete Turbulenz. ZAMM 5, 136–139 (1925) 15. Schlichting, H.: Boundary Layer Theory, 7th edn. McGraw-Hill, New York (1979) 16. Wilcox, D.: 1993, Turbulence Modeling for CFD. DCW Industries, Inc., 5354 Palm Drive, La Ca. nada, California 91011 (1993) 17. Müller, T.: Untersuchungen von Geschwindigkeitsprofilen und deren Entwicklung in Strömungsrichtung in zweidimensionalen transitionalen Grenzschichten anhand eines Geschwindigkeitsmodells. Dissertation, Technische Hochschule Darmstadt, Germany D 17 (1991)

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18. Van Driest, E.R.: Turbulent Boundary Layer in Compressible Fluids. Journal of Aeronautical Sciences 18, 145–160, 216 (1951) 19. Kays, W.M., Moffat, R.J.: The behavior of Transpired Turbulent Boundary layers. Studies in Convection, vol. 1: Theory, Measurement and application. Academic Press, London (1975) 20. Smith, A.M.O., Cebeci, T.: Numerical solution of the turbulent boundary layer equations. Douglas aircraft division report DAC 33735 (1967) 21. Klebanoff, P.S.: Characteristics of Turbulence in Boundary Layer with zero Pressure gradient. NACA TN 3178 (1954) 22. Baldwin, B.S., Lomax, H.: Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows. AIAA Paper 78-257 (1978) 23. Kolmogorov, A.M.: Equations of Turbulent Motion of an incompressible fluid. Akad. Nauk SSR, Seria Fiz. VI, No. 1–2 (1942) 24. Prandtl, L.: Über ein neues Formelsystem für die ausgebildete Turbulenz. Nachrichten der Akademie der Wissenschaften, Göttingen, Math. Phys, Kl, p.6 (1945) 25. Launder, B.E., Spalding, D.B.: Mathematical Models of Turbulence. Academic Press, London (1972) 26. Chou, P.Y.: On the Velocity Correlations and the Solution of the Equation of Turbulent Fluctuations. Quart. Appl. Math., Vol. 3, 38 (1945) 27. Jones, W.P., Launder, B.E.: The Prediction of Laminarization with a Two-equation Model of Turbulence. International Journal of Heat and Mass Transfer 15, 301–314 (1972) 28. Menter, F.R.: Zonal Two-Equation k-Ȧ Turbulence Models for Aerodynamic Flows, AIAA Technical Paper 93-2906 (1993) 29. Rodi, W., Scheurer, G.: Scrutinizing the k-İ Model Under Adverse Pressure Gradient Conditions. J. Fluids Eng. 108, 174–179 (1986) 30. Menter, F.R.: Influence of Freestream Values on k-Ȧ Turbulence Model Predictions. AIAA Journal 30(6) (1992) 31. Dr. D. Wicox, DCW Industries, Inc., Private communications (March 2008) 32. Menter, F.R.: Zonal Two Equation k-İ Turbulence Models for Aerodynamic Flows. AIAA Paper 93-2906 (1993) 33. Menter, F.R.: Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA Journal 32, 269–289 (1994) 34. Menter, F.R., Kuntz, M., Langtry, R.: Ten Years of Experience with the SST Turbulence Model. In: Hanjalic, K., Nagano, Y., Tummers, M. (eds.) Turbulence. Heat and Mass Transfer, vol. 4, pp. 625–632. Springer, Heidelberg (2003) 35. Dr. F. Menter, CFX, Germany, Private communications (April 2008 36. Schobeiri, M.T., Gilarranz, J., Johansen, E.: Final Report on: Efficiency, Performance, and Interstage Flow Field Measurement of Siemens-Westinghouse HP-Turbine Blade Series 9600 and 5600 (September 1999) 37. Schobeiri, M.T., Gillaranz, J.L., Johansen, E.S.: Aerodynamic and Performance Studies of a Three Stage High Pressure Research Turbine with 3-D Blades, Design Point and Off-Design Experimental Investigations. In: Proceedings of ASME Turbo Expo, 2000-GT-484 (2000)

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617

38. Schobeiri, M.T., Abdelfatta, S., Chibli, H.: Investigating the Cause of CFDDeficiencies in Accurately Predicting the Efficiency and Performance of High Pressure Turbines: a Combined Experimental and Numerical Study, submitted for publication. The document along with the entire experimental and numerical dada are also available at Turbomachinery Performance and Flow Research Laboratory, Texas A&M University 39. Chibli, H., Abedlfatah, S., Schobeiri, M.T., Kang, C.: An Experimental and Numerical Study of the Effects of Flow Incidence Angles on the Performance of a Stator Blade Cascade of a High Pressure Steam Turbine. ASME, Gt-2009-59131, presented at ASME Turbo Expo, June 8-12, Orlando, Florida, USA (2009) 40. ANSYS INC, ANSYS-CFX Release Documentation, 12.0 ed (2009) 41. Denton, J.: The Calculation of Three-Dimensional Viscous Flow Through Multistage Turbomachines, Journal of Turbomachinery 114 (1992) 42. Schobeiri, M.T., Pappu, K.: Optimization of Trailing Edge Ejection Mixing Losses Downstream of Cooled Turbine Blades: A Theoretical and Experimental Study. ASME Journal of Fluids Engineering 121, 118–125 (1999) 43. Schobeiri, M.T., Chakka, P.: Prediction of Turbine Blade Heat Transfer and Aerodynamics Using Unsteady Boundary Layer Transition Model. International Journal of Heat and Mass Transfer 45, 815–829 (2002) 44. Schobeiri, M.T., Radke, R.E.: 1994, Effects of Periodic Unsteady Wake Flow and Pressure Gradient on Boundary Layer Transition along the Concave Surface of a Curved Plate. ASME Paper 94-GT-327, presented at the International Gas Turbine and Aero-Engine Congress and Exposition, Hague, Niederlands, June 13–16 (1994) 45. Schobeiri, M.T., Read, K., Lewalle, J.: Effect of Unsteady Wake Passing Frequency on Boundary Layer Transition, Experimental Investigation and Wavelet Analysis. Journal of Fluids Engineering 125, 251–266 (2003), a combined twopart paper, this paper received the ASME-2004 FED-Best Paper Award 46. Wright, L., Schobeiri, M.T.: The Effect of Periodic Unsteady Flow on Boundary Layer and Heat Transfer on a Curved Surface. ASME Transactions, Journal of Heat Transfer 120, 22–33 (1999) 47. Schobeiri, M.T., Öztürk, B., Ashpis, D.: On the Physics of the Flow Separation Along a Low Pressure Turbine Blade Under Unsteady Flow Conditions. ASME 2003-GT-38917, presented at International Gas Turbine and Aero-Engine Congress and Exposition, Atlanta, Georgia (2003), also published in ASME Transactions, Journal of Fluid Engineering 127, 503–513 (2005) 48. Schobeiri, M.T., Öztürk, B.: Experimental Study of the Effect of the Periodic Unsteady Wake Flow on Boundary Layer development, Separation, and Reattachment Along the Surface of a Low Pressure Turbine Blade (2004), ASME 2004-GT-53929, presented at International Gas Turbine and Aero-Engine Congress and Exposition, Vienna, Austria, also published in the ASME Transactions, Journal of Turbomachinery 126(4), 663–676 (2004) 49. Schobeiri, M.T., Öztürk, B., Ashpis, D.: Effect of Reynolds Number and Periodic Unsteady Wake Flow Condition on Boundary Layer Development, Separation, and Re-attachment along the Suction Surface of a Low Pressure Turbine Blade. ASME Paper GT2005-68600 (2005)

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50. Schobeiri, M.T., Öztürk, B., Ashpis, D.: Intermittent Behavior of the Separated Boundary Layer along the Suction Surface of a Low Pressure Turbine Blade under Periodic Unsteady Flow Conditions. ASME Paper GT2005-68603 (2005) 51. Öztürk, B., Schobeiri, M.T.: Effect of Turbulence Intensity and Periodic Unsteady Wake Flow Condition on Boundary Layer Development, Separation, and Re-attachment over the Separation Bubble along the Suction Surface of a Low Pressure Turbine Blade. ASME, GT2006-91293 (2006) 52. Menter, F.: Zonal Two Equation k- Turbulence Models for Aerodynamic Flows. AIAA Paper 93-2906 (1993) 53. Menter, F.: Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA Journal 32, 1598–1605 (1994) 54. Schobeiri, M.T.: Fluid Mechanics for Engineers. A Graduate Text Book. Springer, Heidelberg (2011), doi:10.1007/978-3-642-11594-3 55. Dr. F. Menter, CFX, Germany, Private communications relative to the CFXtransition model (April 2008) 56. Langtry, R.B., Menter, F.R.: Transition Modeling for General CFD Applications in Aeronuatics. AIAA paper 2005-522 (2005) 57. Traupel, W.: Thermische Turbomaschinen, 3rd edn. Springer, New York (1977) 58. Dzung, L.S.: “ Konsistente Mittewerte in der Theorie der Turbomaschinen für kompressible Medien. BBC-Mitteilung 58, 485–492 (1971) 59. Emunds, R., Jennions, I.K., Bohn, D., Gier, J.: The Computation of Adjacent Blade-Row effects in a 1.5-Stage Axial Flow Turbine. ASME Transactions, Journal of Turbo machinery 121, 1–10 (1999) 60. Day, C.R.B., Oldfield, M.L.G., Lock, G.D.: The Influence of Film Cooling on the Efficiency of an Annular Nozzle Guide Vane Cascade. ASME Transactions, Journal of Turbomachinery 121, 145–151 (1999)

21

Introduction into Boundary Layer Theory

In Chapter 19 we have shown that using the computational fluid dynamics (CFD), flow details in and around complex geometries can be predicted with an acceptable degree of accuracy. The flow field calculation includes details very close to the wall, where the viscosity plays a significant role. In the absence of random fluctuations the (laminar) flow can be calculated with high accuracy. For predicting turbulent flows, however, turbulence models were required to be implemented into the Navier-Stokes equations to account for turbulence fluctuations. One of the more important tasks in turbomachinery fluid mechanics is to predict the drag forces acting on the surfaces such as turbine and compressor blade surfaces, endwalls, inlet nozzles and exit diffusers. As seen in Chapter 6, the drag forces are produced by the fluid viscosity which causes the shear stress acting on the surface. The question that arises is how far from the surface the viscosity dominates the flow field. Prandtl [1] was the first to answer this question. Combining his physical intuition with experiments, he developed the concept of the boundary layer theory. This theory in its differential or integral forms can be applied to a turbomachinery flow to determine the blade profile loss. It delivers accurate results as long as the boundary layer is not separated. In what follows, the concept of the boundary layer theory for two dimensional flow is presented. Utilizing the two-dimensional boundary layer approximation by Prandtl, and for the sake of simplicity, we use the boundary layer nomenclature with the mean-flow component, , as the significant velocities in , and -direction. Based on his experimental observations, Prandtl found that effect of the viscosity is confined to a thin viscous layer that he called, the boundary layer. Prandtl estimated that at any longitudinal position x the boundary layer thickness compared to the position x is small, meaning that . For the flat plate under zero pressure gradient shown in Fig. 20.1 with the length L, we have . If we assume that and , then we may estimate the changes in longitudinal direction compared to the normal one. Furthermore, based on Prandtl’s experimental findings, following order of magnitude comparison holds: (21.1) The above order of magnitude estimation enables a substantial simplification of the Navier-Stokes equations that can be solved relatively easily. Furthermore, the concept of the boundary layer theory allows the separation of a flow field into the boundary

620

21 Boundary Layer Theory

Outside the boundary layer: 

U

U



Steady uniform inlet velocity

Inside the boundary layer:

xV = 0

Gx

GL

y x L Flat plate boundary layer development at a high Re-number GL 0, dU/dx (m/s)

8

y (mm) 10.0

(a) region ake core

External region

w Vortical

(m/s)

12

1.5

1.0

0.5

3

(b) 10.0

0.4 0.1

0.0

s/so=0.26, nr= 3, :=1.033 (a)

0

1

2

t/W

3

Fig. 21.25: Ensemble-averaged velocity and fluctuation-rms distribution as a function of non-dimensional time at different y-locations at s/s0 = 0.26 for 3-rods

21 Boundary Layer Theory

663

As the hot wire probe moves toward the plate surface, the velocity continuously reduces because of the increasing viscosity effect but maintains its pattern. A change of pattern is observed, when the probe is at a position y < 1mm. While the external region reduces drastically, the fluctuation within the vortical core increases leading to a much higher turbulence intensity. Figure 21.25(b) shows the ensemble averaged velocity fluctuation rms with a highly vortical wake core region which is occupied by small vortices. For a y-range between 0.1-10 mm, the velocity fluctuations within the vortical core have maximums that appear periodically. Very close to the wall at y = 0.1 mm the maximum fluctuation rms is about 1.8m/s that corresponds to a turbulence intensity of about 52%. The increase in turbulence intensity caused by the impingement of the unsteady wake flow on the boundary layer is one of the mechanisms that suppresses the flow separation under unsteady wake flow condition as seen in the next section. The periodic unsteady turbulence activities along the curved plate at y = 0.1 mm at different instant of time is shown in a time space diagram, Fig. 21.26. A set of ensemble-averaged data is utilized to generate the ensemble-averaged turbulence intensity contour plot for two a lateral positions y = 0.1 mm presented in Fig. 21.26. As shown, for ȍ = 1.033, (3 rods), the boundary layer is periodically disturbed by the high turbulence intensity wake strips. These strips are contained between the wake leading edge and the wake trailing edge that move with two different velocities undisturbed low turbulence regions are observed with significantly lower intensity namely 0.88 and 0.5 as marked in the figure. Outside the wake strips levels indicating the absence of any visible wake interaction. As seen, whenever the wake strip with high turbulence intensity passes over the plate, the boundary layer

2.0

for :=1.033 (3-rod spacing) at y = 0.1 mm 40

t/W

36

0

0.8

1.0

32

V 0> < 5 .

28 24 20 16

> 8

0.7 0.6 0.5 0.4 0.3 0.2

1.0 ed end ext

0.0

0.2

g io d re e m al bec

0.4

0.1

n

0

0.6

0.8

s/s0

Fig. 21.27: Ensemble-averaged intermittency in temporal-spatial domain at y = 0.1mm, wake passing frequency ȍ = 1.033 (3 rods) becomes turbulent. However, the flow state changes from turbulent to laminar one as soon as the wake strip passes by. Thus, the flow state changes intermittently from laminar to turbulence and vice versa. The intermittent character of the boundary layer flow subjected to a periodic unsteady wake flow condition is shown in a time-space intermittency contour plot as shown in Fig.21.27. The frequency as well as the y-locations along the plate correspond to those shown in Fig. 21.26. Figure 21.27 exhibits three distinguished flow zones: (a) a periodic laminar flow zone with the intermittency close to zero, (b) a periodic turbulent zone occupied by the wake vortical core denoted by and (c) an extended becalmed region marked with a dashed triangle. To highlight the effect of the unsteady wake flow impingement on the transition behavior, the vertical dashed line marks the position of the transition start of the boundary layer under steady inlet flow condition shown in Fig. 21.23. As Fig. 21.27 shows, the wake passing has caused a delay in transition start resulting in a becalmed region mentioned above. 21.7.2.1 Experimental Verification: To compare the unsteady velocity distribution with the results from boundary calculation, the periodic unsteady velocities are time averaged. Considering a case with ȍ = 1.725 (5 rod), the results are presented in Fig. 21.28. Solid lines represent the calculation results using the differential method described in section 21.5 with the mixing length turbulence model and the timeaveraged intermittency function as detailed in Chapter 19. The symbols represent the velocity distribution experimentally obtained at longitudinal positions , and that correspond to laminar, transitional and turbulent states, respectively. Details of the transition process that corresponds to the Fig. 21.28

21 Boundary Layer Theory

665

is shown in time-space contour plot Fig. 21.29. Compared to Fig. 21.27 with ȍ = 1.033, (3 rods), the wake strips occupied by wake vortices and high intermittency values have moved closer together. As a consequence, the time averaged turbulence intensity has increased causing the transition start to move towards the leading edge.

Fig. 21.28: Boundary layer velocity distribution in lateral direction at three longitudinal locations for ȍ= 1.725 (5-rods). Computation (solid lines), Experiments (symbols), Chakka and Schobeiri [38] for : = 1.725 (5- rod spacing) at y = 0.1 mm 1 0.9

t/W

2.0

0.8 0.7

trs at :=0

0.6 0.5

1.0

lm ca e B

0.0 0.0

0.2

ed

gi re

0.4

on

0.3 0.2 0.1 0

0.4

0.6

0.8

s/s0

1.0

Fig. 21.29: Ensemble-averaged intermittency in temporal-spatial domain at y = 0.1mm, wake passing frequency ȍ = 1.725 (5 rods), [38]

666

21 Boundary Layer Theory

Given the fact that the method described in section 21.5 is a steady state calculation method, the time averaged experimental results plotted in Fig. 21.28 are in satisfactory agreement with the calculation. They allow predicting the boundary layer parameters and thus the skin friction coefficient. 21.7.2.2 Heat Transfer Calculation, Experiment: For unsteady flow cases with a dimensionless frequency value of ȍ= 5.166 (15 rods), calculated Stanton numbers are compared with the experimental results and shown in Fig. 21.30, where the experimental results are represented by symbols. Furthermore, three curves are plotted in each diagram representing the calculation results. The upper dashed curve represents the streamwise Stanton number distribution when the plate is subjected to an inlet flow intermittency state of max. On the other hand, if the plate is subjected to min (See Chapter 19, section 19.4.4), the lower dashed-dot curve depicts its Stanton number distribution. However, because of the periodic character of the inlet flow associated with unsteady wakes, the plate would experience a periodic change of heat transfer represented by upper and lower Stanton number curves (dashed line and dashed-dot line) as an envelope. The liquid crystal responds to this periodic event with time averaged signals. This time-averaged result is reflected by the solid line, which gives a corresponding time averaged intermittency. As the experimental results show, the increased dimensionless frequency of ȍ = 5.166 (15 rods) compared to the steady case, has caused the transition point to shift towards the leading edge. The shift is a result of a combined effect of wake mixing and the increased impinging frequency of the wake strips that introduce an excessive turbulent kinetic energy transport to the boundary layer that causes a shift of transition start toward the leading edge. Figure 21.30 exhibits a reasonably good agreement between the theory and experiment in the transition and turbulent regions with Rex > 1.2x105. In the laminar region, however, the theory slightly over predicts

0.015 Periodic Unsteady Flow: : = 5.166 0.010 St



Experiment

0.005

0.000 0.0

2.E5

Rex

4.E5

6.E5

Fig. 21.30: Stanton number as a function of local Reynolds number for ȍ 5.166 (15rods) " experiment, dashed line: prediction with dashed-dot line: prediction with , solid line: prediction with

21 Boundary Layer Theory

667

the heat transfer resulting in marginally higher Stanton numbers. In this region, better agreement can be reached by utilizing the minimum intermittency < Ȗ(t)>min.

21.7.3 Application of ț-Ȧ Model to Boundary Layer As we saw in the preceding sections, using the Prandtl-mixing length model in conjunction with the transition model presented in Chapter 8 delivers satisfactory results for two-dimensional boundary layers. It can also be used for moderate adverse pressure gradient as the study by Schobeiri and Chakka [46] shows. The Prandtl mixing length model may be replaced by any of the models described in Chapter 9. As an example, a channel flow case computed using ț-Ȧ -model by Wilcox [47] shows a good agreement between the computation and the experiment in sublayer and logarithmic layer.

25

u+

20

N-Z- Wilcox Laufer Anderson

15

Wieghardt

10 5 0 0 10

101

2 y+ 10

103

Fig. 21.31: Channel flow calculation using Wilcox ț-Ȧ turbulence model, computation solid line, experiments: symbols

21.8 Parameters Affecting Boundary Layer In this section, the effects of major parameters on boundary layer development, separation and re-attachment will be discussed. The discussions are based on experimental findings rather than computational simulation. The parameters are: Unsteady inlet flow condition, pressure gradient, Reynolds number and the inlet turbulence intensity. There are certainly other parameters that my affect the boundary layer development and heat transfer, however, their effects are of secondary relevance compared to the parameters mentioned above. One of the areas of engineering applications, where the above parameters interact with each other is the turbomachinery aerodynamics. Figure 21.32 displays an aircraft gas turbine engine with the components listed in the caption. Within these compo-

668

21 Boundary Layer Theory

nents, particularly in the low pressure (LP) turbine, the pressure gradient, the unsteady wake interaction, Re-number and turbulence intensity determine the development of the boundary layer, its separation and re-attachment on the blade surfaces. In an engine like the one shown in Fig. 21.32, its is very difficult, almost impossible to investigate the effect of these parameters individually. First, there are a number of parasitic effects that negatively influence the extraction of individual data. Furthermore, the high operating temperature and the limited accessibility to the engine inner structure do not allow a systematic measurements of the desired flow parameters. This circumstance compels aerodynamicists to design research facilities for extracting detail information about the particular parameters they wish to investigate.

Fan

LP-Compressor

LP-Turbine

HP-compressor HP-Turbine

Fig. 21.32: An aircraft gas turbine engine with the fan stage, low pressure (LP) and high pressure (HP) compressor stages, HP- and LP-turbine stages As an example, Figure 21.33 shows a multi-purpose, large-scale, subsonic research facility designed to investigate the effect of the parameters mentioned above on boundary layer and heat transfer. Since the facility is described in detail in [48], [49] and [50], only the parameter variation capability of this facility is discussed. A two-dimensional periodic unsteady inlet flow is simulated by the translational motion of an unsteady wake generator (see Figure 21.33) with a series of cylindrical rods attached to two parallel operating timing belts driven by an electric motor.

21.8.1 Parameter Variations, General Remarks Variation of Unsteady Wake Frequency: To investigate the effect of periodic unsteady inlet flow condition on the boundary layer behavior, rods with a constant diameter of 2 mm are attached to the belts, as shown in Fig. 21.34. To simulate different frequency, the rods may be then subsequently attached to the belts at spacings of SR = 160 mm, SR = 80 and SR = 40. The spacing SR =  (no rod)

21 Boundary Layer Theory

669 Air supply unit

S1

Wake generator 8

S2

6 5

Spacing S3

1 2

7

Spacing S4

Test section

1

15

16 4

16

End View

12

10

14

9 11

Adjustable height: y =130 mm

1 Static pressure blade

5

Traversing system

9

Hydraulic platform

13

2 Blade with hot film sensors 6

Inlet nozzle

10

Hydraulic cylinders

14

3 Wake generating rods

7

Straight transition duct

11

Pivot point

15

Honeycomb flow straightener

4 Wake generator

8

Timing belts with rod attachments

12

Wake generator e-motor

16

Traversing slots

Large silence chamber with honeycom and five screens Telescope supprt

Fig. 21.33: The TPFL-Turbine cascade research facility with the components and the adjustable test section

S3= 80 m m

S2=160 m m

S1=

Fig. 21.34: Wake Generator with two timing belts with rods attached, velocity distributions generated with three different dimensionless frequencies ȍ = 0 (steady), 1.59, and 3.18. Location of the data measured: 30 mm upstream of the leading edge

670

21 Boundary Layer Theory

represents the steady state case. Figure 21.23 exhibits the time dependent velocities for SR = 80 mm, SR = 160 mm and SR = . As seen, the steady state case is characterized by a constant velocity. The frequency of the unsteady case with SR = 80 mm is twice as high as with SR = 160 mm. To accurately account for the unsteadiness caused by the frequency of the individual wakes and their spacings, the flow velocity, and the cascade parameters, a dimensionless frequency ȍ is defined that includes the cascade solidity ı, the flow coefficient Q, the blade spacing SB, and the rod spacing SR. (21.141) Te dimensionless frequency ȍ defined in Eq. (21.141) incorporates the rod spacing SR and the blade spacing SB, in addition to the inlet velocity and wake generator speed. For the rod spacings of SR = 80 mm, SR = 160 mm and SR =  (no rod, steady case), the corresponding dimensionless frequencies are ȍ=3.18, 1.59 and 0.0. Figure 21.35 exhibits the cascade with the blade geometry and position relative to the wake generator geometry.

UR

2

Static pressure blades

ax

c

s

R

D2= 145°

V in

J

s

B

55 °

L

R

PS

1

D1=

c

SS DR

Wake generator rod Fig. 21.35: Turbine cascade test section with the blade geometry and flow angles, SS= Suction Surface (convex), PS=Pressure Surface (concave), two blade instrumented with static pressure taps Variation of Turbulence Intensity, Length Scale: A characteristic quantity that describes the intensity of the random fluctuations of incoming flow velocity is the turbulence intensity Tu. In engineering applications, turbulence intensity may assume values from low 1% to high 15%. To simulate certain level of Tu, turbulence grids can be used as detailed in Chapter 9, Section 9.3. For the purpose of investigating the

21 Boundary Layer Theory

671

Tu-effects, three different turbulence grids are used for producing inlet turbulence intensities of Tu = 3.0%, 8.0%, and 13.0%. Similar to Section 9.3, the grids consist of square shaped aluminum rods with the thickness GT and opening GO, given in Table 9.3. The grids were subsequently installed upstream of the wake generator and the resulting turbulence intensity measured with the distance form cascade leading edge GLE defined in Table 9.3. Variation of Pressure Gradient: By choosing a low pressure turbine as a representative example, an aerodynamic system is introduced that inherently generates a continuous distribution of negative, zero and positive pressure gradients. In steady operation mode the rods are removed thus the velocity at the cascade inlet is fully uniform. At the inlet, the velocity vector is tangent to the camberline of the blade such that an incidence free inlet flow condition is established. The Reynolds number is built with the suction surface length S0 and the exit velocity. Figure 21.36 displays the dimensionless pressure distribution along the suction and pressure surfaces of the LP-turbine blade with pressure p1 as the static pressure of the first tap. As Fig. 21.36 shows, the suction surface (convex), exhibits a strong negative pressure gradient. The flow accelerates at a relatively steep rate and reaches its maximum surface velocity that corresponds to the minimum at . Passing through the minimum pressure, the fluid particles within the boundary layer encounter a positive pressure gradient that causes a deceleration until has been reached. This point signifies the beginning of the laminar boundary layer separation and the onset of a separation zone. As seen in the subsequent boundary layer discussion, the part of the separation zone characterized by a constant cp-plateau extends up to .

Separation plateau

-4 Re=150,000, TuI=1.9% (no Tu-grid)

-3

PS

0

SE:0.702

Cp

-1

SS:0.553

SS

-2

1 2

0

0.2

0.4 S/S 0.6 0

0.8

1

Fig. 21.36: Pressure distribution along the LP-turbine blade, SS= Suction Surface, PS=Pressure Surface

672

21 Boundary Layer Theory

La mi

na rB l.

Passing the over the plateau, the flow first experiences a second sharp deceleration indicative of a process of re-attachment followed by a further deceleration at a moderate rate. On the pressure surface, the flow accelerates at a very slow rate, reaches a minimum pressure coefficient at and continues to decelerate until the trailing edge has been reached. Unlike the suction surface, the pressure surface boundary layer does not encounter any adverse positive pressure gradient that triggers separation.The process of flow acceleration, separation and re-attachment is shown in Fig. 21.37, where the measured velocity distribution normal to the surface is plotted along the suction surface.

Suction surface

a Sep ra t e dB

Pressure surface

l. tt Rea l ed B ach

Fig. 21.37: Boundary layer development along the LP-turbine blade

21.8.2 Effect of Periodic Unsteady Flow Generally, in engineering applications, the flow velocity is associated with certain fluctuations, whose degree can be expressed in terms of turbulence intensity Tu. This is true for statistically steady as well as unsteady flow situations. The fluctuations have usually high frequencies that require highly sensitive probes to capture them. For low temperature applications, hot wire anemometry is used, whose data acquisition frequency can be adjusted. For statistically steady flow, the time dependent velocity is measured, from which time averaged and the fluctuation components can be extracted. Boundary Layer Development, Separation, Re-attachment: To identify the streamwise and normal extent as well as the temporal deformation of the separation bubble on the suction surface of the LP-turbine blade under unsteady wake flow condition, detailed boundary layer measurements in normal as well as in streamwise directions are required. The steady state case serves as the reference configuration. The experimental program includes the boundary layer information that covers 11 streamwise locations on the suction surface upstream, within and downstream of the separation bubble. Aerodynamics measurements are performed for the Reynolds number of 110,000, for four different turbulent intensity Tu of 1.9%, 3.0%, 8.0% and

21 Boundary Layer Theory

673

13.0%, and three different dimensionless frequency values of ȍ = 0.0 (SR = 8), ȍ = 1.59 (SR =160 mm) and ȍ =3.18 (SR = 89 mm). For each case, ensemble and time averaged velocity and turbulence fluctuation, turbulence intensity, and unsteady boundary layer parameters are generated. The discussion of the results are centered on the combined effects of the unsteady wakes and the freestream turbulence intensity and their mutual interaction. Thus, only those results are presented that are essential for understanding the basic physics describing the combined effects mentioned above. Time Averaged Velocity and Fluctuation Distributions: The distribution of time averaged velocity and turbulence fluctuations are presented for the above Tu-levels. Figure 21.38 (a,b) display the velocity and fluctuation distributions at one streamwise position upstream, three positions within and two positions downstream of the separation bubble using single hot-wire probes. The diagrams include one steady state data for reference purposes, ȍ = 0.0 (SR = ) and two sets of unsteady data for ȍ = 1.59 (SR = 160 mm) and ȍ = 3.18 (SR = 80 mm). Re=150,000, Tu=1.9% (no grid), s/so=0.57

Re=150,000, Tu = 1.9% (no grid), s/so=0.61 20

20

Re=150,000, Tu=1.9% (no grid), s/so=0.49 no rod 160 mm 80 mm

(a)

0 0

10

Re=150,000, Tu=1.9% ( no grid), s/so=0.73 no rod 160 mm 80 mm

10

no rod 160 mm 80 mm

10

5

0 0

Re=150,000, Tu=1.9% ( no grid), s/so=0.77

15

4

6

8

0 0

2

4

6

V(m/s)

no rod 160 mm 80 mm

4

6

8

0 0

0.25 0.5 0.75

v(m/s)

0 0

1

no rod 160 mm 80 mm

15

4

6

8

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5

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v(m/s)

1

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1.5

2

0 0

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v(m/s)

1

no rod 160 mm 80 mm

15

10

5

0.5

0.25

Re=150,000, Tu=1.9% ( no grid), s/so=0.85 20

no rod 160 mm 80 mm

15

0 0

1

10

5

2

10

Re=150,000, Tu=1.9% (no grid), s/so=0.77 20

10

5

2

0 0

no rod 160 mm 80 mm

15

5

Re=150,000, Tu=1.9% ( no grid), s/so=0.73 20

Re=150,000, Tu=1.9% (no grid), s/so=0.85 20

15

5

8

10

5

2

0 0

10

V(m/s)

y(mm)

y(mm)

15

20

5

5

10

y(mm)

5

V(m/s)

5

no rod 160 mm 80 mm

y(mm)

0 0

no rod 160 mm 80 mm

(b)

10

10

Re=150,000, Tu=1.9% (no grid), s/so=0.61 20

no rod 160 mm 80 mm

15

y(mm)

no rod 160 mm 80 mm

y(mm)

10

5

20

y(mm)

10

15

15

y(mm)

15

y(mm)

15

Re=150,000, Tu=1.9% ( without grid), s/so=0.57 20

y(mm)

20

y(mm)

Re=150,000, Tu=1.9% (no grid), s/s0=0.49

y(mm)

20

5

0.5

1

v(m/s)

1.5

2

0 0

0.5

1

v(m/s)

1.5

2

Fig. 21.38: Distribution of time-Averaged velocity (a) and turbulence fluctuation rms (b) along the suction surface for steady case ȍ=0 (SR=) and unsteady cases ȍ=1.59 (SR=160 mm) and ȍ=3.18 (SR=80 mm) at Re=110,000 and free-stream turbulence intensity of 1.9% (without grid) Effect of Unsteady Wake Frequency: As Figures 21.38(a) and (b) indicate, in the upstream region of the separation bubble at s/so= 0.49, the flow is fully attached. The velocity distributions inside and outside the boundary layer experience slight decreases with increasing the dimensionless frequency, Fig. 21.38(a). At the same positions, however, the time averaged fluctuations shown in 21.38(b) exhibits substantial changes within the boundary layer as well as outside it. The introduction of the periodic unsteady wakes with highly turbulent vortical cores and subsequent mixing has systematically increased the free stream turbulence intensity level from 1.9% in steady case, to almost 3% for ȍ = 3.18 (SR = 80 mm). This intensity level

674

21 Boundary Layer Theory

is obtained by dividing the fluctuation velocity at the edge of the boundary layer, Fig. 21.38(b) by the velocity at the same normal position Fig. 21.38(a). Comparing the unsteady cases ȍ = 1.59 and 3.18, with the steady reference case ȍ = 0.0, indicates that, with increasing ȍ, the lateral position of the maximum fluctuation shifts away from the wall. This is due to the periodic disturbance of the stable laminar boundary layer upstream of the separation bubble. Convecting downstream, the initially stable laminar boundary layer flow experiences a change in pressure gradient from strongly negative to moderately positive causing the boundary layer to separate. The inflectional pattern of the velocity distribution at s/so= 0.57 signifies the beginning of a separation bubble that extends up to s/so= 0.85, resulting in a large sized closed separation bubble. As opposed to open separation zones that are encountered in compressor blades and diffuser boundary layers, the closed separation bubbles are characterized by a low velocity flow circulation within the bubble as shown in Fig. 21.38(a). Measurement of boundary layer also with single wire probes along the suction surface of the same blade is reported, among others, in [51] and [52] reveal exactly the same pattern as shown in Fig. 21.38. In contrast, the single wire measurement in an open separation zone exhibits a pronounced kink at the lateral position, where the reversed flow profile has its zero value. Despite the fact that a single wire probe does not recognize the flow direction, the appearance of a kink in a separated flow is interpreted as the point of reversal with a negative velocity. The effect of unsteady wake frequency on boundary layer separation is distinctly illustrated in Fig. 21.38(a) at s/so= 0.61, 0.73, and 0.77. While the steady flow case (no rod, ȍ = 0.0) is fully separated, the impingement of wakes with ȍ = 1.59 on the bubble has the tendency to reverse the separation causing a reduction of the separation height. This is due to the exchange of mass, momentum and energy between the highly turbulent vortical cores of the wakes and the low energetic fluid within the bubble as shown in Fig. 21.38(b). Increasing the frequency to ȍ = 3.18 has moved the velocity distribution further away from the separation, as seen in Fig. 21.38(a) at s/so= 0.77. Passing through the separation regime, the reattached flow still shows the unsteady wake effects on the velocity and fluctuation profiles. The fluctuation profile, Fig. 21.38(b) at s/so= 0.85, depicts a decrease of turbulence fluctuation activities caused by unsteady wakes ( ȍ = 1.59 and 3.18) compared to the steady case ( ȍ = 0, no rod). This decrease is due to the calming phenomenon extensively discussed by several researchers ([53], [54],[55] and [56]). Combined Effects of Unsteady Wakes and Turbulence Intensity: Increasing the turbulence level from 1.9 % to 3% that is produced by the turbulence grid TG1(Table 9.3), shows that the time-averaged velocity, Fig. 21.39(a), as well as the fluctuation distribution Fig. 21.39(b), hardly experience any noticeable changes with increasing the dimensionless frequency from ȍ = 0.0 to 3.18. This is the first indication that the higher turbulence intensity of 3% generated by TG1 is about to dictate the boundary layer development from leading edge to trailing edge. While the high frequency stochastic fluctuations of the incoming turbulence seem to overshadow the periodic unsteady wakes and the lateral extent of the separation bubble, they are not capable of completely suppressing the separation. A similar situation is encountered at higher turbulence intensity levels of 8% produced by grid TG2, Fig. 21.40(a,b).

21 Boundary Layer Theory Re=110,000,Tu=3%,s/so=0.49 Re=110,000,Tu=3%, s/so=0.57 Re=110,000,Tu=3%, s/so=0.61

Re=110,000, Tu=3% , s/so=0.49 Re=110,000, Tu=3% , s/so=0.57 Re=110,000, Tu=3%, s/so=0.61

20

20

(a)

5

0 0

2

4

2

4

V(m/s)

6

2

4

V(m/s)

6

0 0

8

Re=110,000,Tu=3%, s/so=0.73 Re=110,000, Tu=3%, s/so=0.77 Re=110,000, Tu=3% , s/so=0.85 20

y(mm)

no rod 160 mm 80 mm

10

5

0 0

15

y(mm)

no rod 160 mm 80 mm

10

2

4

2

4

5

0.2 0.4 0.6 0.8

1

2

3

V(m/s)

4

0 0

1

v(m/s)

0.2 0.4 0.6 0.8

no rod 160 mm 80 mm

0 0

1

v(m/s)

20

20

20

15

no rod 160 mm 80 mm 15

no rod 160 mm 80 mm 15

10

5

0 0

6

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6

V(m/s)

no rod 160 mm 80 mm

10

5

no rod 10 160 mm 80 mm

0.5

1

1.5

v(m/s)

Re=110,000, Tu=3%, s/so=0.73 Re=110,000, Tu=3%, s/so=0.77 Re=110,000, Tu=3%, s/so=0.85

20

15

y(mm)

15

15

no rod 10 160 mm 80 mm

y(mm)

20

15

5

0 0

8

20

y(mm)

10

5

0 0

6

V(m/s)

no rod 160 mm 80 mm

10

20

y(mm)

5

y(mm)

no rod 160 mm 80 mm

10

15

y(mm)

no rod 160 mm 80 mm

10

15

y(mm)

15

y(mm)

15

(b)

y(mm)

20

10

5

0 0

5

0.5

1

v(m/s)

1.5

2

no rod 160 mm 80 mm

y(mm)

20

675

5

0 0

0.5

1

v(m/s)

1.5

2

0 0

0.5

1

v(m/s)

1.5

2

Fig. 21.39: Distribution of time-averaged velocity (a) and turbulence fluctuation rms (b)along the suction surface for steady case ȍ=0 (SR=) and unsteady cases ȍ=1.59 (SR=160 mm) and ȍ=3.18 (SR=80 mm) at Re=110,000 and Tu=3% Re=110,000, Tu=8%, s/so=0.49 Re=110,000, Tu=8%, s/so=0.57 Re=110,000, Tu=8%, s/so=0.61 20

20

20

Re=110,000, Tu=8%, s/so=0.49 Re=110,000, Tu=8%, s/so=0.57 Re=110,000, Tu=8%, s/so=0.61 20

6

8

0 0

5

2

4

V(m/s)

6

0 0

8

20

2

4

6

no rod 160 mm 80 mm

y(mm)

y(mm)

no rod 160 mm 80 mm

10

5

10

5

2

4

V(m/s)

6

0 0

no rod 160 mm 80 mm

15

5

2

4

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6

0 0

0 0

8

V(m/s)

20

15

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4

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6

y(mm)

y(mm)

no rod 160 mm 80 mm

10

5

0.25

0.5

0.75

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0 0

1

no rod 160 mm 80 mm

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5

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1

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0 0

1.

0.5

1

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v(m/s)

2

Re=110,000, Tu=8%, s/so=0.73 Re=110,000, Tu=8%, s/so=0.77 Re=110,000, Tu=8%, s/so=0.85 20

20

20

15

15

15

y(mm)

20

no rod 160 mm 80 mm

10

Re=110,000, Tu=8%, s/so=0.73 Re=110,000, Tu=8%, s/so=0.77 Re=110,000, Tu=8%, s/so=0.85

10

15

y(mm)

4

V(m/s)

no rod 160 mm 80 mm

10

5

2

20

y(mm)

5

y(mm)

no rod 160 mm 80 mm

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15

y(mm)

no rod 160 mm 80 mm

10

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y(mm)

15

y(mm)

15

0 0

20

(b)

(a)

10

no rod 10 160 mm 80 mm

no rod 10 160 mm 80 mm

5

5

5

0 0

0.5

1

v(m/s)

1.

0 0

0.5

1

v(m/s)

1.

0 0

no rod 160 mm 80 mm

0.5

1

v(m/s)

1.

Fig.21.40: Distribution of time-averaged velocity (a) and turbulence fluctuation rms (b) along the suction surface for steady case ȍ=0 (SR=) and unsteady cases ȍ=1.59 (SR=160 mm) and ȍ=3.18 (SR=80 mm) at Re=110,000 and Tu=8%

The time averaged velocity as well as fluctuation rms do not exhibit effects of unsteady wake impingement on the suction surface throughout. In contrast to the above 3% case, the 8% turbulence intensity case, Fig. 21.40(a), seems to substantially reduce the separation bubble, where an inflection velocity profile at s/s0 = 0.61 is still

676

21 Boundary Layer Theory

visible. An almost complete suppression is accomplished by utilizing the turbulence intensity of 13% that is produced by turbulence grid TG3.. In both turbulence cases of 8% and 13%, the periodic unsteady wakes along with their high turbulence intensity vortical cores seem to be completely submerged in the stochastic high frequency free-stream turbulence. Combined Effects of Wake and Turbulence Intensity on Bubble Kinematics: For better understanding the physics, the ensemble averaged velocity contours are presented for Tu = 1.9 and 8.0, respectively. Thus, the contour plots pertaining to Tu = 3.0% and 13% that are very similar to the ones with Tu = 8% will not be discussed. The combined effects of the periodic unsteady wakes and high turbulence intensity on the onset and extent of the separation bubble are shown in Fig. 21.41 and 21.42 for the Reynolds number of 110,000. These figures display the full extent of the separation bubble and its dynamic behavior under a periodic unsteady wake flow impingement at different t/IJ. For each particular point s/so on the surface, the unsteady velocity field inside and outside of the boundary layer is traversed in normal direction and ensemble-averaged at 100 revolution with respect to the rotational period of the wake generator. To obtain a contour plot for a particular t/IJ, the entire unsteady ensemble-averaged data traversed from leading to trailing edge are stored in a large size file (of several giga bites) and sorted for the particular t/IJ under investigation. Variation of Tu at ȍ = 1.59: Figure 21.41 with a cascade Tu = 1.9% exhibits the reference configuration for ȍ = 1.59 (SR = 160 mm), where the bubble undergoes periodic contraction and expansion as extensively discussed in [57] and [58].

y(mm)

6 4 2

SB 0.5

0.6

s/so

0.7

y(mm)

4 SB 0.5

0.6

s/so

0.7

2

(d) 10 V(m/s) 8.25 7.73 7.21 6.70 6.18 5.66 5.14 4.63 4.11 3.59 3.07 2.55 2.04 1.52 1.00

6

2

4 SB 0.5

Re=110,000, :=1.59, t/W = 0.75, Tu = 1.9%, (no grid)

V(m/s) 8.25 7.73 7.21 6.70 6.18 5.66 5.14 4.63 4.11 3.59 3.07 2.55 2.04 1.52 1.00

6

0.8

8

Re=110,000, :=1.59, t/W = 0.50, Tu = 1.9%, (no grid)

8 y(mm)

V(m/s) 8.12 7.62 7.11 6.60 6.10 5.59 5.08 4.58 4.07 3.56 3.06 2.55 2.04 1.54 1.03

8

(c) (a) 10

(b) 10

Re=110,000, :=1.59, t/W = 0.25, Tu = 1.9%, (no grid)

0.6

s/so

0.7

0.8

Re=110,000, :=1.59, t/W = 1.0, Tu = 1.9%, (no grid) V(m/s) 8.25 7.73 7.21 6.70 6.18 5.66 5.14 4.63 4.11 3.59 3.07 2.55 2.04 1.52 1.00

8 y(mm)

(a) 10

6 4 2 SB

0.8

0.5

0.6

s/so

0.7

0.8

Fig. 21.41: Ensemble-averaged velocity contours along the suction surface for different s/s0 with time t/IJ as parameter for ȍ=1.59 (SR=160 mm) at Re=110,000 and free-stream turbulence of 1.9% (without grid). White area identifies the separation bubble SB location and size.

21 Boundary Layer Theory

677

During a rod passing period, the wake flow and the separation bubble undergo a sequence of flow states which are not noticeably different when the unsteady data are time-averaged. Starting with Re = 110,000 and ȍ = 1.59, Fig. 21.41 (a) exhibits the separation bubble in its full size at t/IJ=0.25. At this instant of time, the incoming wakes have not reached the separation bubble. The kinematic of the bubble is completely governed by the wake external flow which is distinguished by red patches traveling above the bubble. At t/IJ = 0.5, the wake with its highly turbulent vortical core passes over the blade and generates high turbulence kinetic energy. At this point, the wake turbulence penetrates into the bubble causing a strong mass, momentum and energy exchange between the wake flow and the fluid contained within the bubble. This exchange causes a dynamic suppression and a subsequent contraction of the bubble. As the wake travels over the bubble, the size of the bubble continues to contract at t/IJ = 0.75 and reaches its minimum size at, t/IJ = 1.0. At t/IJ = 1, the full effect of the wake on the boundary layer can be seen before another wake appears and the bubble moves back to the original position. Increasing the turbulence level to 3%, 8%, and 13% by successively attaching the turbulence grid TG1, TG2, and TG3 (detail specifications are listed in Table 9.3) and keeping the same dimensionless frequency of ȍ = 1.59, has significantly reduced the lateral extent of the bubble. The case with Tu = 8%, Fig. 21.42, is an appropriate representative of dynamic changes among the turbulence levels mentioned above. As shown in Fig. 21.42, the instance of the wake traveling over the separation bubble, which is clearly visible in Fig. 21.41, has diminished almost entirely. Increasing the turbulence intensity to 8%, Fig. 21.42(a to d), and 13% respectively, has caused the bubble height to further reduce (the corresponding figure for 13% is very similar to the one with 8%). Although the higher turbulence level has, to a great extent,

6 4 2 0.5

0.6

s/so

0.7

y(mm)

6 4 2 0.6

s/so

0.7

4 2 0.5

0.8

0.6

s/so

0.7

0.8

(d) Re=110,000,:=1.59, t/W=1,Tu=8% (with grid) 10 V(m/s) 7.67 7.21 6.74 6.28 5.82 5.35 4.89 4.42 3.96 3.49 3.03 2.56 2.10 1.63 1.17

0.5

6

0.8

(c) Re=110,000,:=1.59, t/W=0.75,Tu=8% (with grid) 10 8

V(m/s) 7.860 7.382 6.904 6.426 5.947 5.469 4.991 4.513 4.035 3.557 3.079 2.600 2.122 1.644 1.166

8 y(mm)

V(m/s) 7.780 7.305 6.830 6.354 5.879 5.404 4.928 4.453 3.978 3.503 3.027 2.552 2.077 1.601 1.126

8 y(mm)

(b) Re=110,000,:=1.59, t/W=0.50,Tu=8% (with grid) 10

Re=110,000,:=1.59, t/W=0.25,Tu=8% (with grid)

V(m/s) 7.778 7.306 6.834 6.363 5.891 5.419 4.948 4.476 4.004 3.533 3.061 2.589 2.117 1.646 1.174

8 y(mm)

(a) 10

6 4 2 0.5

0.6

s/so

0.7

0.8

Fig. 21.42: Ensemble-averaged velocity contours along the suction surface for different s/s0 with time t/IJ as parameter for ȍ=1.59 (SR=160 mm) at Re=110,000 and Tu=8% (with grid)

Re=110,000,:=3.18, t/W=0.25 Free stream turbulence with 1.9%

10

(b) V(m/s 7.92 7.43 6.94 6.44 5.95 5.46 4.97 4.47 3.98 3.49 3.00 2.50 2.01 1.52 1.03

8 y(mm)

6 4 2 0.5

(c)

0.6

s/so

0.7

10

y(mm)

8 6 4 2 0.5

0.6

s/so

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4 2 0.5

V(m/s) 7.89 7.40 6.91 6.42 5.93 5.44 4.95 4.46 3.97 3.48 2.99 2.50 2.01 1.52 1.03

V(m/s) 8.16 7.65 7.15 6.64 6.13 5.62 5.11 4.61 4.10 3.59 3.08 2.57 2.07 1.56 1.05

6

0.8

Re=110,000, :=3.18, t/W=0.75 Free stream turbulence with 1.9%

Re=110,000, :=3.18, t/W=0.50 Free stream turbulence with 1.9%

10 8

y(mm)

(a)

21 Boundary Layer Theory

(d)

0.6

s/so

0.7

0.8

Re=110,000, :=3.18, t/W=1 Free stream turbulence with 1.9%

10

V(m/s) 8.04 7.54 7.04 6.54 6.04 5.54 5.04 4.54 4.03 3.53 3.03 2.53 2.03 1.53 1.03

8 y(mm)

678

6 4 2

0.8

0.5

0.6

s/so

0.7

0.8

Fig. 21.43: Ensemble-averaged velocity contours along the suction surface for different s/s0 with time t/IJ as parameter for ȍ=3.18 (SR=80 mm) at Re=110,000 and free-stream turbulence of 1.9% (without grid)

(a) Re=110,000,:=3.18, t/W=0.25,Tu=8% ( with grid)

y(mm)

6 4 2 0.5 10

0.6

s/so

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y(mm)

4 2 0.6

s/so

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4

0.5

V(m/s) 7.80 7.33 6.86 6.40 5.93 5.46 4.99 4.52 4.05 3.58 3.11 2.64 2.17 1.70 1.24

0.5

6

0.8

6

0.8

V(m/s) 7.66 7.20 6.73 6.27 5.81 5.34 4.88 4.41 3.95 3.49 3.02 2.56 2.09 1.63 1.17

2 0.6

s/so

0.7

0.8

(d) Re=110,000,:=3.18, t/W=1, Tu=8% (with grid)

(c) Re=110,000,:=3.18, t/W=0.75,Tu=8% (with grid)

8

(b) Re=110,000,:=3.18, t/W=0.50,Tu=8% (with grid)

8 y(mm)

V(m/s) 7.65 7.18 6.72 6.25 5.79 5.32 4.86 4.39 3.93 3.46 3.00 2.53 2.07 1.60 1.13

8

10

10 V(m/s) 7.83 7.36 6.88 6.41 5.93 5.45 4.98 4.50 4.02 3.55 3.07 2.59 2.12 1.64 1.16

8 y(mm)

10

6 4 2 0.5

0.6

s/so

0.7

0.8

Fig. 21.44: Ensemble-averaged velocity contours along the suction surface for different s/s0 with time t/IJ as parameter for ȍ=3.18 (SR=80 mm) at Re=110,000 and free-stream turbulence of 8.0% (with grid TG2)

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suppressed the separation bubble as Fig.21.42 clearly shows, it was not able to completely eliminate it. There is still a small core of separation bubble remaining. Its existence is attributed to the stability of the separation bubble at the present Renumber level of 110,000. Variation of Tu at Higher Wake Frequency: Figures 21.43 and 21.44 represent the dynamic behavior of the separation bubble at Tu = 1.9%, but at a higher dimensionless frequency of ȍ = 3.18. Similar to Fig. 21.41, the case with the Tu = 1.9% presented in Fig.21.43 exhibits the reference configuration for ȍ = 3.18 (SR = 80 mm) where the bubble undergoes periodic contraction and expansion. The temporal sequence of events is identical with the case discussed in Fig. 21.41, making a detailed discussion unnecessary. In contrast to the events described in Fig. 21.41, the increased wake frequency in the reference configuration, Fig. 21.43, is associated with higher mixing and, thus, higher turbulence intensity that causes a more pronounced contraction and expansion of the bubble. As in case with ȍ = 1.59, applying turbulence levels of 3%, 8%, and 13% by successively utilizing the turbulence grids TG1, TG2, and TG3 and keeping the same dimensionless frequency of ȍ = 3.18, has significantly reduced the lateral extent of the bubble. Again, as a representative example, the case with Tu = 8% is presented in Fig. 21.44 (a to d), which reveals similar behavior as discussed in Fig. 21.42. Further increasing the turbulence intensity to 13% has caused the bubble height to further reduce. Although the higher turbulence level has, to a great extent, suppressed the separation bubble, it was not able to completely eliminate it. There is still a small core of separation bubble remaining. As in Fig. 21.42, its existence is attributed to the stability of the separation bubble at the present Re-number level of 110,000. Quantifying the Combined Effects on Aerodynamics: Figures 21.41 to 21.44 show the combined effects of turbulence intensity and unsteady wakes on the onset and extent of the separation bubble. Detailed information relative to propagation of the wake and the turbulence into the separation bubble is provided by Fig. 21.45(a,b, c, and d), where the time dependent ensemble averaged velocities and fluctuations are plotted for Re = 110,000 at a constant location s/s0 = 3.36 mm inside the bubble for different intensities ranging from 1.9% to 13%. As Fig. 21.45(a) depicts, the wake has penetrated into the separation bubble, where its high turbulence vortical core and its external region is clearly visible. Lowest turbulence fluctuations occur outside the vortical core, whereas the highest is found within the wake velocity defect. Increasing Tu to 3%, Fig. 21.45 (b), reduces the velocity amplitude of the periodic inlet flow and its turbulence fluctuations. Despite a significant decay in amplitude, the periodic nature of the impinging wake flow is unmistakably visible. Further increase of Tu to 8%, Fig. 21.45(c), shows that the footprint of a periodic unsteady inlet flow is still visible, however the deterministic periodicity of the wake flow is being subject to the stochastic nature caused by the high turbulence intensity. Further increase of turbulence to Tu =13% causes a degradation of the deterministic wake ensemble averaged pattern to a fully stochastic one. Comparing Figs. 21.45(a) and 21.45(c) leads to the following conclusion: The periodic unsteady wake flow definitely determines the separation dynamics as long as the level of the time averaged turbulence fluctuations is below

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the maximum level of the wake fluctuation vmax, shown in Figure 21.45(a). In this case, this apparently takes place at a turbulence level between 3% and 8%. Increasing the inlet turbulence level above vmax causes the wake periodicity to partially or totally submerge in the free-stream turbulence, thus, downgrading into stochastic fluctuation, as shown in Figs. 21.45(c) and (d). In this case, the dynamic behavior of the separation bubble is governed by the flow turbulence that is responsible for the suppression of the separation bubble. One of the striking features this study reveals is, that the separation bubble has not disappeared completely despite the high turbulence intensity and the significant reduction of its size which is reduced to a tiny bubble. At this point, the role of the stability of the laminar boundary layer becomes apparent which is determined by the Reynolds number.

References 1. Prandtl, L., 1904, Über Flüßigkeitsbewegung bei sehr kleiner Reibung. Verh. 3. Intern. Math. Kongr. Heidelberg (1904), p. 484–491, Nachdruck: Ges. Abh. p. 575–584, Springer, Heidelberg (1961) 2. Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1–37 (1908) 3. Falkner, V.M., Skan, S.W.: Some approximate solutions of the boundary layer equations. Phil. Mag. 12, 865–896 (1930) ARC RM 1314 4. Hartree, D.: On an Equation Occurring in Falkner and Skan’s Approximate Treatment Of the Equation of the Boundary Layer. Proc. Camb., Phi. Soc. 33, Part II, 223–239 (1937) 5. Schlichting, H.: Boundary Layer Theory, 7th edn. McGraw-Hill, New York (1979) 6. Spurk, J.: Fluid Mechanics. Springer, New York (1997) 7. Pohlhausen, K.: Zur näherungsweisen Lösung der Differentialgleichungen der laminaren Reibungsschicht. ZAMM 1, 252–268 (1921) 8. Von Kármán, T.: Über laminare und turbulente Reibung. ZAMM 1, 233–253 (1921) 9. Ludwieg, H., Tillman, W.: Untersuchungen über die Wandschubspannung in turbulenten Reibungsschichten. Ingenieur Archiv 17, 288–299 (1949), summary and translation in NACA-TM-12185 10. Coles, D.E.: The Law of the Wake in Turbulent Boundary Layer. Journal of Fluid Mechanics 1, 191–226 (1956) 11. White, F.M.: Viscose Fluid Flow. McGraw-Hill, New York (1974) 12. Lehman, K.: Untersuchungen turbulenter Grenzschichten. Dissertation, D17, Technische Hochschule Darmstadt (1979) 13. Rotta, J.C.: Turbulente Strömungen. B.C. Teubner-Verlag, Stuttgart (1972) 14. Truckenbrodt, E.: Fluidmechanik, vol. 2. Springer, Heidelberg (1980) 15. Pfeil, H., Sticksel, W.H.: Influence of the Pressure Gradient on the Law of the Wall. AIAA Journal 20(3), 342–346 (1981) 16. Pfeil, H., Amberg, T.: Differing Development of the Velocity Profiles of Threedimensional Turbulent Boundary Layers. AIAA Journal 27, 1456–1459 (1989) 17. Prandtl, L.: Über den Reibungswiderstand strömender Luft. AVA, Göttingen (1927), 3. Liefg. Seite 1–5, 4. Liefg. Seite 18–29

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18. Prandtl, L.: Zur turbulenten Strömung in Rohren, und längs Platten. Vierte Serie. AVA, Göttingen (1932) 19. Nikuradse, J.: Gesetzmäßigkeit der turbulenten Strömung in glatten Rohren. Forsch. Arb. Ing.-Wesen. No. 356 (1932) 20. Launder, B.E., And Spalding, D.B.: Mathematical Models of Turbulence. Academic Press, London (1972) 21. Reichardt, H.: Gesetzmäßigkeiten der freien Turbulenz, 2nd edn. VDI-Forsch.Heft 414. VDI-Verlag, Düsseldorf (1950) 22. Laufer, J.: The Structure of Turbulence in Fully Developed Pipe Flow. NACA Report 1174 (1952) 23. Anderson, P.S., Kays, W.M., Moffat, R.J.: The Turbulent Boundary layer on a Porous Plate: An experimental Study of the Fluid Mechanics for Adverse Free Stream Pressure Gradients. Report No. HMT-15, Department of Mechanical Engineering, Stanford University, CA (1972) 24. Wieghardt, K.: Über die turbulente Strömungen in Rohr und längs einer Platte. Zeitschrift für angewandte Mathematik und Mechanik, ZAMM 24, 294 (1944) 25. Sticksel, W.J.: Theoretische und experimentelle Untersuchungen turbulenter Grenzschichtprofile. Dissertation D 17, Technische Hochschule Darmstadt, Fachbereich Maschinenbau (1984) 26. Coles, D.E., Hirst, E.A.: Computation of turbulent Boundary Layers. AFSORIFP, -Stanford Conference, Vol. II, Thermoscience Division, Stanford University, CA (1968) 27. Coles, D. E.: Law of the Wake in Turbulent Boundary Layer. Journal of Fluid Mechanics 1(2), 191–226 (1956) 28. Clauser, F.H.: Turbulent Boundary Layers in Adverse pressure Gradients. Journal of Aeronautical Sciences 21, 91–108 (1954) 29. Spurk, J.H.: Fluid Mechanics. Springer, New York (1997) 30. Mellor, G.I., Gibson, D.M.: Equilibrium Turbulent Boundary Layers. Journal Fluid Mechanics 24, 225–256 (1966) 31. Prandtl, L., Schlichting, H.: 1934, Werft, Reederei, Hafen 15 1-4. Nachdruck: Gesammelte Abhandlungen, p. 649–662, Springer, Heidelberg (1961) 32. Falkner, V.M.: The Resistance of a Smooth Flat Plate with Turbulent Boundary Layer. Aircr, Eng 15, 65–69 (1943) 33. Blasius, H.: Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten. Forschung. Arb. Ing-Wes 134 (1913) 34. Crawford, M.E., Kays, W.M.: NASA CR-2742 (1976) 35. Cebeci, T., Bradshaw, P.: Physical and computational Aspects of Convective Heat Transfer. Springer, Heidelberg (1974) 36. Kays, W.M., Crawford, M.E.: Konvective Heat and Mass Transfer. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill, New York (1980) 37. Schobeiri, M.T., Chakka, P.: Prediction of turbine blade heat transfer and aerodynamics using unsteady boundary layer transition model. International Journal of Heat and Mass Transfer 45, 815–829 (2002) 38. Chakka, P., Schobeiri, M.T.: Modeling of Unsteady Boundary Layer Transition on a Curved Plate under Periodic Unsteady Flow Condition: Aerodynamic and Heat Transfer Investigations. ASME Transactions, Journal of Turbomachinery 121, 88–97 (1999)

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39. Schobeiri, M.T., Read, K., Lewalle, J.: Effect of Unsteady Wake Passing Frequency on Boundary Layer Transition, Experimental Investigation and Wavelet Analysis. ASME Transactions, Journal of Fluids Engineering 125, 251–266 (2003), a combined two-part paper 40. John, J., Schobeiri, M.T.: A Simple and Accurate Method of Calibrating XProbes. ASME Transactions, Journal of Fluid Engineering 115, 148–152 (1993) 41. Durst, F., Zanoun, E.-S., Pashstrapanska, M.: In-situ calibration of hot wires close to highly heat-conducting walls. Experiments in Fluids 31, 103–110 (2001) 42. Brun, H.H.: Hot-Wire Anemometry. Oxford University Press, Oxford (1995) 43. Hippensteele, S.A., Russell, L.M., Stepka, S.: Evaluation of A Method for Heat Transfer Measurements and Thermal Visualization Using a Composite of a Heater Element and Liquid Crystals. ASME Journal of Heat Transfer 105, 184–189 (1981) 44. Wright, L., Schobeiri, M.T.: The Effect of Periodic Unsteady Flow on Boundary Layer and Heat Transfer on a Curved Surface. ASME Transactions, Journal of Heat Transfer 120, 22–33 (1999) 45. Schobeiri, M.T.: Advances in Unsteady Aerodynamics and Boundary Layer Transition. Flow Phenomena in Nature, vol. 2, pp. 573–605. W.IT. Book, United Kingdom (2006) 46. Schobeiri, M.T., Chakka, P.: Prediction of turbine blade heat transfer and aerodynamics using unsteady boundary layer transition model. International Journal of Heat and Mass Transfer 45, 815–829 (2002) 47. Wilcox, D.: 1993, Turbulence Modeling for CFD. DCW Industries, Inc., 5354 Palm Drive, La Ca. nada, California 91011 (1993) 48. Schobeiri, M.T., Öztürk, B.: Experimental Study of the Effect of the Periodic Unsteady Wake Flow on Boundary Layer development, Separation, and Reattachment Along the Surface of a Low Pressure Turbine Blade. ASME 2004GT-53929, presented at International Gas Turbine and Aero-Engine Congress and Exposition, Vienna, Austria, June 14-17, also published in the ASME Transactions, Journal of Turbomachinery 126(4) (2004) 49. Schobeiri, M.T., Öztürk, B., Ashpis, D.: “Effect of Reynolds Number and Periodic Unsteady Wake Flow Condition on Boundary Layer Development, Separation, and Re-attachment along the Suction Surface of a Low Pressure Turbine Blade,” ASME Paper GT2005-68600 (2005) 50. Private communication with Dr. A. Ameri, NASA Glen Research Center, Cleveland OH 51. Kaszeta, R.W., Simon, T.W.: Experimental Investigation of Transition to Turbulence as Affected by Passing Wakes. NASA/CR-2002-212104 December (2002) 52. Roberts, S.K., Yaras, M.I.: Effects of Periodic-Unsteadiness, Free-Stream Turbulence and Flow Reynolds Number on Separation-Bubble Transition, ASME GT2003-38262 (2003) 53. Herbst, R.: Entwicklung von Strömungsgrenzschichten bei instationärer Zuströmung in Turbomaschinen. Dissertation D-17, Technische Hochschule Darmstadt, Germany (1980) 54. Pfeil, H., Herbst, R., Schröder, T.: Investigation of the Laminar-Turbulent Transition of Boundary Layers Disturbed by Wakes. ASME Journal of Engineering for Power 105, 130–137 (1983)

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55. Schobeiri, M.T., Radke, R.: Effects of Periodic Unsteady Wake Flow and Pressure Gradient on Boundary Layer Transition Along The Concave Surface of A Curved Plate. ASME Paper No. 94-GT-327 (1994) 56. Halstead, D.E., Wisler, D.C., Okiishi, T.H., Walker, G.J., Hodson, H.P., Shin, H.-W.: Boundary Layer Development in Axial Compressors and Turbines: Part 3 of 4. ASME Journal of Turbomachinery 119, 225–237 (1997) 57. Schobeiri, M.T., Öztürk, B., Ashpis, D.: On the Physics of the Flow Separation Along a Low Pressure Turbine Blade Under Unsteady Flow Conditions. ASME 2003-GT-38917, presented at International Gas Turbine and Aero-Engine Congress and Exposition, Atlanta, Georgia, June 16-19, 2003, also published in ASME Transactions, Journal of Fluid Engineering 127, 503–513 (May 2005) 58. Schobeiri, M.T., Öztürk, B.: Experimental Study of the Effect of the Periodic Unsteady Wake Flow on Boundary Layer development, Separation, and Reattachment Along the Surface of a Low Pressure Turbine Blade. ASME 2004GT-53929, presented at International Gas Turbine and Aero-Engine Congress and Exposition, Vienna, Austria, June 14-17, also published in the ASME Transactions, Journal of Turbomachinery 126(4) 663–676 (2004)

A Vector and Tensor Analysis in Turbomachinery Fluid Mechanics

A. 1 Tensors in Three-Dimensional Euclidean Space In this section, we briefly introduce tensors, their significance to turbomachinery fluid dynamics and their applications. The tensor analysis is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics. Once the basics of tensor analysis are understood, the reader will be able to derive all conservation laws of fluid mechanics without memorizing any single equation. In this section, we focus on the tensor analytical application rather than mathematical details and proofs that are not primarily relevant to engineering students. To avoid unnecessary repetition, we present the definition of tensors from a unified point of view and use exclusively the three-dimensional Euclidean space, with N = 3 as the number of dimensions. The material presented in this chapter has drawn from classical tensor and vector analysis texts, among others those mentioned in References. It is tailored to specific needs of turbomchinery fluid mechanics and is considered to be helpful for readers with limited knowledge of tensor analysis. The quantities encountered in fluid dynamics are tensors. A physical quantity which has a definite magnitude but not a definite direction exhibits a zeroth-order tensor, which is a special category of tensors. In a N-dimensional Euclidean space, a zeroth-order tensor has N0 = 1 component, which is basically its magnitude. In physical sciences, this category of tensors is well known as a scalar quantity, which has a definite magnitude but not a definite direction. Examples are: mass m, volume v, thermal energy Q (heat), mechanical energy W (work) and the entire thermo-fluid dynamic properties such as density ȡ, temperature T , enthalpy h, entropy s , etc. In contrast to the zeroth-order tensor, a first-order tensor encompasses physical quantities with a definite magnitude with components and a definite direction that can be decomposed in directions. This special category of tensors is known as vector. Distance X, velocity V, acceleration A, force F and moment of momentum M are few examples. A vector quantity is invariant with respect to a given category of coordinate systems. Changing the coordinate system by applying certain transformation rules, the vector components undergo certain changes resulting in a new set of components that are related, in a definite way, to the old ones. As we will see later, the order of the above tensors can be reduced if they are multiplied with each other in a scalar manner. The mechanical energy

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W = F.X is a representative example, that shows how a tensor order can be reduced. The reduction of order of tensors is called contraction. A second order tensor is a quantity, which has definite components and definite directions (in three-dimensional Euclidean space: ). General stress tensor Ȇ, normal stress tensor *, shear stress tensor ȉ, deformation tensor D and rotation tensor ȍ are few examples. Unlike the zeroth and first order tensors (scalars and vectors), the second and higher order tensors cannot be directly geometrically interpreted. However, they can easily be interpreted by looking at their pertinent force components, as seen later in section A.5.4

A.1.1 Index Notation In a three-dimensional Euclidean space, any arbitrary first order tensor or vector can be decomposed into 3 components. In a Cartesian coordinate system shown in Fig. A.1, the base vectors in x1, x2, x3 directions e1, e2, e3 are perpendicular to each other and have the magnitude of unity, therefore, they are called orthonormal unit vectors. Furthermore, these base vectors are not dependent upon the coordinates, therefore, their derivatives with respect to any coordinates are identically zero. In contrast, in a general curvilinear coordinate system (discussed in Appendix A) the base vectors

Fig. A.1: Vector decomposition in a Cartesian coordinate system

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do not have the magnitude of unity. They depend on the curvilinear coordinates, thus, their derivatives with respect to the coordinates do not vanish. As an example, vector A with its components A1, A2 and A3 in a Cartesian coordinate system shown in Fig. A.1 is written as: (A.1) According to Einstein's summation convention, it can be written as: (A.2) The above form is called the index notation. Whenever the same index (in the above equation i) appears twice, the summation is carried out from 1 to N (N = 3 for Euclidean space).

A.2 Vector Operations: Scalar, Vector and Tensor Products A.2.1 Scalar Product Scalar or dot product of two vectors results in a scalar quantity . We apply the Einstein’s summation convention defined in Eq.(A.2) to the above vectors: (A.3) we rearrange the unit vectors and the components separately: (A.4) In Cartesian coordinate system, the scalar product of two unit vectors is called Kronecker delta, which is: (A.5) with įij as Kronecker delta. Using the Kronecker delta, we get: (A.6) The non-zero components are found only for i = j, or įij = 1, which means in the above equation the index j must be replaced by i resulting in: (A.7) with scalar C as the result of scalar multiplication.

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A.2.2 Vector or Cross Product The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors. Example: (A.8) with C as the resulting vector. We apply the index notation to Eq. (A.8): (A.9) with Jijk as the permutation symbol with the following definition illustrated in Fig. A.2:

Jijk = 0 for i = j, j = k or i = k: (e.g. 122) Jijk = 1 for cyclic permutation: (e.g. 123) Jijk = -1 for anti-cyclic permutation: (e.g.132)

Fig. A.2: Permutation symbol, (a) positive , (b) negative permutation Using the above definition, the vector product is given by: (A.10)

A.2.3 Tensor Product The tensor product is a product of two or more vectors, where the unit vectors are not subject to scalar or vector operation. Consider the following tensor operation: (A.11)

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The result of this purely mathematical operation is a second order tensor with nine components:

(A.12) The operation with any tensor such as the above second order one acquires a physical meaning if it is multiplied with a vector (or another tensor) in scalar manner. Consider the scalar product of the vector C and the second order tensor ĭ. The result of this operation is a first order tensor or a vector. The following example should clarify this: (A.13) Rearranging the unit vectors and the components separately: (A.14) It should be pointed out that in the above equation, the unit vector ek must be multiplied with the closest unit vector namely ei (A.15) The result of this tensor operation is a vector with the same direction as vector B. Different results are obtained if the positions of the terms in a dot product of a vector with a tensor are reversed as shown in the following operation: (A.16) The result of this operation is a vector in direction of A. Thus, the products is different from

A.3 Contraction of Tensors As shown above, the scalar product of a second order tensor with a first order one is a first order tensor or a vector. This operation is called contraction. The trace of a second order tensor is a tensor of zeroth order, which is a result of a contraction and is a scalar quantity. (A.17) As can be shown easily, the trace of a second order tensor is the sum of the diagonal element of the matrix ĭij. If the tensor ĭ itself is the result of a contraction of two second order tensors Ȇ and D:

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(A.18) then the Tr(ĭ) is: (A.19)

A.4 Differential Operators in Fluid Mechanics In fluid mechanics, the particles of the working medium undergo a time dependent or unsteady motion. The flow quantities such as the velocity V and the thermodynamic properties of the working substance such as pressure p, temperature T, density ȡ or any arbitrary flow quantity Q are generally functions of space and time: During the flow process, these quantities generally change with respect to time and space. The following operators account for the substantial, spatial, and temporal changes of the flow quantities.

A.4.1 Substantial Derivatives The temporal and spatial change of the above quantities is described most appropriately by the substantial or material derivative. Generally, the substantial derivative of a flow quantity Q, which may be a scalar, a vector or a tensor valued function, is given by: (A.20) The operator D represents the substantial or material change of the quantity Q, the first term on the right hand side of Eq. (A.20) represents the local or temporal change of the quantity Q with respect to a fixed position vector x. The operator d symbolizes the spatial or convective change of the same quantity with respect to a fixed instant of time. The convective change of Q may be expressed as: (A.21) A simple rearrangement of the above equation results in: (A.22) The scalar multiplication of the expressions in the two parentheses of Eq. (A.22) results in Eq. (A.21)

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A.4.2 Differential Operator / The expression in the second parenthesis of Eq. (A.22) is the spatial differential operator / (Nabla, del) which has a vector character. In Cartesian coordinate system, the operator Nabla / is defined as: (A.23) Using the above differential operator, the change of the quantity Q is written as: (A.24) The increment dQ in Eq. (A.24) is obtained either by applying the product , or by taking the dot product of the vector dx and /Q. If Q is a scalar quantity, then /Q is a vector or a first order tensor with definite components. In this case ,/Q is called

Fig. A.3: Physical explanation of the gradient of scalar field

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the gradient of the scalar field. Equation (A.24) indicates that the spatial change of the quantity Q assumes a maximum if the vector /Q (gradient of Q ) is parallel to the vector dx. If the vector /Q is perpendicular to the vector dx, their product will be zero. This is only possible, if the spatial change dx occurs on a surface with Q = const. Consequently, the quantity Q does not experience any changes. The physical interpretation of this statement is found in Fig. A.3. The scalar field is represented by the point function temperature that changes from the surfaces T to the surface T + dT. In Fig. A.3, the gradient of the temperature field is shown as /T, which is perpendicular to the surface T = const. at point P. The temperature probe located at P moves on the surface T = const. to the point M, thus measuring no changes in temperature (Į = ʌ/2, cosĮ = 0). However, the same probe experiences a certain change in temperature by moving to the point Q, which is characterized by a higher temperature T + dT ( 0 < Į < ʌ/2 ). The change dT can immediately be measured, if the probe is moved parallel to the vector /T. In this case, the displacement dx (see Fig. A.3) is the shortest (Į = 0, cosĮ = 1). Performing the similar operation for a vector quantity as seen in Eq.(A.21) yields: (A.25) The right-hand side of Eq. (A.25) is identical with: (A.26) In Eq. (A.26) the product can be considered as an operator that is applied to the vector V resulting in an increment of the velocity vector. Performing the scalar multiplication between dx and / gives: (A.27) with /Vas the gradient of the vector field which is a second order tensor. To perform the differential operation, first the / operator is applied to the vector V, resulting in a second order tensor. This tensor is then multiplied with the vector dx in a scalar manner that results in a first order tensor or a vector. From this operation, it follows that the spatial change of the velocity vector can be expressed as the scalar product of the vector dx and the second order tensor /V, which represents the spatial gradient of the velocity vector. Using the spatial derivative from Eq. (A.27), the substantial change of the velocity is obtained by: (A.28) where the spatial change of the velocity is expressed as : (A.29) Dividing Eq. (A.29) by dt yields the convective part of the acceleration vector:

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(A.30) The substantial acceleration is then: (A.31) The differential dt may symbolically be replaced by Dt indicating the material character of the derivatives. Applying the index notation to velocity vector and Nabla operator, performing the vector operation , and using the Kronecker delta, the index notation of the material acceleration A is: (A.32) Equation (A.32) is valid only for Cartesian coordinate system, where the unit vectors do not depend upon the coordinates and are constant. Thus, their derivatives with respect to the coordinates disappear identically. To arrive at Eq. (A.32) with a unified index i, we renamed the indices. To decompose the above acceleration vector into three components, we cancel the unit vector from both side in Eq. (A.32) and get: (A.33) To find the components in xi -direction, the index i assumes subsequently the values from 1 to 3, while the summation convention is applied to the free index j. As a result we obtain the three components:

(A.34)

A.5 Operator / Applied to Different Functions This section summarizes the applications of nabla operator to different functions. As mentioned previously, the spatial differential operator / has a vector character. If it acts on a scalar function, such as temperature, pressure, enthalpy etc., the result is a vector and is called the gradient of the corresponding scalar field, such as gradient of temperature, pressure, etc. (see also previous discussion of the physical interpretation of /Q). If, on the other hand, / acts on a vector, three different cases are distinguished.

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A.5.1 Scalar Product of / and V This operation is called the divergence of the vector V. The result is a zeroth-order tensor or a scalar quantity. Using the index notation, the divergence of V is written as: (A.35) The physical interpretation of this purely mathematical operation is shown in Fig. A.4. The mass flow balance for a steady incompressible flow through an infinitesimal volume dv = dx1dx2dx3 is shown in Fig. A.4. V in

V2 dx x2

V2 x

2

2

n S V V3

ds

V

V 1 dx

V1

1

1

n ds

x

1

V2 V3 x

V 3 dx

3

S

3

V

out

3

nout

Fig. A.4: Physical interpretation of /.V We first establish the entering and exiting mass flows through the cube side areas perpendicular to x1- direction given by dA1 = dx2dx3: (A.36) (A.37) Repeating the same procedure for the cube side areas perpendicular to x2 and x3 directions given by dA2 = dx3dx1 and dA3 = dx1dx2 and subtracting the entering mass flows from the exiting ones, we obtain the net mass net flow balances through the infinitesimal differential volume as: (A.38)

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The right hand side of Equation (A.38) is a product of three terms, the density ȡ, the differential volume dv and the divergence of the vector V. Since the first two terms are not zero, the divergence of the vector must disappear. As result, we find: (A.39) Equation (A.39) expresses the continuity equation for an incompressible flow, as we will saw in Chapter 3.

A.5.2 Vector Product This operation is called the rotation or curl of the velocity vector V. Its result is a first-order tensor or a vector quantity. Using the index notation, the curl of V is written as: (A.40) The curl of the velocity vector is known as vorticity, . As we will see later, the vorticity plays a crucial role in fluid mechanics. It is a characteristic of a rotational flow. For viscous flows encountered in engineering applications, the curl is always different from zero. To simplify the flow situation and to solve the equation of motion, as we discusseded in Chapter 3 later, the vorticity vector , can under certain conditions, be set equal to zero. This special case is called the irrotational flow.

A.5.3 Tensor Product of / and V This operation is called the gradient of the velocity vector V. Its result is a second tensor. Using the index notation, the gradient of the vector V is written as: (A.41) Equation (A.41) is a second order tensor with nine components and describes the deformation and the rotation kinematics of the fluid particle. As we saw previously, the scalar multiplication of this tensor with the velocity vector, resulted in the convective part of the acceleration vector, Eq. (A.32). In addition to the applications we discussed, / can be applied to a product of two or more vectors by using the Leibnitz's chain rule of differentiation: (A.42) For U = V, Eq. (A.42) becomes

or (A.43)

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Equation (A.43) is used to express the convective part of the acceleration in terms of the gradient of kinetic energy of the flow.

A.5.4 Scalar Product of / and a Second Order Tensor Consider a fluid element with sides dx1, dx2, dx3 parallel to the axis of a Cartesian coordinate system, Fig.A.5. The fluid element is under a general three-dimensional stress condition. The force vectors acting on the surfaces, which are perpendicular to the coordinates x1, x2, and x3 are denoted by F1, F2, and F3,. The opposite surfaces are subject to forces that have experienced infinitesimal changes F1 + dF1, F2 + dF2, and F3 + dF3. Each of these force vectors is decomposed into three components Fij according to the coordinate system defined in Fig. A.5. The first index i refers to the axis, to which the fluid element surface is perpendicular, whereas the second index j indicates the direction of the force component. We divide the individual components of the above force vectors by their corresponding area of the fluid element side. The results of these divisions exhibit the components of a second order stress tensor represented by Ȇ as shown in Fig. A.6.

Fig. A.5: Fluid element under a general three-dimensional stress condition

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Fig. A.6: General stress condition

As an example, we take the force component area component obtain

results in

and divide it by the corresponding

. Correspondingly, we divide the force on the opposite surface by the same area

and

. In a similar way we find the remaining stress

components, which are shown in Fig. A.6. The tensor has nine components ʌij as the result of forces that are acting on surfaces. Similar to the force components, the first index i refers to the axis, to which the fluid element surface is perpendicular, whereas the second index j indicates the direction of the stress component. Considering the stress situation in Fig.A.6, we are now interested in finding the resultant force acting on the fluid particle that occupies the volume element . For this purpose, we look at the two opposite surfaces that are perpendicular to the axis as shown in Fig. A.7.

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Fig. A.7: Stresses on two opposite walls

As this figure shows, we are dealing with 3 stress components on each surface, from which one on each side is the normal stress component such as and . The remaining components are the shear stress components such as and

and in

. According to Fig. A.7 the force balance in

-direction, we find

-directions is:

A Vector and Tensor Analysis in Turbomachinery

Similarly, in

699

, we obtain

Thus, the resultant force acting on these two opposite surfaces is:

In a similar way, we find the forces acting on the other four surfaces. The total resulting forces acting on the entire surface of the element are obtained by adding the nine components. Defining the volume element , we divide the results by

and obtain the resulting force vector that is acting on the volume element.

(A.44)

Since the stress tensor Ȇ is written as: (A.45) it can be easily shown that: (A.46) The expression is a scalar differentiation of the second order stress tensor and is called the divergence of the tensor field Ȇ. We conclude that the force vector acting on the surface of a fluid element is the divergence of its stress tensor. The stress tensor is usually divided into its normal- and shear stress parts. For an incompressible fluid it can be written as

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(A.47) with Ip as the normal and T as the shear stress tensor. The normal stress tensor is a product of the unit tensor and the pressure p. Inserting Eq. (A.47) into (A.46) leads to (A.48) Its components are (A.49)

A.5.5 Eigenvalue and Eigenvector of a Second Order Tensor The velocity gradient expressed in Eq. (A.41) can be decomposed into a symmetric deformation tensor D and an antisymmetric rotation tensor ȍ: (A.50) A scalar multiplication of D with any arbitrary vector A may result in a vector, which has an arbitrary direction. However, there exists a particular vector V such that its scalar multiplication with D results in a vector, which is parallel to V but has different magnitude: (A.51) with V as the eigenvector and Ȝ the eigenvalue of the second order tensor D. Since any vector can be expressed as a scalar product of the unit tensor and the vector itself , we may write: (A.52) Equation (A.52) can be rearranged as: (A.53) and its index notation gives: (A.54) or

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701

(A.55) Expanding Eq. (A.55) gives a system of linear equations,

(A.56)

A nontrivial solution of these Eq. (A.56) is possible if and only if the following determinant vanishes: (A.57) or in index notation: (A.58) Expanding Eq. (A.58) results in

(A.59)

After expanding the above determinant, we obtain an algebraic equation in Ȝ in the following form (A.60) where

and

are invariants of the second order tensor D defined as: (A.61) (A.62) (A.63)

The roots of Eq. (A.60) are known as the eigenvalues of the tensor D. The superscript refers to the roots of Eq. (A.60) - not to be confused with the component of a vector.

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References 1. Aris, R.: Vector, Tensors and the Basic Equations of Fluid Mechanics. PrenticeHall, Englewood Cliffs (1962) 2. Brand, L.: Vector and Tensor Analysis. John Wiley and Sons, New York (1947) 3. Klingbeil, E.: Tensorrechnung für Ingenieure. Bibliographisches Institut, Mannheim (1966) 4. Lagally, M.: Vorlesung über Vektorrechnung, 3rd edn. Akademische Verlagsgesellschaft, Leipzig (1944) 5. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960)

B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

B.1 Change of Coordinate System The vector and tensor operations we have discussed in the foregoing chapters were performed solely in rectangular coordinate system. It should be pointed out that we were dealing with quantities such as velocity, acceleration, and pressure gradient that are independent of any coordinate system within a certain frame of reference. In this connection it is necessary to distinguish between a coordinate system and a frame of reference. The following example should clarify this distinction. In an absolute frame of reference, the flow velocity vector may be described by the rectangular Cartesian coordinate xi: (B.1) It may also be described by a cylindrical coordinate system, which is a non-Cartesian coordinate system: (B.2) or generally by any other non-Cartesian or curvilinear coordinate system ȟi that describes the flow channel geometry: (B.3) By changing the coordinate system, the flow velocity vector will not change. It remains invariant under any transformation of coordinates. This is true for any other quantities such as acceleration, force, pressure- or temperature gradient. The concept of invariance, however, is generally no longer valid if we change the frame of reference. If, for example, the flow particles leave the absolute frame of reference (such as turbine or compressor stator blade channels) and enter a relative frame, for example rotating turbine or compressor blade channels, its velocity will experience a change as seen in Chapter 3. In this Chapter, we will pursue the concept of quantity invariance and discuss the fundamentals needed for understanding the coordinate transformation.

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

B.2 Co- and Contravariant Base Vectors, Metric Coefficients As we saw in Appendix A, a vector quantity is described in Cartesian coordinate system xi by its components: (B.4) with ei as orthonormal unit vectors (Fig. B.1 left).

Fig. B.1: Base vectors in a Cartesian (left ) and in a generalized orthogonal curvilinear coordinate system (right) The same vector transformed into the curvilinear coordinate system ȟk (Fig. B.1 right) is represented by: (B.5) where gk are termed covariant base vectors and Vk the contravariant components of V with respect to the base gk. For curvilinear coordinate systems, we place the indices diagonally for summing convenience. Unlike the Cartesian base vectors ei, that are orthonormal vectors (of unit length and mutually orthogonal), the base vectors gk do not have unit lengths. The base vectors gk represent the rate of change of the position vector x with respect to the curvilinear coordinates ȟi. (B.6) Since in a Cartesian coordinate system the unit vectors e i, are not functions of the coordinates xi, Eq. (B.6) can be written as:

B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

705

Fig. B.2: Co- and contravariant base vectors (B.7) Similarly, the reciprocal or contravariant base vector gk is defined as: (B.8) As shown in Fig. B.2, the covariant base vectors g2, g2, and g3 are tangent vectors to the mutually orthogonal curvilinear coordinates ȟ1, ȟ2, and ȟ3. The reciprocal base vectors g1, g2, g3, however, are orthogonal to the planes described by g2 and g3, g3 and g1, and g1 and g2, respectively. These base vectors are interrelated by: (B.9) where gk and gj are referred to as the covariant and contravariant base vectors, respectively. The new Kronecker delta įk j from Eq. (B.9) has the values:

The vector V written relative to its co- and contravariant base is: (B.10)

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

with the components Vk and V k as the covariant and contravariant components, respectively. The scalar product of covariant respectively contravariant base vectors results in the covariant and covariant metric coefficients: (B.11) The mixed metric coefficient is defined as (B.12) Using in curvilinear coordinate systems, its is often necessary to express the covariant base vectors in terms of the contravariant ones and vice versa. To do this, we first assume that:

(B.13)

Generally the contravariant base vector can be written as (B.14) To find a direct relation between the base vectors, first the coefficient matrix Aij must be determined. To do so, we multiply Eq. (B.14) with gk scalarly: (B.15) This leads to which means:

. The right hand side is different from zero only if j = k, (B.16)

Introducing Eq. (B.16) into (B.14) results in a relation that expresses the contravariant base vectors in terms of covariant base vectors: (B.17) The covariant base vector can also be expressed in terms of contravariant base vectors in a similar way: (B.18) Multiply Eq. (B.l8) with (B.17) establishes a relationship between the covariant and contravariant metric coefficients: (B.19) Applying the Kronecker delta on the right hand side results in: (B.20)

B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

707

B.3 Physical Components of a Vector As mentioned previously, the base vectors gi or gj are not unit vectors. Consequently the co- and contravariant vector components Vj or Vj do not reflect the physical components of vector V. To obtain the physical components, first the corresponding unit vectors must be found. The covariant unit vectors are determined from: (B.21) Similarly, the contravariant unit vectors are: (B.22) where gi*, represents the unit base vector, gi the absolute value of the base vector. The expression (ii) denotes that no summing is carried out, whenever the indices are enclosed within parentheses. The vector can now be expressed in terms of its unit base vectors and the corresponding physical components: (B.23) Thus the covariant and contravariant physical components can be easily obtained from: (B.24)

B.4 Derivatives of the Base Vectors, Christoffel Symbols In a curvilinear coordinate system, the base vectors are generally functions of the coordinates itself. This fact must be considered while differentiating the base vectors. Consider the derivative: (B.25) The comma in Eq. (B.25) followed by the lower index j represents the derivative of the covariant base vector gi with respect to the coordinate ȟj. Similar to Eq.(B.7), the unit vector ek can be written as: (B.26) Introducing Eq. (B.26) into (B.25) yields: (B.27) with īijn, and īi jn as the Christoffel symbol of first and second kind, respectively with the definitions:

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

(B.28a) It follows from (B.28a) that the Christoffel symbols of the second kind are related to the first kind by: (B.28b) Because of (B.28b) and, for the sake of simplicity, in what follows, we will use the Christoffel symbol of second kind . Thus, the derivative of the contravariant base vector reads: (B.29) The Christoffel symbols are then obtained by expanding Eq. (B.28a): (B.30a) The fact that the only non-zero elements of the matrix

are the diagonal elements

allows the modification of Eq. (B.30a) to: (B.30b) In Eq. (B.30a), the Christoffel symbols are symmetric in their lower indices. Again, note that a repeated index in parentheses in an expression such as g(kk) does not subject to summation.

B.5 Spatial Derivatives in Curvilinear Coordinate System The differential operator / is in curvilinear coordinate system defined as: (B.31) In this Chapter, the operator / will be applied to different arguments such as zeroth, first and second order tensors.

B.5.1 Application of / to Zeroth Order Tensor Functions If the argument is a zeroth order tensor which is a scalar quantity such as pressure or temperature, the results of the operation is the gradient of the scalar field which is a vector quantity:

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709

(B.32) The abbreviation “,i ” refers to the derivative of the argument, in this case p, with respect to the coordinate ȟi .

B.5.2 Application of / to First and Second Order Tensor Functions Scalar Product: If the argument is a first order tensor such as a velocity vector, the order of the resulting tensor depends on the operation character between the operator / and the argument. If the argument is a product of two or more quantities, chain rule of differentiation is applied. Thus, the divergence of the vector filed V is:

(B.33a)

Implementing the Christoffel symbol from (B.27), the results of the above operations are the divergence of the vector V. It should be noticed that a scalar operation leads to a contraction of the order of tensor on which the operator is acting. The scalar operation in (B.33a) leads to: (B.33b) Cross Product: This vector operation yields the rotation or curl of a vector field as:

(B.34a)

Because of the antisymmetric character of a cross product and the symmetry of the Christoffel symbols in their lower indices, the second term on the right-hand side of (B.34a), namely . As a result, we find

(B.34b) with as the permutation symbol that functions similar to the one for Cartesian coordinate system withe the difference that:

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

(B.35)

with

as the determinant of the matrix of covariant metric coefficients.

Tensor Product: The gradient of a first order tensor such as the velocity vector V is a second order tensor. Its index notation in a curvilinear coordinate system is: (B.36) Divergence of a Second Order Tensor: A scalar operation that involves / and a second order tensor, such as the stress tensor Ȇ or deformation tensor D, results in a first order tensor which is a vector: (B.37) The right hand side of (B.37) is reduced to: (B.38) By calculating the shear forces using the Navier-Stokes equation, the second derivative, the Laplace operator ǻ, is needed: (B.39) This operator applied to the velocity vector yields: (B.40)

B.6 Application Example 1: Inviscid Incompressible Flow Motion As the first application example, the equation of motion for a steady, incompressible and inviscid flow is transformed into a cylindrical coordinate system, where it is decomposed into components in r, ș and z-directions. Neglecting the gravitational force , the coordinate invariant version of the equation is written as:

B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

711

(B.41) The transformation and decomposition procedure is shown in the following steps.

B.6.1 Equation of Motion in Curvilinear Coordinate Systems The second order tensor on the left hand side can be obtained using Eq. (B.35): (B.42) The scalar multiplication of (B.42) with the velocity vector V leads to: (B.43) Introducing the mixed Kronecker delta into (B.43), we get: (B.44) For an orthogonal curvilinear coordinate system the mixed Kronecker delta is: (B.45) Taking this into account, Eq. (B.43) (B.46) and rearranging the indices, we obtain the convective part of the Navier-Stokes equations as: (B.47) The pressure gradient on the right hand side of Eq. (B.40) is calculated form Eq. (B.32): (B.48) Replacing the contravariant base vector with the covariant one using Eq. (B.17) leads to: (B.49)

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

Incorporating Eqs. (B.46) and (B.49) into Eq. (B.40) yields: (B.50) In i-direction, the equation of motion is: (B.51)

B.6.2 Special Case: Cylindrical Coordinate System To transfer Eq. (B.40) into any arbitrary orthogonal curvilinear coordinate system, first the coordinate system must be specified. The cylindrical coordinate system is related to the Cartesian coordinate system by: (B.52) The curvilinear coordinate system is represented by: (B.53)

B.6.3 Base Vectors, Metric Coefficients The base vectors are calculated from Eq. (B.7). (B.54) Equation (B.54) decomposed in its components yields:

(B.55)

Performing the differentiation of the terms in Eq. (B.55) yields: (B.56)

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The co- and contravariant metric coefficients are:

(B.57)

The contravariant base vectors are obtained from:

(A.57a)

Since the mixed metric coefficient are zero, (B.57a) reduces to: (A.57b)

B.6.4 Christoffel Symbols The Christoffel symbols are calculated from Eq. (B.30b) (B.59) To follow the calculation procedure, one zero- element and one non-zero element are calculated:

(B.60)

All other elements are calculated similarly. They are shown in the following matrices:

(B.61)

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Introducing the non-zero Christoffel symbols into Eq. (B.50), the components in gl, g2, and g3 directions are: (B.62) (B.64) (B.63)

B.6.5 Introduction of Physical Components The physical components can be calculated from Eqs. (B.21) and (B.24):

(B.65)

The Vi -components expressed in terms of V*i are: (B.66) Introducing Eqs.(B.65) into (B.61), (B.62), and (B.63) results in: (B.67)

(B.68)

(B.69) According to the definition: (B.70) the physical components of the velocity vectors are:

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(B.71) and insert these relations into Eqs. (B.66) to (B.68), the resulting components in r, Ĭ, and z directions are:

(B.72)

B.7 Application Example 2: Viscous Flow Motion As the second application example, the Navier-Stokes equation of motion for a steady, incompressible flow with constant viscosity is transferred into a cylindrical coordinate system, where it is decomposed into its three components r, ș, z. The coordinate invariant version of the equation is written as: (B.73) The second term on the right hand side of Eq. (B.73) exhibits the shear stress force. It was treated in section B.5, Eq. (B.39) and is the only term that has been added to the equation of motion for inviscid flow, Eq. (B.40).

B.7.1 Equation of Motion in Curvilinear Coordinate Systems The transformation and decomposition procedure is similar to the example in section B. 6. Therefore, a step by step derivation is not necessary.

(B.74)

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B Tensor Operations in Orthogonal Curvilinear Coordinate Systems

B.7.2 Special Case: Cylindrical Coordinate System Using the Christoffel symbols from section B.6.4 and the physical components from B.6.5, and inserting the corresponding relations these relations into Eq. (B.74), the resulting components in r, Ĭ, and z directions are:

(B.75)

(B.76)

References 1. Aris, R.: Vector, Tensors and the Basic Equations of Fluid Mechanics. PrenticeHall, Inc., Englewood Cliffs (1962) 2. Brand, L.: Vector and Tensor Analysis. John Wiley & Sons, New York (1947) 3. Klingbeil, E.: Tensorrechnung für Ingenieure. Bibliographisches Institut, Mannheim (1966) 4. Lagally, M.: Vorlesung über Vektorrechnung, 3rd edn. Akademische Verlagsgesellschaft, Leipzig (1944) 5. Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines. John Wiley & Sons, Chichester (1960)

Index

1/7-th boundary layer displacement thickness 631 energy deficiency thickness 631, 633 integral equation 635 length scale 670 momentum deficiency thickness 631 re-attachment 661, 664, 667, 668, 670 separation 667 wall influence 655 Algebraic Model 585 Baldwin-Lomax 585 Cebeci-Smith 584 Prandtl Mixing Length 578 Anemometer 519 Averaging 560 conservation equations 561 continuity equation 561 mechanical energy equation 562 Navier-stokes equation 561 total enthalpy equation 565 Axial vector 24 Base vectors 704, 707 contravariant 704 covariant 704 reciprocal 704 Becalming 535 Bernoulli equation 39, 584 Bio-Savart law 195 Blade design 267 Assessment of 289 Bézier 285 camberline coordinates 285

conformal transformation 267 Blade design using Bézier function 283 Blade force, inviscid flow circulation 145–149, 162–164, 168, 171 lift coefficient 149, 153, 155–158, 166, 168 lift force 145–149, 153 Blade force, viscous flow drag coefficient 153, 157 drag force 152, 153 drag-lift ratio 157, 158 lift coefficient 149, 153, 155–158, 166, 168 lift force 145–149, 153 lift-solidity coefficient 153–155, 162–164, 166–171 no-slip condition 150 optimum solidity 158, 160 periodic wake flow 150 total pressure loss coefficient 150 Blending function 590 Boundary layer 580 buffer layer 580 bypass 517 experimental simulation of unsteady 527 intermittency function 519 logarithmic layer 580 natural 535 outer layer 581 transitional 581 unsteady Boundary 525 viscous sublayer 580 von Karman constant 583

718

Index

wake function 581 wake induced 535 Boundary layer theory Blasius solution 624 concept of 619 viscous layer 619 Boussinesq 577 relationship 577 Bypass transition 517 Calmed region 532 Cascade process 546 Cauchy-Poisson law 35 Circulation function 389 Co- and contravariant Base Vectors 703 Combustion chamber 80, 84, 341, 346, 348, 372 chamber heat transfer 376 mass flow transients 373 temperature transients 374 Compound lean 309–311, 313–315 Compression waves 100 Compressor corrections blade thickness effect 408 circulation function 389–396, 408 Mach number effect 394, 407 maximum velocity 386, 389–391, 393, 395 optimum flow condition 389 profile losses 385, 396, 407, 408 Reynolds number effect 408 secondary flow losses 385, 403, 406 shock losses 384, 385, 397, 400, 403 Compressor blade design 2, 265, 272, 273, 275, 279 base profile 2, 265, 276–279, 281, 285, 286 DCA, MCA 2, 267, 272, 273, 279, 281 double circular arc (DCA) 272 for High Subsonic Mach Number 279 graphic design of camberline 284 induced velocity 274, 275

Joukowsky transformation 267 maximum thickness 273, 277, 278, 285 multi-circular arc (MCA) profiles 272 NACA-67 profiles 2, 272, 274 subsonic compressor 271, 273, 279 summary, simple compressor blade design steps 279 superimposing the base profile 276, 278 supersonic compressor 282 vortex distribution 274 Compressor diffusion factor compressibility effect on, 393, 395, 396, 403 diffusion factor 385–388, 391–393, 395–397, 403–408, 415, 416, 420, 422 modified diffusion factor 385, 391, 403–405, 415, 416 Compressor stage 341, 348, 353 Conformal transformation circle-cambered Airfoil Transformation 271 circle-ellipse Transformation 268 circle-symmetric Airfoil Transformation 269 Convergent nozzle 91, 92 Cooled turbine 60, 82 Coriolis acceleration 51 Correlations autocorrelation 549 coefficient 548 osculating parabola 553 single point 548 tensor 548 two-point correlation 548 Covariance 549 Critical state 88, 97, 103 Cross-section change 93 Curvilinear coordinate Systems 703, 712 base vectors 712 metric coefficients 712 spatial Derivatives 708

Index

Deformation 34, 700 Density change 91 Derivative 690 material 690 substantial 690 temporal 690 Description 15 Lagrangian description 16 material 15 spatial 15 Diabatic expansion 351 Diabatic processes 351 Diabatic turbine module 467 Differential operator / 691 Differential operators 690 Diffusers 220 concave and convex 225 configurations 221 conversion coefficient 224 induced Velocity 194 opening angle 226 pressure recovery 220, 222 recovery factor 224 vortex generator edges 226 Diffusivity 546 Direct Navier-Stokes simulations 577 Dissipation 575, 576, 577 dissipation 564 energy 545 equation 554 exact derivation of 577 function 43 kinetic Energy 576 parameter 557 range 556 turbulence 554 turbulent 564 viscous 564 Eddy viscosity 578 Effect of stage parameters on degree of reaction 130, 131, 133–135, 137, 139, 140 flow coefficient 129, 137, 139, 140, 142 load coefficient 129, 130, 136–143 Efficiency 452

719

Efficiency of multi-stage turbomachines 233 heat recovery 233, 234, 237, 239 infinitesimally small expansion 235 isentropic efficiency 239–242 polytropic efficiency 233, 235, 239–242 recovery factor 236–239 reheat 233, 234, 240, 241 reheat factor 240, 241 Einstein summation convention 37 Einstein’s summation 687 Energy balance in stationary frame 42 dissipation function 43 mechanical energy 42 thermal energy balance 45 Energy cascade process 546, 547 Energy spectral function 558 Energy spectrum 555 dissipation range 556 large eddies 555 Energy transfer 123–125 compressor stage 123, 125, 127–129, 137, 139, 143 Euler turbine equation 127 turbine row 127, 128, 136 turbine stage 124, 127, 128, 135–137, 139 Ensemble averaging 530 Entropy balance 49 Equation 570 turbulence kinetic Energy 570 Euler equation of motion 38 Expansion waves 98, 100, 121 Expansion adiabatic 348 Falkner-Skan equation 629 Fanno 101, 106–109 Fanno line 101, 106–109 Faulkner-Skan equation 628 Flow coefficients 453 Fluctuation kinetic energy 566 Frame indifferent quantities 34 Free turbulent flow 545 Friction stress tensor 35 Frozen turbulence 551

720

Index

Gas dynamic functions 97 Gas turbines components 473 compressed air storage 473 design 473 dynamic 473 multi-spool 475 performance 473 power generation 473 process 473 shutdown 495 single spool 500 startup 495 three-spool 493 twin-spool 473 ultra high efficiency 485 Gaussian distribution 534 Generalized lift-solidity coefficient 162, 166 lift-solidity coefficient 153–155, 162–164, 166–171 turbine rotor 163, 168, 170 turbine stator 159, 162–164, 166 Generic component configuration adiabatic compressor 348 adiabatic turbine 349, 350 compressor 341, 342, 346, 348, 350–355 diabatic turbine 350 heat exchangers 346 inlet, exhaust, pipe 345 schematic 350, 353, 355 turbine 341–343, 347–356 Generic modeling components 2 heat exchangers 346 inlet, exhaust, pipe 345 Geometry parameter 458 Heat transfer 376 conduction 376 convection 373 Nusselt number 660 radiation 376 Stanton number 660 thermography 660 Hot wire anemometry 653 aliasing effect 656

analog/digital converter 656 constant current (CC) mode 654 constant-temperature (CT) mode 654 cross-wire 654 folding frequency 656 Nyquist-frequency 656 sample frequency 656 sampling rate 656 signal conditioner 656 single wire 653 single, cross and three-wire probes 653 single-wire 653 three-wire 654 Incidence and deviation 241 cascade parameters 255 complex plane 242 complex velocity potential 243 conformal transformation 241, 242, 244, 250, 261 deviation 241, 255, 256, 258–261, 263 infinitely thin circular arc 250 optimum incidence 256, 257, 259 singularities 244, 247 transformation function 244, 247 Incompressibility condition 31 Index Notation 686, 687 Integral balances 59 axial moment 70 energy 72, 73, 75, 77, 78 linear momentum 59, 61 mass flow 60 moment of momentum 66, 68, 70, 71, 73 reaction force 65, 66, 68 shaft power 75, 78, 101 Intermittency 533 averaged 534 curved plate 521 ensemble-averaged 533 flat plate 524 function 653 Intermittency into Navier Stokes Equations 536

Index

maximum 533 minimum 534 modeling 524 Irreversibility 72, 79, 84 total pressure loss 79, 86, 87 Isentropic row load coefficients 452 Isotropic turbulence 560 Isotropy 547 Jacobian 16–21, 25, 26 function 16–19, 21, 25, 26 functional determinant 19 material derivative 17, 19, 26 transformation vector 16, 17, 18,19–21, 25 Joukowsky transformation 265 Kinematics of fluid motion 15 deformation tensor 21 material volume 17, 18, 20, 25 rotation tensor 24, 25 rotation vector 24, 25 translation 21, 23 Kinetic energy 556, 560, 566 Kolmogorov 546 eddies 546 first hypothesis 556 hypothesis 546 inertial subrange 555, 556 length scale 555 scales 555 second hypothesis 557 time scale 555 universal equilibrium 546 velocity scale 555 Kronecker delta 705 Kronecker tensor 35 Kutta-Joukowsky Transformation 268 Laminar Boundary 624 Blasius equation 625 Faulkner-Skan equation 628 Hartree 628 Polhausen Approximate 629 Laval nozzle 95, 96, 98, 109 Linear wall function 580 Losses due to blade profile diffusion factor 185

721

effect of Reynolds number 186 integral equation 179, 181 momentum equation 187–189, 205, 212, 214 profile loss 176–178, 182, 184–186, 192, 217 total pressure loss 176, 182, 183, 190, 194, 197, 198, 200, 201, 203, 205, 223, 227 Losses due to exit kinetic energy, effect of degree of reaction 208–212 exit flow angle 176, 177, 203, 211–213, 216, 222 stage load coefficient 209–211 Losses due to leakage flow in shrouds 204 induced drag 194, 196–199, 201 load function 198–200 losses due to secondary flows 192 mixing loss 190–192, 217–219 Losses due to trailing edge ejection mixing loss 190–192, 217–219 optimum mixing losses 219 Mach number 88 Material acceleration 692 Material derivative 17 Metric coefficients 703 contravariant 705 covariant 705 mixed metric coefficient 705 Mixing length hypothesis 578 Modeling 372 combustion chambers 372 Modeling of inlet, exhaust, and pipe system 357 continuity 358 inlet 357 momentum 358 physical and mathematical 357 pipe 357, 358, 361, 362 total pressure 358, 359 Modeling the compressor dynamic performance active surge prevention 430

722

Index

module level 2: row-by-row adiabatic calculation 429 module level 3: row-by-row diabatic compression 436 simulation example 427, 431 simulation of compressor surge 432 Modeling turbomachinery components combustion chambers 341, 346, 348 high pressure compressor 342, 351, 354, 355 high pressure turbine 341, 342 low pressure compressor 342, 353, 355 low pressure turbine 342, 348, 354 modular configuration 342, 351, 353, 355, 356 recuperators 346 twin-spool engine 342 Modular configuration 2, 342, 351, 353, 355, 356 control systems 342, 352 shafts 352, 355 valves 342, 352 Momentum balance in stationary frame 32 Natural transition 517 Navier-Stokes 569 equations 547, 548, 552, 561, 569, 570 operator 569 Navier-Stokes equation for compressible fluids 36 Newtonian fluids 35 Nnormal shock 99, 109–115, 117 Nonlinear dynamic simulation 323 numerical treatment 339 one-dimensional approximation 331 Predictor-corrector 339 Runge-Kutta 339 thermo-fluid dynamic equations. 328

time dependent equation of continuity 331 time dependent equation of energy 334 time dependent equation of motion 332 Numerical treatment 330, 339 Nusselt number 660 Oblique shock 109, 115, 116, 118, 121 Orthonormal vectors 704 Osculating parabola 553 Over expansion 98 Peclet number 651 effective 652 turbulent 651 Physical components of a Vector 706 Physical components of vector 706 Plenum 343 Pohlhausen 629 profiles 630 slope of the velocity profiles 630 velocity profiles 630 Polytropic load coefficients 452 Power law 647 Prandtl 619 1/7–th power law boundary layer 619 boundary layer approximation 619 boundary layer theory 624 effective Prandtl number 652 experimental observations 619 mixing length 650 mixing length model 653 molecular Prandtl number 652 number 651 turbulent Prandtl number 651 Prandtl mixing length hypothesis 580 Prandtl number 651 effective 652 turbulent 651 Preheater 367 Principle of material objectivity 34

Index

Radial equilibrium 291, 295, 298–300, 309, 316 derivation of equilibrium equation 292 free vortex flow 291, 316, 317 Rayleigh 101, 103–106, 108, 109 Recuperator 346, 369 heat Transfer Coefficient 371 recuperators 367 Reynolds transport theorem 25, 27 Richardson energy cascade 547 Rotating frame continuity equation in 52 energy equation in 55 equation of motion in 53 Rotation 700 Rotation tensor 34 Scalar product of / 693 Scales of turbulence length 547 of the smallest eddy 547 time 547 time scale 546 time scale of eddies 548 Shear viscosity 36 Shock tube, dynamic behavior 359 mass flow transients 361, 364 pressure transients 361, 362 shock tube dynamic behavior 361 temperature transients 362–364 Shocks 98, 101, 113, 115, 117 detached shock 117 Hugoniot relation 111–113 normal shock wave 99, 109, 115 oblique shock wave 115 Prandtl-Meyer expansion 118, 121 strong shock 112, 114, 115, 117 weak shock 114, 117 Similarity condition 627 Spatial differential 691 Spectral tensor 558 Stage parameters 129, 138, 140 circumferential velocity ratio 139, 141

723

degree of reaction 130, 131, 133–135, 137, 139, 140 flow coefficient 129, 137, 139, 140, 142 load coefficient 129, 130, 136–143 meridional velocity ratio 139, 141 Stage-by-stage and row-by-row adiabatic compression 409 calculation of compression process 409 generation of steady state performance map 418 off-design efficiency calculation 415 rotating stall 420–423, 430–433 row-by-row adiabatic compression 409, 411, 423 surge 385, 420–424, 428–434 surge limit 420, 422, 423, 428, 434 Stanton number 660 Streamline curvature method 291, 292, 300, 301, 306, 309, 310, 312, 313 application of 292, 300, 310 derivation of equilibrium equation 292 examples 306 forced vortex flow 317 free vortex flow 291, 316 special cases 292, 316 step-by-step solution procedure 302 Stress tensor 34 apparent 547 Substantial change 690 Taylor 553 eddies 551 frozen turbulence 551 hypothesis 551 micro length scale 553 time scale 553 Temporal change 690 Tensor 568, 685 contraction 686, 689

724

deformation 573 eigenvalue 700 eigenvector 700 first-order 685 friction 569 friction stress 563 principal Axis 700 product 687, 678 Reynolds stress 568 rotation 585 second order 686 zeroth-order 685 Tensor product of / and V 695, 702 Thermal efficiency of gas turbines 477, 478, 482–484, 508, 509 improvement of 483 UHEGT 508 Transition 535 Turbine 123–143, 445 adiabatic design 447 adiabatic expansion 449 axial 123, 124, 128, 129, 131, 132, 137–140, 142 blade design 283 blade design using Bezier curve 285 design and off-design behavior 459 expansion process 448 extreme low mass flows 456 modeling 462 off-design efficiency calculation 454 off-design performance 447 off-design primary loss coefficient 454 radial 128, 131, 132, 134, 137, 139, 140, 143 row-by-row diabatic expansion 465 stage 123–125, 127–131, 133–143, 342, 348–350 stage-by-stage 447 velocity parameter 461 Turbulence 547 anisotropic 555 convective diffusion 575

Index

correlations 548 diffusion 546 free turbulence 545 homogeneous 547 isotropic 547, 556 kinetic energy 571 length and time Scales 547 production 575 production of turbulence kinetic energy 575 type of 545, 547 viscous diffusion 546 viscous dissipation 575 wall turbulence 545 Turbulence length 578 Turbulence model 545, 578 algebraic model 578 Baldwin-lomax algebraic model 585 Cebeci-smith algebraic model 584 one-equation model by prandtl 586 Prandtl mixing length 578 standard k-İ vs k-Ȧ 595 two-equation k-İ model 587 two-equation k-Ȧ-model 589 two-equation k-Ȧ-sst-model 590 Turbulent flow 581 fully developed 581 Turbulent kinetic energy 571 Universal equilibrium 546 Unsteady boundary layer 657 Strouhal number 661 wake generator 661 Variation of 670 Length Scale 670 Pressure Gradient 671 Vector 685 cross product 687 scalar product 687 tensor product 688 Vector product 695 Velocity scales 578 Velocity spectrum tensor 558

Index

Ventilation power 457 Viscous diffusion 546 Viscous sublayer 580 von Kármán 635 integral equation 635 von Kármán constant 583 Vortex 547 Vorticity, 695

725

Wave number space 558, 559 Weinig, conformal transformation 241 Zweifel number 161