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dissertation entitled

FLOW ANALYSIS AND MODELING OF CENTRIFUGAL

Ph.D

COMPRESSOR IMPELLERS

presented by
ZEYAD A. AL-SUHAIBANI

has been accepted towards fulfillment
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2/05 eye—incm""eom_.mms"

FLOW ANALYSIS AND MODELING .
OF CENTRIFUGAL COMPRESSOR IMPELLERS

By

Zeyad A. Al-Suhaibani

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering
2005

ABSTRACT

FLOW ANALYSIS AND MODELING OF CENTRIFUGAL COMPRESSOR
IMPELLERS

By
Zeyad A. Al-Suhaibani

Centrifugal compressors performance is heavily affected by impeller design because
of the nature of the flow in the impeller. Improvement and modiﬁcation of a centrifugal
impeller is a major task for designers in the turbomachinery ﬁeld.

A series of impellers’ experimental test data was used for the comparison of different
conﬁgurations and rotational speeds. Four different centrifugal impellers and their
modified versions were tested at Solar Turbines. These impellers cover a wide area of
speciﬁc speeds from 58 to about 135. The comparative study of these test results
indicated noticeable compressor stage and impeller performance differences between the
original and the modiﬁed impellers’ conﬁgurations. Details are discussed to give a
broader understanding for the effect cutting an impeller on its performance as well as the
compressor performance.

Comparing the compressor stage and impeller performance of four different rotational
speeds has been also done. Since the impellers cover a wide range of the speciﬁc speeds,
the effect of speciﬁc speed on the impeller and stage performance was also another task
of this study.

CFX TASCflow, which is a CFD package, was used to simulate the ﬂow. The primary
grid generation was done using a FORTRAN code. The CFD and 1-D code results were
used to compare with the experimental result. More details will be given throughout this

dissertation to help understand the flow in such device in order to improve it.

With Love to My Mother and Father

iii

ACKNOWLEDGEMENTS

At ﬁrst I thank ALLAH (SWT) my Lord who enabled me to complete this work
and I ask him to always help me in participating to make the future of humanity much
better. Then I want to express my thanks to the people and organization that have helped
me throughout my study. The ﬁrst person whom I thank is Professor Abraham Engeda
who was always guiding me in my course work as well as my research. He was always
helping, supporting, guiding, and encouraging me throughout my study in the Department
of Mechanical Engineering at Michigan State University. I am also proud of the other
members of my guidance committee, Professor Chichia Chiu, Professor Tom Shih, and
Professor Craig Somerton for their help and effort. Special thanks go to Dr. Somerton
whom I bothered the most with my questions and he was always doing his best to help
and give valuable advices.

I want also to thank Mike Cave and Tim David from Solar Turbines (a leading
company in compressor and Turbine industry) who showed a great help during this study.
I would also like to specially acknowledge Mr. Craig Gunn at Michigan State University
for his help in preparing this dissertation. Another close friend who helped me so much in
preparing this dissertation is Mahmoodul Haq. I also appreciate the assistance and great
friendship from Dr. Faisal Mahroogi, Dr. Yinghui Dai, Dr. Donghui Zhang, and all other
students in Turbomachinery lab during my Ph.D. program at Michigan State University.
Doug Neal is another student at the Department of Mechanical Engineering who should
be acknowledged for his help in dealing with CFX installation. I should also not forget to

thank the other faculty and staff at my beloved Department (the Department of

iv

Mechanical Engineering at Michigan State University) which I hope I will keep a good
contact with in the future.

After thanking the people who showed a great help academically I want to thank
who was behind all the ﬁnancial support that is the Saudi Arabia government. A great
portion of that thanks goes to the Department of Mechanical Engineering at King Saud
University in Riyadh. They took care of all ﬁnancial issues helping me to concentrate on
my study.

The last but most important for me is thanking my family members whom I gave
hard time during my study because of the effort that I have to put to ﬁnish my studies.
My mother Aishah Abraham Al-Suhaibani and My father AbdulRahman Muhammad Al-
Suhaibani are the ones who have kept me patient all the time and have encouraged me to
ﬁnish this journey of learning. I also appreciate their way of educating me which has
helped me a great deal throughout my academic life. I must not forget my strong
supporter, my wife, Reem Abdullah Al-Suhaibani who was patient helping me in every
way she could. She stood by me and was a very trusted and compassionate friend in all
my life. Her family was also supportive to me to enable us completing this degree. Last
but not least I am grateful to my children AbdulRahman, Abdullah, Abeer, Ghadeer, and
Tariq for the joy they have brought into our life and their love that has been a great

source that has motivated me.

TABLE OF CONTENTS

TABLE OF CONTENTS ................................................................................................... vi
LIST OF FIGURES ......................................................................................................... viii
LIST OF TABLES ........................................................................................................... xiii
NOMENCLATURE USED ............................................................................................. xiv
Chapter 1 INTRODUCTION .......................................................................................... l
1 . 1 Turbomachinery .................................................................................................. l
l .2 Compressor ......................................................................................................... 1
1.2.1 Background ................................................................................................. 2
1.2.2 Compressor Classiﬁcation .......................................................................... 3
1.3 Demands on this Study ....................................................................................... 8
1.4 Objective and Organization of this Study ........................................................... 9
Chapter 2 CENTRIFUGAL COMPRESSOR AND IMPELLER DESIGN
CON SIDERATIONS ........................................................................................................ 1 1
2.1 Centrifugal Compressor .................................................................................... 1 1
2.1.1 Development Need and Applications ....................................................... 11
2.1.2 Centrifugal Compressor Design ................................................................ 15
2.1.3 Compression Process ................................................................................ 24
2.1.4 Centrifugal Compressor Components ....................................................... 26
2.1.4.1 Inlet: ...................................................................................................... 27
2.1.4.2 Impeller (General): ................................................................................ 28
2.1.4.3 Diffuser: ................................................................................................ 29
2.1.4.4 Return Channel: .................................................................................... 32
2.1.4.5 Volute: ................................................................................................... 32
2.2 Centrifugal Impeller .......................................................................................... 33
2.2.1 Inducer ...................................................................................................... 33
2.2.2 Dimensionless Parameters ........................................................................ 36
2.2.3 Incidence ................................................................................................... 37
2.2.4 Impeller Thermoﬂuids Relations and Blade Geometry ............................ 40
2.2.5 Efﬁciency, head Coefﬁcient and Flow Coefﬁcient .................................. 43
2.2.6 Head Rise .................................................................................................. 47
2.2.7 Operating Range between Surge and Choke ............................................ 48
2.2.8 Slip Factor ................................................................................................. 51
Chapter 3 CFD ANALYSIS .......................................................................................... 55
3.1 Need and Development of CFD ........................................................................ 55
3.1.1 CFD and Industry ...................................................................................... 57
3.2 CFD Mathematics ............................................................................................. 58
3.3 Engineering Approaches ................................................................................... 61
3.4 CFD in Current Practice for Centrifugal Compressor Design .......................... 63
3.4.1 CFD Model ............................................................................................... 64
3.4.2 CFD Process Components ........................................................................ 66
3.4.2.1 The Pre-processor: ................................................................................ 67
3.4.2.2 The Solver: ............................................................................................ 67
3.4.2.3 The Post-processor: ............................................................................... 68

vi

3.5 The Simulation Process ..................................................................................... 69

3.5.1 Grid Generation ........................................................................................ 71
3.5.1.1 Geometry from Drawing: ...................................................................... 71
3.5.1.2 CFX GRID: ........................................................................................... 73
3.5.1.3 CFX creates its Grid: ............................................................................ 77
3.5.1.4 Grid Quality: ......................................................................................... 79

3.5.2 Turbulence Modeling ................................................................................ 81

3.5.3 Boundary Conditions Used in Simulation ................................................ 83

3.5.4 Use of Initial Conditions ........................................................................... 87

3.5.5 Control Parameters .................................................................................... 87

3.5.6 Solver ........................................................................................................ 87

Chapter 4 EXPERIMENTAL DATA ANALYSIS ....................................................... 88
4.1 Background ....................................................................................................... 88
4.2 Effect of Speciﬁc Speed on Impeller Characteristics ....................................... 90
4.3 Rotational Speed Effects ................................................................................... 97
4.4 Effect of Trimming an Impeller ...................................................................... 106

Chapter 5 CFD RESULTS AND VALIDATION ....................................................... 112
5.1 Background ..................................................................................................... 1 12
5.2 Primary Check ................................................................................................ 113
5.3 CFD Validation ............................................................................................... 113

Chapter 6 CONCLUSIONS ........................................................................................ 125

APPENDIX ..................................................................................................................... 127

BIBLIOGRAPHY ........................................................................................................... 129

vii

LIST OF FIGURES

Figure 1-1 Classiﬁcation of Compressors ........................................................................... 4
Figure 2-1 Application areas of turbocompressors ........................................................... 12
Figure 2-2 A Single Stage Centrifugal Compressor .......................................................... 14
Figure 2-3 A Multistage Centrifugal Compressor ............................................................. 14
Figure 2-4 Cordier Line (Adapted from Balje 1981) ........................................................ 16
Figure 2-5 Selection of Compressor Type with Speciﬁc Speed (Adapted from Balje 19818)
................................................................................................................................... 1
Figure 2-6 P-v and T-s for a compression process ............................................................ 24
Figure 2-7 Atypical Example of h-s diagram for a centrifugal compressor stage ............ 25
Figure 2-8 Total pressure, temperature change and Entropy rise (loss) in centrifugal
compressors ............................................................................................................... 26
Figure 2-9 Ideal and actual head characteristic of a compressor ...................................... 27
Figure 2-10 Key Components of centrifugal compressors ............................................... 29
Figure 2-11 T-s diagram of diffusion process ................................................................... 30
Figure 2-12 Flow path and Velocity triangle in a vaneless diffuser ................................. 31
Figure 2-13 Change of the relative Mach number with the inducer eye tip diameter ....... 35
Figure 2-14 Incidence Variations Upon Mass Flow Change and Boundary Layer
Separation .................................................................................................................. 38
Figure 2-15 Inlet and Exit Velocity Triangles of Impeller ................................................ 40
Figure 2-16 Effect of Exit Blade Angle on the Head Rise and the Flow Range ............... 42
Figure 2-17 Velocity Triangles Velocity Proﬁle for Different Blade Shapes ................... 42
Figure 2-18 Application Areas of Typical Industrial Centrifugal Compressors ............... 45
Figure 2-19 Representation of a compression process in a T-S diagram .......................... 47

viii

Figure 2-20 A sample performance curve. of a centrifugal compressor ............................ 49
Figure 2-21 Slip effect on velocity triangle at impeller exit for back-swept blade ........... 51
Figure 2-22 Inﬂuence of the blade number and the exit blade angle on the slip factor.... 53
Figure 3-1 Flow chart - Process of the CFD Simulation ................................................... 70
Figure 3-2 Flow chart for initial geometry ﬁles generated by FORTRAN code .............. 73

Figure 3-3 Eight-noded hexahedral ﬂux element. (Adapted from CFX- TASC ﬂow) ..... 75

Figure 3-4 A Flux Element Divided into Octants (Adapted from CFX Manual) ............. 76
Figure 3-5 CFX-TASCgrid Data Flow Diagram .............................................................. 78
Figure 3-6 Blade to Blade grid that is used in the simulation ........................................... 80
Figure 3-7 Inlet to Exit Grid that is Used in the Simulation ............................................ 81
Figure 3-8 Inlet Boundary Condition ................................................................................ 84
Figure 3-9 Outlet Boundary Condition ............................................................................. 84
Figure 3-10 Wall (stationary) Boundary Condition .......................................................... 85
Figure 3-11 Wall (moving) Boundary Condition .............................................................. 85
Figure 3-12 Periodic Boundary Condition ........................................................................ 86
Figure 3-13 Symmetric Boundary Condition .................................................................... 86
Figure 4-1 The Tested Impellers Before and After the Modiﬁcation ............................... 88
Figure 4-2 Variation Of The Stage Efﬁciency For The Wide Impellers .......................... 90
Figure 4-3 Variation Of The Stage Efﬁciency For The Narrow Impellers ....................... 91
Figure 4-4 Stage Head Coefﬁcient Comparison For The Wide Impellers ........................ 91
Figure 4-5 Stage Head Coefﬁcient Comparison For The Narrow Impellers .................... 92
Figure 4-6 Work Factor Variations for The Wide Impellers ............................................ 93
Figure 4—7 Work factor variations for the narrow impellers ............................................. 93

ix

Figure 4-8 DeltaT/T 1 variations for the wide impellers ................................................... 94

Figure 4-9 DeltaT/T 1 variations for the narrow impellers ................................................ 94
Figure 4-10 The change of the relative diffusion (W2/W ltip) for the wide impellers ..... 95
Figure 4-11 The Change of the Relative Diffusion (W 2/W ltip) For the Narrow Impellers
................................................................................................................................... 95
Figure 4-12 The Change of the Impeller Exit Flow Angle (Alpha2) for the wide impellers
................................................................................................................................... 96
Figure 4-13 The Change of the Impeller Exit Flow Angle (Alpha2) for the narrow
impellers .................................................................................................................... 97
Figure 4-14 Variation of the Stage Efﬁciency for B Wide at Variable Rotational Speeds
................................................................................................................................... 98
Figure 4-15 Variation of the Stage Efﬁciency for B Narrow at Variable Rotational Speeds
................................................................................................................................... 98
Figure 4-16 Variation of the Stage Efﬁciency for E Wide at Variable Rotational Speeds
................................................................................................................................... 99
Figure 4-17 Variation of the Stage Efﬁciency for E Narrow at Variable Rotational Speeds
................................................................................................................................... 99
Figure 4-18 Variation of the Impeller Efﬁciency for B Wide at Variable Rotational
‘ Speeds ...................................................................................................................... 100
Figure 4-19 Variation of the Impeller Efﬁciency for B Narrow at Variable Rotational
Speeds ...................................................................................................................... 100
Figure 4-20 Variation of the Impeller Efﬁciency for N Wide at Variable Rotational
Speeds ...................................................................................................................... 101
Figure 4-21 Variation of the Impeller Efﬁciency for N Narrow at Variable Rotational
Speeds ...................................................................................................................... 101

Figure 4—22 Variation of the head coefﬁcient for B wide at variable rotational speeds. 102

Figure 4-23 Variation Of The Head Coefﬁcient For B Narrow At Variable Rotational
Speeds ...................................................................................................................... 102

Figure 4-24 Variation of The Head Coefﬁcient for E Wide at Variable Rotational Speeds
................................................................................................................................. 103

Figure 4—25 Variation Of The Head Coefﬁcient For E Narrow At Variable Rotational
Speeds ...................................................................................................................... 103

Figure 4—26 Variation of the Work Factor for B Wide at Variable Rotational Speeds 104

Figure 4-27 Variation Of The Work Factor For B Narrow At Variable Rotational Speeds
................................................................................................................................. 104

Figure 4-28 Variation of the work factor for E wide at Variable Rotational Speeds ...... 105

Figure 4-29 Variation of the work factor for E narrow at Variable Rotational Speeds .. 105

Figure 4-30 Impellers Efﬁciencies Before and After Trimming ..................................... 106
Figure 4-31 Stage Efficiencies for All Impellers before and After Trimming ................ 107
Figure 4-32 Impellers Head Coefﬁcients Before and After Trimming ........................... 108

Figure 4-33 Stage Head Coefﬁcients for All Impellers Before and After Trimming ..... 108
Figure 4-34 Work Factors for All Impellers Before and After Trimming ...................... 109
Figure 4-35 Variation of Deltat/T 1 For All Impellers Before and After Trimming ....... 109
Figure 4-36 Variation of erms/W 2 for All Impellers Before and After Trimming ..... 110
Figure 4-37 Variation of W2/W1tip for All Impellers Before and After Trimming ....... 110

Figure 4-38 Variation of The Impellers Exit Flow Angles Before and After Trimming 111

Figure 5-1 Experimental, CFD, and 1-D Stage Efﬁciency for B Wide .......................... 114
Figure 5-2 Experimental, CFD, and 1-D Stage Efﬁciency for B Narrow ....................... 115
Figure 5-3 Experimental and CFD Stage Efﬁciency for E Wide .................................... 115
Figure 5-4 Experimental and CFD stage efﬁciency for E narrow .................................. 116
Figure 5-5 Experimental, CFD, and 1-D Impeller Efﬁciency for B Wide ...................... 117
Figure 5-6 Experimental, CFD, and 1-D impeller efﬁciency for B narrow .................... 117
Figure 5-7 Experimental and CFD Impeller Efﬁciency for E Wide ............................... 118
Figure 5-8 Experimental and CFD impeller efﬁciency for E narrow ............................. 118

xi

Figure 5-9 Experimental, CFD, and 1-D head coefﬁcient for B wide ............................ 119

Figure 5-10 Experimental, CFD, and 1-D head coefﬁcient for B narrow ...................... 119
Figure 5-11 Experimental and CFD head coefﬁcient for E wide ................................... 120
Figure 5-12 Experimental and CFD head coefﬁcient for E narrow ................................ 120
Figure 5-13 Experimental, CFD, and 1-D Work Factor for B Wide .............................. 121
Figure 5-14 Experimental, CFD, and 1-D Work Factor for B Narrow ........................... 121
Figure 5-15 Experimental and CFD Work Factor for E Wide ........................................ 122
Figure 5-16 Experimental and CFD Work Factor for E Narrow .................................... 122

Figure 5-17 Experimental, CFD, and l-D Impeller Exit Flow Angle for B Wide .......... 123
Figure 5-18 Experimental, CFD, and l-D Impeller Exit Flow Angle for B Narrow ...... 123

Figure 5-19 Experimental and CFD Prediction of Impeller Exit Flow Angle for E Wide
................................................................................................................................. 124

Figure 5-20 Experimental and CFD Prediction of Impeller Exit Flow Angle for E Narrow
................................................................................................................................. 124

xii

LIST OF TABLES
Table 3-1 Comparison of approaches (adapted from Tannehill) ..................................... 62

Table 4—1 Major stages dimensions in dimensionless form .............................................. 89

xiii

AR

AS

0"

<

O

NOMENCLATURE USED

Area

Area ratio

Aspect ratio

Speed of sound

Blockage

Passage width from hub to shroud
Absolute velocity

Speciﬁc heat at constant pressure or Pressure recovery coefﬁcient
Speciﬁc heat at constant volume

Constant or speed of sound or chord length
Diameter or Dean number

Diameter

Head

enthalpy

Incidence angle

Equivalent roughness

Flow passage length or blade loading
Mach number

Impeller tip Mach number at Impeller exit
Impeller tip Mach number at Impeller inlet (shroud)
Mass ﬂow rate

Impeller rotating speed [rpm]

xiv

PS

Re

SS

N~<€=<<

N

GREEK

Pressure

Pressure side

Volume ﬂow rate
Radius of cross-section

Reynolds number (= umd / v)

Radial location or radius of curvature of curved section
Suction side

Space or pitch

Temperature

Throat

Blade peripheral speed at impeller inlet

Blade peripheral speed at impeller tip

Average velocity of ﬂow regime

 

Speed (=x/u2 + v2 + wz)
Normal velocity to cross-section
Relative velocity or width

Total pressure loss coefﬁcient
Vane number

Axial coordinate

Flow angle with reference to radial or axial direction
Relative ﬂow angle with reference to radial direction

Flow coefﬁcient

XV

7‘s

7‘:

SUBSCRIPT

3V6

Speciﬁc heat ratio
Efﬁciency

Work coefﬁcient or dynamic viscosity
Kinematic viscosity
Static pressure ratio
Total pressure ratio
Angle

Density

Slip factor

Total temperature ratio
rotating speed [rad/s]

Head coefﬁcient

Station at upstream or the Total value
Station at pipe exit or impeller inlet
Station at impeller exit

Station at diffuser leading edge
Station at diffuser trailing edge
Station at volute inlet

Station at compressor discharge

Actual

Averaged quantity

xvi

DP

tip
[-1

1-8

Design point.

Friction

ideal

Meridional direction
Number of vanes or normal direction
Radial component
Reference

Static or isentropic
Total

Station 2 (impeller tip)
Total-to-total
Total-to-static
Tangential component
Axial component

Tangential component

xvii

Chapter 1 INTRODUCTION
1.1 Turbomachinery

Turbomachinery is the term that refers to rotating machinery that either extracts or
adds energy to ﬂuids by the dynamic action of one or more moving blade rows, called
stages. The rotor changes the kinetic energy, stagnation enthalpy, and stagnation pressure
of the ﬂuid. This deﬁnition covers everything from classic fans to turbopumps of the
space shuttle main engines. Turbines, Pumps, Compressors and Fans are major categories
of turbomachinery. Turbomachines are major components in many applications such as
aerospace and marine vehicle applications; energy applications; land propulsion systems;
hydraulic; gas and steam turbines; industrial pipeline; and processing equipment
applications such as gas, petroleum, and water pumping plants. Moreover,
turbomachinery has a wide variety of other applications such as heart-assist pumps,

industrial compressors, and refrigeration plants.

1.2 Compressor

A compressor is a machine that compresses air (or a gas) from a lower intake
pressure to a higher exhaust pressure through reduction in volume. A compressor needs a
driving unit that provides power to operate it and that may be a gasoline engine, diesel
engine, electric motor, or a turbine. The major ﬂuid mechanics problem in
turbomachinery shows up in the design of compressors because the ﬂow is going from
lower to higher pressure. The design of a turbine, with its ﬂow going from higher to

lower pressure, always works and with fair care it would have a high efﬁciency.

Compressors are found in one stage to multi stages when it is needed. The
compression ratio varies depending on the type and application of the compressor. Since
most turbines need a compressor to provide a compressed gas, the compressor

development is then more demanding.

1.2.1 Background

The major ﬂuid-mechanics problem in turbomachinery encountered during the
design of compressors. The design of a turbine almost always works because it operates
from a high to a low pressure. Actually, if the design of a turbine is given reasonable care
it works at high efﬁciency. On the other hand a compressor, particularly the axial-ﬂow
type, may not build a pressure at all. Until almost the beginning of the twentieth century,
the isentropic efﬁciencies of the compressors did not exceed 50 percent.

After Papin's centrifugal blowers, which were invented in 1705, there seems to be
not much development until 1884, when Charles Parsons patented an axial-ﬂow
compressor. In 1887 be constructed and sold a three-stage centrifugal compressor that
was used for ship ventilation. Ten years later he returned to experiments on axial-ﬂow
compressors and made an eighty-one-stage machine in 1899 that reached about 70
percent efﬁciency. This number of stages was very high and could be an all-time record.
At the beginning of the twentieth century around 1907 his company had made or had on
order forty-one stage, axial-ﬂow compressors, but they were plagued by poor
aerodynamics, and consequently he stopped production in 1908. Based on some of their
publications, Parsons seems to use far too high a spacing/chord ratio for the rotor-blade
settings, and all blade rows would likely be stalled over much of the operating range.

Some years later, Parsons returned to making radial-ﬂow compressors.

Auguste Rateau is another major pioneer who was working on compressors at
same time. In 1892 he published a major paper on turboblowers. About ten years later he
designed a turbocompressor to give a pressureratio of 1.5 at 12,000 rpm; but when
tested, it gave a low isentropic efﬁciency of only 56 percent. He did not stop there but
kept designing and building compressors of increasing pressure ratio and mass ﬂow, and
gradually their efﬁciencies increased.

The development of compressor design did not stop there but rather kept going

because there is still need for it.

1.2.2 Compressor Classiﬁcation
A compressor uses the driving power to increase the pressure of a ﬂuid. The gas
enters a compressor at relatively low pressure and exits at a higher pressure. In the
industry there is a very wide range of service requirements so, there are many different
types of compressors that are used. Before selecting a compressor type for an application,
some basic information related to its performance requirements should be at hand. This
includes some parameters (such as pressure ratio, ﬂow rate and cost consideration) and
could include other special characteristics like space limitation and nose. One can then
consider the type of machine needed from a range of types of compressors that are
available.
In general, there are two basic compressor types: positive displacement
compressors and continuous ﬂow compressors. Each type is then classiﬁed further as

shown in the following (Figure 1-1).

 

 

Compressors

 

 

 

[ Positive Displacement ] [ Continuous Flow ]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Reciprtlicau'ng] [ Rotary ] L Ejector ] [ Dynamic ]

 

 

 

 

 

[ Double-Acting ] [Single-Acting J

 

 

 

 

 

 

 

 

{Liquid Piston ]

 

Straight Lobe? [ Sliding Vane] [ Helical Lobe J

 

 

 

 

 

 

 

 

 

 

 

H_ll[ Axial ] [Reemmﬁvej LMixed]

Turbocompressors

 

 

 

 

 

 

 

 

Figure 1-1 Classiﬁcation of Compressors

In the positive displacement type, a given quantity of gas is trapped in a
compression chamber and the occupied volume is mechanically reduced, causing a
corresponding rise in the gas pressure before it is discharged. At constant speed, the air
ﬂow remains basically constant while the discharge pressure varies. These compressors
are available in two types: reciprocating and rotary. A reciprocating compressor can be
thought of like a bicycle pump. A piston reduces the volume in the cylinder occupied by
the gas, which results in compressing it to a higher pressure. This piston is driven through
a crankshaft and connecting rod by an electric motor. In single-acting compressors a
compression stroke is in one direction only, while the double-acting ones allow a
compression stroke as the piston moves in both directions. Large industrial reciprocating
air compressors are double-acting and they use water for cooling. Multi-stage double-
acting compressors are very efﬁcient compressors compared to rotary units and are
typically larger, noisier, and more costly. The size of reciprocating compressors varies
from less than 1 hp to more than 600 hp.

The other type of the positive displacement compressors is the rotary compressor.
Recently, rotary compressors have gained more popularity in American industry. The
compression process in rotary compressors is caused by the positive action of rotating
elements. Normally, they are used in sizes from 30 up to 200 hp. The helical lobe, also
known as rotary screw, is the most popular type of rotary compressor. Male and female
screw-rotors mesh, trap air, and reduce the volume of the air along the rotors, causing a
pressure rise of the air at the discharge point. Rotary screw compressors have the
advantage of low initial cost, compact size, and low weight and are easy to maintain. The

other types of rotary compressors are less popular, including sliding vane, liquid piston,

and straight lobe. Both reciprocating and rotary compressors will not be further discussed
in this study.

The continuous ﬂow compressors are classiﬁed into ejector and dynamic type
compressors. Ejectors are designed to convert the pressure energy of a moving ﬂuid to
velocity energy to entrain the suction ﬂuid. The mixed ﬂuids will then be recompressed
by converting velocity energy back to pressure energy. This is based on the theory that a
properly designed nozzle followed by a properly designed throat will economically make
use of the high pressure ﬂuid to compress from a low pressure region to a higher
pressure. This change from pressure head to velocity head is the basis of the jet vacuum
principle. Ejectors are reliable and economical to produce vacuum. Some advantages of
the ejector design are its low initial cost, lack of moving parts, and simplicity of
operation.

The dynamic compressor is the second type of continuous ﬂow compressors in
which air or gas is compressed by the mechanical action of rotating vanes or impellers
imparting velocity and pressure to the ﬂowing medium. In other words they raise the
pressure of the air by accelerating the ﬂuid to a high velocity then convert the energy
from the velocity of the air to pressure. The dynamic compressors are subdivided into
regenerative, axial, mixed, and centrifugal (or called radial) type compressors. These four
categories are also called “Turbocompressors”.

In a regenerative type compressor the gas moves helically in the casing and
reenters the impeller many times in its peripheral path from inlet to discharge. The main
advantage of regenerative turbomachines is the ability to generate high head but at low

ﬂow rates. At certain tip speed the regenerative compressors have the ability to develop

much higher heads than any other type of compressors. Other advantage of these types of
compressors is the advantage of no surge or stall instability. (Moffat 1987)

In axial compressors the ﬂow comes and leaves the rotor axially. This type of
compressor is made up from rows of airfoil cascades. One half of these rows are called
rotors that are connected to the central shaft of the compressor. The other rows are called
stators, which are ﬁxed to the outer casing and do not rotate. Stators are meant to increase
pressure while keeping the ﬂow from spiraling around the axis by bringing the ﬂow back
parallel to the axis. For many aircraft applications it is preferred to have axial
compressors. In an average single-stage centrifugal compressor the pressure ratio is about
4. In a similar single-stage axial compressor the pressure ratio is 1.2. Axial compressors
are preferred over others is some cases because it is easier and more efﬁcient to link
several stages together and produce a multistage axial compressor. For a typical
multistage axial compressor of 8 stages the pressure ratio is about 4. In a multistage
centrifugal compressor the ﬂow needs to be ducted back to the axis at each stage, which
is less efﬁcient for aircraft because greater cross-sectional area is needed. [NASA web]

In centrifugal compressors the ﬂow comes axially and leaves radially
(perpendicular to the axis of rotation). The centrifugal compressors give a high pressure
ratio per stage compared to the axial compressors. This type will be discussed in more
details in the next section.

In the mixed ﬂow compressors the axial and radial components of the velocity are
considerably present at the rotor outlet. They are designed to have an advantage of both

axial and centrifugal compressors.

Even though fans and blowers create a pressure difference like compressors, they
normally are not called compressors. For fans, a typical Pressure Ratio (1: is between 1.0
and 1.1. For blowers it is slightly higher, which is between 1.1 and 2.0. When the

pressure ratio is higher than 2.0, it is normally called a compressor. (Engeda 2004)

1.3 Demands on this Study

A centrifugal compressor is a commonly used machine in petroleum and gas
industry. Due to the limitations of the operating range of many centrifugal compressors,
in many cases the compressor needs to be shutdown to prevent damage. Shutting down a
compressor even for a short time will cost millions of dollars and may cause the whole
plant to be shutdown. Due to the current oil prices, the loss will be even more. Therefore,
the importance of having wide operating range centrifugal compressors for such
applications can not be ignored. The centrifugal compressor manufacturers attempt to
offer a standard line of compressors, which would provide a large ﬂow range for a variety
of inlet conditions and process gases. Modifying an existing design is another common
practice that is preferred by the centrifugal compressor manufacturers rather than a new
design. The operating range of centrifugal compressors depends heavily on their impeller
design. Due to these reasons the study of the effect of centrifugal impeller design on the
ﬂow range is essential for the development of these machines to meet the current gas

industry requirement.

1.4 Objective and Organization of this Study

The objective of this study is to analyze and model the ﬂow of centrifugal
compressor impellers. The compressor industry makes money by one of three ways:
producing new machines, solving a current problem, or modifying a previous design to
meet customer satisfaction. This work is related to the third reason where centrifugal
compressor impellers are modiﬁed so the compressor could have a better performance or
wider operating range. The following steps explain how the goals of this work were
systematically achieved.

1. Experimental testing of the compressor with four original and four modiﬁed
impellers has been done at SOLAR TURBINES Laboratories. The impellers’
geometries drawings and experimental data were provided to the Turbomachinery
Lab at MSU to be analyzed and studied.

2. The experimental data for all the eight impellers, which cover a wide area of
speciﬁc speeds, were analyzed and studied as part of this work.

3. The experimental data were also compared with results obtained from a One-
dimensional (l-D) code done at the Turbomachinery Lab at MSU.

4. A CFD study was added to model four extreme impellers that have the lowest and
highest speciﬁc speed for both the original and the modiﬁed.

i. The grid from the available drawings for the four impellers was
generated.
ii. Then the numerical simulation was done using CFX-TASCFlow

and results were obtained.

iii. The results were then compared with the experimental data and
one-dimensional results and then discussed.

Based on the previous steps, this work was carried out and presented in this
dissertation. It consists of six chapters starting with the general introduction to
turbomachinery and compressors in Chapter 1. Chapter 2 presents a general and
theoretical discussion of a centrifugal compressor, its components, and the centrifugal
impeller in particular. This Chapter, also, includes some of the previous research work
related with the current study. Chapter 3 deals with CFD analysis in general and gives
more details about the CFX-TASCFlow package used in this work. It also discusses the
major steps in CFD analysis, which are the pro-processing, the solver, and the post-
processing. Grid generation, boundary conditions, and the turbulence modeling are
discussed in more details in this chapter. Chapters 4 mainly presents and discusses the
experimental data for all the impellers and compares them with the results obtained from
the 1-D analyses. Chapter 5 shows the CFD results and compare it with the experimental
data. Chapter 6 draws the conclusions focusing mainly on the effect of the speciﬁc speed,
the rotational speed and the cut the impeller on the compressor stage performance and the

operating range.

10

Chapter 2 CENTRIFUGAL COMPRESSOR AND
IMPELLER DESIGN CONSIDERATIONS

2.1 Centrifugal Compressor

2.1.1 Development Need and Applications

Centrifugal Compressors have a very wide area of applications. Because of the
advantages of the higher-single stage pressure ratio and the wider stable operating range
over axial compressors, the centrifugal compressor has been widely used. Centrifugal
compressors are found in small gas turbine engines, turbochargers, and refrigeration
systems and are used extensively in the petrochemical and process industry. Although
they are not used in high-thrust devices most low-to-moderate thrust turboshaft,
turboprop, and turbojet engines have centrifugal compressors. Sometimes in aircraft
engines the centrifugal compressor is combined with an axial one to achieve better
performance. Industrial facilities use compressed air for too many operations. Almost
every industrial facility has at least a centrifugal compressor. Their applications go even
beyond that use, like health applications and much more. Although all the previous
applications are important, the driving force for centrifugal compressors development is
their involvement in airspace and power generation applications. Therefore, the demand
on efﬁcient and reliable centrifugal compressors is increasing because of their association
with many industries, which are on demand today.

In the previous applications and much more compressors can be found in small .
scale as well as large ones. They also differ in the pressure ratio that they built and the

driving device. So, it is clear that for a certain application there will be a certain

11

centrifugal compressor that will be preferred over others. Looking at the application type
carefully the centrifugal compressor type can then be selected.

Compared to the other types of compressors, centrifugal compressors are reliable,
compact, and robust; they have better resistance to foreign object damage and are less
affected by performance degradation due to fouling. Centrifugal compressors are mainly
used in areas where the requirements are high pressure ratio and low to moderate mass
ﬂow (Figure 2-1). They also have the advantage of a wide stable operating range
compared to some other types of compressors. In centrifugal compressors the gas ﬂows

in a radial direction perpendicular to the axis of rotation, which makes them more

 

 

compact.
2,0-
15 Flow Impeller
. or
High Flow Coefﬁcient
1.0 , Radial Impeller
3 0.8

0.6

Common Application
0'], Area of the
Radial Impeller

Axial Flow Compressor

0.2

0.1
0.01. 0.02 0.04 am 008 01 0.2 0.4 0.6 0.8 1.0

'i’

 

 

 

Figure 2-1 Application areas of turbocompressors

12

Since the centrifugal compressor ﬁnds a wide variety of application each
application places its own demands on the design of the compressor. For these reasons it
is a common practice by the centrifugal compressor manufacturers to offer a standard line
of compressors that would provide large ﬂow range for a variety of inlet conditions and

process gases.

The main four components in centrifugal compressors are shown in Figure 2-2, as
follows: (1) a stationary inlet casing, (2) a rotating impeller, (3) a stationary diffuser of
the vaneless or vaned type, and (4) the collector or volute. In a multistage centrifugal

compressor the return channel is another essential component.

The ﬂuid is drawn in through the inlet casing into the eye of the impeller of the
ﬁrst stage (in multistage compressor) parallel to the axis of rotation. In order to add
angular momentum, the impeller whirls the ﬂuid outwards and turns it into a direction
perpendicular to the rotation axis. As a result, the energy level is increased, resulting in
both higher pressure and velocity. The gas then proceeds through a vaned or a vaneless
diffuser, which convert some of the kinetic energy of the ﬂuid into static pressure. In a
single stage centrifugal compressor the diffuser is followed by the volute whose function
is to collect the ﬂow from the diffuser and deliver it to the discharge pipe. It is possible
for the ﬂuid to gain a further deceleration in the volute and thereby an additional pressure
rise. In a multistage centrifugal compressor the diffuser is followed by a return channel
and the volute is placed after the last stage (Figure 2-3). Multistaging some times is
needed for centrifugal compressor to achieve higher pressure ratio. The contribution of

each component of the compressor is shown in Figure 2-7.

13

 

 

Flow path within
vaned diffuser

Diffuser vanes

-
' -.o....-
.0 o.
o .

,.'-' Flow path within
vaneless diffuser

4? f'v

f ._..\ '41:. ‘. .-._.--
vv :7 \

\///

\

 

 

 

 

 

 

 

 

 

Figure 2-2 A Single Stage Centrifugal Compressor

 

 

return channel

 

 

 

   

 
   

     

 

First Stage
Diffuser gecond __ -
u... tage
Diffuser
First Stage 0. Second
Impeller l 9 Stage " ” 9
, ~ Impeller /
Eye / ; f
s—_—-——— / /
Flow /
4 -o

/

 

 

Figure 2-3 A Multistage Centrifugal Compressor

l4

 

 

2.1.2 Centrifugal Compressor Design

The ﬂow in a centrifugal machine is highly complex and not easy to explain. It is
three dimensional, highly turbulent, viscous, and unsteady in many cases. This makes the
design of the centrifugal compressor more difﬁcult than other machines. Dimensional
analysis and similarity principles are very useful in designing centrifugal compressors
like designing many other machines. The dimensionless parameters state that the
machines, which are geometrically similar, have similar velocity triangles. Some of the
dimensionless parameters, which are commonly used in designing centrifugal
compressors, are the head coefﬁcient, the work factor, the ﬂow coefﬁcient, the
efﬁciency, the speciﬁc speed, the speciﬁc diameter, and Mach number. The speciﬁc
speed and speciﬁc diameter are a group of dimensionless parameters that are used for
stage selection applications. The other group of dimensionless parameters (the head
coefﬁcient and the ﬂow coefﬁcient) are generally used to predict the off-design
characteristics of the machine. Dimensionless parameters will be discussed more in the
impeller section.

Common Approach:

A widespread method that a designer would follow to design a simple centrifugal
stage starts from the non-dimensional speciﬁc speed verses the speciﬁc diameter chart.
This chart was presented by Balje (1981) and was calculated at the design point of
different geometrically of different machines. He then produced what is termed the
Cordier line, which has a single operating point for each machine type. Balje (1981)

made a good use of these nS — (13 charts (Figure 2-4).

15

 

Pelton turbines

   
  
 

Axial machines

Specific Diameter ds _
h
1

Radial machines

 

 

1 .—

0.8 -

0.6 -

0.4 l I I l I l l I 11 l I
0.1 0.2 0.4 0.6 0.8 1 2 4 6 8 10 20 40

Speciﬁc Speed n5

 

 

 

Figure 2—4 Cordier Line (Adapted from Balje 1981)

To design a centrifugal stage some data should be available. These data include
the mass ﬂow rate, the inlet total pressure and temperature, the pressure ratio, and
rotational speed, which depend on the driving device. Based on having these data, the
designer can use the relation of the non-dimensional ns — (15 charts. The designer knows
that ns and (1S have different deﬁnitions but all are dimensionless and need to be
consistent. Balje chart can be used to estimate at2 and other parameters. A quick

estimation of the size of the tip impeller diameter can be found from the following

equation:

d =d2AhOSI/4

s J5 (2-1)

where: d2 is the impeller tip diameter (at the impeller outlet)

Ahm is the total isentropic enthalpy given as:

AhOS : CPTOI E]— —1 (2_2)

Q is the volume ﬂow rate which can be found from continuity equation:

m = pAc = pQ (2-3)
where:

A is the cross sectional area

m is the mass ﬂow rate
C is the axial velocity
,0 is the gas density

For Balje chart to be used, either ns or (15 need to be known. The equation of the

speciﬁc speed is given as

_ 2— \/Q
s — 3/4 2.4
Ah... < >

The speciﬁc speed above is a non-dimensional quantity, when the rotational

11

speed a), is given in rad/s, and Q and Ahos are in a consistent set of units such as: m3/s
and J/kg respectively.

Balje (1981), also, presented another graph that relates the speciﬁc speed to the

machine efﬁciency for radial, axial, and mixed machines (Figure 2-4). Therefore, for a

17

certain machine type the design point will be near the peak efﬁciency, which gives

estimation for the speciﬁc speed. Then from the known data Q can be found from the

speciﬁc speed equation. After that the Speciﬁc diameter equation can be used to ﬁnd d2.

The other parameters at inlet and exit of each component at every stage can be found
using ideal gas relations and general thermo-ﬂuid formulas. Balje (1981) gave a
systematic way to ﬁne the other parameters of the centrifugal compressor. Throughout
the design, some suitable assumption need to be done and iterations may also be required

to get a reasonable design.

 

1.0 —

Mixed ﬂow

I:
b
I

 

Maximum Efficiency
0
in
l

e
h.
l

 

o.2_h_.I_I_I_L__I__H_I_I_I__.I

0.1 0.2 0.4 0.6 0.. 2 4 6 a 20 40
1 10
Speciﬁc Speed

 

 

Figure 2-5 Selection of Compressor Type with Speciﬁc Speed (Adapted from Balje 1981)

18

 

Japikse (1996) gave a more detailed design procedure including some solved

examples and some computer codes as well. In addition Rogers (1974), Dean (1976),

Whiteﬁeld (1990), and Aungier (2000) discussed the design of the centrifugal

compressor. Additional people discussed a speciﬁc component design as it can be seen in

the following section.

Current Centrifggal Compressor Design Trends:

The design of bladed components typically begins in simplified codes such as
one—dimensional and streamline—curvature codes.

This one-dimensional ﬂow design calculations use empirical data from the
literature and test results from a previous design to give preliminary results about
the compressor performance, as well as impeller and diffuser geometrical
parameters. The calculation routine is sequential, iterative and quick with modern
personal computers.

For the one-D calculations, input data includes the compressor pressure ratio, inlet
temperature, inlet pressure, ﬂuid properties, rotational speed and mass ﬂow.
Simple one-D codes includes variables that can be altered later: speciﬁc speed,
different geometrical ratios, number of blades in the impeller and diffuser, loss
constants, inlet flow angles, volute pressure recovery factor, etc.

One—D calculation of compressor efﬁciency is mostly based on measured data.
The radial, vaneless diffuser is calculated analytically, including viscous effects
and also the volute is calculated analytically. Between the diffuser and the volute

there is a semi-empirical loss coefﬁcient model, based measurements in the

19

literature. Reynolds number correction for efﬁciency is done by referring to

literature.

One-D calculation results include:

1.

Pressure, temperature, enthalpy, density at inlet and outlet of the impeller,
and inlet and outlet of the diffuser and volute.

Pressure, temperature and enthalpy must be given both in static and
total/stagnation values.

The main geometry parameters for the impeller, diffuser, and volute, as
well as compressor speciﬁc speed and power consumption.

Isentropic efﬁciency and pressure ratio in each point and pressure
recovery in the diffuser and volute.

Velocity triangles, deviation angle, and relative Mach number in the
impeller wheel inlet and outlet

Advanced one-D codes also estimate off-design performance and surge

point and produce the compressor performance map.

Next follows, the generation of the impeller blade geometry. The main geometry

data of the impeller, such as the main dimensions and the blade angles obtained

from one-D ﬂow design are used as input data for three-dimensional modeling.

The blade design is started with deﬁning the camber lines for the hub and shroud

of an impeller blade. A three-D modeling software provides a graphical blade

design environment for modifying the camber line geometry. Such a program is

exclusively used for geometrical design, No ﬂow calculation is done at this stage.

20

Geometry speciﬁcation usually takes place in special-purpose codes, as opposed
to full CAD packages.

The next stage is to deﬁne the thickness distributions for the hub and shroud of
the blade, and they can be modiﬁed, graphically, as well. Based on the camber
lines and the thickness distributions, the suction and pressure side co-ordinates for
the hub and shroud are ﬁnally calculated. The blade can be displayed using
various projections, and enlargements of blade details can be displayed, as well.
When the blade designer accepts the blade geometry, the numerical geometry data
can then be generated for compressor ﬂow CFD analysis: ﬁnite element method
(FEM) based stress and vibration analysis (FEM) and computer aided
manufacturing (CAM). If any non-conformity is found in the blade geometry, or
the results from either the CFD or FEM analysis do not meet the requirements, the
design procedure must be repeated partially or totally, even starting from one-
dimensional ﬂow design.

Specialized grid generators are now available which take data directly from the
geometry generation codes and provide automated templates for rapidly building
grids.

Non-axisymmetric components such as inlets and volutes are often developed in
CAD packages or specialized codes. The CAD packages can export the data in
IGES or in native CAD format, which can now be read into some CFD pre-

processors and grid generators.

On the post-processing side, customized automated tools are now available. Post-

processors contain post-processing “macros” (a sequence of post-processor

21

commands), which can provide both quantitative information and graphical
images. Performance predictions for individual components or for the entire
model can be obtained.

Available information includes averaged total and static temperature and
pressures, loss coefﬁcients, efﬁciency, ﬂow distortion etc. In addition, useful
circumferential]y-averaged meridional proﬁles of a variety of quantities can be
obtained.

Qualitative plotting of fully 3-D, blade-to-blade and circumferentially-averaged
quantities can be automated, providing the analyst a detailed view of the ﬂow
ﬁeld, enabling visual assessment of its quality.

The high complexity of the ﬂow, especially in the rotating impeller, makes the
CFD modeling very difﬁcult. Much research has been done in analyzing the ﬂow
and the different viscous phenomena. The selection of the turbulence model has a
great inﬂuence on simulation accuracy, particularly if there is instability in the
real ﬂow. When time-dependent calculation of the whole compressor becomes a
routine practice (at the moment it requires a lot of computation time), the
accuracy of modeling obviously increases.

The performance of the ﬁnal compressor geometry is tested in the test facility.
Industrial designers and analysts require CFD software to provide accurate
solutions to their applications of interest in a cost-effective and timely manner.
With the widespread availability of affordable workstation class computing, the
main issues are therefore accuracy and productivity (productivity also being

directly related to cost). The information provided by the CFD must be of

22

sufﬁcient accuracy to allow the designer to make appropriate decisions based on
the available information, and it must be available rapidly enough to ﬁt within the
time scales of the design cycle
0 Productivity in the usage of CFD has improved signiﬁcantly over the past ﬁve
years, for several reasons, including:
1. More powerful computing, including PC’s.
2. The development of parallel processing.
3. Links between CFD codes and other design tools and with CAD.
4. Customized and semi-automated grid generation tools.
5. Specialized pre- and post-processing tools and environments.

The more powerful computing now available enables the analysis of larger grids
and more components together in a shorter period of time. The recent availability of
parallel computing further expands the size of model that can be analyzed, and reduces
turnaround time. The results are not only of higher quality but obtainable sufﬁciently
quickly to be useful within the design cycle.

The centrifugal compressor is not an easy device to be designed. Its design
combines many design considerations, which are mainly based on three major ﬁelds:
stress, rotordynamics, and aerodynamics. Stress problems are clearer on impeller blades
and diffuser vanes. They are caused by the material strength limitations that are limited
by the aerodynamic performance requirement and the manufacturing cost. Rotordynarnics
is another ﬁeld of design considerations. These include the purely mechanical vibration

of the machine caused by the impeller rotation as well as the ﬂow induced vibratory

23

effect. They give some restrictions on the design of compressor components to prevent

the entire machine from signiﬁcance vibrations.

2.1.3 Compression Process

Compression of gasses tends to behave according to the following equation
P.vn = Constant, where P is the pressure and v is the speciﬁc volume (Volume/unit
mass).

Compressing a gas from P1 to P2 is shown in the T-s (h-s) and P-v diagrams

depending on the value of n (Figure 2-6):

 

 

 

 

 

 

 

 

 

p A T A
28 2b 26 2d 2d
2C
2b
(— P un=constant 9 2a A
1
> J
.0 8

 

Figure 2-6 P-v and T-s for a compression process

where:

1-2a: Isothermal compression, with highest cooling n=1

l-2b: Polytropic compression with cooling, n<k

1-2c: Reversible, adiabatic compression, n=k

1-2d: Polytropic compression, with friction effect and heat, rejection, n>k

Figure 2-7 shows a typical example of h-s diagram for a centrifugal compressor stage.

The numbers shown correspond to the locations shown in Figure 2-2. The upper line in

24

Figure 2-7 represent the stagnation (Total) pressure values (the dashed line) while the

other one is representing the static values (the solid line).

 

 

 

 

 

 

 

 

Diffuser Volute

Inlet : Impeller
I

 

p---
p-----

L------

 

V

 

 

 

Figure 2-7 Atypical Example of h-s diagram for a centrifugal compressor stage

The change of the stagnation (total) properties for the ﬂow inside the centrifugal

compressor components with no cooling are given in Figure 2-8.

25

 

 

APO ~<
AT

A. \
\ APO

+ev

 

 

 

Inlet Impeller Diffuser Volute

 

 

 

 

 

 

 

 

 

Figure 2-8 Total pressure, temperature change and Entropy rise (loss) in centrifugal
compressors

2.1.4 Centrifugal Compressor Components

The wide range of demands on centrifugal compressors brings many design
considerations. Most of the design requirements need solutions to two major problems:
stress and aerodynamics. The stress problems are caused by the material strength
limitations and the capability to accurately predict blade and impeller steady state and
vibrational stress for complex impeller shapes at high rotational speeds. The aerodynamic
problem is to efﬁciently accomplish large air deﬂections and diffusion at high ﬂow
velocity, with the added difﬁculty of very small passage ﬂow areas required to get good
efﬁciency and high pressure ratio. Even though the individual components of the
compressor are capable of achieving high efﬁciency, it is the efﬁciency of the whole
stage that is of great importance. Thus, component matching is an essential aspect of
design. It is often required to redesign one or more components of the compressor due to
improper matching and sometimes the efﬁciency of a component is sacriﬁced to achieve

good matching.

26

The actual and ideal head characteristic are different due to the internal losses in
the compressor (Figure 2-9). These losses include mainly the incidence loss as well as the
friction loss. If the inlet ﬂow is highly distorted or swirling, it affects the incidence losses.
On the other hand, the friction loss is a function of the square of the ﬂow velocity and the

passage length which the ﬂuid particles follow.

 

     
  
 

 

 

 

“g ideal
d)
.C
A!“
-..' friction
actual " loss
incidence
loss
flow

 

 

Figure 2-9 Ideal and actual head characteristic of a compressor

2.1.4.1 Inlet:

In centrifugal compressors, like many other machines the inlet plays an important
role. There are more than one type of inlet that are in used. These include a straight inlet
and a curved inlet. The objective of the inlet device is to bring the ﬂow as uniform as
possible to the impeller’s eye with or without a preferred level of inlet swirl. A good

design for the inlet would give a better efﬁciency as well as a better operating range for

27

the compressor. This can be achieved when the design parameters are matched between
the inlet and the impeller. Therefore, the ﬂow properties after an inlet component have a

great affect on the impeller performance and then on the entire compressor.

2.1.4.2 Impeller (General):

Figure 2-7 shows that the major rise in static enthalpy is in the impeller as well as
the diffuser. But, looking at Figure 2-8 we can see clearly the importance of designing an
efﬁcient impeller. This ﬁgure show that the impeller is the only component in the
compressor where a signiﬁcant rise in the stagnation (total) temperature and pressure
occurs. In other words the impeller is the heart of the compressor where the energy is
transferred from the shaft to the ﬂuid by increasing the velocity and pressure of the ﬂuid.
The impeller is an essential element of the centrifugal stage because of the pressure it
builds (Figure 2-7) In centrifugal compressors it is the only major rotating part, which
makes the ﬂow more complex than other compressor components. Two main types “of
impellers are available: unshrouded (open), which does not have a front cover, and
shrouded (closed), which has a front cover. When a shrouded impeller is used, the
secondary ﬂow arising, because of the tip clearance, can be avoided. On the other hand
when the unshrouded impeller is used, the cover friction losses can be avoided so, clearly
a trade-off must be made. Another disadvantage of shrouded impeller is the high stress
generated by the front cover, which limits the use of this type of impeller to relatively
low rotational speed machines. A detailed discussion of the centrifugal impeller will be

done later in this chapter.

28

Air Out

Plenum located A" collected,

   
  

ﬂm routed to
discharge
Air In pipe
0)
ROTOR
(Transfers
energy
to fluid) DIFFUSER
(Transforms kinetic
energy into static
pressure)

Figure 2-10 Key Components of centrifugal compressors

2.1.4.3 Diffuser:

The diffuser is used to convert (transform) the kinetic energy available in the gas
after it exits the impeller into static pressure by decelerating the fluid (Figure 2-6). The
process of diffusion or the decrease of velocity to gain static pressure is another essential
process of a centrifugal compressor. Diffusers may be found as vaned or vaneless. Since
there is a possibility of separation, the rate of diffusion must be limited and the process is
inevitably accompanied by a loss of energy. For compressible ﬂow, the diffuser
effectiveness is deﬁned by the ratio of the isentropic enthalpy change corresponding to
the actual rise of pressure to the enthalpy change corresponding to the isentropic rise of
pressure caused decelerating the gas. Figure 2-11 shows that the actual rise of static
pressure is less than that of the isentropic change of velocity, which makes it less than

unity.

29

 

 

 

 

 

 

 

 

 

 

T P2
p 20
2 28/
/3
1—(v 2 — v 2)
2 I 2 PI
1
s
Figure 2-11 T-s diagram of diffusion process
So the diffuser effectiveness of compressible ﬂow can be deﬁned as
’13 ' hr
= — 2-5
77.1 hz __ hl ( )
where 123 — 121 is the isentropic enthalpy change from pI to 192,,
T 7_“
h3 —h, = Cp(T3 —T,) = CpT1(—3—1) = gang) 7 —1] (2-6)
TI PI
and h2 — h1 = %(V12 — v22) (27)
which eventually gives
7_-1
P2
CpT1[( “) 7 -11

P1

77d: 1 2 2
5071 *V2 )

(2-8)

30

Vaneless diffusers are easy to design and manufacture. They are used in process
compressors, turbocharger compressors, and refrigeration compressors. A vaneless
diffuser gave a reasonable static pressure recovery compared to their prices. This type of
diffuser is simple to design and manufacture because it consists of two parallel walls
forming an open radial annular passage from the impeller exit to the diffuser outer radius.
Another advantage of the vaneless diffuser is that it does not have a throat, which makes
choking not likely to happen. This leads to a possibility of a wide operation range. Figure

2-12 shows the ﬂow path and velocity triangle in a vaneless diffuser.

 

 

 

 

 

Figure 2-12 Flow path and Velocity triangle in a vaneless diffuser

The vaned diffuser is the other type of diffuser. It consists of diffusing channels that are
between vanes, in order to have more diffusion compared to a vaneless diffuser with the
same radial space. Even though, the losses are quite high in a vaned diffuser, high
efﬁciency may still be reached with a vaned diffuser because of the increased diffusion

that increases the efﬁciency for the next component such as a volute. The vaned diffuser

31

throat is the smallest area between the leading edge of one vane and the other side of the
next vane. Depending on the throat area of the impeller blade passage, the vaned diffuser
may control the compressor maximum ﬂow (“choke”) or the minimum ﬂow (“surge”). A
critical part of the aerodynamic design is matching the vaned diffuser to the impeller,
which is dependent on the vaned diffuser area and the vane setting angle. This will affect
the overall efﬁciency as well as the ﬂow range of the compressor. Locating the vanes at
certain distances from the impeller exit (impeller tip) will reduce the vibration, the noise,
and the value of Mach number before reaching the diffuser vane leading edge.

Because of their importance and wide range of applications, vaned diffusers vary from
the simple to design and manufacture to the 3-D diffuser, which are used in advanced

applications.

2.1.4.4 Return Channel:

For a multistage machine, as the ﬂow exits the diffuser, a bend for the ﬂow path is
required to turn the ﬂow to the next stage. The return channel connects each stage to the
next one and carries the ﬂow from radial back to the axial direction in multistage

centrifugal compressors.

2.1.4.5 Volute:

In a single stage centrifugal compressor, the diffuser discharges into a volute that leads
the ﬂuid to the exit to the piping system. For a multistage centrifugal compressor this
happens in the last stage. It consists of a circular channel that is located circumferentially
around the diffuser exit, with a gradually increasing area to get hold of the increasing
mass ﬂow rate as the ﬂow collection process goes on. The full-collection plane is the

location where all the mass ﬂow from the diffuser has been collected by the volute to be

32

discharged to the piping system. What divides the collected and uncollected ﬂow is called
the volute tongue. An exit cone may be connected to the volute exit to deliver the
compressed gas to the piping system. Having a circumferential uniform static pressure
over the impeller outlet, is the major objective in designing a good volute. If this goal is
not accomplished, the ﬂuid ﬂow in the impeller circulate with every rotation of the shaft
leading to a development of vortexes, which results in vane vibration and lower

compressor efﬁciency or even damages the compressor.

2.2 Centrifugal Impeller

As was mentioned earlier, the impeller is the most important part in the centrifugal
compressor and actually it is the part that makes the compressor centrifugal. Since it is
mostly the only rotating part in the compressor, the energy transfer takes place there.

As a result of the impeller rotation, the energy level of the ﬂuid increases giving
more pressure and higher velocity, which is the main objective of the centrifugal
impeller. Depending on the throat area of the impeller blade passage, the impeller could
control the compressor maximum ﬂow (choke) or the minimum ﬂow (surge). So,
matching the design of the impeller inlet to the compressor inlet as well as impeller exit
to the diffuser inlet is needed to get rid of some aerodynamic problems. It is now clear

how important to design an efﬁcient impeller, which is a wide area of study.

2.2.1 Inducer

The ﬂuid enters the centrifugal impeller through the inducer, which is the inlet
section of the impeller, in which the relative ﬂow is turned from the inlet direction which
is parallel to the machine axis to almost radial. So, the objective of the inducer is to guide

the ﬂow from axial to radial direction, while increasing its angular momentum. The

33

designer tries to keep the ﬂow subsonic by minimizing the inlet relative Mach number.
The design of the inducer has a very obvious effect on the impeller performance and
hence on the overall compressor efﬁciency.

A number of design considerations should be taking care of to determine the eye-
hub diameter (dbl) of the inducer. It can be dictated by the minimum shaft and retaining
nut diameter if inboard bearings are used. The size of the shaft and bearing diameter
dictate the eye-hub diameter to be large for compressors with outboard bearings. Other
factors that affect the choice of the eye-hub diameter and the inducer stress/vibration
considerations, the impeller manufacturing techniques, and the number of blades
selected. The inducer, eye—tip diameter (dsl) at the shroud is another parameter that the
designer would take care of because of its importance. Normally, the highest inlet relative
Mach number occurs in that area; and, therefore, a special attention should be taken to
choose that parameter. Assuming no pre-whirl and a uniform axial velocity the relative

velocity at the inducer eye can be driven from the velocity triangle as

2 2 2
W51 =Cxl +Usl (2-103)

 

W... = ‘\/er2 + (”N(1 “)2 (2.10m

where: N is rotational speed of the impeller.

The relative Mach number at the eye tip (Mwsl) can then be written as

 

M = W3] Wsl _ \ICx12 +(7st1)2

wsl a] =W— W (2.10c)

Ignoring the width of the blades we can approximate the mass ﬂow rate to be

 

 

34

 

. 2 _ 2
m : ”(dsl dhl )Cxlpl

 

 

 

 

 

2.10d
4 ( )

_ 4 m
This gives x] _ 2 2 (2.10c)

”(dsl _ dhl )pr
Substituting for C“ in eq.2.10c the relative Mach number at the inducer becomes
, 2
4m
2 2 + (nNds, )2

M _ ”(dsr _dh1 )pr
wsl — T (2.100

W I

For a certain mass ﬂow rate and impeller speed, N, dsl should be chosen to give
the minimum relative Mach number according to eq 2.10f. Plotting Mwsl versus dsl would

give a ﬁgure similar to the following.

Mws

_-———-—

 

 

 

b
dsl

Figure 2-13 Change of the relative Mach number with the inducer eye tip diameter

35

The relative Mach number (Mwsl) may be reduced more if a pre-while is provided
for the ﬂow (inlet guide vanes). Doing so will reduce the energy transfer in the impeller.
The designer may add inlet guide vanes when designing high-pressure ratio compressors
because the inlet Mach number exceeds unity, which allows shock waves to form
reducing the impeller efﬁciency. A trade-off should be made between improving the
stage performance from the impeller and a reduction in the stage performance from the
diffuser. So, a careful evaluation for each application needs to be taking care of when
using inlet guide vanes. Thus, a variable inlet guide vanes may a valid choice to be enable

to change the compressor ﬂow characteristic as needed.

2.2.2 Dimensionless Parameters

When designing a centrifugal impeller, the designer needs to use dimensional
analysis and similarity roles. For a long time, dimensionless parameters were used to
describe the ﬂow and they are still in use. They are used in comparing data from a certain
machines or to predict the performance of a different machine. They are, also, used to
predict the performance at different rotating speed or operating ﬂow point (different
operating point). In testing a scale-model, dimensional analyses are carried out. Some
important dimensionless parameters like ﬂow coefﬁcient, isentropic head coefﬁcient,
work coefﬁcient, slip factor, and efﬁciency will be discussed in next few paragraphs.

The ﬂow coefﬁcient based on inlet volume ﬂow rate can be deﬁned as

Q

¢=———
71' 2
—DU
42 2

(2.12)

The isentropic head coefficient is given as

36

 

 

 

P
Ah .
ll’=1 "’2 = 1 2 J (2.13)
—U —-U
2 2 2 2

Work coefﬁcient or some time called the actual head coefﬁcient is another dimensionless

parameter given by

 

 

 

Aha = Cp(TO2 ‘T01)= UzCuz —U1Cu,

ﬂ = (2.14)
1 2 I 2 1 2
—U —U —U
2 2 2 2 2 2
Efﬁciency can then be deﬁed as
W
77 = — (2.15)
p

Machine Mach number or called the impeller tip Mach number is another important

dimensionless parameter deﬁned as

 

(2.16)

2.2.3 Incidence

The angle of attack of the ﬂow at the impeller blade leading edges causes the
incidence loss for the impeller. If the relative ﬂow angle does not match the blade angle,
there will be a tangential component of relative velocity that is not used and appears as a
head (pressure) loss. If the incidence is high, additional losses takes place due to the
boundary layer separation (Figure 2—14). For the impeller blade-leading edge, the

incidence is deﬁned to be the difference between the relative ﬂow angle ( the angle

37

between the relative velocity and the axial direction) at impeller inlet and the actual blade

angle (the angle that the blade leading edge make with the axial direction) (Figure 2-15)

i=a—a. (ma

pressure

 

 

 

 

 

ml

Figure 2-14 Incidence Variations Upon Mass Flow Change and Boundary Layer
Separation

Getting a little positive incidence at the design point in addition to a suitable
choice of the throat area would improve the design. On the other hand, a bad choice for
the throat area affects the mass ﬂow rate and may damage the compressor. The impeller
rotates the ﬂuid and turns it into a direction perpendicular to the axis of rotation, which
consequently adds a momentum to the ﬂuid.

Positive incidence (angle) corresponds to the incidence that causes the ﬂow to

acts on the pressure side of the blade, which is the side pushing the ﬂow in the direction

38

of rotation. On other hand, negative incidence acts on the suction side of the blade as
shown in Figure 2.14. At the design point, the incidence should be kept close to zero and
the efﬁciency is the maximum provided that the inlet system of the compressor has
uniform ﬂow condition before entering the impeller inducer.

In case of off-design points the high positive and negative incidence will cause
separation in the boundary layer (Figure 2-14). Prescribed swirl at impeller inlet is
another cause of such a problem which would be solved by having inlet guide vanes.
From the velocity triangle at impeller inlet, the relative ﬂow angle decreases with the
increased mass ﬂow rate, which causes negative incidence and the boundary layer
separation on the pressure side of the impeller blade at the extreme condition. However,
the decreased mass ﬂow rate increases the relative ﬂow angle and, hence, causes positive
incidence and boundary layer separation on the suction side of the impeller at the extreme
condition. Because, ﬂow in the impeller has less momentum on the suction side than on
the pressure side, it is beneﬁcial to have a small negative incidence. This increases the
impeller performance since the ﬂow with the small negative incidence makes up for the
momentum deﬁcit on the suction side.

Typical velocity triangles at inlet and exit of impeller are shown in Figure 2-17 for a case

of prewhirl at the impeller inlet.

39

 

 

 

 

 

 

U
at 8 Cu Wu
8 :53:
§' .2 C...
C (I B w

 

 

 

 

 

 

A
V
-—-_——-- _— ——-——-—-———-—

 

 

Figure 2-15 Inlet and Exit Velocity Triangles of Impeller

2.2.4 Impeller Thermoﬂuids Relations and Blade Geometry

The head change in a compressor can be related to the impeller inlet and exit
velocities using Euler equation. The Euler equation is derived from the conservation of
linear momentum, which is the rate of change in linear momentum of a volume moving
with the ﬂuid is equal to the surface forces and body forces acting on the ﬂuid. By the
conservation of momentum principle, the change of angular momentum can be obtained
by equating the change in the tangential velocities to the summation of all forces acting

on the rotor (i.e. the net torque of the rotor), that can be written between inlet and exit as
T = m(r2Cu2 — rICul) (2-18)
where C,” and Cuz are the tangential velocity at inlet and exit of the impeller respectively.

The rate of change of energy transfer is the product of the torque and the angular

velocity, which gives the total energy transferred to be

E = 70) = ”102506142 - rleu1)= ’11(U2Cu2 _U1Cul) (2-19)

40

where U1 and U; are the tip velocity at r1 and r2 respectively.
The energy transferred per unit mass ﬂow is equal to the change in total

(stagnation) enthalpy ho and hence
Aha = U2Cu2 — UlCul = hoz " hor = Cp (T02 ‘ T01) (2-20)

Then, it is also possible to relate the enthalpy change to the exit blade angle.
Using the velocity triangle at the impeller exit and assuming no prewhirl (Cu1=0), the

enthalpy rise is given by
Cu2 =U2 +Cmz 1311,32 (2-21)
with the convention that ,82<0 for a backward swept impeller which means that the

blades of an impeller are inclined backward at outlet.

Introducing the theoretical head rise

 

_ Aha
9” - 2 (222)
U 2
and the ﬂow coefﬁcient
(0 : Cm2
U2 (2-23)

to make a non-dimensional representation of the enthalpy rise versus mass ﬂow rate, the

following relation is obtained.

1” = 1 + (P tan .52 (2-24)

Figure 2-16shows the velocity triangles and ﬂow angle change upon the increased mass

ﬂow rate for three different impeller blade shapes.

41

 

1.8~
1.6-
1.4-

1.2 «
[32:00

 

0.8 s
0.6 i
0.4 i
0.2 .

Bz=‘45°

 

 

 

Figure 2-16 Effect of Exit Blade Angle on the Head Rise and the Flow Range

 

 

 

 

 

 

 

 

 

 

 

 

(a) forward blade (b) radial blade (c) backward blade

Figure 2-17 Velocity Triangles Velocity Proﬁle for Different Blade Shapes

42

By studying equation (2-24), if mass ﬂow rate is increased, the actual head is
increased for a forward-leaned blade impeller, is independent for a radially-ending
impeller, and is decreased for a backward-swept impeller which can be seen clearly in
Figure 2-16. For backward-swept impeller, the increasing head with decreasing mass-
ﬂow rate gives more stable compressor stage characteristics and consequently a greater
operating range.

The enthalpy change in the machine can also be expressed in the following way.
1
Aho =§[(U22 ‘U12)+ (W12 ‘W22)+(C22 ‘C12)] (2-25)

and W 2 = C2 + U 2 — 2UCu (2—26)
Equation (2.25) shows that the enthalpy rise in a centrifugal compressor is due to

three different contributions. The ﬁrst term in that equation represents the change of
energy of a particle as it moves from one radius to another. Casey and Marty, 1986,
presented that the effect of these centrifugal forces are about one-half of the work input
and can be accomplished without loss. The rise in enthalpy due to the diffusion of the
relative ﬂow is represented by the second term. This term is approximately one-ﬁfth of
the total work input. The last term in that equation represents the diffusion of the absolute
velocity, assuming that the velocity at the inlet and exit of the stage is equal; and it is

approximately about one-third of the work input.

2.2.5 Efﬁciency, head Coefﬁcient and Flow Coefﬁcient

Centrifugal compressors, as in any type of turbomachinery, there exist an
optimum efﬁciency over a certain range of ﬂow. The ﬂow coefﬁcient (D of a centrifugal

stage can be correlated to the efﬁciency, and is most commonly deﬁned by Equation 2-

43

12. The optimum ﬂow coefﬁcient range has been well established to be between 0.06 and
0.11, as shown in ﬁgures 1.1 and 1.18. Even though low ﬂow coefﬁcient stages, d><
0.02, tend to have very poor efﬁciency, there are strong demands from current
applications to extend the application area to as low as (0:0.004. In a multistage machine,
the volume-ﬂow rate decreases as the pressure of the ﬂow is increased through the stages.
As the last stages are reached, the ﬂow coefﬁcient (1) is, therefore, smaller than at the inlet
of the machine. The impeller and diffuser channel are narrower to handle the smaller
volume ﬂow rate. The friction losses are thereby increased, and a smaller efﬁciency is to
be expected. Also, the disk friction losses and the leakage losses are known to be
inversely proportional to the volume ﬂow rate. Hence, low-ﬂow coefﬁcient compressors
are inherently inclined to have lower efﬁciency than the earlier stage.

If efﬁciency is directly related to ﬂow coefﬁcient, one would think that the only
way to improve the performance of such compressors is to increase their ﬂow coefﬁcient.
Looking back at the above equation, this would imply decreasing D2 or reducing U2. In
both cases, that would lead to a decrease in the work input of the machine, which is not
desirable. Centrifugal impeller design is often based on a series of compromises to obtain
an optimum conﬁguration for a particular application. Currently, industrial centrifugal
compressors are designed in the ﬂow coefﬁcient range (I) = 0.01 to 0.16. But often the
need arises, as opposed to the low ﬂow coefﬁcient, to design high ﬂow coefﬁcient
centrifugal stages; but the aerodynamic challenges are not easy to satisfy, therefore, a

designer usually decides on a mixed ﬂow impeller stage.

 

I
Design Point Efﬁciency

   

Disc Friction Losses
- Leakage Losses

‘ .
’ n
V I
l

 

 

 

 

 

 

 

 

 

 

 

 

 

0.24.» r-—- . -~--
I l ’ l Ale'rodynamic Lossesl
e i I z a e e ,
0 _ 0.05 (p 0.1 ' 0.15

 

Figure 2-18 Application Areas of Typical Industrial Centrifugal Compressors

In the cylindrical coordinate system, the energy transfer through turbomachinery
is governed by the rate of change of the angular momentum (UCu). The function of
turbomachinery involves the exchange of signiﬁcant levels of kinetic energy. In
turbomachinery, a pump uses liquids for a working ﬂuid, while a compressor uses gases.
For a compressor, three different terms (a fan, a blower, and a compressor) may be used
depending on the pressure ratio or the pressure rise achieved. Furthermore, depending on
the discharge ﬂow direction, a compressor can be classiﬁed as axial, centrifugal (or
radial), or mixed ﬂow.

For a steady ﬂow the work input and exchanged heat can be related to the change

in total enthalpy and elevation by using the ﬁrst law of thermodynamics as follows:

45

Q+W

m

 

1 _
=(h.—h.)+5(c:—c.2)+g(z.-z.) (2-30)

Assuming ideal gas behavior, no change in altitude, and adiabatic (no heat
exchange with surrounding) the change in energy is directly related to the change in total

temperature as

W - 1
Z=W :(hz—h1)+_2'(C22-C12)=(h02‘h01)=Cp'(T02_T01) (231)

As was mentioned earlier the compression process can be represented in a T-s
(h-s) diagram. A typical compression process is shown in Figure 2.19. The isentropic
efﬁciency compares the actual enthalpy change during the compression process to the
one that would have taken place for an isentropic process. It is deﬁned by
77- = hozs ‘hor : Tozs "T01
13 hoz — hm T02 ‘T01 (2-32)

 

 

State 1, here, refers to the total (stagnation) conditions at the inlet. State 2 corresponds to
the exit conditions. Using the total conditions at state 2 gives the total-to-total isentropic
efﬁciency deﬁnition, while using the static conditions at the exit gives the total-to-static
isentropic efﬁciency. From equation (2-31), if stages having a large exit velocity then the
difference between these two efﬁciencies will be large. Either one of these deﬁnitions are
used depending on the type of application. Normally, the total-to-total efﬁciency is used
if the exit velocity is used. On other hand the total-to-static efﬁciency is used if the exit

velocity is not used. For a perfect gas with constant speciﬁc heat, the above equation

becomes

46

y_-I
—— —1
Cp(T02is —T01)_ P01

”'s = " (2—33)
I Cp(TO2-TOI) E14
T01

 

 

 

1- ..........................................................
°” 02s

 

 

 

 

 

 

 

Figure 2-19 Representation of a compression process in a T-S diagram

2.2.6 Head Rise

The head rise (enthalpy) is an essential need in the centrifugal compressor design.
The isentropic head can be deﬁned as the energy transferred through a compressor per

unit mass of ﬂuid. For example, If a compressor produces l m of head, it means that one

47

kg-force is needed to elevate one kg of gas mass to a height of one meter. The head

cannot be measured directly but for a compressible ideal gas ﬂow it is given by

Aha“ = CpAT

01's

c—iR

where p 7 _ 1

and for an isentropic compression process

7—1

7
AT,“ =T01 (522—) -1

P01

 

(2-27)

(2-28)

1

(2-29)

I

The head rise as well as the total pressure ratio are used to express the energy exchange

into the impeller and can be used interchangeably.

2.2.7 Operating Range between Surge and Choke

The performance of the compressor can be represented with a curve(s) that shows

the pressure ratio (on the y-axis) versus the mass-ﬂow rate (on the x-axis) for different

rotational speeds, or the head versus the volume-ﬂow rate. In a dimensionless form it is

represented by the head coefﬁcient versus the ﬂow coefﬁcient. Figure 2.20 shows a

sample performance curve for a centrifugal compressor stage.

48

 

 

Presemeretio

 

Rotational
speed
Increasing

Surge
Region
Choked Flow
Operating Range Region
/

I

Constant Impeller
, ’ Speed Lines

 

Mass Flow Rate

 

Figure 2-20 A sample performance curve of a centrifugal compressor

Looking at the above ﬁgure, there are three different regions that exist. On the

right side is the choke limit line, which is obtained when sonic conditions are reached in
the minimum section of the compressor. This could happen in the impeller throat or
diffuser throat, whichever is more critical. In the compressor map, the choke limit
appears as a vertical line because the efﬁciency and the pressure ratio drop signiﬁcantly.
On the other side is the stall (or surge) region and line. This region is unstable. Rotating
stall is an unstable phenomenon, often prior to surge. It may happen in the impeller or the
diffuser. Rotating stall is a local instability in which the separated ﬂow rotates in one

component of the compressor. These regions of separated ﬂow are called cells. If the

49

 

compressor runs at high pressure, the pressure change caused by the rotating stall
generates noticeable mechanical vibration that can cause a big damage to the machine.
Therefore, the compressor should not operate in this region. For a vaneless diffuser, the
diffuser inlet ﬂow angle has been linked to the occurrence of this phenomenon (Senoo
1978).

Throughout the compressor surge, oscillations of the mass ﬂow rate occur
between the exit and inlet which may damage the machine. For high density gas, this has
more effect on the compressor. Van den Braembussche (1984) related the value of d‘P/dd)
to the occurrence of this phenomenon. He explain that for a negative value of d‘I’ldd) (i.e.
increasing pressure for decreasing mass ﬂow) any disturbance in the ﬂow will be damped
out, resulting in a stable operating point. But when d‘P/dd> becomes positive, the ﬂow
would likely become unstable. This, on the global system, will affect the compressor as
well as the inlet and the discharge pipe. So operating the compressor in this region should
be avoided. It should be operated at a medium ﬂow coefﬁcient, called the operating range
between the choke and the surge regions, in which a high efﬁciency can be obtained that
is normally close to the design point.

Because of the importance of this topic many other studies have been made to
prevent surge. Sorokes from Dresser-Rand Company combined a good summery of
literature review plus Dresser-Rand experience. Surge control sensors are also devolving
to help in this area. The development in this area will allow compressors to have wider
operating ranges and will avoid costly and unnecessary recycling of ﬂow (McKee 2003).
There is active work in this area to develop a dependable surge control to operate

centrifugal compressors reliably and safely, closer to the surge limit. (McKee 2000).

50

2.2.8 Slip Factor

Slip occurs in centrifugal compressor because the relative ﬂow leaving the
impeller is not totally guided by a ﬁnite number of blades. This leads to a modiﬁed
velocity triangle (Figure 2.21). As a result, the magnitude of the tangential component of
velocity will be reduced, which cause a reduction of the pressure ratio and more power
consumption. So, with slip, more rotating speed is needed to get the same pressure ratio

as if there was no slip (ideal).

 

m2

 

 

 

 

 

 

 

 

 

Figure 2-21 Slip effect on velocity triangle at impeller exit for back-swept blade

51

As a result of the increase in the rotational speed of the impeller, it will
experience more stress and higher velocities that cause an increase in friction losses and a

reduction in efﬁciency. A common deﬁnition for the slip factor is

 

 

Cslip :1_ C9200 " C62
U2 U2 (234)

0:1-

From the velocity triangle shown in Figure 2.8, the tangential component of

velocity is, now, rewritten as

C62 : JUZ + Cm2 tan :BZb (235)

Because of the need to account for the losses due to slip in design many

correlations became available in the literature to predict the value of the slip factor.

Wiesner (1967) gave a correlation that is one of the most commonly used
correlations as

x/ 005 .5212

a: ‘ 20.7 (2-36)

 

This equation where obtained by ﬁtting data of the some theoretical curves that
was found by Buseman (1928) for inviscid ﬂow. This correlation was validated by over
60 compressor and pumps with various blade angles and blade numbers and compared to
other correlations, by Wiesner. This shows a good agreement with exception of some
machinesthat was off by few percent.

The effect of both the blade number and blade angle on the slip factor is shown in

the following ﬁgure. It can be seen that slip factor increases with an increase in the blade

52

number. In addition, the slip factor is less for a radial impeller compared to a back-swept

impeller.

 

1.0

0.95

 

P
(0
GI

 

factoro
.0
on
or

 

pSIIP
on
o

 

P
\l
. <41

 

 

 

0.70
Exit blade angle 02,,

Figure 2-22 Inﬂuence of the blade number and the exit blade angle on the slip factor

The concept of a relative eddy was used by Stodola (1927) to derive the slip
velocity and the slip factor. The ﬂow into the impeller inlet has no rotary motion; but
after passing through the impeller passage, the relative ﬂow will have an angular
velocity, that is equal in magnitude and the opposite in direction, to the impeller rotating
speed. The actual relative velocity is then formed by the vector summation of the eddy

and the radial component of velocities.

C _ ”U2 COS ﬁzz)
slip _ Z (2-37)

 

53

7r cos 162b
' Z (2-38)

 

0:1

where B21, is the impeller blade exit angle and Z is the number of blades.
Another slip-factor correlation was also provided by Stanitz (1952). For blade

angles in the range of —45° to +45°, this correlation is considered to be acceptable.

0' = "—_ (2-39)

54

Chapter 3 CFD ANALYSIS
3.1 Need and Development of CFD

Computational Fluid Dynamics (CFD) is a computer based tool that is used to
simulate the behavior of ﬂuid ﬂow and heat transfer. Simulating such processes needs to
be done by solving the governing equations of ﬂuid ﬂow and heat transfer (in a particular
form) over the region of interest, with known boundary conditions of that region. (CFX-5
manual)

For many years, computers have been used to solve many ﬂuid ﬂow problems. It is
believed that the automatic digital computer was built by Atanasoff in the late 19305 and
was used from the beginning to solvesome problems in ﬂuid dynamics; but the explosion
in computational activity did not begin until a third ingredient, the general availability of
high-speed digital computers, that occurred in the 19608(T B). Too many programs have
been written to solve either a particular problems or particular type of problem. About 30
years ago general purpose CFD solvers were developed based on the understanding of
some complex mathematics required to generalize some key algorithms. In the early
19808 powerful computing machines, as well as more understanding of the ﬂuid
dynamics came together to help the development of CFD. (CFX-5) The availability of
previously unthinkable computing power has motivated many changes. The need for such
changes was more visible in industry and research laboratories, where the need to solve
complex problems was urgent and critical. Recently, changes brought about by the
computer have become noticeable in many of our daily lives. In the educational process,

computers are found to be used widely at all levels. (Tannehill and others 1997)

55

CFD was used as a tool almost exclusively in research. Recently, development in
computing power, together with the advanced graphics and the interactive 3-D
management of models, allowed the process of making a CFD model and analyzing the
results to be done in much less time and therefore saving money. Nowadays, as a
consequence of this development, Computational Fluid Dynamics has become a
dependable design tool helping to reduce design time and cost. Improving the design
processes throughout the engineering world is the overall result of such development.
Comparing CFD with scale model testing, it gives an accurate alternative as well as cost-
effective. Being able to vary the parameters on the simulation very quickly is another
clear advantage of CFD. (CFX-5)

Engineers and scientists are using CFD more often in a wide range of ﬁelds. In the
process industry, for example, CFD is used in turbines, compressors, pumps, mixing
vessels, and chemical reactors. It is also used in building construction and in design for
air conditioning and ventilation. Health and safety is another area where CFD is used to
investigate the effects of ﬁre and smoke. CFD plays a large role in the development of
the motor industry where modeling of combustion and car aerodynamics is an issue. In
the ﬁled of electronics CFD is used to simulate the heat transfer within and around circuit
boards. For environmental applications CFD is use to solve many problems such as the
dispersion of pollutants in air or water. In medicine simulating a blood ﬂow through
grafted blood vessels is another area where CFD can be used. Power, energy and
airplanes industry is a very wide area of application for CFD. All these application drive

the development of CFD. (CFX-5)

56

3.1.1 CFD and Industry

Even though the development of CFD mostly takes place in an educational environment,
industry is the major motivation of it. The industrial manufacturers are doing their best to
reduce the design time and cost. So, they are putting more efforts on developing
numerical models that physically describe the behavior of their hardware as accurately as
possible. This is even truer for manufacture is producing large machinery. First, the
excessive high construction costs for one-of-a-kind hardware. Second, the long time that
is required for constructing the environment of the experiment. Third, the high risk of
some machines that deals with human lives. Under these circumstances, design engineers
are no longer able to rely only on past experience and the use of the traditional l-D
empirical based tools. Getting the design correct from the ﬁrst time would save time and
money. This requirement also applies for diagnosing aerodynamic problems that occur in
machines that are already installed. If there is a problem with a running machine, the
shut-down time is often very limited in order to minimize the customer’s production loss.
So, there is normally little opportunity for testing that may take time. (T annehill and
others 1997)

Experimental in addition to theoretical methods were the traditional methods used to
develop designs for equipment and vehicles involving ﬂuid ﬂow and heat transfer. After
the exploration of the digital computer, a third method (the numerical approach) was
found. Even though the experimental method continues to be important (especially when
the ﬂows are complicated) the demand is clearly toward greater dependence on

computer-based predictions in design. (T annehill and others 1997)

57

The use of CFD provides a cost-effective and time saving way to design and investigate
ﬂuid ﬂow hardware. The success of this method, CFD, depends strongly on the user as
well as the models in the code. Even though CFD are used for the initial design to solve
current problems, it may also be used to minimize experimental work or to validate the
results that were found using other techniques. (Tannehill and others 1997)

Applying CFD to turbomachinery has its own beneﬁts as well as challenges. The
experimental setup cost much money and time to get results. A single problem, for
example, could cost a shut down for a whole plant for weeks because a turbine or a
compressor needs to be disassembled and taken somewhere for testing. These problems
are common in industrial plants and may cost hundreds of thousands or even millions of
dollars.

In CFD, the turbulence models, the grid density, quality of the model, and the
numerical schemes are traditional topics that need to be taken care of when solving
problems. In addition to these, for modeling turbomachinery cases, more problems would
arise such as handling the rotation of the ﬂow and the interface between the rotating and
stationary components. For example, for a centrifugal compressor stage, a CFD model
typically needs two interfaces. The ﬁrst one is to couple the stationary inlet region to the
rotating impeller. The second one is to couple the rotating impeller to the downstream

stationary components such as diffuser, return bend, and return channel or volute.

3.2 CFD Mathematics
Navier-Stokes equations are the set of equations that describe the processes of
momentum, heat, and mass transfer. These equations are partial differential equations that

were derived by two scientists in the nineteenth century. Up to this date, there is no

58

known general analytical solution for those equations. As an alternative, they can be
discretised and numerically solved. In addition to Navier-Stokes equations, other
equations that describe more processes, such as combustion, may also be solved in
conjunction. Mostly, these additional equations may be driven using an approximating
model. Turbulence models as an example of particular importance. (CFX-5 manual)
There are three major numerical discretisation techniques that are used to discretise
partial differential equations: the ﬁnite difference method, the ﬁnite element method, and
the ﬁnite volume method. All of these methods are used to transform a partial differential
equation into its numerical analogue. (Shaw 1992)

The ﬁnite difference method is based upon the use of the Taylor series to construct the
equations that describe the derivatives of a variable as the differences between two values
of the variable at two points in space or time. Smith (1985) wrote a comprehensive
reference for this method. The second method is the ﬁnite element method. It is named so
because the domain over which the partial differential equation applies is divided into a
ﬁnite number of smaller domains that are known as elements. The Zienkiewicz and
Taylor book (1989) is assumed to be standard reference for this method. The most
common method used in CFD is the third method, which is known as the ﬁnite volume
method. In some ways this method is similar to the ﬁnite difference method, but some
implementations of it also come from the ﬁnite element method. This method is very
popular in CFD because it was developed speciﬁcally for solving the equations of heat
transfer and ﬂuid ﬂow. Patankar (1980) gave a well detailed description for this method.
Since all of these methods are used to discretise partial differential equations they have

some similarities. They produce equations for the values of the variable at a ﬁnite number

59

of points in the domain. They also require that we know a set of initial conditions as well
as boundary conditions of the problem so that we can solve the equations. Explicit or
implicit forms may be produced from these techniques. (Shaw 1992)

Even though the above methods are similar in some ways they have differences as well.
Both the ﬁnite difference method and the ﬁnite volume method generate the numerical
equations for a certain point based on the values at the adjacent points. However, the
ﬁnite element method generates the equations for each element independently of the
other elements. In this method the interaction between elements is taken into
consideration when forming the global matrices. Another difference between these
methods is that in the ﬁnite element method the derivative boundary conditions is taken
care of when forming the element equations then the ﬁxed values of variables must be
applied to the global matrices. The other two methods, however, can apply the ﬁxed-
value boundary conditions easily by inserting the values into the solution while the
derivative boundary conditions require modiﬁcation of the equations. (Shaw 1992)

The development of these methods may help to overcome their disadvantages and make
them more efﬁcient.

The method of particular importance to us here is the ﬁnite volume method because it is
the one that is mostly used in CFD codes. The main idea in this method is to divide the
region of interest into small subregions, called control volumes. Then for each control
volume the equations are discretised and solved iteratively. A solution can then be
obtained by approximating the value of each variable at speciﬁc points throughout the
domain. This will drive a complete full picture of the behavior of the ﬂow. (CFX-5

manual)

60

Even though the main method used in such applications is the ﬁnite volume method some
times other approaches may be associated to get better results. For example, CFX-
TASCﬂow, the software used in this study, is using the ﬁnite element approach to
represent the geometry. Thus, this way of associating the ﬁnite element approach with the
ﬁnite volume method maintains much of the geometric ﬂexibility of ﬁnite element
methods as well as the important conservation properties of the ﬁnite volume method.

(CFX-TASCﬂow Theory Documentation Version 2.12)

3.3 Engineering Approaches

As it has been discussed before, the three methods available for a designer are the
Experimental, Theoretical, and Computational methods. Each Method has its advantages
and disadvantages (Table 3-1). If a good understanding of the problem is obtained and
faster computer is used, then CFD would save more time than the experimental and
theoretical methods. Saying that does not mean that computational methods will soon
completely replace experimental testing to gather information for design purposes.
Rather, it is believed that CFD will be used even much more in the future. In many ﬂuid
ﬂow and heat transfer design situations experimental testing may be still be necessary.
However, CFD can be used to reduce the range of conditions where testing is required.

In some ﬁelds such as turbomachinery, experiments will remain for some time in
applications because of the involvement of turbulent ﬂow and the complex geometry.
This situation is destined to change in the long run, because it has become clear that the
time-dependent Navier-Stokes equations can be solved numerically to give accurate
details of turbulent ﬂow. Thus, as computer hardware and algorithms improve, the limit

will be pushed back continuously allowing ﬂows of increasing practical interest to be

61

computed using CFD. Even though, there was a progress in simulating turbulent ﬂow, the
work still needs more development compared to the status of the models used for laminar

single-phase ﬂow over aerodynamic bodies. (Tannehill and others 1997)

Table 3-1 Comparison of approaches (adapted from Tannehill)

 

Approach Advantages Disadvantages

 

Experimental 1. Capable of being most realistic 1. Equipment required
2. Scaling problems
3. Tunnel corrections
4. Measurement
difﬁculties

5. Operating costs

 

 

Theoretical 1. Clean, general information, 1. Restricted to simple
which is usually in formula geometry and physics
form 2. Usually restricted to

linear problems

Computational 1. No restriction to linearity 1. Truncation errors

(CFD) 2. Complicated physics can be 2. Boundary condition
treated problems

3. Time evolution of ﬂow can be 3. Computer costs
obtained

 

 

 

 

 

62

Commercial codes are available for some time and many compressors’ companies started
using them more often which help the development in this area. More demand on codes

that can deal with turbulence is noticeable.

3.4 CFD in Current Practice for Centrifugal Compressor Design

The emergence of Computational Fluid Dynamics (CFD) in the last twenty-ﬁve
years provided a major movement to solve the Euler and N avier-Stokes equations, which
also govern the ﬂow ﬁelds in turbomachines. This has been possible mainly due to
advances in, grid generation, turbulence modeling, boundary conditions, pre- and
post-data processing, and computer architecture. Most of the techniques used for the
solution of the Navier-Stokes equations can be classiﬁed as ﬁnite difference, ﬁnite
area/volume ﬁnite element, and spectral methods. Only the ﬁrst two techniques are
widely used in turbomachinery.

Compressor designers and analysts require CFD software to provide accurate
solutions to their applications of interest in a cost-effective and timely manner. With the
widespread availability of affordable workstation class computing, the main issues are
therefore accuracy and productivity. The information provided by the CFD must be of
sufﬁcient accuracy to allow the designer to make appropriate decisions based on the
available information, and it must be available rapidly enough to ﬁt within the time scales
of the design cycle. For the designer, “accuracy” means providing reliable information of

the following type:

- Qualitative: correctly reproduces the important ﬂow features, such as swirl,
boundary layers, shocks, wakes, separation zones, stagnation points, mixing

layers, etc.

63

0 Quantitative: efﬁciency, work input, pressure rise, blade proﬁles, component loss
coefﬁcients, ﬂow distortion parameters, incidence, deviation or slip, etc.

While it is desirable that such features are perfectly reproduced, this expectation is
unrealistic given limitations due to mesh size, turbulence modeling and other modeling
assumptions such as steady state ﬂow. To be used conﬁdently, it is therefore of
importance that CFD provide:

0 A “Sufﬁcient” level of accuracy

0 Repeatability and consistency

The two factors that inﬂuence accuracy most for compressor applications are the
discretization accuracy of the ﬂow solver and the computational mesh. Only after these
two are adequately treated do other issues such as the turbulence model become of
primary signiﬁcance. In particular, the turbulence model is the usual scapegoat for poor
CFD predictions, while in reality other causes are often more signiﬁcant. Hence, it is

often the case that “numerical errors” exceed “model errors”.

3.4.1 CFD Model

TASCﬂow-CFX is a general purpose, commercially available CFD code, widely
used among turbomachinery industry. For the present study, TASCﬂow-CFX is used for
steady-state compressor stage numerical simulation. TASCﬂow-CFX solves three-
dimensional, Reynolds-stress-averaged, Navier-Stokes equations with mass-averaged
velocity and time-averaged density and pressure and energy equations. The mean form of
the governing equations, expressed in a ﬁnite-volume formulation that is fully
conservative, include the following.

1. Conservation of mass

27%;: = <3»

2. Conservation of momentum

a a _ 8p iI
EWI)+‘ax—'(P“jui)—‘_+ i

[8141. auj 1]]? + S
#e —' + ' ' ui (3.2)
I axi GXj II If I

dx- 3x

1
where ,ueﬂ = ,u+/1,. Sm. is the momentum source term for the impeller in the rotating
frame of reference. The effect of Coriolis and Centripetal forces are modeled in the code
by including

Sm- = —2DXU —§2x(flx ‘r’) (3.3)

3. Energy equation

_a_(pH)_iI:+_a_ .H)=.i[/13T + ’u’ ah J+SE

 

a: at dxj I 3x] dxj Pr, Bx-
3.4
+i[u. ﬂ ai+ai]_3’u 2 a_ul ( )
axj' II I t axj axl- Max x, JUIjIIII

l
where the total enthalpy is deﬁned byH = Iii-Elli“; +k. In the rotating frame of

2 2

 

reference, the rothalpy I = H - is advected in the energy equation in place of the

total enthalpy. (0 is the rotation rate, and R is the local radius.

For the sliding interfaces between stationary (inlet and diffuser) and rotating (impeller)
components, two models are available: one is the “frozen rotor model” and the other is
the “stage model”. The stage model circumferentially averages the ﬂuxes at the interface

before the interpolation of the ﬂow variables across the different frame of reference

65

although the pressure distortion caused by any perturbation (for example the impeller and
the diffuser leading edge) is still allowed. For an axisymmetric inlet ﬂow condition, the
stage model can be used with the advantage of modeling one or two passages having a
periodic boundary condition for the tangentially-neighbored blade passages instead of
modeling fully 360 degree passages, which is signiﬁcantly economical for computation
time and effort.

On the other hand, the frozen-rotor model achieves a frame change across the interface
without a relative position change over time as well as without any interface averaging of
flow variables. This model is an exact representation of the case when the Strouhal
number is zero; in which case, either the sound speed is inﬁnite or the impeller rotating
speed is zero. Therefore, when the Strouhal number is small enough such as the
compressor stage simulation presented here (St is between 0.1423 and 0.1431, depending
each inlet models), the predicted simulation result is an approximation of the real
situation. In the case of the frozen rotor model, all of the passages have to be modeled
and this model is adequate to investigate the inﬂuence of the distorted ﬂow caused by the
bend inlet along the compressor stage flow passages since local ﬂow features are allowed
to transport across the interface; and thus, the non-uniformities of flow variables among
the passages can be predicted, which results in different ﬂow conditions at the

compressor stage exit

3.4.2 CFD Process Components
Performing a single CFD simulation is a process that involves many steps. These
steps can be grouped into three major components: pre-processing, solving, and post-

processing. Each one of these components involves many steps. The following

66

paragraphs will go over them in more detail and give the speciﬁc subprograms for each

one under the CFX-TASCﬂow.

3.4.2.1 The Pre-processor:

The pre-processor, as it appears from its name, is the ﬁrst component in the CFD
process. It is used to generate the input for the solver. Pre-processing involves the
following steps:

0 Deﬁning the geometry of the region of interest where the ﬂow simulation will

take place. This step may also be called the grid generation

0 Choosing the physical models that are needed to be integrated in the simulation

0 Stating the ﬂuid properties

0 Stating the boundary conditions

0 Stating the initial condition

0 Generating a mesh of control volumes

In most commercial codes it is now common to have automated Pre-processing
operations. The original geometry may be imported from major CAD packages using
native format or the CFD code may have its own package. A mesh of control volumes is

then generated automatically. (CFX-5)

3.4.2.2 The Solver:

The solver is the second component in the CFD process. It is the heart of the CFD
process because it solves the CFD problem, generating the required results. This is done

through the following steps:

67

0 Integrating the partial differential equations over all the control volumes in the
simulation region. In other words, applying a basic conservation law (e.g. for
mass or momentum) to each control volume in the region of interest.

0 Transferring these integral equations into a system of algebraic equations. This
can be done by producing a set of approximations for the terms in the integral
equations.

0 Solving iteratively the non-linearity of these algebraic equations.

0 The solution is said to converge when it approaches the exact solution. As a
measure of the overall conservation of the ﬂow properties, an error (or residual) is
reported for each step. There are number of factors that affect how close the ﬁnal
solution is to the exact solution. These factors include the size and shape of the
control volumes and the size of the ﬁnal residuals. Combustion and turbulence are
often modeled using empirical relationships because of their complexity. This
includes more approximations that contribute to the difference between the
solution obtained by CFD and the real ﬂow. This component of the CFD process
(the solver) does not require user interaction and is therefore usually carried out as
a batch process. Finally, the solver creates a complete ﬁle of results that will be

passed to the post-processor. (CFX-5)

3.4.2.3 The Post-processor:
The post-processor is the last component of the CFD process or may be called the end

result of the whole process. It helps the user to get and analyze the results. Post-

processing could include anything from obtaining values at certain points to some

68

complex animated sequences. Some important features of post-processors that the user

needs to get are:

Visualizing the geometry as well as the control volumes.

Plotting the direction and magnitude of the ﬂow ﬁeld.

Visualizing of the variation of scalar variables (such as temperature) through the
domain.

Calculating quantities that are not available if needed.

Animating of the ﬂow ﬁeld.

Plotting charts that shows graphical plots of variables

Printing to ﬁle as well as hardcopy output.

As it may be noticed that pre-processing as well as post-processing are interactive

processes while the solver in a non-interactive one. (CFX-5)

3.5 The Simulation Process

The following ﬂow chart (Figure 3-1) explains the process of the CFD simulation

that was followed for this work and includes the CFX-TASC ﬂow subprograms.

69

 

Pre-processor

 

 

Generating the Grid from the available drawings
Creating the geometry ﬁles (CFX-TASCgrid)
Checking the quality of the grid (CFX-TASCtool)

 

ll

 

 

 

- Stating the boundary conditions (CFX-TASCbobBD)
Stating the initial conditions (CFX-TASCtool)

Stating the extra parameters (e. g. the turbulence
models) (CFX-TASCbob3D)

 

 

 

 

 

 

 

 

V

 

SOLVER

 

 

0 Stating the criteria (Control Parameters or
RPM ﬁle)

0 Solving the equations (CFX-TASCﬂow3D)
0 Monitoring the convergence plots

 

 

 

 

 

 

«My

 

 

Post-processor

 

- Getting the results at certain points
(GUI)

0 Showing the result charts
Showing the result plots

 

 

 

 

 

 

 

J.

0 Modifying the code if applicable
(CFX-TASCtool) or need to start
over

 

 

 

 

Figure 3-1 Flow chart - Process of the CFD Simulation

7O

 

3.5.1 Grid Generation

Grid generation is the ﬁrst essential part in the CFD process. Since CFD works
numerically to solve the needed ﬂow and heat equations, so it needs to discretiz the
domain into small volumes to be able to perform the calculations. In other words grid
generation is the process of breaking the solution domain up into discrete sub-domains.
Many methods and software were developed to do this for simple objects. Grid
generation becomes more difﬁcult when the domain of interest is complex. Since a
centrifugal compressor already has a very complicated shape, it involves more attention
to develop the grid.

In many cases having the grid is not enough because during the simulation
process it may be needed to reﬁne the mesh, which also needs some time if the object is
complicated. When the object is complicated, many steps are then required to generate
the grid. Some CFD solvers do not have a grid generator combined with them and they

require the user to build macros to import the grid. Grid generator software vary in

difﬁculties.

3.5.1.1 Geometry from Drawing:

The ﬁrst step in the grid generation process is to transfer the data from the
hardware to the software. This is a non-automated step that needs to be done manually. It
is the most time consuming step that needs to be done before any computational work.
For this step to be done the geometry was transformed from a non-standard coordinate
system to a standard one. There were 20 points given in the drawing for each edge of the
impeller’s blade. The edges of the blade are at the hub-pressure side, the hub-suction

side, the shroud-pressure side, and at the shroud-suction side.

71

The standard X, Y, Z system was chosen because of it is easiness to visualize and
for compatibility issue. Since the blades of the impeller are identical, getting the data for
one of them is enough to create the geometry of the whole impeller. It is also wise to do
the simulation for one blade to reduce the simulation time. After having the X, Y, and Z
coordinates for each point in the drawing, the blade geometry should be visualized using
one of the CAD software to check its smoothness and accuracy. AUTOCAD was used for
this purpose.

After making sure the blade shape looks right, Excel spread sheets were created to
manipulate the data and ﬁnd the angles and transfer the data to the cylindrical coordinate.
Then, the data were tabulated in a certain format in two ﬁles. Those ﬁles are imp.mnl,
which include the mean line data and imp.srf, which include the surface data. A third ﬁle
was created (impup.mnl) to represent the up stream segment (e. g. the inlet section of the
impeller). The Fourth ﬁle needs to be created is a data ﬁle that includes some input
parameters such as the general dimensions and number of blades. This ﬁle is named
in.dat A FORTRAN code, which was made originally by SOLAR TURBINES named,
was then used to create the geometry ﬁles that will be read by CFX-TASCﬂow.

The FORTRAN code generates six data ﬁles (named bld_sb.dat, bld_st.dat,
bld_nb.dat, b1d_nt.dat, interpst.dat, and interpsb.dat). These ﬁles are used as an input ﬁles

to CFX-TASCﬂow.

72

 

 

 

 

 

 

 

 

 

 

 

 

1mpup.mnl bl d sb.dat
I) bld_st.dat
1mp.mnl
_> bld_nb.dat

 

 

RADIALF

lmp'srf > b1d_nt.dat
:9 interpst.dat
in.dat
interpsb.dat

Figure 3-2 Flow chart for initial geometry ﬁles generated by FORTRAN code

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.5.1.2 CFX GRID:

Since Grid generation is the process of dividing the solution domain into discrete
sub-domains, TASCﬂow like other CFD package need a grid to work with. CFX-
TASCﬂow accepts grids that are boundary-ﬁtted, generally non-orthogonal and
curvilinear.

CFX-TASCgrid is the grid generator in TASCﬂow, and it generates three-
dimensional structured computational grids that are suitable for use with the rest of the
CFX-TASCﬂow software in the numerical solution of ﬂuid ﬂow and heat transfer
problems.

The computational domain is deﬁned as a three-dimensional (for 3-D problems) volume
completely enclosed by a three-dimensional surface which is called the exterior boundary
surface. Within the fully enclosed domain, there may be one region or more that are to be
excluded from the computational domain. Each of these excluded regions are called

internal objects. These objects must be enclosed fully by a single two-or three

73

dimensional interior boundary surface. A grid normally given in terms of the x, y and 2
location of grid nodes distributed throughout the computational domain. At each node in
the domain, the code should be able to determine values for all dependent variables
including pressure, velocity components, the turbulence quantities, temperature and
others. The nodes should be distributed throughout the volume enclosed by the exterior
boundary surface of the domain such that they form a complete three-dimensional matrix
of nodes. Each node in the matrix will have an index (i; j; k) with the range of i, j and k
given by:

l
1
1

IA

i

IA IA 'A

ID
j JD
k KD

IA IA

where ID, JD, and KD are the dimensions of the grid in the I, J and K respectively.

This type of node layout is often named structured or logically Cartesian.

It is useful to introduce the concept of a ﬂux element, in order to completely describe the
distribution of nodes in the computational domain. A ﬂux element, used in this work, is a

linear, hexahedral element deﬁned by eight nodes as shown in Figure 3-3.

74

 

i,i+1,k / i+1,i+1,k

   
 
 

i,i+1,k+1

i+1,i,k

 

 

i+1,i,k+1

 

Figure 3-3 Eight-noded hexahedral ﬂux element. (Adapted from CFX- TASC ﬂow)

As was mentioned earlier, TASCﬂow code adopts a strongly controlled volume
approach. To deﬁne a control volume within the node plan and ﬂux element deﬁnition,
already mentioned, the concept of an octant is introduced. An octant is deﬁned such that

eight octants forms a ﬂux element with an octant associated with each of the eight nodes

(Figure 3.3)

75

 

     

Integration Point
surfaces

7%

Figure 3-4 A Flux Element Divided into Octants (Adapted from CFX Manual)

Figure 3-45hows a ﬂux element divided into octants and one octant removed. The
six sides of each octant are divided into two groups. Some are coincident with the ﬂux
element sides and others are in the interior of the ﬂux element. These will form the
surfaces of the control volume over which surface integrals will be evaluated; they are
referred to as integration point surfaces.

Furthermore, the control volume over which mass, momentum, energy, etc. will
be conserved is deﬁned by all octants. The following notes give more understanding for

the interaction between the nodes and control volume.

0 Interior nodes are completely surrounded by control volumes (8 octants).

76

o Nodes that lie on a control volume side, edge or comer are called boundary and
they are not completely surrounded by the control volumes.

0 Control volumes, which are on exterior boundary surfaces, do not include more
than four octants.

0 Control volumes, which are on exterior boundary edges, do not include more than
two octants.

0 Control volumes, which are on exterior boundary comers, include only one

octant.

3.5.1.3 CFX creates its Grid:

After creating the initial geometry ﬁles, the work will be carried out through
many stages. TASCﬂow has CFX-TASCgrid as its grid generator. It generates three-
dimensional structured computational grids that are suitable to be with the other CFX-
TASCﬂow software in the numerical solution of ﬂuid ﬂow and heat transfer problems.
Figure 3-5is a ﬂow chart shows that the goal of the CFX-TASCgrid is to create the GRD
(Grid) ﬁle, which contains the coordinates of the ﬁnished grid. For this to be done a
series of steps must be followed. The CFX-TASCgrid had four subprograms which need
to be compiled in a certain order. Each subprogram corresponds to a phase in the grid

generation process.

77

    
  

1 m ' ’ 7
II 1' Condo
1 , 4,

 

   

 

  

TASthida

IL—

I IASthidi 1
w mscmma and nscwal II Gm) rue

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3-5 CFX-TASCgrid Data Flow Diagram

The ﬁrst phase is the Geometry Phase, which deﬁnes the geometry using a
subprogram called CFX-TASCgridg. It gives a means to describe the physical domain.
As can be seen from Figure 3-5, CFX-TASCgridg requires a ﬁle named GDF (Geometry
Deﬁnition File) to be its input. The GDF ﬁle itself needs other ﬁles to supply CFX-
TASCgridg the needed data. These ﬁles are the ones created by RADIAL.F plus two
macro ﬁles that will supply enough data for CFX-TASCgridg to create two ﬁles IGT
(Internal GeomeTry ﬁle) and CDT (Curve Distribution Template ﬁle).

After doing the ﬁrst phase, the second one may be done. The second phase is the

Curve Phase in which the nodes are distributed on curves using CFX-TASCgridc. The

78

ﬁrst grid nodes are attached to comers of the physical domain during this phase. The
numbers of grid nodes (ID, JD, and KD) are deﬁned as an input to this program. This
means that deciding if the grid is ﬁne or coarse is to be taken care of at this time. It is a
common practice in CFD to have the grid ﬁner where the ﬂuid properties change the
most and coarser other places (Figure 3-6 and Figure 3-7). For the current work the ID,
JD, and KD where chosen to be 25, 25, 115 respectively. This will give 25*25*115 grids.
These numbers were chosen based on industrial experience (Solar Turbines). Moreover,
the nodes will lie on the edges of the domain. Figure 3-5 shows that the output of the ﬁrst
phase was used as inputs to the second phase plus the CDF (Curve Distribution File).

The third phase is the surface phase where CFX-TASCgrids is used to distribute
nodes on surfaces. This program uses the output ﬁles of the pervious phase, CND (Curve
Node Distribution ﬁle) and SDT (Surface Distribution Template ﬁle), to be its input ﬁles
in addition to one ﬁle from the ﬁrst phase (IGT).

The fourth phase is the interior phase in which the CFX-TASCgridi program is
used to distribute nodes in the interiors of regions. This program also uses an output ﬁle
of the pervious phase, SND (Surface Node Distribution ﬁle), to be its input ﬁle in
addition to one ﬁle from the ﬁrst phase (IGT) and [DP (Interior Distribution File). The
output of this phase is the ﬁnal output of CFX-TASCgrid which is GRD ﬁle (GRID

coordinates ﬁle).

3.5.1.4 Grid Quality:
At this stage the grid is ready for CFX-TASCﬂow, but its quality needs to be
tested before going any farther. This can be done using CFX-TASCtool to do three major

checks for the grid. The ﬁrst check is for the volumes to know if all the volumes (and

79

Octants) are positive if there will be even one negative the code will not start. The second
check is for the skew to know the maximum and minimum angles in the grid. The angles
should be above 20 to have no warnings but still the code may give good results even if
there are some angels above 15. The third check that needs to be performed is the aspect
to be between 1 and 100 in value. If there is a problem with the grid, the original input
geometry ﬁles need to be reviewed and it may require some adjustments for ﬁles such as
CDF. If the quality check did not turn the way it should be, then trying to ﬁx it may

consume a lot of time.

@FX

 

 

 

 

 

Figure 3-6 Blade to Blade grid that is used in the simulation

80

 

 

 

 

 

 

Figure 3-7 Inlet to Exit Grid that is Used in the Simulation

3.5.2 Turbulence Modeling

Modeling the turbulences in the ﬂow is what makes modeling the ﬂow in the
impeller very difﬁcult. This is because of the high rotation of the impeller and high
velocities of the ﬂuid and the compressibility affects. Up to this date, no one turbulence
modeling can handle all the ﬂuid ﬂows. Some of them are preferred in some cases others
are better in other cases. Experienced people suggest that for ﬂow in a centrifugal
impeller Ronald’s Stress Model works the best because it takes care of the centrifugal
motion.

In theory, the Navier-Stokes equations describe both laminar and turbulent ﬂows

without the need for additional information. But turbulent ﬂows at high Reynolds

81

numbers experience a large range of turbulent length- and time scales. The direct
numerical simulation (DNS) of these ﬂows would require a very high computing power.
This power is many orders of magnitude higher than what could be available in the
foreseeable future. It is therefore necessary to reduce the complexity of the simulations
by the introduction of the Reynolds averaged equations, which describe the mean ﬂow,
without a need for the resolution of the turbulence. The averaging procedure introduces
additional terms into the equations that need to be modeled in order to achieve what is
called closure, to have a sufﬁcient number of equations for all the unknowns, including
the Reynolds-Stress tensor resulting from the averaging procedure.

The Reynolds-averaged Navier-Stokes (RANS) equations correspond to the
transport equations for the mean ﬂow quantities only. This approach greatly reduces the
time required for computational when compared with a direct numerical simulation and is
generally adopted for practical engineering calculations. Closure of the equations is
achieved by the introduction of additional equations to model turbulent correlations, like
the turbulent kinetic energy. The Reynolds-Stress tensor can be related to the mean ﬂow
variables by the mean of these variables.

The following (RANS) turbulence models are available in CFX-TASCﬂow:
o Two-Equation Models
0 Standard b- amodel
o k- e RNG model
0 Standard k-w model
0 ESL and SST zonal k-w based models

0 Algebraic-Stress Models (ASM)

82

 

o k- 8 based ASM model
0 k-w based ASM model
0 Zonal k-w based ASM model
0 Reynolds-Stress Models (RSM)
o Launder, Reece and Rodi model (LRR)
o Speziale, Sarkar and Gatski (SSG)
This gave a variety of choices for TASCﬂow to choose the most suitable one

depending on the time, grid, and ﬂow type. (TASCﬂow User Documentation).

3.5.3 Boundary Conditions Used in Simulation

Boundary conditions speciﬁcation is an essential part in using CFD. They vary
from symmetric to periodic, from wall to another, and from inlet to outlet. As was
mentioned earlier specifying the boundary conditions is a pre-processing process. CFX-
TASCbob3D is a subprogram in CFX-TASCﬂow that was used to specify the boundary
conditions. BCF (Boundary Conditions File) is the ﬁle where the boundary conditions
are stored. When specifying the boundary conditions, the location must be given with
respect to the nodes.

The boundary conditions used in this work are as follows

- Inlet: the pressure and Temperature is given.

83

 

@FX

 

 

 

 

Figure 3-8 Inlet Boundary Condition

0 Outlet: the mass ﬂow rate is given.

 

"@TFX

 

 

 

 

 

Figure 3-9 Outlet Boundary Condition

84

0 Wall (stationary) which is the none moving boundary.

 

Em

 

 

 

 

Figure 3-10 Wall (stationary) Boundary Condition

0 Wall (moving) which is either the hub or shroud.

 

@1323.

 

 

 

 

Figure 3-11 Wall (moving) Boundary Condition

85

0 Periodic: this boundary is making the problem easer to solve because the number

of nodes is small compared to the whole impeller.

 

@[FX

 

.......

 

 

 

Figure 3-12 Periodic Boundary Condition

0 Symmetric: this boundary condition is some times wanted to take care of the slip.

@IFX

 

 

 

 

 

Figure 3-13 Symmetric Boundary Condition

86

3.5.4 Use of Initial Conditions

The initial ﬂow conditions need to be set using CFX-TASCtool before starting the
solver. These initial conditions are stored in GCI (General Command Initialization). For
this study the initial conditions were Pressure, Temperature, and turbulence model

parameters.

3. 5.5 Control Parameters
Before running the solver, a ﬁle containing parameters that control the solution of
the ﬂuid ﬂow problem should be created (Parameter ﬁle containing job control

parameters). These parameters include the rotational speed, the memory needed.

3.5.6 Solver
The ﬂuid ﬂow solver that was used is CFX-TASCﬂow3D or “ﬂow code” of CFX-
TASCﬂow. The governing equations of ﬂuid ﬂow are coupled, non-linear, and partial

differential equations.

When it starts running, CFX-TASCﬂow3D requires no interactive input from the
user because it is a batch-oriented program. Selecting input parameters enables the code
to produce a converged solution for a certain problem. The convergence criterion was to
have the maximum residual to be 0.00005. The user is able to see the convergence of his
problem while the simulation is running and some times it will save some time.

After completing the simulation, the user will be able to look at the results in many

forms some of them will be shown and discussed in chapter 5.

87

Chapter 4 EXPERIMENTAL DATA ANALYSIS
4.1 Background

A series of compressor rig tests were done between 1976 and 1977 at Solar Turbine
Laboratories. Four different impellers with two conﬁgurations each were tested. The
wide term was used refer to the original impeller tip width, while the narrow is used to
refer to the impellers after the trim. The impeller exit tip widths were reduced by 25%
each. The tested stages’ speciﬁc speed ranged from 59 to 135. The four impellers were

named B, C, D, and E starting from lower to higher speciﬁc speeds (Figure 4-1).

 

Shroud after cut

 

 

 

 

Figure 4-1 The Tested Impellers Before and After the Modiﬁcation

A turbocharger with shop air was used to drive the compressor rig. The test gas
used was the ambient air in an open loop. The total (stagnation) pressures and the static

pressures instrumented from the compressor inlet to the exit scroll. The diffusers were

88

unpinched in all cases. Table 4-1 shows all major dimensions with respect to a reference

dimension (Rref) for the eight impellers and their vaneless diffusers. All the tested

impellers were open-faced (unshrouded) impellers.

Table 4-1 Major stages dimensions in dimensionless form

 

B(0rig)

B(T rim)

C(Orig)|C(Trim)

D(Orig)

D(Trim)

E(01‘ig)

E(Trim)l

 

Impeller

 

Inlet Shroud
Radius
(R 1 s/Rref)

6.818

6.818

7.182 7.182

8.000

8.000

9.091

9.091

 

Inlet Hub
Radius
(R 1 h/Rref)

4.545

4.545

4.545 4.545

4.545

4.545

4.545

4.545

 

Exit Tip Radius
(R2/Rref)

12.727

12.727

12.727 12.727

12.727

12.727

12.727

12.727

 

Exit blade width
(B2/Rref)

1.000

0.749

1.273 0.956

1.716

1.356

2.545

1.909

 

Vaneless
diffuser

 

Inlet Radius
(R3/Rref)

14.182

14.182

14.182 14.182

14.182

14.182

14.182

14.182

 

Inlet width
(B3/Rref)

0.909

0.909

1.164 1.164

1.636

1.636

2.364

2.364

 

Exit Radius
(R4/Rref)

19.091

19.091

19.091 19.091

19.091

19.091

19.091

19.091

 

Exit width
(B4/Rref)

 

0.909

 

0.909

 

1.164 1.164

 

 

1.636

 

1.636

 

2.364

 

2.364

 

 

 

89

 

4.2 Effect of Specific Speed on Impeller Characteristics

The speciﬁc speeds for the original B, C, D, and E impellers are 58, 82, 115, and
126 respectively; while the speciﬁc speeds for the trimmed B, C, D, and E impellers are
68, 84, 117, and 133 respectively. So, for both conﬁgurations the speciﬁc speed increases
from the B through C then D to E. Having two sets of impellers gives a great
understanding of the effect of changing the speciﬁc speed on the impeller characteristics.
Looking carefully at the general trend of ﬁgure 4.2 and 4.3, a simple conclusion such as
stages with higher impeller speciﬁc speed gives higher stage efﬁciency can be made. This
can be said about the wide impellers as well as the narrow ones.

Figure 4.4 and 4.5 shows how the operation range changes from a narrow one for

the lower speciﬁc speed impeller to wider operation range as the speciﬁc speed increases.
An additional factor needs to be considered in this case which, is that each impeller of

these impellers has its own range of ﬂow coefﬁcients.

 

Stage Efﬁciency

nT/nT rel

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

 

(11¢ ref

 

 

Figure 4-2 Variation Of The Stage Efﬁciency For The Wide Impellers

90

 

 

 

nT/nT ref

Stage Efficiency

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0

43/0) ref

 

Figure 4-3 Variation Of The Stage Efﬁciency For The Narrow Impellers

 

WT/LPT ref

Stage Head Coefficient

1.20
1.00
0.80
0.60
0.40

0.20

 

0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

CD/CD ref

 

Figure 4—4 Stage Head Coefﬁcient Comparison For The Wide Impellers

91

 

 

 

Stage Head Coefficient

1.20
1.00
0.80
0.60

llJT/KPT ref

0.40
0.20

 

0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0

 

 

¢l¢ ref

 

Figure 4-5 Stage Head Coefﬁcient Comparison For The Narrow Impellers

The variation of the work factor is shown in ﬁgures 4.6 and 4.7. Looking at each
series alone, one may notice that the change is almost liner. Another valid notice is that
the slopes of these lines are all negative. If the four lines in each ﬁgure are compared, it is
clear that the slope tents to be less negative as the speciﬁc speed increases.

Almost exactly the same thing may be said about ﬁgures 4.8 and 4.9, which show the
variation of DeltaT/1‘ 1 with the ﬂow coefﬁcient. DeltaT refer to the change of the total

temperature between the inlet and exit of each impeller.

92

 

 

q/q ref

Work Factor

1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
(1)/CD ref

 

Figure 4-6 Work Factor Variations for The Wide Impellers

 

 

q/q ref

Work Factor

1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00

 

CID/CD ref

 

Figure 4-7 Work factor variations for the narrow impellers

 

 

 

 

 

 

 

 

 

Delta T/T1
2‘3
E
5..
S,
E
3
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
¢/¢D ref
Figure 4-8 DeltaT/T1 variations for the wide impellers
Delta T/T1
1.40
“9:, 1.20
’E‘ 1.00
i— 0.80
S
a 0.60
E 0.40
3 0.20
0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0
cp/cp ref

 

 

 

Figure 4-9 DeltaT/T1 variations for the narrow impellers

The change of the relative diffusion (W2/Wltip) with respect to the ﬂow
coefﬁcient is shown in ﬁgures 4.10 and 4.11. Looking at the individual impeller, one may

notice that all the variations are almost linear. The positive slope is another characteristic

94

of these lines. Comparing the four lines in each ﬁgure shows that as the speciﬁc speed

increases the slope tends to be less.

 

Relative Diffusion (W2/W1tip)

 

(W2NV1tip)/(W2/W1tip) ref

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
CD/CD ref

 

 

 

Figure 4-10 The change of the relative diffusion (W2/W1tip) for the wide impellers

 

Relative Diffusion (W2/W1tip)

(W2/W1tip)/(W2NV1tip) ref

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0

(DIG) ref

 

 

 

Figure 4—11 The Change of the Relative Diffusion (W 2/Wltip) For the Narrow Impellers

95

Another important characteristic of any impeller ﬂow is the impeller exit ﬂow
angle (Alpha2). The normalized angles are given for both the wide and the narrow
impellers in ﬁgure 4.12 and 4.13 respectively. These ﬁgures show that as the speciﬁc
speed gets higher, the curve tends to ﬂatten over the operating range and have more

negative slope.

 

Impeller Exit Flow Angle (Alpha2)

1.20
1.00
0.80
0.60

02/02 ref

0.40
0.20

 

0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

¢/¢ ref

 

 

 

Figure 4-12 The Change of the Impeller Exit Flow Angle (Alpha2) for the wide impellers

96

 

Impeller Exit Flow Angle (Alpha2)

1.20
1.00
0.80
0.60
0.40
0.20
0.00

02/02 ref

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0
(1)/CD ref

 

 

Figure 4-13 The Change of the Impeller Exit Flow Angle (Alpha2) for the narrow
impellers

4.3 Rotational Speed Effects

This part discusses the effect of changing the compressor rotational speed on the
performance of the four impellers. These impellers are B and E with both conﬁgurations
wide and narrow. These four were chosen from the eight impellers speciﬁcally because
they are the extreme ones in their speciﬁc speed and operation range. These experimental
results refer to four different rotational speeds 22000, 28000, 34000, and 40000 rpm for
the four impellers.

Figures 4.14 to 4.17 shows that stage efﬁciency stays almost constant at the design
point in all rotational speeds for the low speciﬁc speed impellers (B wide and narrow),
while it clearly decreases with increasing the rotational speed for the high speciﬁc speed
impellers (E wide and narrow). The same thing can be said about the impeller efﬁciency
for the high speciﬁc speed impellers (Figure 4.20 and 4.21). For the low speciﬁc speed
impellers (spatially the narrow one), the impeller efﬁciency is clearly more for the higher

rotational speed (Figure 4.19).

97

 

 

O 0.5 1 1.5 2

 

 

0/0 ref

 

Figure 4-14 Variation of the Stage Efﬁciency for B Wide at Variable Rotational Speeds

 

Sta eEfficienc -BN

    

1.2
1.0

'- 0.6

WI]

0.4
0.2
0.0

OIO ref

 

 

 

Figure 4-15 Variation of the Stage Efﬁciency for B Narrow at Variable Rotational Speeds

98

 

Sta e Efﬁcienc -EW

    

1 .2
1 .0
0.8
0.6
0.4
0.2
0.0

nTInT ref

 

 

4
43/0 ref

 

Figure 4-16 Variation of the Stage Efﬁciency for E Wide at Variable Rotational Speeds

 

nTInT ref
.0 0 .0 .-‘ .-‘
A m on o lo

.0
N

 

.0
o

3
0l¢ ref

 

 

Figure 4-17 Variation of the Stage Efﬁciency for E Narrow at Variable Rotational Speeds

99

 

 

0 0.5 1 1.5 2
$10 ref

 

 

Figure 4-18 Variation of the Impeller Efﬁciency for B Wide at Variable Rotational
Speeds

 

 

OIQ ref

 

 

 

Figure 4-19 Variation of the Impeller Efﬁciency for B Narrow at Variable Rotational
Speeds

 

    

lm eller Efficienc -EW

 

3 4
¢I¢> ref

 

 

Figure 4—20 Variation of the Impeller Efﬁciency for N Wide at Variable Rotational
Speeds

 

 

 

3
¢I¢ ref

 

 

Figure 4-21 Variation of the Impeller Efﬁciency for N Narrow at Variable Rotational
Speeds

Another interesting point worth mentioning is that the referenced head coefﬁcient
increases at the design point with the increase of the rotational speed for the low speciﬁc

speed impellers (ﬁgures 4.22 and 4.23) while it clearly increases with decreasing the

101

rotational speed for the high speciﬁc speed ones (ﬁgures 4.24 and 4.25). Looking back to
these four ﬁgures provides fair conﬁdence to correlate the wideness of the operating

range to the lowering the rotational speed in the given range of the rotational speeds.

 

Sta e Head Coefficient-8W

       

§ 0.8
l-

0.6
E

3 0.4

0.2

o

o 0.5 1 1.5 2
we ref

 

 

 

Figure 4-22 Variation of the head coefﬁcient for B wide at variable rotational speeds

 

 

 

 

 

Figure 4-23 Variation Of The Head Coefﬁcient For B Narrow At Variable Rotational
Speeds

102

 

WTNT ref

 

0/0 ref

 

 

Figure 4-24 Variation of The Head Coefﬁcient for E Wide at Variable Rotational Speeds

 

Stage Head Coefficient-EN

    

E 0.8

I-
0.6
E

9 0.4

 

0 1 2 Oléref 4 5 6

 

 

Figure 4-25 Variation Of The Head Coefﬁcient For E Narrow At Variable Rotational
Speeds

Figures 4.26-4.29 show that the work factor is higher for the higher rotational
speed, it then goes down when the rotational speed decreases. This is true for most of the
operation range except for a relatively high ﬂow coefﬁcient where the work factor

decreases more rapidly for higher rotational speed.

103

 

Work Factor-BW

    
    

o
.0
or

1 1.5
0/0 ref

1'0

 

 

 

Figure 4-26 Variation of the Work Factor for B Wide at Variable Rotational Speeds

 

1.4 Work Factor-8N

1.2

§ 0.8
g 0.6
0.4
0.2

o
.0
or

1 1.5 2
0/0 ref

 

 

 

Figure 4—27 Variation Of The Work Factor For B Narrow At Variable Rotational Speeds

104

 

Work Factor-EW

    
    

1.4
1.2

‘5 0.8
g 0.6
0.4
0.2

o
.5
N

 

3 4
¢I¢ ref

 

 

Figure 4—28 Variation of the work factor for E wide at Variable Rotational Speeds

 

Work Factor-EN

0.8
a. 0.6
0.4
0.2

O
—L
N

 

 

3
tit/OD ref

 

Figure 4-29 Variation of the work factor for E narrow at Variable Rotational Speeds

105

4.4 Effect of Trimming an Impeller

The tested impellers include four original impellers in addition to four trimmed
ones, which cover a wide area of speciﬁc speeds. Comparing the original to the trimmed
impeller is another task of this study to know the effect of trimming an impeller on its
performance. Industry companies prefer to modify an existing design rather than
designing new equipment. Even though the impellers differ in some of their dimensions,
all the original ones were smoothly cut 25% at their impeller exit width from the shroud
side.

Trimming all four impellers showed better impeller efﬁciency (Figure 4.30).

Looking at ﬁgure (4.31) shows also an increase in the stage efﬁciency after trimming the

 

 

impellers.
Impeller Efficiency
1.20
1.00
._ 0.80
2
24 0.60
N
c 0.40
0.20
0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
<D/<D ref

 

 

Figure 4-30 Impellers Efﬁciencies Before and After Trimming

106

 

Stage Efficiency

1.20
1.00
0.80
0.60

nT/nT ref

0.40
0.20

 

0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

 

41/4) ref

Figure 4-31 Stage Efﬁciencies for All Impellers before and After Trimming

Impellers’ head coefﬁcients (Figure 4.32) had decreased for all four impellers
after trimming them. Stage head coefﬁcients (Figure 4.33) also experienced a slit increase
after the modiﬁcation. Work factors at all points for all impellers were clearly decreased
after trimming them (Figure 4.34). That decrease was almost constant for every impeller
after the cut, which is represented by almost a parallel straight line for the trimmed
impeller. The variation of the temperature across the impellers divided by the inlet

temperature for was given in ﬁgure 3.35 before and after trimming.

107

 

 

 

 

 

    

Impeller Head Coefficient
”é
or
3-
R
3-
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
¢/<I> ref
Figure 4-32 Impellers Head Coefﬁcients Before and After Trimming
Stage Head Coefficient

120 ,

1.00 _
~03 0.80 g, ' '
5' 0.60
r: . .
3 0.40 W

0.20

0.00 '
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

 

cp/o ref '

 

Figure 4-33 Stage Head Coefﬁcients for All Impellers Before and After Trimming

108

 

Work Factor

    
    

q/q ref

0.0 1 .0 2.0 3.0 4.0 5.0 6.0 7.0

¢l¢ ref

 

 

Figure 4-34 Work Factors for All Impellers Before and After Trimming

 

Delta T/T1

(AT/T1 )/(AT/T1 ) ref

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

(1)/(D ref

 

 

Figure 4-35 Variation of Deltat/T 1 For All Impellers Before and After Trimming

The relative diffusion, Wlnns/W 2, is one of the considerations in determining the
tip width of an impeller design. This relative diffusion for all impellers was clearly

reduced as shown in ﬁgure 4.36 because of the 25% reduction in the tip width.

Another non-dimensional quantity of interest to the designer is the WZIWltip relative
diffusion. Normalized values for these quantities are given in ﬁgure 4.37. Obviously, the

values of these quantities had increased after the impellers were cut.

 

Relative Diffusion (W 1 rmsNV2)

(W1 rms/W2)/(W1 rms/W2) ref

 

¢/¢> ref

 

 

Figure 4-36 Variation of erms/W 2 for All Impellers Before and After Trimming

 

Relative Diffusion (W2/W1tip)

(W2NV1 tip)/(W2NV 1 tip) ref

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

¢l¢> ref

 

 

Figure 4-37 Variation of W2/W1tip for All Impellers Before and After Trimming

110

The variation of the impellers’ exit ﬂow angles are given in the following ﬁgure.
Looking at this ﬁgure, provides great conﬁdence that trimming such impellers will
reduce the impeller exit ﬂow angle. This is sometimes desired to satisfy some design
considerations, such as matching the diffuser design. This ﬁgure also shows that the

decrease is more for all impellers when the ﬂow coefﬁcient gets higher.

 

Impeller Exit FIow Angle (Alpha2)

    

02/02 ref
0
a:
O

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

 

(1)/d) ref

Figure 4-38 Variation of The Impellers Exit Flow Angles Before and After Trimming

lll

Chapter 5 CFD RESULTS AND VALIDATION
5.1 Background

This chapter deals mainly with the results of the CFD model. Experimental and l-D
code results are available for validation. The Experimental results were provided by Solar
Turbines Company while the l-D results were obtained from a code that was developed
by the MSU Turbomachinery Lab. CFD models were created for four impellers to
simulate the ﬂow to match the test data from the original rig tests. The primary reason to
analyze these stages is to understand the performance of relatively low and high speciﬁc
speeds stages before and after impellers cut.

There are two important facts about CFD results obtained. The ﬁrst one is that
output of CFD model depends primarily on the inputs that the user supplies to the
computer. These inputs may be divided into two main categories. The first category is
supplying the equations and models used to simulate the ﬂow problem, which was mainly
taking care of by CFX TASCﬂow that is a known CFD package in turbomachinery
simulation. The other category is supplying the parameters and values, like the grid
generation and boundary condition, used to run the model. The second important fact
about the CFD results is that it gives much more information about the ﬂow than what
can be obtained from the test data or the 1-D results. More speciﬁcally, the data needed
may be obtained at locations where the experimental setup or l-D analysis would not

help because of the some difficulties.

112

 

5.2 Primary Check

The CFD simulation that was done for this study includes running about 50 cases.
Running these cases took variable times using a Pentium 4 computer with 526 Ram.
Some took about 2; hours others took tens of hours to be done. The convergence criterion
was chosen to be that the maximum residual is less than 5*10'5. The major sets of these
cases refer to the wide B, narrow B, wide E, and narrow E impeller. After doing the
required preprocessing and running the code, the results were obtained. Being able to get
the results doesn’t necessarily mean that they are correct. So, it is required to check the
output of the code. This may be primarily done by checking the output with common
behavior of such ﬂow. The results then need to be validated with the other methods of
getting the result.

For this study the CFD results, obtained from CFX TASCﬂow, were primarily
compared with the common behavior of the ﬂuid ﬂow in centrifugal compressor. Figure
2-7 and Figure 2-8 show that the pressure and temperature trend in centrifugal
compressors. The variation of the total pressure, static pressure, total temperature, and
static temperature matched what was expected as in Figure 2-7 and Figure 2-8. Other
parameters like the density were also check and the CFD output where matching what

was expected .

5.3 CFD Validation

In general, the CFD results give a good trend match compared with the test and l-D
results. A more in-depth look at these results shows some carves with almost exact
matching; other show some differences in the values obtained for some of the

dimensionless parameters.

113

Figures 5.5 and 5.6 show the stage efficiency curves for the low, speciﬁc-speed
impellers (B wide and narrow), while ﬁgures 5.7 and 5.8 show it for the high speciﬁc
speeds impellers (E wide and narrow). In all cases, CFD over predicts the stage
efﬁciency. For the low speciﬁc speeds impellers the over prediction is constant with a
value about 20% from the design point to surge point (the lowest ﬂow coefﬁcient) while
it is more above the design point. For the same impellers, the stage efﬁciency was well
predicted using the l—D code above the design point (higher ﬂow coefﬁcient) but almost
as good as the CFD for low ﬂow coefﬁcient. For the high speciﬁc-speeds impellers (E
wide and narrow) the over prediction is minimum at the design point (less than 10%), but

it increases as the ﬂow coefﬁcient gets higher or lower.

 

Stage Head Coefficient_BW

 

 

1.4
-e—CFD
1.2 «+I1E>g’-——A 4

 

 

1.0 -~ --____,
0.8 -m, —
0.6 -v -—
0.4 —-———-t~
0.2

0.0 . . . r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0/0 ref

 

 

WT/‘PT ref

 

 

 

 

 

 

 

 

 

 

Figure 5-1 Experimental, CFD, and l-D Stage Efficiency for B Wide

114

 

 

Stage Efflcie

ncy_BN

      

0.0 0.2 0.4 0.6 1.0 1.2 1.4 1.6

0.8
OIO ref

 

 

Figure 5-2 Experimental, CFD, and l-D Stage Efﬁciency for B Narrow

 

 

Stage Efficiency_EW

    

3 4
@l0 ref

 

Figure 5-3 Experimental and CFD Stage Efﬁciency for E Wide

115

 

Stage Efficiency_EN

9 0.8

nT

p 0.6
C

 

 

3
QIO ref

 

 

Figure 54 Experimental and CFD stage efﬁciency for E narrow

The following four ﬁgures (5.9-5.12) show the impeller efﬁciency for the four
impellers. For all of them, CFD also over predicts the impeller efﬁciency while the 1-D
code gave accurate values for the low speciﬁc-speeds impellers. The over prediction of
CFD for low speciﬁc-speed impellers stays constant (about 10%) when the ﬂow
coefﬁcient gets lower than the design point. On the other hand, for high speciﬁc-speed
impellers the over prediction is minimum (about 5%) at the design point then increases as

ﬂow coefﬁcient increases or decreases.

116

 

 

Impeller Efficiency_EW
1.2 ’ .— < i .1 lip/,2}: : I“ if"? 777477“- '17,??? ' , , .7 T :74

 

    
   

 

 

 

 

   

iri .
Xl‘r

III

 

.I .,
In. ‘III‘ ...II

 

 

 

 

 

 

 

1.2 - -
1.0 "
“£0.23

“0.4
0.2 W
0.0

 

0.0 0.2 0.4 1.0 1.2 1.4 1.6

0:19er

 

Figure 5-6 Experimental, CFD, and l-D impeller efﬁciency for B narrow

117

 

 

 

 

Impeller Efficiency_EW

 

1.2

1.0 _..
20.8 T

h

c0.6

 

 

 

 

 

 

0.4

 

 

0.2

 

 

 

0.0

 

 

Figure 5-7 Experimental and CFD Impeller Efficiency for E Wide

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Impeller Efficiency_EN
1'2 —e—CFD H i w
W\

1.0 r -I— EXP PM \'
“950.8 ~--- — — —- -~ M v -_ 4
N I
50-6 -- —

t{0.4 .._ — —— — — a
0.2 -1”. — —————— —————~
0.0 . .

2 4
0 Old? ref 5 6

 

 

Figure 5—8 Experimental and CFD impeller efﬁciency for E narrow

Another important characteristic of any stage is the head coefﬁcient. It gives good

indication of the surge and choke limes of the ﬂow. Figures 5.13 to 5.16 shows that CFD
over predict the head coefﬁcient. For the low speciﬁc-speeds impellers (Figure 5.13 and

5.14), the over prediction is constant from the lowest ﬂow coefﬁcient to the design point

118

then it increases as the ﬂow coefﬁcient increases. For the same impellers the 1-D code
predicts the head coefficient well at the higher ﬂow coefﬁcient but over predict it at
lower ﬂow coefﬁcient.

For the high speciﬁc-speeds impellers (Figure 5.15 and 5.16) the best CFD
prediction was at the design point (less than 20%), but it increases as the ﬂow coefﬁcient

increases 01' decreases.

 

Stage Efficiency_BW

    

nT/nT ref

0.0 0.2 0.4 0.6 1.0 1.2 1.4 1.6

0.3
we ref.

 

 

 

Figure 5-9 Experimental, CFD, and l-D head coefﬁcient for B wide

 

Stage Head Coefficient_BN

WTI‘PT ref

 

 

 

 

Figure 5-10 Experimental, CFD, and l-D head coefﬁcient for B narrow

119

 

 

 

 

 

 

Stage Head Coefficient_EW
’E
E
'—
3-
0 1 2 3 4 5 6
0/0 ref
Figure 5-11 Experimental and CFD head coefﬁcient for E wide
Stage Head Coefficient_EN
’2'
.-
E
3-
0 1 2 3 5
01¢ ref

 

 

Figure 5—12 Experimental and CFD head coefﬁcient for E narrow

Figure 5-13 to Figure 5-16 show comparison of the work factor curves. In
general, the ﬁrst two ﬁgures show excellent match, between all three methods for the low
speciﬁc speed impellers. In addition, the match is exact at the design point. For the high

speciﬁc speed impellers (ﬁgures 5.19 and 5.20), there is a good match between the

120

experimental and CFD results at the lowest ﬂow coefﬁcient; but as it increases, the CFD

over predicts the values.

 

Work Factor_BW

 

0.0 0.2 0.4 0.6 1.0 1.2 1.4 1.6

 

0.8
010 ref

 

 

Figure 5-13 Experimental, CFD, and l-D Work Factor for B Wide

 

Work Factor_BN

    

0.0 0.2 0.4 0.6 .8 1.0 1.2 1.4 1.6
CID/g ref

 

 

 

Figure 5-14 Experimental, CFD, and l-D Work Factor for B Narrow

121

 

 

Work Factor_EW

 

3 4
@IO ref

 

Figure 5-15 Experimental and CFD Work Factor for E Wide

 

 

Work Factor_EN

    

3
0M ref

 

Figure 5-16 Experimental and CFD Work Factor for E Narrow

 

 

The variation of the impellers’ exit-ﬂow angles are given in the following four

Figure 5-17 to Figure 5-20. Comparing these curves, the ﬁrst two ﬁgures show excellent
match, between all three methods for the low speciﬁc speed impellers. Moreover, the

match is exact at the design point. For the high speciﬁc speed impellers (ﬁgures 5.23 and

122

5.24), there is a good match between the experimental and CFD results near the design

point, but clearly the values get off for higher and lower ﬂow coefﬁcients.

 

Impeller Exit Flow Angle (AIpha2)_BW
1.2 .. . ,.. ,, ,,,, -,_ ,,,, ,,

 

 

 

 

 

 

 

 

 

 

 

 

 

. .2 .4 0.6 . 1. 1. . .
00 0 0 M8?“ 0 2 14 16

 

 

 

Figure 5-18 Experimental, CFD, and 1-D Impeller Exit Flow Angle for B Narrow

123

 

Impeller Exit Flow Angle (Alpha2)_EW

1.0
Eon
30.6
N
0.4
0.2

 

0.0

 

 

30/0 ref 4

 

Figure 5-19 Experimental and CFD Prediction of Impeller Exit Flow Angle for E Wide

 

Impeller Exit Flow Angle (Alpha2)_EN

    

1.0
50.8
30.6
0.4
0.2
0.0

0 1 2 ¢/¢3ref 4 5 6

 

 

 

Figure 5-20 Experimental and CFD Prediction of Impeller Exit Flow Angle for E Narrow

124

Chapter 6 CONCLUSIONS

This study mainly consisted of analyzing experimental data as well as creating a
complete CFD models and validating the results with experimental and 1-D results. to
have reliable data then to be able to analyze and validate it are important parts of a
successful study. Analyzing the experimental data for a verity of impellers and different
rotational speeds provided conﬁdence to generalize the results. This study included more
than 200 data points, which cover a variety of impeller conﬁgurations at different speeds,
which provide more convenience to generalize the results.

CFD is another tool that was used in this study. CFD analysis could be the future
of design in turbomachinery world. With the development of computers and CFD
packages, the designer needs to do much less work to design and analyze compressors.
CFD models give a much deeper understanding of the ﬂow inside centrifugal
compressors and enable the user to solve many problems easier and much faster. To
ensure accurate CFD models, extensive simulations and validation of the results was
done.

Analyzing the experimental data shows the effect of changing the speciﬁc speed, the
rotational speed, and 25% impeller trimming on the stage and impeller dimensionless
parameters.

Four CFD models were created for impellers with variable speciﬁc speeds. These
impellers are two original impellers with two extreme speciﬁc speeds (58 and 125) in
addition to the two modiﬁed (trimmed) impellers.

Overall the analysis shows logical results, a reasonable agreement, and an

agreement in the trend with most of the 1-D and CFD results.

125

More CFD analysis for the other four impellers (C original, C trimmed, D
original, and D trimmed) will give more convincing results. Having more experimental
data about different impellers’ outer diameters (before and after cut) will be a great help

to generalize the results even more.

126

APPENDIX

Table A: The experimental referenced values of all the pointes for both the wide and
narrow impellers. The shaded values are the design point for each impeller.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B-Original
Phl 5m -2IEff1-efo PsI2 Psl1- q (Work) Del rm W2/W1fip W1rms/W2 Alpha2 Ns
0.6411015 0.895 1.1911050 1.167 1.179 0.644 1.527 1.047 46
0.7921020 0.960 1.1141048 1.086 1.091 0.798 1.242 1.030 51
9711.000 1.000 0.99 0.995 0.990 1.000 0.993 1.002 0.999 58
1.1840932 0.966 0.833 0.862 0.889 0.897 1.208 0.828 0.949 72
1.3800734 0.784 0.5710609 0.774 0.781 1.455 0.692 0.875 101
B-Trimmed
Phi Effr- Efff-efaﬁsfz Psit-elq(Work)DeIT/T11W2/W1fip W1rms/W2 Alpha2 Ns
0.5711013 0.852 1.1760987 1.155 1.168 0.707 1.397 1.033 45
07431.019 0.937 1.0610975 1.036 1.042 0.946 1.047 1.000 52
0.9101013 0.979 0.964 0.930 0.947 0.956 1.137 0.873 0.959 60
1.069 0.987 1.000 0.8510861 0.858 0.863 1.330 0.750 0.908 68
1.2160929 0.978 0.727 0.765 0.778 0.783 1.507 0.664 0.850 80
1.38 0.782 0.854 0.533 0.581 0.678 0.682 1.736 0.580 0.757 104
1.47 0.582 0.662 0.352 0.400 0.601 0.606 1.916 0.527 0.673 143
C-Original
Phi Eff1-2Etf1-eto P312 P311- q (Work) Del T/T1 W2/W1tlp W1rms/W2 Alphaz Ns
.922 0.984 0.895 1.1371033 1.149 1.159 0.638 1.528 1.035 55
1.2410983 0.965 1.0331013 1.045 1.053 0.831 1.180 1.002 65
1.4630979 0.995 0.962 0.977 0.978 0.987 0.964 1.023 0.972 73
1.6820970 1.014 0.903 0.942 0.926 0.934 1.067 0.930 0.941 80
1.785 1.045 0.943 0.898 82
20940929 1012 0.7610828 0.815 0.823 1.278 0.786 0.864 98
2.5550638 0.702 0.416 0.458 0.649 0.656 1.594 0.641 0.711 170
2.6410515 0.428 0.322 0.267 0.622 0.628 1.652 0.620 0.673 258
C-Trimmed
th Eft1- Effr-eforl Psi2 17311-qu (Work) Del T/T1 wzxwm; W1rms/W2 Alpha2 Ns
08611.020 0.917 1.1611043 1.133 1.143 0.708 1.376 1.017 53
09731023 0.965 1.1111045 1.079 1.090 0.809 1.206 1.001 56
1.1351027 1.024 1.0401036 1.008 1.018 0.950 1.030 0.973 61
1.6781016 1.129 0.839 0.931 0.822 0.832 1.338 0.741 0.854 81
1.763 1.130 0.916 0.807 84
2.020 0.961 1.077 0.735 0.822 0.760 0.767 1.476 0.679 0.776 97
23630.770 0.892 0.495 0.573 0.640 0.647 1.736 0.584 0.642 138

 

 

127

 

Table A contd.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D-Origlnal
Phi Eﬂ1-ZEﬁ1-etotl Psi2 Psif-elq (Work) Del Tﬂ' 1 W2/W1fip W1rms/W2 Alpha2 Ns
1.110 0.972 0.813 1.212 1.022 1.241 1.246 0.475 1.989 1.019 61
1.453 0.980 0.905 1.116 1.029 1.133 1.143 0.597 1.589 1.017 70
1.784 1.003 0.964 1.055 1.014 1.047 1.057 0.739 1.290 0.994 78
2.2331019 1.004 0.992 0.976 0.968 0.974 0.883 1.088 0.954 90
2.5841034 1.024 0.934 0.925 0.899 0.907 1.006 0.962 0.916 101
9551.028 1.034 0.845 0.850 0.818 0.826 1.147 0.851 0.865 115
3.3761031 1.008 0.770 0.751 0.742 0.751 1.274 0.775 0.803 135
4.000 0.983 0.869 0.646 0.570 0.653 0.661 1.417 0.708 0.704 180
D-Trimmed
Phi Eff1-2 Efft-etod Psi2 Psi1- q (Work) Del T/T1 W2/W1tip W1rms/W2 Alpha2 Ns
1.212 0.971 0.873 1.139 1.023 1.167 1.177 0.580 1.630 1.008 64
1.624 0.995 0.965 1.056 1.022 1.056 1.066 0.764 1.244 0.975 74
2.0121026 1.029 0.995 0.997 0.964 0.973 0.934 1.024 0.930 84
2.551 1.055 1.090 0.908 0.937 0.856 0.867 1.143 0.846 0.855 99
8.0861041 1.118 0.793 0.851 0.758 0.766 1.334 0.734 0.764 117
3.931 1.022 0.901 0.614 0.540 0.597 0.603 1.621 0.618 0.591 186
E-Original
Phi Eff1-2'Eff1-eta Psi2 Psi1oel {(Work) Def rm W2/W1tlp W1 rms/W2 Alpha2 Ns
1.927 0.966 0.891 1.097 1.012 1.130 1.140 0.511 1.802 1.023 81
2.433 0.959 0.968 1.001 1.010 1.038 1.050 0.645 1.433 1.001 91
3.122 0.968 1.079 0.891 0.992 0.916 0.926 0.840 1.108 0.953 105
.996 0.985 1.124 0.803 0.915 0.811 0.818 1.013 0.930 0.882 126
52371.004 0.984 0.676 0.661 0.670 0.678 1.230 0.783 0.758 184
5.837 0.717 0.714 0.431 0.428 0.597 0.604 1.363 0.715 0.657 270
E-Trimmed
Phi Eff1-2Eff1-eto PsIZ Psi1- {(Work) Del T/T1 W2/W1tip W1rms/W2 Alpha2 Ns
2.220 0.933 0.952 0.949 0.966 1.011 1.023 0.725 1.272 0.971 90
2.869 0.956 0.997 0.882 0.919 0.918 0.929 0.889 1.043 0.913 107
3.5631021 1.122 0.838 0.920 0.816 0.827 1.063 0.880 0.843 119
.148 1.125 0.878 0.777 133
4.2941050 1.116 0.810 0.860 0.767 0.775 1.168 0.811 0.775 137
5.0451017 0.989 0.682 0.662 0.667 0.675 1.343 0.714 0.672 181
5.490 0.865 0.703 0.526 0.427 0.605 0.614 1.480 0.656 0.576 262

 

 

 

128

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