Table Of ContentTurbomachinery Flow Physics
and Dynamic Performance
Meinhard T. Schobeiri
Turbomachinery Flow Physics
and Dynamic Performance
Second and Enhanced Edition
With433Figures
ABC
Author
Prof.Dr.-Ing.MeinhardT.Schobeiri
DepartmentofMechanicalEngineering
TexasA&MUniversity
USA
ISBN978-3-642-24674-6 e-ISBN978-3-642-24675-3
DOI10.1007/978-3-642-24675-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012935425
(cid:2)c Springer-VerlagBerlinHeidelberg2005,2012
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Preface to the Second Edition
The first edition of this book appeared in 2005 and was concentrated on turbo-
machinery 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 appropri-
ately 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, turbul-
ence 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 turbul-
ence 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
tschobeiri@mengr.tamu.edu
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 turbo-
machinery 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 turbo-
machinery 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 calcula-
tion 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 sug-
gestions 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
tschobeiri@mengr.tamu.edu
Contents
I Turbomachinery Flow Physics
1 Introduction, Turbomachinery, Applications, Types ....... 3
1.1 Turbine ................................................ 3
1.2 Compressor ............................................. 7
1.3 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 .............. 15
2.1.1 Material Description................................. 15
2.1.2 Jacobian Transformation Function and Its Material Derivative 17
2.1.3 Spatial Description .................................. 20
2.2 Translation, Deformation, Rotation .......................... 21
2.3 Reynolds Transport Theorem ............................... 25
References .................................................. 27
3 Differential Balances in Turbomachinery ................ 29
3.1 Mass Flow Balance in Stationary Frame of Reference ............ 29
3.1.1 Incompressibility Condition ........................... 31
3.2 Differential Momentum Balance in Stationary Frame of Reference . 32
3.2.1 Relationship between Stress Tensor and Deformation Tensor 34
3.2.2 Navier-Stokes Equation of Motion...................... 36
3.2.3 Special Case: Euler Equation of Motion ................. 38
3.3 Some Discussions on Navier-Stokes Equations ................. 41
3.4 Energy Balance in Stationary Frame of Reference............... 42
3.4.1 Mechanical Energy.................................. 42
3.4.2 Thermal Energy Balance ............................. 45
XII Contents
3.4.3 Total Energy ........................................ 48
3.4.4 Entropy Balance ..................................... 49
3.5 Differential Balances in Rotating Frame of Reference ............ 50
3.5.1 Velocity and Acceleration in Rotating Frame ............. 50
3.5.2 Continuity Equation in Rotating Frame of Reference ....... 52
3.5.3 Equation of Motion in Rotating Frame of Reference ........ 53
3.5.4 Energy Equation in Rotating Frame of Reference .......... 55
References .................................................. 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 Energy Transfer in Turbomachinery Stages ................... 123
5.2 Energy Transfer in Relative Systems ........................ 124
5.3 General Treatment of Turbine and Compressor Stages .......... 125
5.4 Dimensionless Stage Parameters ........................... 129
5.5 Relation between Degree of Reaction and Blade Height ........ 131
5.6 Effect of Degree of Reaction on the Stage Configuration ........ 134
Contents XIII
5.7 Effect of Stage Load Coefficient on Stage Power .............. 136
5.8 Unified Description of a Turbomachinery Stage ............... 137
5.8.1 Unified Description of Stage with Constant Mean Diameter . 137
5.8.2 Generalized Dimensionless Stage Parameters ............ 138
5.9 Special Cases .......................................... 140
5.9.1 Case 1, Constant Mean Diameter ...................... 141
5.9.2 Case 2, Constant Mean Diameter and
Meridional Velocity Ratio ........................... 141
5.10 Increase of Stage Load Coefficient, Discussion ................ 140
References ................................................. 144
6 Turbine and Compressor Cascade Flow Forces .......... 145
6.1 Blade Force in an Inviscid Flow Field ....................... 145
6.2 Blade Forces in a Viscous Flow Field ....................... 150
6.3 The Effect of Solidity on Blade Profile Losses ................ 156
6.4 Relationship Between Profile Loss Coefficient and Drag ........ 156
6.5 Optimum Solidity ....................................... 158
6.5.1 Optimum Solidity, by Pfeil........................... 158
6.5.2 Optimum Solidity, by Zweifel ........................ 158
6.6 Generalized Lift-Solidity Coefficient ........................ 162
6.6.1 Lift-Solidity Coefficient for Turbine Stator .............. 164
6.6.2 Turbine Rotor ..................................... 168
References ................................................. 171
II Turbomachinery Losses, Efficiencies, Blades
7 Losses in Turbine and Compressor Cascades ............ 175
7.1 Turbine Profile Loss ..................................... 176
7.2 Viscous Flow in Compressor Cascade ....................... 178
7.2.1 Calculation of Viscous Flows......................... 178
7.2.2. Boundary Layer Thicknesses ......................... 179
7.2.3 Boundary Layer Integral Equation ..................... 181
7.2.4 Application of Boundary Layer Theory to Compressor Blades 182
7.2.5 Effect of Reynolds Number .......................... 186
7.2.6 Stage Profile Losses ................................ 186
7.3 Trailing Edge Thickness Losses ............................ 186
7.4 Losses Due to Secondary Flows............................ 192
7.4.1 Vortex Induced Velocity Field, Law of Bio-Savart ........ 194
7.4.2 Calculation of Tip Clearance Secondary Flow Losses ...... 197
7.4.3 Calculation of Endwall Secondary Flow Losses .......... 200
7.5 Flow Losses in Shrouded Blades ........................... 204
7.5.1 Losses Due to Leakage Flow in Shrouds ................ 204
7.6 Exit Loss ............................................. 211
