r 4-3.?!an .

7 »
5:11..
2:35.. i
24%....
‘36:. .h

549.,

.3-

 

 

 

 

 

 

 

 

 

1.2!!
#32:}; .
v {3.111.

 

 

 

.1947 Hi... p. 4h . I.

 

l '3?
LIBRARY
90% Michigan State
University

This is to certify that the
thesis entitled

MULTIPOINT OPTIMIZATION OF A 30 RADIAL
COMPRESSOR IMPELLER

presented by

Steven C. Kowalski Jr.

has been accepted towards fulﬁllment
of the requirements for the

MS. degree in Mechanical ELqineering

 

 

AL?”

Major‘i-J’refjssor’s Signature
I .2 / ’ ‘r / o 3/

 

Date

MS U is an Afﬁrmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 c:/ClRC/DateDue.indd-p.15

 

 

ML’LTIF

 

 

MULTIPOINT OPTIMIZATION OF A 3D RADIAL COMPRESSOR IMPELLER
By

Steven C. Kowalski Jr.

A THESIS

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

MASTER OF SCIENCE
Department of Mechanical Engineering

2005

ML’LTIPOI

In this M
geometry at I
concepts. In '

numerical tool.

° AmﬁcI
° Genetic

. NliVler-

With Ihls 5'
is an Aﬂiﬁcia
3CCUIHI€ bm $10

ABSTRACT
MULTIPOINT OPTIMIZATION OF A 3D RADIAL COMPRESSOR IMPELLER
By

Steven C. Kowalski Jr.

In this work, one has attempted to ﬁnd an optimum radial compressor impeller
geometry at both single and multi-point operation by utilizing advanced numerical
concepts. In particular the Optimization that was attempted is based on advanced

numerical tools applied to aerodynamics such as:

0 Artiﬁcial Neural Network (ANN)
- Genetic Algorithm (GA)

0 Navier-Stokes Solver (NS)

With this system the performance is predicted using a fast approximate model known
as an Artiﬁcial Neural Network (ANN), and then subsequently veriﬁed by a very
accurate but slow computing Navier-Stokes Solver, the core of this knowledge based-on
design being a Genetic Algorithm (GA).

Optimum geometries were found for both the single point case and multiple point
case which were different and have improved efﬁciencies relative to the baseline
geometry. Thus, the main conclusion is that the geometry that works well for one

operating point will not necessarily work well for multiple point operation.

Copyﬁght by
Steven C. Kowalski Jr.
2005

 

 

 

 

This

 

 

This thesis is dedicated to my parents, Steve and Sue Kowalski.

iv

 

 

[would 1
an the SUPP"r
for rccommcn'
Karma) InSIlI
following WW

I would ll
gratitude for thc
mpmision. hi:
academic year.

I would all»
advice that 1 YOU
adoubl that l m
Thinks as well
Flooded me.

Next, so as it

C0:
716 I0 kHOW 31'

 

Wk enl‘ironme»

Lost, but bx

 

Support.

 

ACKNOWLEDGMENTS

I would like to thank Professor A. Engeda of Michigan State University (MSU) for
all the support and advice he has given me during my academic years at MSU and also
for recommending me to be a participant of this prestigious institution known as the von
Karman Institute (VKI). It has deﬁnitely been an opportunity of a lifetime. The
following people that are greatly acknowledged are all from the world famous VKI.

I would like to express Professor R.A. Van den Braembussche all my sincere
gratitude for the opportunity he has given me to work on this project, for his help and his
supervision, his knowledge and wisdom, which he shared with me throughout the
academic year.

I would also like to thank Dr. Z. Alsalihi for his kindness, patience and the great
advice that I received countless times throughout the duration of this project. It is without
a doubt that I would have never been able to complete this project without his guidance.
Thanks as well to Mr. J. Prinsier and Mr. T. Verstraete for all their help that they
provided me.

Next, so as to not leave anyone out, I am very thankful for all the students that I have
come to know and appreciate during my period of study here at the VKI, for the positive
work environment they have created, and for their friendships.

Last, but by no means least, I would like to thank my parents for all there loving

support.

Steven C. Kowalski Jr.

 

 

llSI OF TAB}
LIST OF F IGL'
KEY TO SYM
IVTRODL' C III

CHAPTER I

BACKGROL'N
1.1 Air C OI
1.2 Limitin
1.3 Technit
1.4 Need fo

CHAPTER 2
OPIIMIZATIOT
ll General
3-3 Databas
2.3 Artiﬁcia
3-4 Genetic
2'5 Oblt‘cllx

CHAPTER 3
BASELINE GU.
3'] Geometl
3.2 Fmdamc
33 MCShInQ
3.4 Xm'kpgl
3.4.] 80'
3.4.2 Bo
34'3 PO:

 

 

 

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii

LIST OF FIGURES ............................................................................................................ ix

KEY TO SYMBOLS AND ABBREVIATIONS ............................................................ xiii

INTRODUCTION .............................................................................................................. 1
CHAPTER 1

BACKGROUND AND MOTIVATION ............................................................................ 3

1.1 Air Conditioning Fundamentals ............................................................................ 3

1.2 Limiting factors of ﬂow range ............................................................................ 10

1.3 Techniques to extend the ﬂow range .................................................................. 13

1.4 Need for Multi-point optimization ...................................................................... 20
CHAPTER 2

OPTIMIZATION METHOD ............................................................................................ 24

2.1 General Philosophy ............................................................................................. 24

2.2 Database .............................................................................................................. 26

2.3 Artiﬁcial Neural Network ................................................................................... 31

2.4 Genetic Algorithm .............................................................................................. 36

2.5 Objective Function .............................................................................................. 38
CHAPTER 3

BASELINE GEOMETRY ................................................................................................. 41

3.1 Geometric Fitting ................................................................................................ 41

3.2 Fundamentals of the TRAF 3D Program ............................................................... 45

3.3 Meshing approach and parameters ...................................................................... 46

3.4 Navier-Stokes Solver Parameterization ............................................................... 54

3.4.1 Solution Parameters .................................................................................... 54

3.4.2 Boundary Conditions .................................................................................. 56

3.4.3 Post-Processing Parameters ........................................................................ 59

3.4.4 Baseline Performance ................................................................................. 61
CHAPTER 4

SINGLE POINT OPTIMIZATION .................................................................................. 67

4.1 Geometry Deﬁnition ............................................................................................ 67

4.2 Design Space ....................................................................................................... 70

4.3 ANNs Utilized .................................................................................................... 77

4.4 Objective Function and GA Settings .................................................................. 79

4.4.1 Objective Function ..................................................................................... 79

4.4.2 GA Settings ................................................................................................ 84

vi

 

 

4.5 Prescn‘
4.5.1 E
4.5.2 C
4.5.3 r"

4.6 Perforn

CHIPTER 5
MULTIPOINT
5.1 Objecti
5.1.1 0
5.1.2 G
5.2 Present;
5.2.1 B
5.2.2 CI
5.23 AI
5.3 Perform

CHAPTER 6
OVERALL C0)

REFERENCES

 

 

4.5 Presentation of the Optimization ......................................................................... 87

4.5.1 Database ...................................................................................................... 89

4.5.2 Convergence History .................................................................................. 94

4.5.3 Analysis of Optimized Geometries .......................................................... 111

4.6 Performance of Optimum Geometry and conclusion ........................................ 122
CHAPTER 5

MULTIPOINT OPTIMIZATION .................................................................................. 130

5.1 Objective Function and GA Settings ................................................................. 130

5.1.1 Objective Function .................................................................................. 130

5.1.2 GA Settings .............................................................................................. 133

5.2 Presentation of the Optimization ....................................................................... 135

5.2.] Database ................................................................................................... 137

5.2.2 Convergence History ................................................................................ 140

5.2.3 Analysis of Optimized Geometries ........................................................... 146

5.3 Performance of Optimum Geometry and conclusion ........................................ 176
CHAPTER 6

OVERALL CONCLUSIONS AND FUTURE WORK ................................................. 181

REFERENCES ............................................................................................................... 184

vii

 

 

 

Table?"3 I
Table 4.1 (
Table—4.2 V
Table-l3 G
Table 4-4 Di
Iable4-5 N3

Table 4.6 Ch

Tnb1e5.1 We!
T331652 GAI
Table 5.3 NS r:
Iab1e5.4 N5 R
Table 5.5 NS RI
Table 5.6 .\'S Re
Table 5.7 NS Rt

Table 5.8 NS Rt

 

Table 5.9 Chara!

 

LIST OF TABLES

Table 2.1 GA’s Terminology ............................................................................................ 36
Table 3.1 Characteristics of the Baseline Geometry ........................................................ 45
Table 3.2 Impeller Mesh Parameters (number of cells) ................................................... 49
Table 3.3 Near wall parameters ........................................................................................ 52
Table 4.1 Geometric Variables ......................................................................................... 73
Table 4.2 Weights used in Single point Optimization ...................................................... 84
Table 4.3 GA and ANN Parameters ................................................................................. 84
Table 4.4 Database Navier-Stokes Results ....................................................................... 91
Table 4.5 Navier-Stokes calculations of Optimized Geometries ................................... 116
Table 4.6 Characteristics of Single Point Optimum ....................................................... 124
Table 5.1 Weights used in Multiple Point Optimization ................................................ 134
Table 5.2 GA and ANN Parameters ............................................................................... 134
Table 5.3 NS results of the Database at IGV Setting Angle = 0 Degrees ...................... 139
Table 5.4 NS results of the Database at IGV Setting Angle = 40 Degrees .................... 140
Table 5.5 NS Results of the Database at IGV Setting Angle = 60 Degrees ................... 141
Table 5.6 NS Results of the Optimizations at Pre-rotation = 0 Degrees ................................ 152
Table 5.7 NS Results of the Optimizations at Pre-Rotation = 40 Degrees ..................... 153
Table 5.8 NS Results of the Optimizations at Pre-Rotation = 60 Degrees ..................... 154
Table 5.9 Characteristics of the Multi-Point Optimum Geometry ................................. 177

viii

 

 

 

Figrrell Ge
Figire12 Air
Figure13 SCI
Figure 1.4 [m
Figure 1.5 Elli
Figure 1.6 Con
Figure 1.7 \l'oi
Figure 1.8 Perl
Figure 1.9 Perl}
FigurellO lnll
Figurelll Per
TlgurellZ Ma.
FlgurellS lllu

Famu4tm
sePirated flow. 5

Figure 1.15 Nor
113ml] F low
HmQQQm]

1:1qu 2

3An1

Emaigm,

ll‘él’rez ’

11g11l€36 Hm
1'

 

LIST OF FIGURES

Figure 1.1 Generalized phase change process .................................................................... 4
Figure 1.2 Air Conditioning cycle ...................................................................................... 6
Figure 1.3 Schematic of an air conditioning unit ............................................................... 7
Figure 1.4 Impeller Nomenclature. [6] ............................................................................. 10
Figure 1.5 Effect of incidence on boundary layer accumulation. [6] ............................... 12
Figure 1.6 Components that are capable of extending the ﬂow range ............................. 13
Figure 1.7 Working principle of IGVs [6] ........................................................................ 14
Figure 1.8 Performance map with variable inlet swirl using a vaneless diffuser ............. 15
Figure 1.9 Performance map with peripheral speeds using a vaneless diffuser ............... 17
Figure 1.10 Inﬂuence of a diffuser vane on the ﬂow in a radial diffuser. [9] .................. 19

Figure 1.11 Performance map with variable inlet angle of diffuser. Swirl free inﬂow.... 19

Figure 1.12 Mach number distribution illustrating incidence of IGV. [3] ....................... 21
Figure 1.13 Illustration of ﬂow separation of an IGV . [3] .............................................. 21
Figure 1.14 Unbalance of Centrifugal and pressure gradient forces and radial shift of

separated ﬂow. Suction side. ............................................................................................. 22
Figure 1.15 Non-uniform impeller total pressure distribution ......................................... 23
Figure 2.1 Flowchart of optimization system. [2] ............................................................ 25
Figure 2.2 Comparison of DOE and Random database generation. [2] ........................... 30
Figure 2.3 A Three-Layer Artiﬁcial Neural Network. [2] ................................................ 31
Figure 2.4 Sigmoid Transfer function. [2] ........................................................................ 32
Figure 2.5 Properly trained (a) and improperly trained (b) data ...................................... 35
Figure 2.6 Flow Chart of the working principle of a GA. [2] .......................................... 38

ix

Figure 3.1 Deﬁnitions of the meridional contour geometric parameters. [4] ................... 41

Figure 3.2 Baseline Geometry: (a) Meriodinal contours. (b) Blade proﬁles ................... 43
Figure 3.3 Main Grid Parameters. H-type Grid. [5] ........................................................ 48
Figure 3.4 Boundary layer mesh parameters. [5] ............................................................. 50
Figure 3.5 Typical Blade to Blade Plane Grids. (a) Shroud (b) Hub ............................... 53
Figure 3.6 Convergence History ....................................................................................... 55
Figure 3.7 Non-Uniform Inlet Proﬁles. (a). Pressure (b). Flow Angle ............................ 57
Figure 3.8 Post-Processing Planes and numerical pinch .................................................. 60
Figure 3.9 Performance Maps. (a) it, (b) ms, (c) nu ......................................................... 61

Figure 3.10 Mach Number Distributions at various Pre-Rotations: (a) 0 degrees (b) 40

degrees, (c) 60 degrees ..................................................................................................... 64
Figure 4.1 Geometry deﬁnition and baseline. (a) Geometry Deﬁnition, (b) Baseline ..... 67
Figure 4.2 Illustration of design space concept ................................................................ 70

Figure 4.3 Inducer stalling and choking incidence as a function of leading edge angle and

inlet Mach number ............................................................................................................. 72
Figure 4.4 Main Blade and splitter blade contours ........................................................... 75
Figure 4.5 Deﬁnition of (a) outlet lean (b) inlet lean ....................................................... 76
Figure 4.6 Illustration of Splitter channels ....................................................................... 78
Figure 4.7 Neural Network Structure ............................................................................... 79
Figure 4.8 Mach number distribution demonstrating unwanted features ......................... 83
Figure 4.9 General Working Principle of the Genetic Algorithm. [2] ............................. 86
Figure 4.10 The operating point chosen. P2/P01imposed = 1.4 ............................................ 88

Figure 4.11 Mach Number Distribution Database sample 1 ............................................ 92

Figure 4.12 Mach number distribution of Database sample 20 ........................................ 93
Figure 4.13 Mach number distribution of Database sample 24 ........................................ 94
Figure 4.14 Total Penalty. (a) With database. (b) Without database ................................ 96

Figure 4.15 Total-Static Efficiency Penalty. (a) With database (b) Without database ....98
Figure 4.16 Channel mass difference Penalty. (a) With database (b) Without database100
Figure 4.17 Mass Penalty. (a) With database (b) Without database ............................... 102
Figure 4.18 Negative Loading Penalty. (a) With database (b) Without database .......... 104
Figure 4.19 Negative Loading Penalty. (a) With database (b) Without database .......... 106
Figure 4.20 Loading Unbalance Penalty. (a) With database (b) Without database ........ 108
Figure 4.21 Loading Unbalance Penalty. (a) With database (b) Without database ........ 110
Figure 4.22 Comparison of efﬁciencies(T-S and T-T) of the database and iterations ...1 13
Figure 4.23 Meridional Contours (left) and Mach distributions (right) of Iterations ..... 117
Figure 4.24 Baseline Mach number distribution. Illustrates the improvement. ............. 123
Figure 4.25 Performance Maps. (a) it (b) ms (c) 11“ ...................................................... 125

Figure 4.26 Mach Number Distributions at various Pre-Rotations: (a) 0 degrees (b) 40

degrees, (c) 60 degees ..................................................................................................... 128
Figure 5. 1 The operating points chosen for optimization. (a). PI (b). TS ..................... 136
Figure 5.2 ConvergenCe History of Total Penalty at the three operating points. .......... 142

Figure 5.3 Convergence History. Efﬁciency Penalty at the three operating points. ....... 143
Figure 5.4 Convergence History. Mass Penalty at the three operating points ............... 144

Figure 5.5 Convergence History. Unbalance Penalty at the three operating points. ...... 145

xi

Figure 5.6 Convergence History. Neg. Loading Penalty at the three operating points. .146
Figure 5.7 Plots of the efﬁciencies at Various IGV Setting angles. (a) 0° (b) 40° (c) 60° ....... 149
Figure 5 .8 Meridional Contours (left) and blade proﬁles (right) of the Iterations. ........ 155
Figure 5.9 Mach number distributions at various IGV settings of Iterations ................. 162
Figure 5.10 Mach number distributions of the Baseline and Single Point Optimum

geometries. ....................................................................................................................... 171

Figure 5.11 Performance Map: (a) Pressure ratio (b) ms (c) nu ...................................... 178

xii

SYMBOLS:

COP

U

:1: Q:

:3“

KEY TO SYMBOLS AND ABBREVIATIONS

NOTE (SI UNITS ARE USED THROUGHOUT THE TEXT)

: Absolute Flow angle (Degrees)
: Relative Flow angle (Degrees)
: blade height

: Boundary Condition Vector

: absolute velocity

: Speciﬁc heat at Constant Pressure

: Coefﬁcient of Performance

: Diameter

: Geometry Vector

: Enthalpy

: Speciﬁc enthalpy

: Mach number

: Total Mass ﬂow rate

: Mass Flow rate in a splitter channel
: Rotational Speed (RPM)
: Efﬁciency

: Pressure, Penalty

: Performance Vector

: Density

: Heat Transfer Rate

xiii

R : Radius of Curvature, Radial coordinate direction
RE : Reynolds Number

S : Curvilinear length

T : Temperature

W : Weights, Weight Matrix, Relative Velocity

y : Ratio of Speciﬁc Heats

Z : Blade number

SUBSCRIPTS

0 : Value related to the total conditions

1 : Value related to the inlet of the computational domain (impeller inlet)
2 : Value related to the outlet of the computational domain (diffuser outlet)
fb : Full Blade

H : High Temperature Reservoir

hub : Hub

L : Low Temperature Reservoir

Le : Leading edge

lu : Loading unbalance

Min : Minimum Value

Max : Maximum Value

mass_diff : Mass Difference

OF : Objective Function

n1 : Negative Loading

xiv

ps : Pressure Side

Ref : Reference Value

Req : Required value

shr : Shroud

Sb : Splitter Blade

85 : Suction Side

Te : Trailing edge

ts: : Total to Static

tt : Total to Total

U : Value related to the circumferential direction
X : Value related to the axial direction
ABBREVIATIONS

2D, 3D : Two-dimensional, Three-dimensional
ANN : Artiﬁcial Neural Network

CF D : Computational Fluid Dynamics

DOE : Design of Experiment

GA : Genetic Algorithm

IGV : Inlet Guide Vane

NS : Navier-Stokes or Navier-Stokes Solver
RANS : Reynolds averaged Navier-Stokes Equations
TRAF : Transonic Reynolds averaged ﬂow solver

XV

INTRODUCTION

Computational Science has provided the building blocks necessary to design
innovative engineering systems at every level of sophistication.

Nowadays, performances of a given geometry and their associated boundary
conditions can easily and quite reliably be evaluated using computational ﬂuid dynamic
(CFD) codes using state of the art computer systems. Although experimental testing will
never become extinct, the digital evaluation and development of new products and
technologies may eventually begin to overtake experimental testing as the dominant
analysis method. This is mainly due to the fact that state of the art computational systems
save companies much time and money, which attracts many of them to invest more in
these systems every year.

This project combines two recent design tools together with a CF D solver, in order to
get the full beneﬁt of computational methods in the design of turbomachinery
components. The ﬁrst new tool that was applied to the design problem is based on a
method known as function approximation. The expert system used to implement this
concept, an artiﬁcial neural network (ANN), has the capability of analyzing design
variables and of predicting the performance that can be expected from a new proposed
geometry. This is done by creating a network of non-linear relations between inputs and
outputs by utilizing a learning process. This learning process provides the capability of
the ANN to predict the performance accurately for existing, and more importantly, new

geometries with much less computational effort than a full Navier-Stokes computation. A

 

database coma
trainthc ANN.
The other II
geometric modr

I capable of had I
|
to ﬁnd the gen
That is. in a no:
In 1116 follo
case. a 30 rati
Impeller gt’omc
prodded by a]; ll
“lib a fundami
readers. Follon
Will then take 1
clilies. SUbse]

Finally. 501116 0

prodded.

 

database containing full three-dimensional Navier-Stokes (NS) computations is used to
train the ANN, which then classiﬁes the system as one that is self-learning.

The other tool that has been utilized is the genetic algorithm, or GA. This tool makes
geometric modiﬁcations that are based on biological and genetic evolution, making it
capable of ﬁnding the geometry that demonstrates superior performance. Its main goal is
to ﬁnd the geometry that corresponds to maximum performance in a statistical manner.
That is, in a non-deterrninistic way.

In the following chapters, the above mentioned design system is applied to a selected
case, a 3D radial compressor impeller. For this case one will attempt to optimize an
impeller geometry to accommodate multiple operating points. These operating points are
provided by an inlet guide vane (IGV). Before attempting this however, one will start
with a ﬁrndamental background which is aimed speciﬁcally at completely uninitiated
readers. Following this a more complete description of the general optimization system
will then take place, leaving the unique aspects of our particular optimization to later
chapters. Subsequently, the results of the optimizations will be presented and discussed.
Finally, some overall conclusions will be drawn and suggestions for future work will be

provided.

CHAPTER 1

BACKGROUND AND MOTIVATION

In the following subsections the author would like to provide a short overview
regarding the fundamentals of air conditioning cycles, and aerodynamics of compressors
in order to provide a sufﬁcient background for readers of all levels and motivation to

perform an optimization.

1.1 Air Conditioning Fundamentals

Everyone knows from experience that heat ﬂows in the direction of decreasing
temperature, that is, from high temperature regions to the low temperature regions. This
heat transfer process occurs in nature without requiring any devices. The reverse process,
however, cannot occur by itself. This leads us to special devices known as air
conditioners and reﬁigerators which operate on a thermodynamic cycle. These air
conditioners and refrigerators are essentially the same in that the objective of their design
and application is to utilize work to move heat from a cooled space to a hot space using a
reﬁigerant. To do this, one can choose to use many different types of cycles. However,
the one discussed here will be the more common vapor compression cycle.

In order to understand the vapor compression cycle, and other similar cycles, it is
important to understand about the phase changes that the reﬁigerant experiences. This is
best illustrated through the use of the pressure-enthalpy diagram displayed in Figure 1.1

below.

Pressure vs. Enthalpy Diagram

 

] Constant

ME Temperature lines

Supercritical Region

 

 

 

 

 

 

 

C.P.

 

 

 

 

 

    

b c

a
“-I --- O —-------- O --

 

 

 

Subcooled 14

Liquid SHV
<- MLY ~> Region
Region

 

 

 

 

 

 

 

 

 

 

 

 

Saturated
Liquid
line

Saturated
Vapor
line

 

 

 

 

 

 

 

 

Figure 1.1 Generalized phase change process

In this ﬁgure, a refrigerant in the subcooled liquid region (point a), is at a temperature
that is below its boiling point. If one adds heat while maintaining constant pressure, the

reﬁigerants temperature will rise which leads to a corresponding enthalpy rise. This

 

enthalpy rise corresponds to a change of state from point “a” to “b”, where at point “b”
the liquid is about to vaporize. As heat is continually added, the refrigerant will start to
vaporize and its enthalpy will increase, but its temperature will remain constant as it
moves through the mixed vapor and liquid range, which is located between points “b”
and “c”. As the process continues the state of the ﬂuid will soon reach the saturated vapor
condition at point “c”, where there is only reﬁigerant in the vapor phase. Finally, with
more heat addition, the reﬁigerant state moves into the superheated vapor region (point (I)
where the temperature of ﬂuid begins to rise again.

This concept of phase changes is applied to the well known ideal air conditioning
cycle. Figure 1.2 illustrates the ideal vapor compression cycle on a pressure-enthalpy
diagram. The starting point of the cycle is at point 1 in the saturated vapor phase, where
the reﬁigerant enters a compressor and is compressed adiabatically from point 1 to point
2. The compressor is a device which increases the pressure, enthalpy, and temperature of
the refrigerant. Once the refrigerant leaves the compressor, it then enters the condenser.
The condenser is essentially a heat exchanger that transfers heat from the high
temperature reﬁigerant and rejects it to the environment. As heat removal from the
refrigerant begins, the ﬂuid cools until the saturated vapor state is reached which
corresponds to point 2a. As heat continues to be removed, the enthalpy continues to
decrease but the temperature and pressure remain constant. During this part of the
process, the refrigerant begins to condense to the liquid phase. This condensation
continues until the saturated liquid line is reached at point 3, whereupon the refrigerant
consists entirely of liquid. At this point, the reﬁigerant leaves the condenser as a saturated

liquid.

Ideal Vapor Compression Cycle

 

 

 

 

 

 

 

 

 

    
 
 

 

 

 

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

L l Constant
‘er/E Temperature lines
Supercritical Region ’
3 2
Subcooled :
p E Region
:‘COIOOOOOOQOOOCOOOIOOOOO
4
Saturated Saturated
Liquid Vapor
line line
H

Figure 1.2 Air Conditioning cycle

The next part of the cycle consists of an isenthalpic expansion process, which is

usually provided by an expansion valve or capillary tube (see Figure 1.3). This expansion

valve dramatically reduces the temperature and pressure of the refrigerant while holding
its enthalpy level constant. Some of the liquid vaporizes during the expansion process
until the end of the process is reached at state 4. Now the refrigerant is a low quality .
mixture of liquid and vapor at a temperature somewhat below the temperature of the

space to be cooled.

   
  
  
  
 
    

F M t
Condenser Coils an 0 or

(HOT)

Evaporator Coils
(COLD)

Expansion Valve

Inside a Basic Window
Air Conditioner

Compressor

Figure 1.3 Schematic of an air conditioning unit

Next, the refrigerant ﬂows through an evaporator, which is another heat exchanger
that transfers heat from the space to be cooled to the reﬁigerant. The reﬁigerant absorbs
this heat, increasing its enthalpy during this process, although its temperature and
pressure remain constant. Finally the liquid gradually vaporizes until the saturated vapor
line is reached at state I," whereupon the reﬁigerant enters the compressor in order to
repeat the cycle.

The heat load or the amount of heat that an air conditioner must remove from a room
varies throughout the year. More speciﬁcally, a small heat load in the spring and autumn
changes to a larger heat load in the summer. How can one vary this heat load? Well the
way in which the heat load can be controlled without changing the cycle is to vary the
mass ﬂow. This can be seen by applying the steady ﬂow form of the conservation of
energy to the ﬂow devices while neglecting kinetic and potential energy changes of the
ﬂuid. This is because they are small relative to the work input and heat transfer processes.

The rate of heat supplied, rejected, and power input can be written below as

QL = "7011 “’14) [1.1]

Q” = #1012 — ’13) [1.2]

Win = ”.70’2 " hl) [1.3]

From these equations it can be seen that for a ﬁxed cycle, the only way to increase or
decrease the heat transfer rate and the corresponding amounts of heat is to vary the mass

ﬂow. Thus, a designer should select a compressor that is capable of delivering the mass

ﬂow efﬁciently at constant outlet pressure, which will result in a high coefﬁcient of
performance (COP) for the air conditioner. That is, for the same QL, the COP will
increase if the work input is at a minimum. The work input will be a minimum only if the

compressor is able to perform efﬁciently at a particular Operating point.
COP = ——Q'
AC
m" [1.4]

This section ends with a question. How can one vary the mass ﬂow efﬁciently using

a centrifugal compressor without changing the pressure rise?

1.2 Limiting factors of Flow range
The need for multiple point operation of an air conditioner is dictated by the seasons
as discussed above. So the component of major concern is then the ﬂow range of the

compressor. A deﬁnition of a typical impeller can be seen in Figure 1.4.

 

 

TRAILING EDGE DRIVING FACE
(Inpeﬂer Tip) (Pressure Side)
TRAILING FACE
BACK FACE \ . (Suction Side)
' MPELLER
SPLITTER BLADE
VANE '
SHROUD
“3.33“ '
T" A r mucosa
HUB
LEADING
EDGE

 

Figure 1.4 Impeller Nomenclature. [6]

Before discussing how to increase the ﬂow range of the compressor, we will ﬁrst
discuss the limiting factors of the ﬂow range in basic detail. One of the factors is known
as compressor choke, which is deﬁned as the maximum mass ﬂow that the compressor
can swallow and is relatively easy to predict and control. This choking occurs because of

tWO reasons:

10

0 Sonic velocity in the throat of the inducer or vaned diffuser
0 Negative stalling incidence of the inducer. The passage area decreases due to

boundary layer blockage. This can be seen in Figure 1.5.

The other limiting factor is known as surge or stall, which is the complete breakdown of
the ﬂow, or alternatively called separation or return ﬂow. This normally occurs at low
mass ﬂows and is a much more complex and difﬁcult phenomena to predict and control.
There are many types or sources of stall, a few of which are listed here for the sake of
brevity.

However, some basic ideas will be discussed in the following paragraphs.

0 Impeller rotating stall.
0 Stall in the vaneless or varied diffuser

0 Surge

At lower mass ﬂows, the compressor provides a higher pressure ratio, and thus higher
diffusion rates. The impeller channels act as rotating diffusers that generally exhibit large
relative velocity decelerations. These large relative velocity decelerations in the impeller
channels cause large boundary buildup. The thicker the boundary layer, the weaker and
more sensitive to separation it becomes. The Coriolis and curvature effects tend to make
the boundary layer more turbulent and thus more resistant to ﬂow separation on the
pressure side of the blades and at the hub. However, the reverse situation of less ﬂow

stability occurs at the suction side of the blades and at the shroud. That is, the boundary

ll

layer tends to remain laminar which means that it is more susceptible to ﬂow separation
due to adverse pressure gradients. The situation can become more serious when the inlet
relative ﬂow angle does not coincide with the inlet blade angle. This situation can lead to
high velocity peaks which will inevitably cause ﬂow separation. This non-ﬂow alignment

to the blades is known as incidence and effects are illustrated in Figure 1.5.

 

PRESSURE
SIDE
LAYER Y
SIDE SEPARATION
/—‘ /-.
ZERO NEGATIVE POSITIVE
INCIDENCE INCIDENCE INCIDENCE

Figure 1.5 Effect of incidence on boundary layer accumulation. [6]

In this ﬁgure an excessive positive incidence means that the ﬂow impinges on the
pressure side which will in turn lead to a high velocity peak on the suction side. This high
peak has to be deceleratcd with an opposing adverse pressure gradient in order to achieve
a pressure rise, which inevitably leads to ﬂow separation and thus limits the ﬂow range.

Finally, for a more detailed treatment of stall, one should consult references [12-13].

12

1.3 Techniques to increase the flow range

A few ways in which to circumvent this problem of limited ﬂow range is by using
one or more of the following techniques all of which are illustrated in Figure 1.6:

0 Variable inlet guide vanes (IGVs)

0 Variable Speed Motors

0 Variable vaneless or vaned diffusers

0 Active control of surge and stall

 

Figure 1.6 Components that are capable of extending the ﬂow range

Variable [6 Vs:

Installing a ﬁxed or variable inlet guide vanes (IGVS) allows circumventing the
incidence problem by aligning the ﬂow to the blades. In other words, the IGV extends the
ﬂow range by introducing pre-rotation to the ﬂow so as to maintain the optimum
incidence at different mass ﬂows. Figure 1.7 shows the basic working principle of inlet

guide vanes in a self-explanatory manner.

     

INLET GUIDE VANE
. . wl / 1 Up: Cu1>0
With Rotation ‘ (Lower Head)
.. (Positive Prewhirl) I w1 Reduced
‘ Type “1” Q1 Reduced
U1 I U1 '¢—{- C“!
l ,
l
. l Cu1= 0
- m Zero Prewhirl :
. _ Type “2”
Wheel
. . U1X Clll<0
Against Rotation (Higher Head)
(Negative Prewhirl) W1 Increased
Type “3”
U1 Q1 Increased
Inlet Guide Vane U1 C“!

Figure 1.7 Working principle of IGVs [6]

Error!

14

 

 

As one can see above, when positive pre-rotation is introduced it leads to a reduction in
mass ﬂow and in turn to a slightly less enthalpy rise (or pressure ratio). The opposite is
the case for negative pre-rotation. That is, a higher mass ﬂow and a slightly increased
pressure ratio results. One can understand this by studying Figure 1.7 along with
considering the Euler equation. Both effects on performance can be seen in Figure 1.8.

[10]

Euler Equation:

H = A(UCu) = U2Cu2 “ UICul [1.5]

 

1.2 ‘ l

 

 

 

 

 

 

5‘2 . “mats-Io
.c ”mks—ﬁlo ,
s 1.0 ”I"
.5
<1 ‘\ \
08 \

 

o. \

0-“ em = 64° 130° 35' 20 45

1
or. 0.6 0.8 1.0 1.2 _ 1.}. 1.6
V/Vner

 

 

 

 

 

 

 

 

 

Figure 1.8 Performance map with variable inlet swirl using a vaneless diffuser

15

Variable Speed Motor:

Another potential solution is to vary the rotational speed. However this is more
expensive because one needs to use a variable speed motor. Also the pressure ratio varies
in accordance with the Euler equation (equation [1.5]) and consequently produces a
higher pressure ratio mainly due to the change in peripheral speed. This can be a
problem if one considers that the condenser needs to have roughly the same inlet pressure
over a wide ﬂow range. Figure 1.9 shows the effect of varying the rotation speed on the

performance of compressor. [10]

16

 

 

7"
1.1. ‘

«50:0" 1.1 /

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13? 1.2 —
S 7 /I
E “/0 “521.0 /
<‘ 1.0 , 5‘s /
0.9 ' 05“” / a
I 6.”
0.8 — ' / "$0 __
00 ’
0.6 g?
/
]
0.2 V

 

 

 

 

 

 

 

 

 

0.4 0.6 0.8 1.0 1.2 11¢ 1.6
WVREF

Figure 1.9 Performance map with peripheral speeds using a vaneless diffuser

l7

Variable diffusers:

Yet another alternative to vary the ﬂow range is by variable vaneless or vaned
diffusers. If one uses a vaneless diffuser, it is obvious that by changing diffuser width one
can easily vary the ﬂow range. However the problem of the vaneless diffusers is the
amount of friction that is produced along the endwalls due to the low radial velocity
component that is associated with the lower mass ﬂow regime. That is, the ﬂow is very
tangential at low mass ﬂows and results in more accumulated ﬁiction due to the fact that
is simply takes more time for the ﬂow to leave the diffuser. In addition to that, diffuser
rotating stall will occur if the ﬂow becomes too tangential.

The vaned diffuser consists of diffusing channels between vanes, where the diffusion
process is accomplished in a much shorter ﬂow path (see Figure 1.10) with much higher

efﬁciency than in the vaneless diffuser.

The drawbacks are as follows:
0 The maximum mass ﬂow is limited due a throat region.
0 At low mass ﬂows, and at certain distances from the impeller tip, there can be a
lot of noise and vibration
0 At off-design operation incidence is again present and can lead to fully developed
stall. That is, no re-attachment of the boundary layer and therefore no pressure

rise.

18

 

  

 

 

 

.I ‘
, s: x,"
r A ‘,-
.415)" are
‘-. v.‘.. _
~.'.i‘ ..
‘\l-c.\, i‘ll I
. ‘23:,
.,- .7
. \‘d
a
’ I
‘1 -.._‘ ,- d ‘. .,..
.£,:I.\"."- » ,’_ ‘V‘Ja ‘ -.~:'-
' .. nix .. -;.
.. ‘_ , . . . -.-I;
\

STREAMLINE

Figure 1.10 Inﬂuence of a diffuser vane on the ﬂow in a radial diffuser. [9]
The performance of a compressor with a variable vaned diffuser with no pro-rotation is

shown in Figure 1.11. [10]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 I I
E INDUCER STALL i U/UREF =10
. l- l ‘.
.6105 x 1 . am:
$3. . i l T\‘~L “Inaesz‘
.C I f \ I .0 l
<1 3 [ ‘ 0.9a
0.8fr—-r—---~T— i 095 ml
‘ 1 f t I l 0.90
1 I loco :
0.6 ; w T I l I
I
0.1. _ ,. ., ,1 , ,,
“av-11 12,5 15] 20 26 30

 

 

 

 

 

 

 

r L l
0.4 0.6 0.8 1.0 1.2 1A 1.6
V/ VREF

Figure 1.11 Performance map with variable inlet angle of diffuser. Swirl free inﬂow.

l9

Finally, the IGV was chosen, mainly for cost reasons, as the device to vary the ﬂow
range in the compressor in this work. This section leaves with a question. If the IGV
already increases the ﬂow range efﬁciently, then why does one need to consider

optimization?

1.4 Need for MultiPoint Optimization

Although IGVs provide high efﬁciency over a larger mass ﬂow range, a drawback is
the separation that occurs at larger pre-rotation angles (lower mass ﬂows). The ﬂow
comes into the IGV at a high incidence, which results in high velocity peaks that trigger
ﬂow separation. Both phenomena are illustrated in Figures 1.12 and 1.13 respectively.
This separated ﬂow zone within the IGV has a lower tangential velocity and thus a lower

centrifugal force attempting to maintain radial equilibrium.

Radial Equilibrium:
2
C, _ 1 6P

 

R — p OR [1.6]

The separated ﬂow then mixes with the core ﬂow and because of the higher radial
pressure gradient in this region, the lower momentum ﬂow or separated ﬂow is then
forced to the hub of the machine. This situation leads to radially non-uniform
distributions of total pressure. Thus, one must optimize the impeller to be able to adapt to
these very different inlet conditions. Finally, these two effects are seen in Figures 1.14

and 1.15.

20

Mach number distribution

1 2 (1') = 75 Degrees

 

".g_

.xk ‘ u ’.
0.6 » a c
. __ I |

T 0

Mach is

0.4 l- " _, "I. _.

l '1

 

Figure 1.12 Mach number distribution illustrating incidence of IGV. [3]

 

Typical IGV Blade Profile

 

 

0.2 0.4

 

 

X
(12 = 75 Degrees , Velocity Vectors

 

Figure 1.13 Illustration of ﬂow separation of an IGV. [3]

21

   

AlfCI:+75

mepmf

 

 

DQhOJGHQO

 

g.-

“Pressessswssssssssfos

YOOE-Ol
6505—01
6005-01
5505—01
SOOE—Ol
4505-01
4OOE-Ol
350E—Ol
300E—Ol
250E-Ol

-200E-Ol

150E-Ol
lOOE-Ol
OBOE-01
OOOE-Ol
950E-Ol
SOOE-OI
BSOE—Ol
8003—01
750E—01

.IOOE-Ol

 

Figure 1.14 Unbalance of Centrifugal and pressure gradient forces and radial shiﬁ of

separated ﬂow. Suction side.

22

 

25 0993 HEAS URED

 

 

 

‘NWbUIOINOKO

Figure 1.15 Non-uniform impeller total pressure distribution

23

CHAPTER 2

OPTIMIZATION METHOD

2.1 General Philosophy

The basic principle of the method presented here is to mimic the traditional design
procedure in which the designer makes a ﬁrst optimization by means of an approximate,
but fast, geometry generation and analysis method. The evaluation by the approximate
model, being very cheap and fast, allows a large number of performance evaluations
required by the optimization algorithm to be done with ease.

At the end of this ﬁrst optimization phase, an accurate Navier-Stokes calculation
(high ﬁdelity) is performed to verify the prediction quality of the lower-ﬁdelity model.
The optimization, with the simpliﬁed model, is then repeated, but not before the fast-
evaluation method has been recalibrated using the results of the previous computation. In
this way, the design system can be considered self-learning, and the optimum, which is
deﬁned by means of the fast evaluation method, should get closer to the one that would

be found using the more expensive Navier-Stokes method.

24

 

 

01:0me
<—>—
GENETIC
<cwmmfa>‘— ALGORITHM]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PERFORMANCjt . i
’ -» mmrcr .
TRAFM, p - J
LIMITING
DATA ’
BASE
v ‘ ANN ‘
PERFORMANCE T
‘3

 

 

 

Figure 2.1 Flowchart of optimization system. [2]

The blade design algorithm, of which a ﬂowchart is presented in Figure 2.1, starts
from the user-deﬁned aerodynamics and mechanical requirements. They are, depending
on the type of optimization conducted (2D, quasi-3D or 3D), inlet and outlet ﬂow angles,
outlet static pressure, Reynolds number, mass-ﬂow, efﬁciency, speciﬁc work,
maximum/minimum blade cross section and moment of inertia, Mach distribution
uniformity, skewness and so on.

The artiﬁcial neural network (ANN) is used to build an approximate relation between

the geometry (G ), the boundary conditions (B—C ), and the performance vector ( P ).

25

This relation can be represented in function form as:

1°=f(B_C',G) [2.1]

The ANN learning process deﬁnes this relation using the information stored in a
database. The trained ANN is then able to predict the performance of newly encountered
geometries using the speciﬁed boundary conditions.

The optimizer makes use of a genetic algorithm (GA) whereby the ANN is used to
evaluate the performance. The resulting geometry, which is normally optimum according
to the ANN predictions, is then veriﬁed by means of the Navier-Stokes ﬂow-solver. The
geometry and its performance are added to the database as a new sample and the ANN is
recalibrated or re-taught.

This iterative optimization cycle is repeated until the performance check conﬁrms
that an optimum geometry was found. As the database gows after each iteration, the
approximate relation (f in equation 2.1) is expected to become more accurate and the

optimizer should ﬁnally lead to the real optimum solution.

2.2 Database
The core of the design system is a database containing the input and output results of
Navier-Stokes computations performed during past computations. The following

information about each sample is stored:

0 EC : Flow-ﬁeld boundary conditions used by the NS solver.

26

o 6: Parameters deﬁning the geometry. Every type of turbomachine (axial, radial,
turbine or compressor) will have its own geometry deﬁnition and this part of the
database has to be adapted accordingly. This will be further explained when we
will come to discuss our application.

0 P : Aerodynamic performance. The amount of data computed by a two
dimensional NS solver can be as large as 800000(4 variables for 200000cell
nodes). Due to the ANN intrinsic architecture and to obvious limitations in the
database size, their number has to be drastically restricted. This allows a
reasonable ANN training and a processing time considerably shorter than the time
required for a N8 computation. The performance vector usually contains the mass

ﬂow, efﬁciencies, and the Mach number distributions.

The main purpose of the database is to provide information about the impact of the
geometrical changes on performance. The more accurate and general this information is,
the more accurate may be the ANN and the closer the ﬁrst optimum geometry, deﬁned by
the GA, will be to the real optimum. Hence less iteration loops will be required.

The making of this database is an expensive operation because it requires a large
number of 3D Navier-Stokes calculations and one is interested in making the smallest
possible database. It is however important that the information contained in the database
covers the whole design domain, that it is relevant and with a minimum of redundancy. In
other words, that the impact of every design parameter is included at most one time.

Any information missing in the database may introduce an erroneous ANN that could

drive the GA into a geometry that is not optimum. This is not a problem because the

27

Navier-Stokes analysis of that geometry will provide the missing information when it is
added to the database. Contrary to this, an incomplete database could result in an
erroneous extrapolation by the ANN predicting a low performance in that part of the
design space where in reality the performance is high. As a consequence, the
corresponding geometry will never be selected by the GA and the real optimum may
never be found. This second drawback is more difﬁcult to circumvent because no
mechanism is built in to compensate for it and it may continue during the entire
optimization. It is therefore import to assure that the initial database covers the whole
design domain.

Design of experiment (DOE) refers to the process of planning an experiment such
that the appropriate data can be analyzed by statistical methods, resulting in valid
conclusions. It is a statistical approach in which logical changes are made to the input
variables of a process so that one may discover and identify the reasons for the changes
that are observed in the output response. It is used to select the most signiﬁcant
geometries that will be used to construct the database. Figure 2.2, taken from reference
[2], derrnonstrates the loss of information by comparing the ANN predictions, based on
different DOE desigred databases, with the exact values for the test function displayed

below.

R(i)=1-0.001(A-D)3+0.002(C+E)(F-B)-O.06(A-F)2+(F+C)(E+A) [2.2]

28

The DOE constructed databases are also compared to randomly generated databases
in Figure 2.2. This is made possible through equation [2.2] and by deﬁning a global error

as follows:

n_samples R(l) _ AW“)
globalerror = _ 100
I; R ( z) [2.3]

 

Thus, the results show inconsistency of the global error in the random generating
case. It also shows that the error is not much smaller from the 64 to the 32 sample

databases. Thus, it has been well demonstrated that the DOE technique is sufﬁcient.

29

HE .coumbaow 838% :8ng was won mo cemgmﬁoo ~.N 9.5»:—

 

moEEum mo LonEaz

 

 

 

 

 

 

 

ucooemlvcm. .
£52 22 w 2512.... m m Eel use 2 “serves a w 9 mm S
2 c
L L we _
L , L
n L
L L
L no _ L L E: L
. L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L—_ hﬂ?

 

 

 

 

th

moi—can 9:59: ho 33:5: E282: Lo.— .o:o .303 w.zz<

om

ow

om

cow
8..
cc w
om F
on V
CON
ONN
ovw
cow
can

1%] 10m rectors

 

 

30

 

—T

 

23 Artiﬁci
The A.\'\
inputs and or
computer mul
ponerful lllIC‘.
The input
Nader-Stoke

conditions are

used to quanti

 

 

 

2.3 Artificial Neural Network

The ANN is a function approximation method suitable for functions having multiple
inputs and outputs. While the initial motivation for developing an ANN was to create
computer models that could mimic certain brain functions, it can also be thought of as a
powerful interpolator.

The inputs to the ANN, schematically depicted in Figure 2.3 are the same as for the
Navier-Stokes solver. That is, the geometrical parameters and the aerodynamic boundary
conditions are used in both tools. The outputs are the aerodynamic parameters that will be

used to quantify the performance.

W10)

FT. We" FT.

’ b1“) \

 

    
 
 

 

FT3

 

FT3

FT3
W k) ' b3(m-1)
3‘
. P30“)

Figure 2.3 A Three-Layer Artiﬁcial Neural Network. [2]

31

A neural network is composed of several layers. The ﬁrst layer is always the input

layer and the last one the output layer. In between is one (or more) hidden layer(s). The

processing elements in each layer are called nodes, and they are connected to the nodes of

neighboring layers by weight and bias parameters. The ﬁrst layer has as many nodes as

the number of variables of the design problem and the last layer as many as the

performance vector.

If each layer is represented by a vector 5: , and deﬁning the weight matrix W and the

bias vector In, the input-output mapping of the ANN can be presented in compact matrix

form:

 

 

€=F<W..aT+E>

 

 

The transfer function F usually used is a signoid function, F 1 + e-x .

Output

 

 

-2 0 2 4 6

Activation

Sigmoid function

Figure 2.4 Si gnoid Transfer function. [2]

32

[2.4]

Our remaining issue is the number of hidden layers. As the relation becomes more

complex, a larger number of layers are needed. The critical number of hidden nodes is

 

deﬁned by [1] as:
__ l1training°nout _ nout
nh '—
nin + nout

where nmmg is the number of training samples, nm the number of nodes in the input

layer, and no.“ the number of nodes in the output layer.

A11 connections are “feed—forward”. The optimal values of the connection weights
and biases, corresponding to the most accurate prediction for all cases presented, are
deﬁned during the ANN learning. This work has used the back-propagation of error
algorithm. Before the optimization procedure and the ANN learning are launched,
random values are assigned to W and B. The output of the ﬁrst NS calculation is then
compared to that of the ANN, and the error is back propagated accordingly, to update the
weights in the way that minimizes the mean square evaluated error.

Because of the structure of the transfer function F, all the inputs and outputs of the
ANN should be numbers between 0 and 1. To satisfy this criterion, each geometry and
performance parameter is scaled. For the optimum sensitivity of the ANN, the upper and
lower limits of the outputs should be selected in such a way that the intervals remain as
narrow as possible. This is due to the asymptotic behavior of the signoid function shown

in Figure 2.4.

33

The scaling is performed using the following formula. Gscaled is the scale factor, Gama]

the unsealed variable, while Gupper and Glower are the upper and lower variable limits:

 

G _ Gactual _ Glower
scaled — G G [25]
upper _ lower

Finally, it is sometimes possible to end up with an overtrained ANN. This means that “f”
in equation [2.1] is a polynomial ﬁtted to a high degee. This can lead to situations that,
for same input, there are two or more outputs. This is shown Figure 2.5. The steps taken
to properly train ANN are beyond the scope of this project and will not be discussed here.
The reader should consult reference [2] for more details regarding proper and improper

training.

34

 

Output

 

OUIput

 

 

 

Nonlinear
mapping

\ Generalization

Output

 

 

 

 

 

 

Input
(a)

Training data

 
   
    

1

Output?

 

 

“XI/u

Generalization

 

 

Input.
0))

Figure 2.5 Properly trained (a) and improperly trained (b) data

35

2.4 Genetic Algorithm

The goal of the optimizer is to ﬁnd the geometry corresponding to the minimum of an
objective ﬁmction using the simpliﬁed analysis model I3 = f (TB—5,6) , and knowing that

many local optima may exist in the design space for only one global minimum.

The genetic algorithm uses the genetic evolution and Darwin’s theory as a model to
simulate the design evolution and to reach the best solution. The core of this theory is
“the survival of the ﬁttest". Fitness is in general related to the objective function by

Fitness = , . 1 , . It is obvious that a design having a lower penalty (minimizing
Objective Function

 

the objective ﬁrnction) will have better ﬁtness and more chance to survive in the

optimization procedure.

The GA’s terminology is presented in Table 2.1:

Table 2.1 GA’s Terminology

 

 

 

 

 

Gene One design Variable
Chromosome Set of Design Variables
Individual Blade Shape
Population Set of Blades
Fitness Perfornnance Measure of an Individual

 

 

 

 

We are interested in the ﬁtness of the individual. For this purpose, the algorithm
creates a random population. Genetic processes are applied to the population during each

iteration. The dynamics of this mechanism is reproduction. After a reproduction, a new

36

generation of reproduced individuals, takes the place of their parents. This procedure is
repeated until a predeﬁned number of generations are reached. The ﬁttest of the last
generation and earlier generations is selected and proposed to the NS solver. For each
new iteration, beginning with a new and hopefully better approximation of the ANN, a
new population is created and a new evolution cycle started. The ﬁtness of the
individuals is approximated by the ANN.

Reproduction mechanisms have several subroutines. Selection is one of them. There
are different types of selection methods regarding which individuals will survive in the
next generation after a reproduction. The tournament selection is usually used. Two
random individuals are selected from the population and the ﬁttest is used as one of the
parents for the new generation creation. The other individual still survives in the old
population, preserving its chances to be selected again. This is done until the required
number of individuals for the next generation to be created has been reached. The best
individual of the previous generation is also selected and transferred to the next
generation.

Recombination is another sub-mechanism of the reproduction process.
Recombination means exchange of genes. The mutation is the action under which the
gene of an individual is selected with a predeﬁned probability and its value changes to a
random one, within the design range. Mutations attempts to prevent the GA from getting
trapped in a local minimum of the objective function, by creating some disturbance and
increasing the variation of the population.

The operation principle of a standard GA is shown in Figure 2.6. Pairs of individuals

(parents) are selected from an initially random population of n designs, each represented

37

by a binary coded string of length 1. Genetic material is subsequently exchanged between
them (crossover) and altered within the child (mutation). This process is repeated to
create the n individuals of the new generation. The best geometry of the last generation

is the solution of the GA optimization process.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

01101
Initial Population | 10110 n
1: 1+1 selection Sdectionofgood individuals

I I
New Generation crossover (DION 0011 11
l 1 111111 11|ooo
evaluation mutation 11000 :2 11010

L 1

 

Figure 2.6 Flow Chart of the working principle of a GA. [2]

2.5 Objective Function

A global performance measure is needed for the Optimization process, during the
ANN and NS calculation evaluations.

High efﬁciency is Only one of the objectives to meet for an aerodynamic shape
optimization. A good design should also respect the mechanical and manufacturing
constraints, and achieve the required aerodynamic performances (turning, surge margin,
good off-design behavior).

This is achieved by the minimization of an objective function (loss coefﬁcient) in

several variables subject to several constraints. The general approach to this problem is to

38

 

 

transform
simplest v
penalty ter

The global

1

01:115. G.

In equa
their relatit

1161101611 by l

‘i PAM
perfo
lUmir
for In

P PMcca
the m
are W

; PGmrr

01le r

 

’ Psldc' ::

Ol'erhl

 

transform the original constrained minimization problem into an unconstrained one. The
simplest way to do this is to create a pseudo-objective function (OF) by summing up
penalty terms which are increasing when the constraints are violated.

The global objective function takes the following form:

 

OF(P,G,B© = (.),,PAW +mum +(ng.PGm mpg”, ”3“th [2,6]

 

 

 

In equation [2.6] the right hand side terms of the equation are the penalty terms and
their relative performances can be changed by adjusting the weight factors which are

denoted by the symbol 00 :

> PAemperf, = penalty on aerodynamic performance, to realize the required
performance at the inlet and outlet of the computational domain, (i.e. mass ﬂow,
turning, efficiency) and to minimize the losses, i.e. the enthalpy loss coefﬁcient
for turbines.

> PM,“ = penalty for mechanical constraints, to verify if the parameters related to
the mechanical integity of the design, i.e. static and dynamic load of the blade,
are within prescribed limits,

> POW“? penalty for geometrical constraints (blade curvature radius, axial chord,
thickness. . .),

> PSide = penalty for side constraints (acoustic problems, manufacturing problems,

overhaul conditions. . .),

39

 

 

into

C0115

> PMach = penalty for non optimum Mach number distribution (to avoid velocity
distributions known to be far from optimal, such as those with a high probability

of early transition, laminar or turbulent separation, poor off-design performances).
Finally, it is noted that in this work, that only the aerodynamic performance is taken

into account. That is, stress, vibrations, heat transfer, and economic calculations are not

considered.

40

 

 

 

I
i

geor
unit;

test

I].

CHAPTER 3

BASELINE GEOMETRY AND TRAF SETTINGS

3.1 Geometric Fitting

Before the optimization system can be utilized, one must start from a reference
geometry or baseline geometry. The way in which this was done in this work was to
imitate an existing geometry at VKI using Bézier curves to create a pararneterized
version of the geometry. This parameterization is the input used to create the geometry,
gid and stacking of the gid. The deﬁnition of the meridional contour using these

geometric parameters is illustrated below.

out

 

Shroud . A

 

 

 

 

Hub

 

 

 

in f

\

 

 

/
\ LE , J
‘Vv
ULE I m2
I
I
I
I
/
in2 1113 L "J
.. m1
I 50—-
f
................. _> x

Figure 3.1 Deﬁnitions of the meridional contour geometric parameters. [4]

41

 

outlet
ﬁlled
111 :
111161 :
To m
are in
blade
blade

to. the

points
Which
and

leading

The major stations illustrated above are the inlet, leading edge, trailing edge and
outlet which are indicated with black points. The Bézier control points above are the
ﬁlled circles attached to the dashed lines. The inlet section (between the points in and
LE) and the impeller leading edge are parameterized with 3"d order Bézier curves. The
inlet section control points are in2 and in3, and the blade control points are ml and m2.
To make the ﬁgure more readable the point pairs with the same name at hub and shroud
are indicated only at the hub or shroud. The vector ULE, represents the percentage of the
blade meridional length where the splitter blade begins, both at hub and shroud. The
blade meridional length is the curvilinear distance along the black solid line from the LE
to the TE. Although not shown above, one can also vary the metal angle of the blades by
varying the above deﬁned points. Finally, one should consult Pierret [1] for a theoretical
explanation about Bézier curves.

Using the above deﬁnition, a parameterized geometry was created using Bézier
points. This was done through atrial and error process that resulted in the ﬁnal geometry
which shall henceforth be known as the baseline geometry. Figure 3.2a shows the
meridional contour of the baseline geometry. It clearly identiﬁes the location of the

leading and trailing edges of the full blade.

42

Meridional Contour

Baseline Geometry

(21)
250

Baseline Geometry

25

200

175

150

125

Radius (mm)

100

75

50

25

 

0 25 5O 75 100 125 150 175 200 225 250
Axial Distance (mm)

Figure 3.2 Baseline Geometry: (a) .Meriodinal contours. (b) Blade proﬁles

43

Baseline Blade Profiles

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) .
Full blades =10 ; Splitter blades =10 ; Total blades =20
3 ; I i i I L LIL LI A 1
.. =1. ---— Main blade hub ,
2 5 L i f _' -—--- Splitter blade hub LL
g . g ; g —— Main blade shroud
2 § z: i Splitter blade shroud 2'
1.5
1 i f "
wl~ _ " l L
:2 “I“ 1? ' r g
o ; 3 i : ,
.c 0 5 l I 1
I— p _.- i- L..- . .. .,: . . . . é . .. L, LILLL L_E,. , I L
0 i ‘C‘s‘x 1 l
,_.. \. - ,: .LLFL‘ ‘ﬁ'L. :~L‘L .. .L_._.§ c-
; I rj.‘ ;: 5 ‘~ ﬁb‘ ‘ ‘1 “ l
—0 5 i ix 1 T :‘ ,: L‘Cs ~
:~ \I .. . ‘Nb‘x ”‘22.,
T '""?”\. ? K: We, 1?“ " ‘
_1 \’ ’5 i 33:; l" i
1 . 3% re:‘
L L -.....-;.-.-.-_ L L L _ .LLLLs... L
-1.5 g
_2 L l '. i ; j ‘_ L
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
m/Fl
Figure 3.2 (cont’d)

The associated baseline blade proﬁles are shown 3.2b. In Figure 3.2b the blade
proﬁles were generated using a conformal mapping. This means that the blade angles are
preserved. The dashed line proﬁles represent the hub, with the larger blade obviously
being the full blade and the smaller one the splitter blade. The same reasoning applies to
the solid lines. That is, the solid lines represent the shroud with the full blade being the
larger one and other one being the splitter blade. The blade proﬁles can also be

distinguished by noting that the blade angle is larger at the shroud than at the hub.

44

 

in are

I

LIMLILIILI» LELELSLI L \.Lh.,L-N.L.bm

 

 

Some characteristics of the above geometry to be noted are listed in Table 3.1 on the

next page

Table 3.1 Characteristics of the Baseline Geometry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometric Characteristics

Beta-le-hub 21.70 Degrees
Beta-le-hub-sp 18.00 Degrees
Beta-le-shr 64.73 Degrees
Beta-le-shr-sp 53.00 Degrees
Beta-te 30.00 Degrees
Shroud clearance 0.575 mm
Max Hub Thickness 4 mm

Min Hub Thickness 3 mm

Max Shroud Thickness 3 mm

Min Shroud Thickness 2.5 mm
Percent blade chord length where ﬁnishes 25 %

the transition from Max to Min thickness.

The thickness remains constant from this

point all the way to the trailing edge.

Impeller Tip Radius 100 mm

 

3.2 Fundamentals of the TRAF3D program

The TRAF3D code solves the Reynolds-Averaged Navier-Stokes equations (M).
It was developed in a research project on viscous cascade ﬂow simulation by the
University of Florence and NASA [1]. It is able to predict the aerodynamic performance
of turbomachinery blades, working on linear or annular cascades, and considering
rotating blade passages with tip clearance ﬂow.

These RANS equations are solved using an explicit, steady, four-stage Runge-Kutta
scheme, in conjunction with accelerating techniques such as local time stepping, residual

smoothing and FAS (Full Approximation Storage) Multigid. These equations are

45

 

 

 

discn

hall

115cc

Utter
llli SI

Cale;

than
11111111
can 0
‘5 P05
than]
Ulla Id.

0A
”Fatal

discretized using ﬁnite volumes and a cell-centered scheme, with artiﬁcial dissipation.
The turbulence is simulated using a Baldwin-Lomax two-layer mixing length model to
evaluate the turbulent viscosity, tit, and the turbulent kinetic energy, k, under an eddy-
viscosity hypothesis. The transitional criterion of Baldwin-Lomax is adopted on the blade
surface while the shear layer on the hub and shroud contours are assumed fully turbulent,
from the irnlet boundary surface. The transition can also be imposed at a given point on
both suction and pressure sides, which is the way we have been proceeding, considering a
fully turbulent ﬂow on the blade surface (high Reynolds number). In other words,

transition is assumed to start at the leading edge of the blades.

3.3 Meshing approach and parameters

All the meshes were created using an H-type two-block gid, one block devoted to an
upper and lower channel. To avoid excessively long computational times, one has to limit
the size of the mesh using a reasonable number of cells. This raises the problem of the
calculations’ gid dependency and of the accuracy of the results that were obtained.

First, one has to note that the use of CFD will m provide exact results. Indeed, the
discretization itself, the fact that the turbulence is modeled instead of using direct
numerical simulation (DNS), and other assumptions that are made through the process,
can only give approximate results. The point is to make these approximations as accurate
as possible to provide the designer an idea of what to expect, and to provide information
about the ﬂow ﬁeld. Pierret [1] has shown, comparing experimental and computational
data, that the TRAF code gives very good and reliable results for applications that are

comparable to ours, even with relatively coarse meshes. The second point is that the

46

C0

in;

:21

code will be used to make comparisons between different geometries. This is important
because if one can assure to retain the same, or almost the same, mesh for all the entries,
then the studied cases should be entirely comparable, and the gid dependency negligible.

Two identical annular meshes are made, one for the upper and one for lower blade
channels. They will be used as the geometric inputs for the NS calculation. The main
inputs for the mesh creation are deﬁned in Figure 3.3, Table 3.2, and Table 3.3 (see

TRAF Code for Radial T urbomachinery manual for further explanation [5]).

47

 

"Y

"Y

 

  

 

 

Upper channel

 

 

 

 

Lower channel

 

 

 

 

 

 

Figure 3.3 Main Grid Parameters. H-type Grid. [5]

48

Table 3.2 Impeller Mesh Parameters (number of cells)

Parameters Lower
32
72
48
24
36
48

 

The parameters near the blade are listed, depicted, and tabulated below: (see TRAF

manual for a more detailed treatment [5])

Lref = impeller tip radius

d03=size of ﬁrst gid on the blade suction side non-dimensionalized by Lref

d1s= size of last gid on the blade suction side non-dimensionalized by Lref
d0p=size of ﬁrst gid on the blade pressure side non-dimensionalized by Lref
d1p= size of last gid on the blade pressure side non-dimensionalized by Lref

dsn = distance of the ﬁrst gid perpendicular to the wall non-dimensionalized by
Lref

jﬁxs = number of cells near leading edge and/or trailing edge with variable length
along the suction side of the blade

jﬁxp = number of cells near leading edge and/or trailing edge with variable length

along the pressure side of the blade

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4 Boundary layer mesh parameters. [5]

50

 

 

 

 

 

 

Cell nr
jfixs

 

 

 

Figure 3.4 (cont’d).

51

Table 3.3 Near wall parameters

Parameters

 

Finally, the stacking procedure for both upper and lower channels is listed below along
with typical 2D gids.
> The number of cells in the spanwise direction is set to 48, with a ﬁrst and last gid
cell size of 5 x 10'3 (non-dimensionalized by the reference length).
> The number of stretched cells near the endwalls is set to 28.

> The tip clearance is 0.575 mm and 8 cells are used in the clearance

52

 

 

 

 

 

 

 

 

 

 

(b)

Figure 3.5 Typical Blade to Blade Plane Grids. (a) Shroud (b) Hub

53

 

 

 

 

deﬁr

C031

11161

3.4 Navier-Stokes Solver Parameterization

The next step of the work was to set the operating conditions of the ﬂow solver by
deﬁning the boundary conditions and the solution parameters to be used for the
computations.
3.4.1 Solution parameters

For all our calculations, we used residual smoothing and full-multigid accelerating
methods (the multigid level was turned to three). Computations have been performed by
setting the Baldwin-Lomax coefﬁcients and abscissa for the laminar-turbulent transition
to zero, assuming a fully turbulent ﬂow. The total number of iterations was ﬁxed to 700,
100 iterations being spent on the coarse gid (Euler grid) A commonly obtained

convergence history is depicted on Figure 3.7:

54

0
-0.5
_g —1
{U
:1
:9
g —1.5
I
Q)
5 —2
'5
g
>
E? —3
E
O -3.5
c.
O
2 —4
—4.5
-5

Typical Convergence History

 

T ﬁT Y I 1"

T T T T T

I

 

1 n i l

LLLLLJLIIII

Ll l l

 

 

1 00 200

Number of CFD Iterations

Figure 3.6 Convergence History

55

300 i 400 I500 600 700 800

3.4.2

Boundary conditions (Lref = Impeller tip radius = constant = 100 mm)

Listed below are the boundary conditions and ﬂow parameters that were used:

>

>

VV VVV

Reference pressure, P0. = Pref = 101325 Pa,
Reference temperature, T01 = Trey = 293 K,
R = 288 J/kg.K,

y=l.4

,/RT
ML”, = 238909625
.1101

Reynolds Number, Re =

Prandtl Number, Pr = 0.7

Turbulent Prandtl Number, Pr, = 0.9

CF L = 4.0

Shaft rotational speed, N = 28000 RPM

Imposed irnlet ﬂow measured from axial direction on a blade-to-blade surface (01).
This value varies over the span. See inlet proﬁles.

Imposed irnlet ﬂow measured from axial direction on a meridional surface ((3:0).
This value is constant over the span.

Imposed pressure ratios were varied to determine the performance map. That is,

—3-=1.O, 1.1,1.2, and so on. The chosen design points for the optimizations will
01

be discussed in those sections.

The current optimization system sets up the above mentioned conditions

automatically, so that different Reynolds numbers can be obtained for different impeller

56

tip radii. However, in the current work, the impeller tip radius was held constant, and thus
the parameters that contain the reference length remain constant.
As for the inlet, three inlet proﬁles corresponding to various IGV setting angles were

used. These proﬁles are presented below in Figure 3.8

 

 

   
   

 

 

 

 

 

 

 

 

(a) Nondimensional span vs. P01/P01 ref

1 . r I T r f 1 I 4

0.9 L F—-V Setting Angle = 0 Degrees l

Z 0—0 Setting Angle = 40 Degrees I

0.8 "_ A—A Setting Angle = 60 Degrees ;

'o : ‘

8 ' .

E 0-7 r 1

U)

9. Z I

.o 0.6 - a

3 - .

.C a

E 0.5 :- .
“E .

2 L _'

'6 0.4 _ ‘

a ; .

5 0.3 *- -

D. ’ .

‘0 : :

0.2 - -

o.1 -

. : l

0 I n 1 . ‘_. . L A l n . 1 1 I 1 a n 1

0.8 0.85 0.9 0.95 1 1.05 1.1

POl/PO1 ref

Figure 3.7 Non-Uniform Inlet Proﬁles. (3). Pressure (b). Flow Angle

57

(b) Nondimensional span vs. inlet absolute flow angle

Angle 1n the axisymmetric plane measured from axial direction

   
   

 

 

1 ....................... I .
I v—-v Setting Angle: 0 Degrees '
0.9 L O—O Setting Angle: 40 Degrees L

A——-A Setting Angle = 60 Degrees

 

 

 

0.8
f
0.6
0.5 :
0.4

0.3 '

Span position from hub to shroud

0.2

I I T Y—V I V V I

0.1

 

 

l l 1 A l A 14 1 L4 1 l I l

O . n n I 1 .
—20 0 20 40 60 80 100 120 140 160 180 200
Absolute flow angle In degrees.

Figure 3.7 (cont’d)

58

3.4.3 Post-processing parameters
Once the computation is completed, the program uses the ﬂow solution to compute
the mass averaged values such as: efﬁciencies, pressure ratios, mass ﬂows, etc. To do

this, one has to set some basic parameters, which are:

> Position of the impeller inlet-averaging plane : The position of the this plane is set
at an axial position of 115 mm as seen in Figure 3.8

> Position of the stage outlet-averaging plane: The position of this plane was chosen
at the diffuser exit. This allows a sufﬁcient distance for the mixing at the exit of
the impeller to take place.

> The numerical pinch: This pinch is to make sure that one does not get local ﬂow
reversal at the exit of the machine because the CFD code will give wrong results
if this happens. This is because the code does not know the total temperature,
total pressure, and the ﬂow angle of the local return ﬂow at the outlet, which can
cause the code to be unstable. Thus, the pinch is just a method of providing
stability in the code. The length of the pinch is constant at 20 mm and the exit
width of this contraction is always 7.4 mm. This is done so that all geometries
will be comparable. That is, the effect of the pinch on performance should be the

same for all geometries.

59

Radius (mm)

Meridional Contour

Baseline Geometry

250
Baseline Geometry
225
200
Numerical S t' 2
. - ec ron :
1 75 Section 1 : 1331:3160 Avg.
Avg. plane at a lane at
x =115 mm mm p
150 R=150 mm
125
1 00
75
50
25

 

0 25 50 75 100 125 150 175 200 225 250
Axial Distance (mm)

Figure 3.8 Post-Processing Planes and numerical pinch

60

3.4.4 Baseline Peformance:

(a)

P02/P01

1.8

Pressure Ratio vs. Mass Flow Rate

 

Rotational Speed = 28000 RPM

 

 

 

b A—-—AIGV= ODegrees
1 7 : HIGV=40 Degrees
' _ V—-VIGV=60Degrees 1
1.6 - -
1.5 — -
1.4 - -
C
1.3 - -
1
1.2 ~ -
1. .1
' l
1.1 ~ -
l .
1 n m l n 1 l l n n l l l 1 n l 1 l n l
0.5 1.0 1.5 2.0 2.5

Mass Flow Rate (Kg/s)

Figure 3.9 Performance Maps. (a) 7:, (b) ms, (C) Tltr

61

(b) Total to Static Efficiency vs. Mass flow rate

Rotational Speed =28000 RPM

 

Ir‘ VIIIIIIVI [Y TYYYI

0.7 g—
0.6
co
l-| g
< 0.5 E—
i- :
L” z
0.4 ;—
o.3
0.2

v-
p
.—
p-
01-
g»
p
p
,.

 

l

T I ﬂ I

l

 

 

1.5

, . . , a
F—V IGV = 0 Degrees

v—-o IGV = 40 Degrees ‘;
A———A IGV = 60 Degrees i
i
i
4
J t l t 2

2 2.5

Mass Flow Rate (Kg/S)

Figure 3.9 (cont’d)

62

(c) Total to Total Efficiency vs. Mass flow rate
Rotational Speed =28000 RPM

 

 

 

 

1 . . . . . . . . . . . . . . ._ 1
E v———v IGV = 0 Degrees q
i -——o IGV = 40 Degrees 3
0 9 :_ n———t IGV = 60 Degrees ;
' t :
0.8 3- €
l: : ' f
(I 0.7 :- a
l- ~ 3
DJ 1
g i
0.6 :- ‘j
g 2
0.5 r 1
04 : t . . . r . . . . 1 . r . . r . . . . ‘
0.5 1 1.5 2 2.5
Mass Flow Rate (Kg/S)
Figure 3.9 (cont’d)

Finally, the Mach number distributions for the three setting angles are displayed

below in Figure 3.10.

63

Baseline Mach number Distribution

 

 

 

 

 

(a) Pre-Rotation = 0 Degrees
2 . v ' '
. Full blade hub
1 8 l —-—— Splitter blade hub _
- — - - Full blade shroud
1 5 — - - Splitter blade shroud

_.L

.h.
r
1

_.L
N
i
l

 

 

lsentropic Mach Number

 

 

 

’ iv

0.8 \ ' \~ —
\‘ ! \‘\ _________ :----
05 \\~~-J:_ 1“~\_ ,."" ‘~‘/l
:"'"‘\___/’ T ..... = ‘ .~- ‘~
I ' :"‘~—-_.-_.‘_’:_:.—- .-—="" 4
04 l l/ / _
l l.
0.2 . l a
O l l l L l

O 02 0.4 06 08 1

S/C

Figure 3.10 Mach Number Distributions at various Pre—Rotations: (a) 0 degrees (b) 40
degrees, (c) 60 degrees

64

(b) Baseline Mach number Distribution

Pre—Rotation = 40 Degrees

 

 

 

2 T T ' l
—— Full blade hub
1 8 r -— Splitter blade hub -
' —-— Full blade shroud
1 6 --- Splitter blade shroud ‘

 

1.4 — —

1.2 ~ a

lsentropic Mach Number

 

 

 

 

 

 

S/C

Figure 3.10 (cont’d)

65

(0) Baseline Mach number Distribution

Pre-Rotation = 60 Degrees
2 . l . .
L Full blade hub
—— Splitter blade hub
—-— Full blade shroud
—-— Splitter blade shroud

 

 

do
a)
l

 

 

 

lsentropic Mach Number
0 O —L _L _L
b: be _. to "A 3:»
T i l i F ' T
l l L l l l

.0
A

 

 

 

 

 

 

S/C

Figure 3.10 (cont’d)

One can obviously see that there is deﬁnite room for improvement by simply
glancing at these distributions. A designer never likes to see negative loading and would
like to eliminate high incidence. In the following chapters, single and multiple point
optimizations will be attempted to see if it is possible to improve the performance of this

machine.

66

CHAPTER 4

SINGLE POINT OPTIMIZATION

4.1 Geometry Deﬁnition

In order to perform an optimization, one must ﬁrst parameterize the geometry using
Bézier curves. The geometry deﬁnition and the resulting baseline geometry can be seen
in Figure 4.1. A more detailed description of the deﬁnition and how the baseline
geometry was obtained was already described in section 3.1, however some ﬁgures are

repeated here for convenience.

   
 

 

 

A
l
l
I
l
l
l
} m2
in2 in3
m1
'1 ‘0—
................. _’ x

Figure 4.1 Geometry deﬁnition and baseline. (3) Geometry Deﬁnition, (b) Baseline

67

Meridional Contour

Baseline Geometry
250

Baseline Geometry

225

200

175

150

125

Radius (mm)

100

75

50

25

 

0 25 50 75 100 125 150 175 200 225 250
Axial Distance (mm)

(b)

Figure 4.1 (cont’d)

From Figure 4.1a and its associated blade proﬁles, the amount of parameters that can
be changed is something like 50. This is not counting a variable tip clearance and a
variable thickness distribution along the blades, which would add even more variables.

Obviously one must limit the number of variables used both from a computation point of

68

view and from the fact the designer will have an over whelming amount of material to try
and decipher.

From the computational viewpoint, the more variables the designer selects the more
computations that have to be stored in the database. Since the Navier-Stokes
computational time is approximately 2.5-3.5 hours depending on the machine used, one
must limit the computations required. Currently, if the amount of design variables is
between 16 and 30, the theory of DOE says that one should use 33 samples or more per
Operating point. Thus in our case (24 variables), in which three operating points are used,
the minimum amount of samples is 99 which leads to 346.5 hours of computational time
assuming that the slowest machine is used. Also, if one accounts for the fact that the user
and manager of the system is certain to make mistakes or changes along the way, then
this is even more costly (S databases). All of these observations contribute to the fact that
the designer should select a limited and physically appropriate design space. It also
provides motivation to begin with single point optimization because the designer has to
deal with only one third of the computations required for multiple point operation.
Although, one would like to avoid making any mistakes, if this happens to occur, it will
be less disastrous and more easily circumvented, than for the multi-point case. Finally,
for all of the above reasons, the following discussion will be about the single point

optimization and the multi-point operation will be deferred until the next chapter.

69

4.2 Design Space

Now that the design parameters are known, the designer must select the ones that
will be variables and to select the range in which they can vary. That is, one must select a

feasible design space. Figure 4.2 illustrates the concept of a design space.

 

50

DESIGN SPACE

 

8

 

 

 

 

 

 

—-l

..ocanY2h)

radius R (mm)

B

 

i

 

 

 

 

 

 

 

(X1h,Y1 h)

 

 

 

-4o —3oL—2o‘—ioioiio
axial distance x (mm)

Figure 4.2 Illustration of design space concept

Choosing the design space really requires a vast experience with the particular
machine that one wants to optimize. Although one can take the time to try and explain all

the variables used in this work, one will just explain a few of them.

70

The axial position of the shroud trailing edge (xshr-te) was chosen as a variable
because it is known that the wheel efﬁciency depends upon the outlet width. That is, a
narrow exit will have a large portion of the span occupied by the boundary layer and less
diffusion in the impeller. This means that there will be little or no diffusion and only
friction, thus the efﬁciency will drop. On the other hand, a really wide exit will cause too
much diffusion which results in separation with no more diffusion taking place. For a
more in depth discussion about wheel efﬁciency, one should consult VKI course note 134
[7].

Another variable that was selected was the shroud leading edge blade angle. This is
because the inducer has an important impact on the impeller performance. A bad inducer
will accelerate the ﬂuid and the downstream deceleration will cause an early flow
separation at high velocity. The jet velocity will be high and the wake will grow to a
large part of the impeller. The mixing downstream of the impeller will involve high
losses with a subsequent drop in efﬁciency. Thus an important design criteria is therefore
to minimize the inlet relative Mach number and if possible to keep it subsonic.

It has been shown by C. Rodgers [9] that with higher inlet angles the maximum
relative Mach number is lower at a cost of reduced range due to stalling and choking
incidence. This can be seen in Figure 4.3. So in the present work, the value of Beta-le-shr
was chosen to be between 55 and 65 degrees. Values higher than 65 degrees lead to high
blockage due to blade thickness and also to high turning which results in losses in the
inducer. Values lower than 55 degrees lead to an increased range but smaller inlet
section which results in an increased axial velocity and decreased tangential velocity.

This then leads to higher relative tip Mach numbers which results in the impeller having

71

to do more diffusion which provoques separation and can also lead to shock losses if the
Mach number exceeds unity. Also, as illustrated in the following ﬁgure, one can show

that for a given mass flow and RPM that the minimum relative shroud inlet Mach number

is obtained for beta values around 60 degrees.

 

1.2
"m

1.0

 

 

 

 

 

0'3 CHOKING STALL

 

0.6 //i//

0.1» ' V
31 60’ 50° 40°

on1 0.1. as 0.623 0.743
0.2 I l
-20 -10 O 10 20 i 30

 

 

 

 

 

 

 

 

Figure 4.3 Inducer stalling and choking incidence as a function of leading edge angle
and inlet Mach number
Another parameter used was the blade number. Obviously, too many blades can lead
to too much blockage, while few blades can lead to high loading.
All the variables that were chosen in this work and their corresponding upper and
lower limits which deﬁne the design space are tabulated in Table 4.1 on the next page.

Most of these variables have ﬁxed limits. That is, they are hardwired to a speciﬁc value.

72

Two of the variables have limits that are deﬁned in terms of other limits. For example,
xshr-m2 will have limits that are determined by the two other variables near it. That is,
xshr_te is the upper limit and xshr_ml is the lower limit for the variable xshr_m2. This
is termed a ﬂoating variable because it has variable limits or a variable design space. The

other ﬂoating variable is Dshr_m1, however it was negated by a constraint that was

implemented. This will be shown shortly.

Table 4.1 Geometric Variables

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometric Variables Lower Limit Upper Limit
(Meters) (Meters)

Meridional Contour
Xhub-le 0.130 0.140
Xhub-ml 0.150 0.170
Dhub-ml 0.040 0.060
Dhub-mZ 0.090 0. 1 60
Xshr-ml 0.150 0.170
Xshr-m2 0 1
thhr-te 0.175 0.190
Dshr-le 0.130 0.150
Dshr-ml 0 1
Dshr-m2 0.160 0.180
Blade angles (Degrees) (Degrees)
Beta-le-hub 20 30
Beta-m 1 -hub 20 30
Beta-mZ—hub 20 30
Beta-le-shr 55 65
Beta-ml-shr 50 59
Beta-mZ—shr 30 50
Beta-le-hub-sp 20 30
Beta-ml-hub-sp 20 30
Beta-mZ-hub-sp 20 30
Beta-le-shr-sp 55 65
Beta-ml-shr-sp 50 60
Beta-m2-shr-sp 20 50
Splitter position
ULEtip 0.2 0.3
Number of Full Blades 7 11

 

73

 

In addition to the above variables, one can preserve the shape of certain areas if it
looks good to the designer. This can be done by using Bezier curves that are chained.
That is, one can preserve the shape of one area of the contour (inlet) while letting others

vary. The following geometric constraints have been implemented:

1. Xshr_le = Xhub_le

2. Dshr_te = Dhub_te

3. Dshr_ml = Dshr_le

4. Beta_te_hub_sp = Beta_te_hub

5. Beta_te_shr_sp = Beta_te_shr

6. Dshr_in2 = Dshr_le . Axial Shroud at inlet
7. Dshr_in3 = Dshr_le. Axial Shroud at inlet

8. xhub_ out — ”hr ‘0“, = Q = 0.7884 . Preserves diffuser area ratio
xhub _ te — xshr _ te b2

 

One can see that even though Dshr_ml has variable limits, it will always ﬂoat to
Dshr_le because of constraint number 3. Thus one can say that there are really only 23
variables instead of 24. This was done to preserve the shape of the inlet.

Another particular aspect of this optimization is that the splitter blade is constrained to
be at constant axial position(x). That is xhub_le_sp = xshr_le_sp. The reason for this is
because the impeller has shroud clearance which results in a lack of support of the splitter
blade. The fact that the splitter is at constant axial position(x) eliminates the bending
problems that are associated with non-axially aligned splitters (xhub_le_sp > xshr_le_sp).

In Figure 4.4 it is demonstrated that the splitter is at constant axial position(x). That is,

74

the splitter leading edge line indicated in Figure 4.4 is completely vertical. All geometries
in the database and the ones produced during the optimization process have this
characteristic. Perhaps the use of the word constant can be misleading. The splitter blade
in this optimization can still move leﬁ or right, and it does so in such a way that the axial

position of the leading edge at hub and shroud are the same.

 

Baseline Geometry
m Main blade and splitter

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Iso-theta contours
0.1 :-
0.09 :- Splitter Leading Edge
C (Splitter is constrained
0.08 _- to be at constant x)
0.07 _————-"""'"
II30.06 --
i-
005 2. Main Blade
26/ Leading Edge
0.04 _-
0.03 -
M n I I I l I I I I I I I I I l
0.14 0.16 0.18 0.2 0.22
X

 

 

 

Figure 4.4 Main Blade and splitter blade contours

One ﬁnal thing that needs to be mentioned before moving to the next section is that inlet
and outlet lean plays a role in the design process. A large inlet lean results in very high
bending stresses. Thus the inlet lean was constrained to be between i: 8 degrees. With the

negative meaning that the blade is leaned in the opposite direction of rotation. The outlet

75

lean is linked to the inlet lean and it is attempted to keep it below 45 degrees. Anything
that is larger than 45 degrees will amount to high friction. The deﬁnition of lean can be

seen in Figure 4.5

Outlet
Lean:
45 Degrees

Inlet Lean:
Zero
Degrees

 

(b)

Figure 4.5 Deﬁnition of (a) outlet lean (b) inlet lean

76

Finally, some of the constants in the optimization are the Impeller tip radius, and the

blade thickness distribution, both of which were deﬁned in section 3.1

4.3 ANNs Utilized

During the optimization process, certain characteristics of the performance of the
impeller are predicted by multiple artiﬁcial neural networks for each Operating point. For
a single point prediction, seven neural networks were used to predict the performance of
a given geometry using the above mentioned geometric variables and boundary
conditions as inputs. Given the 24 geometric variables, plus BC’s as inputs, the neural
networks give individual outputs. That is, one of the networks is devoted to the mass ﬂow
in a splitter channel “m1”, while another network predicts the mass ﬂow in splitter
channel “m2” of the machine. Figure 4.6 depicts the mass ﬂows in the separate channels
in a clear way. Figure 4.7 indicates that 4 of the networks are dedicated to predicting a
total of 160 Mach numbers. This results in networks 4 to 7 predicting 40 Mach numbers
per blade proﬁle. This breaks down further to 20 Mach numbers on suction side and
pressure sides. Also since there are four blade proﬁles, that is, hub/shroud main blade
and hub/shroud splitter blade, this leads to the grand total of 160 Mach numbers. Also,
one of the networks predicts the total to static efﬁciency.

The reason for using such an implementation is that if just one ANN is used to
predict everything, than the neural network will have too much information, and may try
to predict all of the quantities with the same accuracy. That is mass ﬂow, efﬁciency, and
the mach numbers at each point are all equally important. This means, that there will be

many unknowns that have to be found within the hidden layers. Thus, a way to

77

circumvent this is to have more networks. In this work 7 neural networks are used per
operating point, which means that 21 neural networks were used for the multiple point
optimization discussed in the next chapter. Finally a simple, yet not technical way of
saying the same thing, is that “Two heads are better than one”, or in our case “7 to 21

heads are better than one”. One should consult reference [2] for a more in depth

 

 

 

 

 

 

 

 

discussion.
m2
m1
TRAILING EDGE DRIVING FACE
("HD0997 Tip) (Pressure Side)

   

TRAUNG FACE
_\ (Suction Side)
\.

MPELLER
BLADE

'l SHROUD
'

INDUCER

 

 

LEADING
EDGE

Figure 4.6 Illustration of Splitter channels

78

 

INPUT
DATA

Xshr_te

Beta_le

Dshr_le
Etc.

BCs

 

 

ANNl

 

 

 

 

 

 

 

ANNZ

 

OUTPUT
DATA
m1

 

 

 

 

 

 

 

ANN3

 

OUTPUT
DATA
m2

 

 

 

 

 

 

 

ANN 4-7

 

 

 

V

 

OUTPUT
DATA

"ts

 

 

 

 

 

 

Figure 4.7 Neural Network Structure

4.4 Objective Function and GA Settings

4.4.1 Objective Function

Once the ANN has predicted the performance of an individual, this performance is
then converted to an objective function with which the genetic algorithm will minimize.
The Genetic algorithm mimics biological evolution and thus is a statistical or non-
deterministic process. The purpose is that blades will reproduce other better geometries
and the poor geometries will tend to die out. This is why the term survival of the ﬁttest is

usually used. A somewhat in depth treatment of the operational principle of the GA was

79

 

OUTPUT
DATA
160 Mach
Numbers

 

 

 

given in section 2.3, consequently, we will progress to the speciﬁcs that apply to our
design case.

The objective function used by the genetic algorithm is displayed below:

0F : W P + Wmass_difmeass_diff + W P + PVIuBu + Wannl

mass/10w massﬂow 0:3 ’7 ts

[4.1]

This function is a weighted sum of various penalties used to quantify the
performance. A higher value of the objective function implies a lower performance
impeller, while a lower value represents good impeller performance. In the above
equation, the ﬁrst term represents the penalty for the mass ﬂow in the machine. It is
desired to keep the mass ﬂow within a certain tolerance of the required mass ﬂow. The

penalty for the mass ﬂow of the machine can be seen in equation [4.2]

P _ max {mreq _ m l _ mthreshold 30}
massﬂow — - 4.2
mref [ ]

 

In this equation there is a percentage tolerance on the reference ﬂow, for which there is
no penalty, which is taken as 1 % of the reference mass ﬂow. The reference mass ﬂow is
1.3 Kg/sec for both single and multiple point optimizations.

The second term on the right hand side accounts for the difference in mass ﬂow in

between the separate channels which are created by the splitter blade. Recall Figure 4.6

80

for visualization of the channels and for illustration of the corresponding mass ﬂows
denoted by m. and m2. A designer does not want 90% of the mass ﬂow in one channel
and then the 10 % in the other. As we require constant pitch at the exit for both full and
splitter blades. Having widely different mass ﬂow in the separate channels would lead to
non-periodic outﬂow. Thus the following mass difference penalty was implemented to

account for it.

m1 Tmz

 

P . =
mass_diﬂ m1 + m2 [4.3]

The next term is the obvious one. That is, the designer wants the efﬁciency to be
high. With the current optimization system, one can choose either to optimize total-static
or total-total efﬁciency. The total to static efﬁciency was chosen because the ﬂow is not
directed into another stage, but is pumped to another component. If one sticks to the
application of air-conditioners, the ﬂuid is pumped to the condenser as discussed before,
and does not enter another compressor before going there. The total to static efﬁciency

penalty is as follows:

Pnts 2 ”ts — ntSreq

[4.4]

81

The fourth term in equation [4.1], is the loading unbalance penalty. That is, a
designer does not want most of the load on the splitter or main blade. If this occurs this
can be a source of separation for the highly loaded blade if too much acceleration
precedes the deceleration. It can also cause really high forces acting on individual blades.

The loading unbalance penalty can be expressed with just a few equations.

Area_fb=MSS_ﬂ) —M

ps _ fb 4.5]

Area_sb =M M

55 _ sb _ ps _ sb [4.6]

P _i lArea_fb—Area_sbl
1" _— LE Area_fb+Area_sb
2

 

[4.7]

 

The ﬁnal penalty that was implemented was the negative loading penalty. A
designer certainly does not want his/her impeller behaving like a turbine. Thus, the

following penalty was employed.

R11 = max[(Mps — Mss).dm,0] [4.8]

82

Perhaps the loading unbalance and negative loading penalties can be better illustrated

through the use of a Mach number distribution displayed on the next page.

Baseline Mach number Distribution

Pre-Rotation = 0 Degrees

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 . , . ,
Full blade hub
1.8 _ Negative —— Splitter blade hub A
. Loading -'- Full blade shroud
1 6 —-- Splitter blade shroud
Loading

a; 1'4 h Unbalance ‘
‘3 .. ¢ _
E 12
z ' 7 —
.C
O
('6
2
.9
o.
9
E
a)
_‘L’

 

 

 

 

S/C

Figure 4.8 Mach number distribution demonstrating unwanted features

Associated with each of these penalties is a corresponding weight. The weights are
applied to achieve a desired penalty. That is, if one wants a higher efficiency, the more
weight he/she should put on this penalty. For the single point optimization, approximately
the same order of magnitude was given to the mass and efﬁciency penalty, with the

negative loading and unbalance penalties being just a little higher. This can be seen in the

83

ﬁgures containing the database penalties in the next section, which is devoted to the
convergence history. Finally, the choosing of the weights is rather subjective and it is
unknown to the author if there is an optimum choice of these weights. In any event, the

weights used in this work are tabulated in Table 4.2 on the next page.

Table 4.2 Weights used in Single point Optimization

700
100000
400

100
100
3300
3300

 

4.4.2 GA Settings

Before performing the optimization one must select the GA and ANN settings.
Through the advice of the system manager and 1iterature[2], the following parameters
were used

Table 4.3 GA and ANN Parameters

84

 

Where the parameters in the table are deﬁned as follows (see reference [4]):

GA:

irestrt = 0 for a new GA run ; 1 for a restart continuation of a GA run

npopsiz = The population size of a GA run

pmutate = Jump mutation probability

maxgen = The maximum number of generations to be run by the GA

pcross = Cross over probability

ielite = l ; Irnplies that the best individual is replicated into the next generation
itoumy = l ; Irnplies tournament selection

pcreep = Creep mutation probability

iunifrm = 0 ; Irnplies single point crossover

nchild = 2 children are produced per pair of parents

TRAINING = 80 % of all the samples areused for training of the ANN

85

NANN = Number of neural networks per operating point.

The working principle of the GA was already explained. However the following

ﬁgure should give a basic knowledge of the principle and to make connections to the

values listed in Table 4.3 above.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

01 101
Initial Population 10110 n
t<tm .1.i°1 1
t: t +1 selection Selech‘on of good individuals
1 :41
new generation ﬂower 00|000 [:3 00|111
i 1 111111 111000
evaluation mutation 11000 :9 11010
f l

 

Figure 4.9 General Working Principle of the Genetic Algorithm. [2]

86

4.5 Presentation of the Optimization

In this section one will discuss the database creation, the convergence history, and
ﬁnally compare a few of geometries which were obtained during the optimization.
However, one must ﬁrst mention the operating point that was chosen.

The operating point was chosen near the high efﬁciency plateau shown in Figure

4.10. A designer should optimize at a point with sufﬁcient surge margin.

87

E Y ' ' * I Y 1% a T r

v—w IGV = 0 Degrees
0-9 F 0—0 IGV = 40 Degrees ‘;
A—t IGV = 60 Degrees
0.8
; Operating
0.7 :— / Point
0.6
w E a
t', z a
l- : :
LLI E 3
0.4 g
0.3
0.2
0.1 i,
o ’ I . . . I . . . . I I I . ‘

0.5 1 1.5 2 2.5
Mass Flow Rate (Kg/S)

Total to Static Efficiency vs. Mass flow rate
Rotational Speed =28000 RPM

 

 

 

 

 

 

 

 

Figure 4.10 The operating point chosen. P2/P01imposed = 1.4

88

 

Pressure Ratio vs. Mass Flow Rate

Rotational Speed = 28000 RPM
1.8 ....,.rr.,...-,-

 

 

 

 

 

 

 

 

 

t——A IGV: 0 Degrees 1
1 7 '_ HIGV=40Degrees _:
' _ V—VIGV=60Degrees
r ct
’_ Operating _
1'6 _ M Point .
1.5 - —
é . .
a 1.4 - —
o n .-
O.
1.3 - ~ ,
1.2 ~ -
1.1 ~ -
L
1 p I I I I l I I I I l I I I I l I I I I
0.5 1.0 1.5 2.0 2.5
Mass Flow Rate (Kg/s)

Figure 4.10 (cont’d)

The mass ﬂow required for the operating point was chosen to be 1.3 Kg/sec. The
efﬁciency requirement on total-static efﬁciency was set at 90%. The reason being is that
one does not expect to improve the efﬁciency by more than 25 points. That is, going from
63% to 90% seems a little unreachable, and it has been observed by the system manager
that is it better to put a value that is within expectation, rather than to just put 100%.

However, 90% seems a little high also.

89

 

4.5.] Database

As explained before in chapter 2, the knowledge of the ANN and GA comes from a
reference database. The database contains 34 Navier-Stokes calculations, coming from
different geometries representing the design space within the previously deﬁned limits.
The selection of the geometries can be realized in two different ways, either randomly, or
using the design of experiment method, DOE technique. It can be used in a wide variety
of applications when some parametric analyses are needed. Its aim is to construct a more
representative and systematic database, performing a statistical survey of the design
parameters. This approach was used for 32 of the samples. Indeed, it presents the
disadvantage of considering only two extreme values (25% and 75% over the lower limit)
of the parameter’s range so that in the end not much is known about the center of the
design space. This is why a central point case is run in which the variables are forced to
be at 50% over their lower limit. Cases 1-32 were given by the DOE, while the other
samples that were included were the baseline geometry and the central point run as can

be seen in Table 4.4 below.

90

Table 4.4 Database Navier-Stokes Results

 

 

 

     

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

08 Sample# Mass flow ETAA'IT ETA_TS P02/P01 P2/P01 inlet Leanl(Deg_)_W Outlet Lean De-
1 1" :0 7 0&9 " ' 171.59 -'-»-711'-.' "33111133512337"x" 1.1-.1335 1i". ' ' 25ml»
2 1.102 0.717 0.575 1.564 1.437 -13.496 45.000
3 1.311 0.832 0.660 1.574 1.440 -9.928 45.000
4 1.370 0.820 0.675 1.603 1.481 -1 2.970 45.000
5 1.437 0.826 0.678 1.612 1.486 -6.092 45.000
6 1.350 0.821 0.643 1.585 1.442 -6.751 45.000
7 1.177 0.761 0.617 1.575 1.452 -15.328 45.000
8 1.382 0.852 0.674 1.587 1.448 -5.001 45.000
9 1.187 0.799 0.636 1.563 1.433 -10.028 45.000
10 1.117 0.727 0.585 1.559 1.435 -2.540 45.000
11 1.402 0.851 0.672 1.582 1.443 0.000 42.882
12 1.228 0.729 0.582 1.571 1.441 -2.355 45.000
13 1.037 0.680 0.541 1.542 1.417 -8.831 45.000
14 1.213 0.795 0.631 1.556 1.426 -9.086 45.000
15 1.298 0.791 0.645 1.589 1.465 -20.146 45.000
16 1.413 0.839 0.656 1.600 1.452 -10.480 45.000
17 1.305 0.841 0.670 1.575 1.443 -14.990 45 .000
18 1.017 0.697 0.556 1.554 1.428 -17.209 45.000
19 1.313 0.832 0.662 1.568 1.436 -13.680 45.000
20 ' " .....-. .84 .0” .5932" '
21 1.516 0.836 0.680 1.637 1.501 0.000 14.790
22 1.322 0.818 0.644 1.575 1.437 0.000 43.814
23 1.221 0.777 0.632 1.581 1.458 -6.349 45.000
24 ' ’;_‘;"'"i*"'80“"” ‘ - -
25 1.245 0.802 0.634 1.566 1.432 0.000 34.625
26 0.976 0.682 0.542 1.544 1.418 -0.260 45.000
27 1.442 0.842 0.658 1.583 1.439 0.000 27.734
28 1.079 0.698 0.557 1.547 1.423 -8.991 45.000
29 1.295 0.768 0.617 1.595 1.462 0.000 0.649
30 1 .323 0.810 0.635 1.565 1.428 0.000 30.363
31 1.252 0.776 0.631 1.574 1.452 -4.616 45.000
32 1.248 0.810 0.642 1.573 1.438 -6.748 45.000
Central 1.371 0.821 0.664 1.596 1.466 -6.854 45.000
Baseline 1 .2 57 0.824 0.6 35 1 .562 1 .417 4.336 45 .000

 

 

 

 

 

 

If one surveys the table, one would probably argue that database sample 1 has the
highest total-total efﬁciency and thus why pursue the optimization? Well, the reason is
that this geometry exceeds the lean limit (d: 8 degrees) and thus will have high bending
stresses. Also if one looks at the Mach number distribution (Figure 4.11), it can be easily

seen in the following ﬁgure that it is plagued by negative loading (full blade hub) and

91

 

also by loading unbalance at the shroud. Plus, one could also say that there is a little bit

too much incidence on the splitter blade amongst other things.

 

The reader should keep in mind that the number of blades in Figure 4.11 is really just
the number of full blades and one must double this to get the total number of blades.

Also, that the value of P2/P01 is an imposed pressure, or boundary condition.

DB Sample 1, 8 Blades

P2/P01=1.400000 ; Pre—Rotation = 0 Degrees

 

 

 

 

 

 

2 w r l
Full blade hub
1 8 —— Splitter blade hub - .
' — - — Full blade shroud ‘
1 6 — - - Splitter blade shroud 4

_L
A

_s
N

 

lsentropic Mach Number
.0 .0 .0
.1:- a) on a

.0
to

 

 

 

S/C

Figure 4.11 Mach Number Distribution Database sample 1

Continuing with our survey, one might say, well number 20 has the highest total-

static efﬁciency. Also again, the lean is exceeded. One must also pay attention to the fact

 

that the design requires that the mass ﬂow be at or near 1.3 Kg/sec when making these

 

92

comparisons. The Mach number distribution for sample 20 (Figure 4.12) has negative

loading at the splitter blade hub and shroud.

 

 

 

 

 

DB Sample 20, 8 Blades
P2/P01=1.400000 ; Pre—Rotation = 0 Degrees
2 ' f r f
Full blade hub .
1 8 _ --— Splitter blade hub _
' _ --- Full blade shroud
1 6 - - — Splitter blade shroud

_a
A

I V
I

_II
N
Y i
I

lsentropic Mach Number

 

 

 

 

I\ {X

08 l“\~ I- _,—:::::::-‘ -«
l T '''' 4“ ". 7‘:=:~-
l ‘‘‘‘‘ \ “:‘\

06 if T“l\~\\. “-‘§.
' I ~\-;‘§\ 7

""'—=.*:— ———————

0.4 r! ‘7 —
l

0.2 -

o I I I I I l I I I
0 0.2 0.4 0.6 0.8 1
S/C

Figure 4.12 Mach number distribution of Database sample 20

Continuing with this process of analyzing the Table 4.4 and the Mach number
distributions, the best database sample in the authors opinion is number 24(Figure 4.13).
Well, in terms of negative loading, it is better. There is still too much incidence at the full
blade shroud and still loading unbalance. The mass ﬂow is around 1.3 Kg/Sec and the

lean is just a touch over the limit (-8.861 Degrees). The total-static efﬁciency is 4 points

93

 

 

higher than the baseline, while the total-total efﬁciency is 3 points higher. However, it

should be possible to obtain even more efﬁciency improvement than this.

DB Sample 24, 8 Blades
P2/P01=1.400000 ; Pro-Rotation = 0 Degrees

 

 

 

 

 

2 1 I
_ Full blade hub ,
1 8 _ — Splitter blade hub _
. - - - Full blade shroud
1 6 L - - — Splitter blade shroud

—L

o

A

I 1
l

—l
N
7

I

.0
on

lsentropic Mach Number

.0
o:

P
A

0.2

 

 

 

 

S/C

Figure 4.13 Mach number distribution of Database sample 24

Finally, none of the samples in the database has the optimum combination of
parameters. This is why we will now progress to the results of the optimization in hopes

that the system can ﬁnd the optimum.

94

 

4.5.2 Convergence History

In theory, the NS and ANN penalties should overlap each other in order to say that
the system has found the optimum geometry, or converged. One can see in Figure 4.14
that the NS and ANN penalties slowly start to come together. A possible reason why the
lines do not come fully together could be that other penalties are offsetting each other.
For instance, in Figure 4.15 the ANN and NS actually come completely together because
it is only one penalty, instead of the sum of all the penalties.

Figure 4.14b shows that between iterations 50-58 the geometries are quite similar.
One possible explanation for this could bethat the solution is converged and thus there is
not anything better in the design space. Another possible explanation is that since the GA
always starts from the previous solution, it eventually gets saturated with the similar
geometries, which leads to offspring that are very similar. The penalties still vary slightly
due to the fact that mutations are taking place. At the around iteration 60 the penalty
starts to increase, probably because the mutations are producing worse geometries. The
penalty continues to increase up to Iteration 62, where ﬁnally a particular geometry is
produced that gives a negative temperature. The author tried to rerun the program again
without restarting the GA from scratch, however, the same result was obtained. That is, a
negative temperature. The only way to begin the optimization process again was to restart
the GA from scratch. That is, a new randomly created population of 50 members is
introduced into the GA. This leads to an increased penalty at Iteration 63 because the
geometries are randomly generated at the beginning of the GA process. Finally, the
penalty starts to decrease again as the GA is reconﬁgured to start from the previous

solution and the penalties appear to be converging again.

95

As can be see in Figure 4.14b, Iteration 12 had the lowest total penalty and will be
one of the geometries that will be analyzed in the next section. “Why can it be that the
penalty is already so low at Iteration 12?”, one might ask. Well, the reason is that it can

happen because of chance. That is, the system is statistical, or non-deterministic, which

means that sometimes the system can be lucky.

(a) Convergence History

34 Sample DATABASE USED

1500111
t———A Iterations -ANN Penalty
0—0 Iterations -NS Penalty

I—I DATABASE —NS Penal

 

 

 

 

 

l

1 000

 

Total Penalty
:1.
:.I=..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 I I , I I I I I I I I I
—40 —30 —20 -10 0 10 20 30 40 50 60 70
Iteration Number

Figure 4.14 Total Penalty. (a) With database. (b) Without database

96

(b) Convergence History

 

 

 

 

 

34 Sample DATABASE USED
400 . . . . . . . . . .
Ar—A Iterations -ANN Penalty
o—o Iterations -NS Penalty
300 ~ ~

 

 

200 — ﬂ -

 

Total Penalty

 

 

mama-WWW-

 

 

 

 

 

 

 

 

 

0 I I I J I
0 1 0 20 30 40 50 60 70

Iteration Number

Figure 4.14 (cont’d)

In Figure 4.15 one can see that the ANN and NS come together at around Iteration

50. It appears that Iteration 13 has the best total-static efﬁciency and will also be

analyzed in the next section.

97

Convergence History

 

 

 

 

(a) .
Weight =400
150. . I . .-IIL...2..
t—t Iterations -ANN Penalty
o—o Iterations -NS Penalty
I—I DATABASE -NS Penal
. -
140 ~ .
- I

 

 

—L
N
O

i
I

 

 

 

 

 

 

 

 

 

 

 

 

A

8

l

I

V.»
w
- .

l

 

 

 

 

Total to Static Efficiency Penalty

' ',‘ ‘HE‘HS i‘

" 4 1461‘ "}

 

on
O
l

 

 

 

 

 

 

 

 

 

 

60 I I I I I I I I I I I
—40 -30 -20 -10 0 10 20 30 40 50 60 7O
Iteration Number

 

Figure 4.15 Total-Static Efﬁciency Penalty. (a) With database (b) Without database

98

Convergence History
(b) Weight=400
120 . . - . . I I I . .
A—A Iterations —ANN Penalty
P o——0 Iterations —NS Penalty

 

 

 

 

 

110 - .

 

é

 

 

 

 

 

8
0"

Total to Static Efficiency Penalty

 

 

 

00
O
l

I

r

 

 

 

 

 

 

 

 

 

70 I I I I I I
0 1 0 20 30 40 50 60 70
Iteration Number

Figure 4.15 (cont’d)

Finally, for the sake of completeness, the rest of the convergence history will be

listed because similar trends can be observed in each of them.

99

(a) Convergence History

Weight =100000
3o . . . . I . . f I I
o——-0 Iterations -NS Penalty
A—-A iterations -ANN Penalty
I -——a Database — NS Penalty

 

 

 

 

 

N
C
i
I

 

 

 

 

 

Channel Mass difference Penalty
8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I - - l

0 . .' . . Lille" ”5131.10 ”iii“
—40 —30 —20 —10 0 10 20 30 40 50 60 70
Iteration Number

Figure 4.16 Channel mass difference Penalty. (a) With database (b) Without database

100

(b) Convergence History

Weight =100000
1o 1 . . r -r 1 . . , ,
o——o Iterations -NS Penalty {

 

 

 

 

t—A Iterations —ANN Penalty '

 

 

 

 

 

 

 

Channel Mass difference Penalty
01

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

O 10 20 30 40 50 60 7O
Iteration Number

Figure 4.16 (cont’d)

101

(a)

Total Mass Penalty

Convergence History
Weight: 700

 

200 r T ' ' Y I ' I ' I ' l ' I

 

0—0 Iterations —NS Penalty
H Iterations —ANN Penalty
* I—I Database -NS Penalty

 

 

 

150

I

 

100 ~

+ ' ~ “It

 

 

 

 

 

l

 

 

 

 

 

 

 

 

 

 

 

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

—4o --30 —20 —1o 0 1o 20 30 4o 50 60 7o

Iteration Number

 

 

 

l

Figure 4.17 Mass Penalty. (a) With database (b) Without database

102

 

(b)

Total Mass Penalty

3.75

2.5

1 .25

Convergence History
Weight = 700

 

I

I

 

0 maAnm-ﬁng-

O

 

 

 

I II ' [LI ‘ I

 

0—0 Iterations —NS Penalty
A—-—A Iterations -ANN Penalty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10

A

20 30 4O 50 60
Iteration Number

 

Figure 4.17 (cont’d)

103

 

 

 

 

7O

Convergence History

Weight =3300
300. . . . .I.ICI.T.[-1.
o—o Iterations -NS Penalty

A——A Iterations —ANN Penalty
-—- Database - NS Penalty

(8)

 

 

 

 

 

 

200- -

 

I
l

 

Negative Loading Penalty at Hub

 

 

100 +

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
o A ' ‘
—40 -30 -20 -10 0 10 20 30 40 50 60 7O
Iteration Number

 

Figure 4.18 Negative Loading Penalty. (3) With database (b) Without database

104

(b)

Negative Loading Penalty at Hub

2.5

1 .25

Convergence History
Weight =3300

 

 

 

 

 

 

 

 

-
v

 

 

I ' I ’ J ' l
0—4 Iterations -NS Penalty
A—A Iterations —ANN Penalty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L

 

 

 

 

 

1O

20 30 4o 50 60 7o
Iteration Number

Figure 4.18 (cont’d)

105

r‘ l '
—" um

(a)

Convergence History

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Weight =3300
30° IILIJIIIIIF
o——o Iterations -NS Penalty
A———A Iterations —ANN Penalty
I—I Database - NS Penalty
'0
3
O
E
‘13 200 — -
(U
*3
C “i
a)
0- I
3’ I i
:6 |
(U
3 I I
.3 100 - -
“iii :
U)
0)
2
F ‘ I
I LI . I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 ' ' - “
—40 -30 -20 -10 0 1O 20 30 4O 50 60 70
Iteration Number

Figure 4.19 Negative Loading Penalty. (3) With database (b) Without database

106

(b)

3.75

2.5

Negative Loading Penalty at Shroud

Convergence History

Weight =3300

 

 

 

 

 

 

 

 

 

 

 

 

10

A
r

 

 

. IT 11 I I I I I I
o——0 Iterations -NS Penalty
A—t Iterations -ANN Penalty

 

 

I

 

 

 

 

 

 

 

 

 

 

 

 

y
r
‘

 

 

 

I II

)
..II I'

 

 

 

 

 

 

 

 

 

 

 

 

 

20 30 40 50 60

Iteration Number

Figure 4.19 (cont’d)

107

 

 

 

 

70

(a) Convergence History

Weight =100
300.....l.,.[,1.I
o——-o Iterations -NS Penalty
‘—-. Iterations -ANN Penalty
-——- Database — NS Penalty

 

 

 

 

 

 

 

 

 

 

Loading Unbalance Penalty at Hub

 

 

 

 

 

 

 

    

 

 

 

 

   

 

 

 

 

 

 

It

070

   
   

o . l- 7 -. 7 I u . . , . r W? . ..
—40 —30 —20 —10 0 1O 20 30 40 50 6
Iteration Number

 

 

Figure 4.20 Loading Unbalance Penalty. (a) With database (b) Without database

108

0))

Loading Unbalance Penalty at Hub

2.5

Convergence History
Weight =100

 

 

 

 

 

 

LLIJ'H I ' III I' T

 

o—o Iterations -NS Penalty
A——-A Iterations —ANN Penalty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”DIEM

 

 

 

 

 

10 20 ab 40 50 60
Iteration Number

Figure 4.20 (cont’d)

109

70

(a) Convergence History

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Weight=100
200 f|-'*"1‘i'rrlrlﬁl
0—0 Iterations -NS Penalty
H Iterations -ANN Penalty
. -——- Database - NS Penalty
‘3 I
.C
a)
a . E
E
«1 + "
C
(D
o. 100
- II
E
(U
.0
5 I + T. T .
O) O
l I ,
'O
I .
. I
_ I
O
I

 

 

 

 

 

 

0 . l I I
—40 -30 -20 -10 0 10 20 30 40 50 60 70
Iteration Number

Figure 4.21 Loading Unbalance Penalty. (a) With database (b) Without database

110

0))

Loading Unbalance Penalty at Shroud

2.5

Convergence History
Weight =100

 

 

 

 

f I ' IIII '1 III ' I

 

0—0 Iterations —NS Penalty
t——-t Iterations —ANN Penalty

 

 

l l

 

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

I

 

 

 

 

 

t

 

1o 20 30 4o 50 60
Iteration Number

Figure 4.21 (cont’d)

111

70

4.5.3 Analysis of Optimized Geometries

If a designer is to select the most important criteria by which to make a selection
from a large amount of geometries, it would obviously be the efﬁciency. In this section,
we will compare a few geometries that were produced during the optimization process
with the aid of Figure 4.22 and Table 4.5

Figure 4.22 shows the value of the total-total and total-static efﬁciency versus the
number of iterations. The negative Iterations represent the NS efﬁciencies of the
database, while the positive values represent the efﬁciencies obtained during the
optimization process. Indeed, this ﬁgure resembles the convergence history, but in a way
that has more physical meaning to the designer. That is, it is easier to understand
efﬁciencies rather than penalties.

Figure 4.22 demonstrates that both the total-total and total-static efﬁciency increase
as the optimization unfolds. Looking at the total-total efﬁciency, it is clearly shown that
database sample number 1 (-1 iteration) has the highest value in the database of 86.5%.
However, with the aid of Table 4.4 in section 4.5.], one can see that the inlet lean is -
21.76 degrees, which exceeds the i8 degree limit. If one examines the Iterations, it is
evident that, Iteration 39 has the highest total-total efﬁciency (87.3%), with a relatively
high total-static efﬁciency (68%). This sample has a lower value of total-static efficiency
than other geometries because it has a narrow channel (Fig. 4.23e), when compared to
other geometries that have wider channels. Recall that the performance is evaluated at the
diffuser exit and thus it explains why the total-static efﬁciency is lower because of the

fact that the boundary layer occupies a large percentage of the span in narrow channels.

112

Efficiences vs. Iterations

 

 

 

 

 

 

 

 

 

 

 

Single Point Optimization
1
>——> DB_NS -EI'A_TS
. v———v ITER_NS — ErA_1's
Baseline <—< DB_NS -ETA_TT
Geometry .——. ITER_NS — ETA_ 'T
18 at 0-9
Iteration I r
I. -I
- I
3“ \ WI rt
0 8 I ‘ A

 

 

 

 

A-=:
A-C:
I
f—gF:

 

Efficiencies

 

 

 

O
\l
k!-

0.5
—40 —30 —20 -10 0 10 20 3O 40 50 60 70
ITEFIATIONS

 

III I! III m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.22 Comparison of efﬁciencies(T-S and T-T) of the database and iterations

Iteration 39 also has an acceptable inlet/outlet lean and mass ﬂow (1.31 Kg/Sec) which
are listed in Table 4.5. Recall that the desired mass ﬂow is 1.3 Kg/sec. The. Mach
number distribution (Figure 4.23f) for this sample illustrates that the machine will make
pressure rise because there is a deceleration on the shroud at both the full and splitter
blades. There is no loading unbalance and negative loading, which is good. However,
there is still a slight velocity peak at the shroud full blade and splitter blade.

Iteration 12 has the lowest total penalty as already demonstrated in Figure 4.14. The
total-total efﬁciency is 84.6 % and the total-static efficiency is 68.4 %. One can say that

the total-total efﬁciency is lower because there is slight loading unbalance (Fig 4.23b)

113

and also because of the fact that there is almost no loading on the main blade shroud at
approximately S/C=0.25. However, it has a medium size exit width, which possibly
explains why it has approximately the same total-static efﬁciency at Iteration 39. The
mass ﬂow is 1.305 Kg/sec and is close to the desired value. The mass ﬂows of Iterations
39 and 12 are approximately the same because even though Iteration 39’s exit width is
more narrow than Iteration 12 (ﬁg. 4.23a), the impeller inlet width is larger in 39(ﬁg.
4.23e) and smaller in 12, which provides the reason for similar mass flows. When
looking at the Mach number distribution Gig. 4.23 b) more carefully, it can be observed
that the high velocity peaks are rounded off which is good in terms of preventing
separation. Finally, the inlet and outlet lean of the impeller are within acceptable limits as
shown in Table 4.5

Figure 4.22 demonstrates that Iteration 13 has the highest total-static efﬁciency. It
also has the widest exit width (Fig 4.230) of the three Iterations which provides the
reason to why it has the highest total-static efﬁciency. The mass ﬂow is 1.451 Kg/Sec,
which is due to the fact that is has larger exit width, but the same impeller inlet width as
the baseline. That is, the outlet width changes, while the impeller inlet height remained
the same as the baseline. Looking at the Mach number distribution (Fig 4.23d), it is
evident that the velocity peaks have been eliminated, however now it appears there is not
a lot of diffusion present at the full blade shroud. Perhaps the diffuser makes up for it,
since of the three samples the total pressure ratio is a litter bit higher than the other
Iterations discussed. This can be seen in Table 4.5. Finally, there is no negative loading
and no loading unbalance, although the loading of the main blade shroud between 0 and

40 % of the blade length (S/C) is a little on the low side.

114

Irvin—‘_-I-I-.-L .' ‘ 1
- . .‘ . IL

 

Finally, all the above mentioned geometries are much better than the baseline. The
baseline Mach number distribution is repeated in Figure 4.24 in order to show the
improvement. Also geometries at the later Iterations (68) have not been considered here
because they are similar to Iteration 39. Also, the author wanted to avoid choosing a
geometry from the last few Iterations because it is unknown as to exactly what is
happening in between Iterations 50-60. In any event, Figure 4.22 shows that these

geometries are not any better than the ones that have been considered above.

115

Table 4.5 Navier-Stokes calculations of ' ' Geometries

1.464 0.870 0.691 1.601 1.460 38.233
1.388 0.845 0. . 1. 45.

. . . 1. . .
1. . . 1. 45.
1. 1. 1

.415
1.

1.4

1.
1.473
1.451

 

116

Meridional Contours

(a) 34 Sample DATABASE USED
250
— ITERATION 12, 7 Blades P2/P01=1

225 Baseline

200
1 75
1 50

125

Radius (mm)

100

75

50

25

 

0
O 25 50 75 100 125 150 175 200 225 250

Axial Distance(mm)

Figure 4.23 Meridional Contours (left) and Mach distributions (right) of Iterations

117

(b) ITERATION 12, 7 Blades
P2/P01=1.39999998

 

 

 

 

 

 

 

 

 

 

 

2 . I - I
. — Full blade hub
1 8 - —— Splitter blade hub _
' —-— Full blade shroud J
--— Splitter blade shroud
1.6 — —
._ 1.4 — —
Q)
.0 .
S 12 -
z ' 7
.C
8 ..
2 1 “
.9.
Q
9
E
Q)
2
o l L I I
0 02 0.4 06 08 1
S/C

Figure 4.23 (cont’d)

118

(C)

Radius (mm)

Meridional Contours
34 Sample DATABASE USED

250
— ITERATION 13, 7 Blades P2/P01=1.

225 — Baseline

200
1 75
1 50
1 25
1 00

75

50

25

 

0 25 50 75 100 125 150 175 200 225 250
Axial Distance(mm)

Figure 4.23 (cont’d)

119

 

(d)

lsentropic Mach Number

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

ITERATION 13, 7 Blades
P2/P01 =1 .39999998

 

 

 

 

 

 

 

 

’ Y ' Full blade hub
- — Splitter blade hub
— - — Full blade shroud
- - —- Splitter blade shroud
!\- ......... /':~_:.-.:.‘.;:_‘.T_- rrrrr
E." . ‘‘‘‘‘ \ .Irlr‘\\ -— - —m\\
Ir ~ ~ -~
. I -
i .
I
O 0.2 0.4 0.6 0.8
S/C

Figure 4.23 (cont’d)

120

 

 

(e)

Radius (mm)

Meridional Contours
34 Sample DATABASE USED

250
— ITERATION 39, 7 Blades P2/P01=1.

225 Baseline

200
1 75
1 5O
1 25
1 00

75

50

25

 

O 25 50 75 100 125 150 175 200 225 250
Axial Distance(mm)

Figure 4.23 (cont’d)

121

ITEFIATION 39, 7 Blades

 

 

 

 

 

(f) P2/P01=1 .39999998
2 I I
. Full blade hub
1 8 _ — Splitter blade hub _ﬁ
' - -- Full blade shroud
1 6 - - - Splitter blade shroud

_L
I
.h
l I
I

_L
C
N
I Y
I

P
on

lsentropic Mach Number

.0
c»

.0
A

 

0.2

 

 

 

S/C

Figure 4.23 (cont’d)

122

 

Baseline Mach number Distribution
Pre—Rotation = 0 Degrees

 

 

 

 

 

2 ' r ' r
Full blade hub .
1 8 _ —— Splitter blade hub q
' — - - Full blade shroud
1 6 - -— Splitter blade shroud

 

1.4 ' 1

1.2 ~ ~

 

 

lsentropic Mach Number

 

 

 

S/C

Figure 4.24 Baseline Mach number distribution. Illustrates the improvement.

4.6 Peformance of O timum Geome and Conclusion

 

All of the geometries that have been analyzed are rather good, when compared to the
baseline. So the question is, how does one select the best one? It is quite difﬁcult to say
that there is the ultimate optimum geometry that is by far better than any other one. In
fact, there are many suitable candidates for the operation of an air conditioner and other
applications. So, the way in which the author chose the optimum was to take the one with
the maximum total-static efﬁciency, because that is the quantity that we are trying to

optimize using the system. Thus, using this criteria, the one that was chosen is Iteration

123

13. The mass ﬂow is 1.451 Kg/sec which is 11.6 % more mass ﬂow than the desired 1.3
Kg/sec. However, it is not likely that this machine will go directly to the market, and that
the mass ﬂow is really strictly required to be 1.3 Kg/sec. Perhaps, Iteration 39 was the
wiser choice, if one considers mass ﬂow and rm. Nevertheless, the author made the
decision and thus single point optimum is Iteration 13. The performance map of this

machine is displayed in Figure 4.25

Table 4.6 Characteristics of Single Point Optimum

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometric Characteristics . ,
Beta-le-hub 21.70 Degrees
Beta—le-hub-sp 20.87 Degrees
Beta-le-shr 64.73 Degees
Beta-le-shr-sp 64.60 Degrees
Beta-te 30.00 Degrees
Shroud clearance 0.575 mm
Max Hub Thickness 4 mm

Min Hub Thickness 3 mm

Max Shroud Thickness 3 mm

Min Shroud Thickness 2.5 mm
Percent blade chord length where ﬁnishes 25 %

the transition from Max to Min thickness.

Thickness remains constant from this

point all the way the trailing edge.

Impeller Tip Radius 100 mm

 

124

(8)

P02/ P01

1.8

1.6

1.4

1.2

Pressure Ratio vs. Mass Flow Rate

 

 

Rotational Speed = 28,000 ; Single Point Optimum Geometry

 

 

 

 

 

 

I ' 7 T I
i v——v Pre—Rotation = 0 Degree .
; o—o Pre-Rotation = 40 Degree 3
i A—A Pre-Rotation = 60 Degree 3
I a
:_ 3
E
L r r I I A r r 1 I I I r g I I n . I I r I r I L I r . .
O 0.5 1 1.5 2 2.5 3
Mass Flow Rate (Kg/s)

Figure 4.25 Performance Maps. (a) 1: (b) ms (0) m:

125

 

Total to Static Efficiency vs. Mass flow rate

 

 

 

 

 

 

 

(b)
Rotational Speed = 28000 RPM ; Single Point Optimum Geometry
1,....,.-..
: v——v Pre-Rotation = 0 Degree g
: o——o Pre—Rotation = 40 Degree 5
0.9 ’— A——A Pre-Rotation = 60 Degree 1:
0.3
0.7
a) E 5
'7 06 : 3
< ° F ‘2
E E i
0.5
0.4 g
' I
0.3
0.2 : I I I I I I I I I I I I I I I An I I I I I I I l I I I I j
0 0.5 1 1.5 2 2.5 3
Mass Flow Rate (Kg/s)
Figure 4.25 (cont’d)

126

Total to Total Efficiency vs. Mass flow rate

 

 

 

 

 

 

 

 

 

(C) Rotational Speed = 28000 RPM ; Single Point Optimum Geometry
1E....,..,.,..,,,..I,.,I.,i
0.9
0.3
0.7
0.6
E 0.5
0-4 g- v———v Pre-Rotation = 0 Degree ‘
E 0—0 Pre—Rotation = 40 Degree ;
0.3 :- t——t Pre-Rotation = 60 Degree é
0.2
0.1 - -
O : #4 I I l I g I 'I i I I 4 I 1 I I I I i I I I I 1
O 0.5 1 1.5 2 2.5
Mass Flow Rate (Kg/s)
Figure 4.25 (cont’d)

Finally, the Mach number distributions of the single point optimum at various Pre-
rotations demonstrate an improvement over the baseline. However there is a problem at
60 degrees Pre—Rotation. That is, there is some negative loading on the full blade hub and
even worse, there is separation at the splitter blade hub, which is shown in Figure 4.26c.
Thus, the geometry that gives an improvement by 7(63.5% to 70%) points of total-static

efficiency for a pre-rotation of 0 degrees, however, will not be suitable for all three

127

operating points. Finally, this gives motivation to pursue the multi-point optimization in
the next chapter.

(a)

Single Point Optimum 7 Blades
P2/P01=1.4000 ; Pre—Rotation = 0 Degrees

 

 

 

 

 

 

2 ' I T
. Full blade hub
1.8 _ — Splitter blade hub ‘
—-— Full blade shroud
16 . —-- Splitter blade shroud ‘

L4+ -

1.2 - -

lsentropic Mach Number

 

 

 

 

 

o8~ ,I‘ ‘
i\‘\ _________ f _________
’n-\ '___ T;--—-—"—'—--'-_-=-_-.—_.\
o6 ‘— ‘~sJu ~ ‘\
H F“‘§‘ .
I. .‘..
04¢ H «
i .
02 _
O I l I L I l I l I
0 oz o4 as as 1
so

Figure 4.26 Mach Number Distributions at various Pre-Rotations: (a) 0 degrees (b) 40
degrees, (c) 60 degees

128

(b) Single Point Optimum, 7 Blades
P2/P01=1.420000 ; Pre-Rotation = 40 Degrees

 

 

 

 

 

2 . I I
Full blade hub
1 8 _ —— Splitter blade hub
' L ---- Full blade shroud
—-- Splitter blade shroud
1.6 ~
1.4 —

1.2 r

lsentropic Mach Number

 

 

 

 

 

0.8 _
0.6 E' f, - \ _______ _-
!.~‘.‘;':_'."_’,:‘." I: '''''''''
b—p' -;>;>~‘.~_=~’/
0.2 '
l
0 , t 1 g 1 l
0 0 2 0.4 0 6 O 8

S/C

Figure 4.26 (cont’d)

129

(C)

1.8

1.6

1.4

1.2

0.8

lsentropic Mach Number

0.6

0.4

0.2

Single Point Optimum, 7 Blades
P2/P01=1.425000 ; Pre—Rotation = 60 Degrees

 

 

 

 

 

 

 

 

 

 

130

F T Full blade hub
_ —— Splitter blade hub
— - — Full blade shroud
— - — Splitter blade shroud
i\-‘ _____ t:::r_‘r_'.—._-=—rr—'—-“"“‘
r -_'.:::-_. I \
_ l ’ ‘ 11¢ xv? _ _
k L...) ‘ ' T;~ - ~ / /
v i - \ — ;T:— '
0 0.2 0.4 0.6 O 8
S/C
Figure 4.26 (cont’d)

-\ II~.L)UI-I.

 

1mg.

CHAPTER 5

MULTIPOINT OPTIMIZATION

In this chapter the results of the multiple point optimization will be discussed. The
design space used is the same as for the single point case and so is the performance
prediction. Consequently, these concepts will not be repeated. However, the reader is
reminded that the number of Artiﬁcial Neural Networks used is now 21, instead of the 7
used for the single point optimization. For more details about the above mentioned, one

should consult chapter 4.

5.1 Objective Function and GA settings
5.1.1 Objective Function

As discussed in the previous chapter, the ANN predicts the performance of an
individual. This performance is then converted to an objective function with which the
genetic algorithm will minimize. The Genetic algorithm mimics biological evolution and
thus is a statistical or non-deterministic process. The purpose is that blades will reproduce
other better geometries and the poor geometries will tend to die out. This is why the term
survival of the ﬁttest is usually used. A somewhat in depth treatment of the operational
principle of the GA was given in section 2.3; consequently, we will progress to the

speciﬁcs that apply to our design case.

The objective function used by the genetic algorithm is displayed below:

131

Numofpo int s=3

0F = ZWIOPI = WIOPI + W20Pz + W303 [5.1]
1

OP, =(W

mass/10w mass/low + Wmass_diﬂ Pmass_dI'/f + Wmsts + Wm Pia + WnIPnI )1 [52]

This function is a weighted sum of various penalties used to quantify the
performance. A higher value of the objective function implies a lower performance
impeller, while a lower value represents good impeller performance. The penalties in
equation [5.2] are the same as discussed in section 4.4.1, and as a consequence will not be
repeated here.

For the multiple point case various overall weights (W1,W2,W3) were used to
represent the various seasons of the year. That is, in the summer one requires more mass
ﬂow to cool the building. Even if the summer is shorter than spring and autumn, the air
conditioner could possibly waste more energy in that season than in the other two seasons
combined. This is simply because more power is required to pump a large amount of
Freon through the building and if this is done inefﬁciently, a lot of energy can be wasted.
Thus, for operating point 1, which represents the summer, the overall weight was chosen
to be W l = 55 %, while the others were W2=3O%, W3=15%. The way in which this was
implemented was through the following procedure. This procedure is presented for
anyone whom will try to re-create and understand how the author calculated the weights.

First of all, there is no input ﬁle to implement these overall weights. Thus, one must

calculate manually these weights and put them in the existing input ﬁles (weightsinl,

132

 

weight.in2, weights.in3). Once the weights are chosen for operating point 1 (Pre-Rotation
= 0 Degrees), as seen in the ﬁrst column of Table 5.1 below, to get the weights for the
other two operating points, the following procedure should be consulted. This procedure
is applied to the ﬁrst row of Table 5.1 Finally, the procedure was repeated for the rest of

the penalties.

1. To calculate 100 % weight, assuming the weights for operating point 1 are already

at 55 %.

5—00- = 909.09
0.55

2. To Calculate 30 % weight.

909.09(0.30) = 276

3. To Calculate 15 % weight.

909.09(0.15) = 136

This was done, because if one multiplies the original values in the ﬁrst column by
0.30 and 0.15, the values of the penalties can be a little on the low side. Also one does not
know what should be the order of magnitude of the Penalty input to the GA. In any event,
it can be seen from Table 5.1 and the convergence history, discussed in section 5.2.2, that

more emphasis is placed on 0P1 (summer) than the other two operating points.

133

 

Table 5.1 Weights used in Multiple Point Optimization

500 273
100000 , 54545
600 327
100 54

100 54

 

5.1.2 GA Settings

Before performing the optimization one must select the GA and ANN settings. The
only parameter that was changed for the multiple point optimization was the mutation
probability. This is because the optimization system was updated such that it calculated
the substring length. Then from GA theory [2], it was advised by the system manager that
the following parameters be tried. In any event, the author is unsure of the impact, if any,
the change in the mutation probability (from 1 % to 0.45%) had on the results.

Table 5.2 GA and ANN Parameters

 

134

 

Where the parameters in the table are deﬁned as follows (see reference [4]):

GA:

ANN:

irestrt = 0 for a new GA run ; 1 for a restart continuation of a GA run

npopsiz = The population size of a GA run

pmutate = Jump mutation probability

maxgen = The maximum number of generations to be run by the GA

pcross = Cross over probability

ielite = 1 ; Irnplies that the best individual is replicated into the next generation
itoumy = l ; Irnplies tournament selection

pcreep = Creep mutation probability

iuniﬁm = 0 ; Irnplies single point crossover

nchild = 2 children are produced per pair of parents

TRAINH‘IG = 80 % of all the samples are used for training of the ANN

NANN = Number of neural networks per operating point.

135

 

5.2 Presentation of the Optimization
In this section one will discuss the database creation, the convergence history, and
ﬁnally compare a few Navier-Stokes results which will lead to the selection of the multi-
point optimum. However, one must ﬁrst mention the operating points that were chosen.
The operating point was chosen near high efﬁciency plateau as can be seen in the

following ﬁgure. A designer should optimize at a point with sufﬁcient surge margin.

 

Pressure Ratio vs. Mass Flow Rate
Rotational Speed = 28,000 ; Single Point Optimum Geometry

(21)

 

 

 

 

 

 

 

 

 

 

 

 

1.8 I ' ‘ I ' r 1 1 L f 4

I . 4

: v——v Pre-Rotatlon = 0 Degree 3

0—0 Pre-Rotation = 40 Degree 3

A—A Pre—Rotation = 60 Degree 3

1.6 r Operating Line ‘

I- E .

a t I

N 1.4 :- €

0 : 3
n.

1.2 I— -I

1 t I I I I l 4 I I I l I I I I l I I I I l I I I I 1 I I I I :

0 0.5 1 1.5 2 2.5 3

Mass Flow Rate (Kg/s)

Figure 5. 1 The operating points chosen for optimization. (a). PI (b). TS

136

Total to Static Efficiency vs. Mass flow rate
b
( ) Rotational Speed = 28000 RPM ; Single Point Optimum Geometry

1 ‘7 Y T f T V V j’ I
I.
-

v——v Pre—Rotation = 0 Degree
: o—a Pre-Rotation = 40 Degree ‘
0.9 E- t—-—A Pre—Rotation = 60 Degree €

F d

l- 1

h ‘ 1

h -4

b 4

F 3 2 <

0 8 d
' 4
v‘

4

l- A ' 4

b 4

> -<

.

r-
» .I
r- <
07.. _
' h .I
D 4
b d
4
4
.

 

 

 

 

 

 

 

Wﬁ]

 

 

 

 

 

 

 

 

 

a) E
l--l g
(06? ‘
l— :
[U I
.
t
0.5:— ~
I
E
0.4} a
:
:
t
0.3} -
I
0'2:IIIIIIIII1IIIItIIIIiIIIItIIII‘
0 0.5 1 1.5 2 2.5 3

Mass Flow Rate (Kg/s)

Figure 5.1 (cont’d)

The mass ﬂow required for the ﬁrst operating point was chosen to be 1.3 Kg/sec.
While for operating point 2 it is 1.0 Kg/sec and 0.7 Kg/sec for operating point 3. The
efﬁciency requirement on total-static efﬁciency was set at 90% as mentioned before in
chapter 4. The reason being that one does not expect to improve the total-static by more
than 25 points. That is, going from 63% to 90% seems a little unreachable, and it has

been observed that is it better to put a value that is within expectation, rather than to just

put 100%.

137

-. T ~. VtWMQo-Vn-

 

 

5.2.1 Database

As explained before in the optimization method chapter, the whole original
knowledge of the ANN and GA comes from a reference database. The database contains
35 Navier-Stokes calculations per operating point, coming from different geometries
representing the design space within the limits deﬁned in chapter 4. The selection of the
geometries can be realized in two different ways, either randomly, or using the design of
experiment method, DOE technique. It can be used in a wide variety of applications when
some parametric analyses are needed. Its aim is to construct a more representative and
systematic database, performing a statistical survey of the design parameters. This
approach was used for 32 of our samples. Indeed, it presents the disadvantage of
considering only to extreme values (25% and 75% over the lower limit) of the
parameter’s range so that in the end not much is known about the center of the design
space. This is why a central point case is run in which the variables are forced to be 50%
over the lower limits. Cases 1-32 were given by the DOE, while the other samples that
were included were the baseline geometry and the single point optimum geometry.

In the Multi-point optimization, the same geometries that were utilized in the single
point case were used for the multi-case. The geometries were simply run at the other pre-
rotation values of 40 and 60 degrees. Finally, tables containing information about the NS
computations of the database are displayed below in Tables 5 .3, 5.4, and 5 .5. Finally, if
one takes the time to survey all the tables in the same manner as was done in chapter 4,
then it is evident that none of the samples in the database have the optimum combination
of characteristics. This is why we will now progress to the results of the optimization in

hopes that the system can ﬁnd the optimum.

138

Table 5.3 NS results of the Database at IGV Setting Angle = 0 Degrees

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EBSamplerr Mass Flow ETA_'TT. ETA_TS 1302/1301 P2/P01 Inlet Lean (o_eg) Outlet Lean @924
1 . 1.456 0.665 0.660 1.599 1.453 -21.766 45.000
2 1.102 0.717 0.575 1.564 1.437 43.496 45.000

' 3 1.311 0.632 0.660 1.574 1.440 -9.928 45.000
. 4 1.370 0.620 0.675 1.603 1.461 42.970 45.000
5 1.437 0.626 0.676 1.612 1.466 -6.092 45.000
6 1.350 0.621 0.643 1.565 1.442 —6.751 45.000
7 1.177 0.761 0.617 1.575 1.452 45.326 45.000
6 , 1.362 0.652 0.674 1.567 1.446 -5001 45.000
9 1.167 0.799 0.636 1.563 1.433 40.026 45.000

- 10 1.117 0.727 0.565 1.559 1.435 -2540 45.000
11 1.402 0.651 0.672 1.562 1.443 0.000 42.662
12 1.226 0.729 0.562 1.571 1.441 -2355 45.000
13 1.037 0.660 0.541 1.542 1.417 -8.831 45.000
14 1.213 0.795 0.631 1.556 1.426 -9.086 45.000
15 1.296 0.791 0.645 1.569 1.465 -20.146 45.000
16 1.413 0.839 0.656 1.600 1.452 -10.480 45.000
17 1.305 0.641 0.670 1.575 1.443 44.990 45.000
16 1.017 0.697 0.556 1.554 1.428 47.209 45.000
19 1.313 0.632 0.662 1.566 1.436 -13.680 45.000
20 1.430 0.640 0.693 1.606 1.466 -25.641 45.000
21 1.516 0.636 0.660 1.637 1.501 0.000 14.790
22 1.322 0.616 0.644 1.575 1.437 0.000 43.814
23 1.221 0.777 0.632 1.561 1.456 -6.349 45.000
24 1.362 0.646 0.673 1.560 1.444 -8.861 45.000
25 1.245 0.602 0.634 1.566 1.432 0.000 34.625
26 0.976 0.662 0.542 1.544 1.416 -O.260 45.000
27 1.442 0.642 0.656 1.563 1.439 0.000 27.734
26 1.079 0.696 0.557 1.547 1.423 -8.991 45.000
29 1.295 0.766 0.617 1.595 1.462 0.000 0.649
30 1.323 0.810 0.635 1.565 1.428 0.000 30.363
31 1.252 0.776 0.631 1.574 1.452 -4.616 45.000
32 1.246 0.610 0.642 1.573 1.438 -6.748 45.000
33 1.371 0.621 0.664 1.596 1.466 -6.854 45.000
Sin le opr 1.451 0.867 0.705 1.597 1.470 0.000 43.539
leaseline, ., .. 1.257 0.624 0.635 1.562 1.417 -4336 45.000

 

 

139

 

Table 5.4 NS results of the Database at IGV 40

 

140

Table 5.5 NS Results of the Database at IGV 60

 

5.2.2 Convergence history

In theory, the NS and ANN penalties should overlap each other in order to say that
the system has found the optimum geometry, or converged. One can see in Figure 5.2 that
the NS and ANN penalties slowly start to come together. A possible reason why the lines
do not come fully together could be that other penalties are offsetting each other. For
instance, in Figure 5.3 the ANN and NS actually come together at various places.
Another reason is that it may not be fully converged because there are a lot of design
variables and 3 operating points. Thus, it can be expected that it may really take 100 or

more Iterations to fully converge. This however was unfortunately unobtainable due to

141

lack of time. In the Figures (5.2-5.6), the upper left hand corner represents operating
point I as deﬁned in Figure 5.1. The other two operating points are also labeled in the
ﬁgure according to Figure 5.1. Finally, for the sake of completeness, the rest of the

convergence histories are listed.

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OPERATING POINT 1 OPERATING POINT 2
Pre-Rotation = 0 Degrees Pro-Rotation = 40 Degrees
400 ' I ' ' 200 ‘ l ‘ '
I—-I ANN 180 H ANN 1
0—0 NS d f 0—0 NS I
160 _— j
g g 120 _ 1
£ 0‘.) 100 _, j
‘73 3 8°. :
1— 20° ‘ 1- 60 _ j
20 f j
1 00 I l I l I I I 0 I I I I I I I
0 1 0 20 30 40 0 10 20 30 4O
ITERATIONS ITERATIONS
OPERATING POINT 3 GRAND TOTAL PENALTY
Pre—Rotation = 60 Degrees
100 . I I I I I 600 I I I I I I I
L H ANN 1 H ANN
90 r o—o NS 1 o——o NS l
I 500 _I
a 7 e
(U (U
a . 6
o. . (L 400 -
E ‘ E
O O
4 I-
'— 300 W I
I I I I 200 I I I t I 4 I
0 1 0 20 30 4O 0 1 0 20 30 40

ITERATIONS ITERATIONS

Figure 5.2 Convergence History of Total Penalty at the three operating points.

142

200

160
I40

Efficiency Penalty

100

80

Efficiency Penalty
m 00 w A is
01 O 01 O 01

N
O

180

120

OPERATING POINT 1

W819i“ = 600

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ II-—I ANN—ETA__ 6
9—0 NS-ETA_TS
O 1 0 20 30 40
ITERATIONS
OPERATING POINT 3
Weight: 164
I ' IL—I min—51,65
F o—o NS—EI'A_TS
0 1 0 20 30 40
ITERATIONS

(D
O

on
O

Efficiency Penalty

A
O

Efficiency Penalty
N
01
O

N
O
O

150

OPERATING POINT 2
Weight = 327

 

8 E‘:

01
O

 

 

 

T

H ANN—ETA_T 3
H NS-EI'A__TS

 

 

 

 

10 20 30 40
ITERATIONS
TOTAL EFFICIENCY PENALTY

 

 

 

 

H ANN-ETA_T
o—o NS-ETA_TS

U)

 

 

 

 

10 20 30
ITERATIONS

Figure 5.3 Convergence History. Efﬁciency Penalty at the three operating points.

143

 

OPERATING POINT 1

80 II

Weight = 500

 

Mass_tot Penalty.
8

 

H ANN
O——O NS

 

 

 

  

 

 

 

“’1

    

 

 

 

 

 

 

 

 

 

 

 

O I.
O 10 20 30 4O
ITERATIONS
OPERATING POINT 3
Weight = 136
20 I I I I
H ANN
o—o NS *
E 15 -
m
C
0)
CL
*5 10 -
:1
S
2 5 7
0 I L A L _ I I I
0 10 20 30 4O

TERATIONS

ty

Mass_tot Penal

Mass_tot Penalty

_L
O

A
C

150

100

50

0)
O

N
O

OPERATING POINT 2

Weight = 273

 

1—

 

.-
‘
b |.

O

I

1111.11. 1

10

H ANN
o—o NS

 

 

 

 

 

 

30 4o
ITERATIONS

GRAND TOTAL PENALTY

 

 

'

l

I

 

H ANN
H NS

 

 

 

 

 

 

J ' A l 4

 

O

1 O 20 30 40

ITERATIONS

Figure 5.4 Convergence History. Mass Penalty at the three operating points.

144

 

OPERATING POINT 1

150 . I

Weight: 100

 

100

50

Loading Unbalance Penalty

 

O 1 0 20 3O 40

HANN
O o—oNS

 

 

 

 

 

 

 

 

ITERATIONS

OPERATING POINT 3

Weight = 27

 

.b
O

I

(A)
O
1

Loading Unbalance Penalty
5 B

 

 

H ANN
H NS

 

 

 

 

 

 

O

0 1 0 20 30 4O

ITERATIONS

Loading Unbalance Penalty

 

 

 

 

 

 

 

 

OPERATING POINT 2
Weight=54

3. 50 . ' - II-—IIANN
E o—ONS
a) 40 T T
0- .
a.)
g 30 I . ~
g . .
c 20 r ‘ ‘ 2
D .
.- 10 ‘- ‘ _
P I
(U
3 0 #4 L) ‘M B d

0 10 20 3O 40

ITERATIONS ..,-.

TOTAL LOADING UNBALANCE PENALTY

200

150

 

 

 

 

 

 

 

 

 

 

 

 

I I III—4 ANN
. o—o NS
0 10 20 3O 40

ITERATIONS

Figure 5.5 Convergence History. Unbalance Penalty at the three operating points.

145

OPERATING POINT 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Weight = 3300
0 i ,
2‘ 5 HANN
E i o—o NS
c 40 r n
a)
Q_ .
.8 30
.0
m .-
O
—| 20
(D
E ..
a 10
a)
z i-
0
O 10 20 30 4O
ITERATIONS
OPERATING POINT 3
Weight = 900
10 r I ' 1 7 1
2. P t—A ANN
76 0—0 NS
c 8 - —
a)
n_ .
E 6 - a
“o
to
O
_J 4 _ _
CD
.2 '
a 2 — -
(D
Z
0 l 1 1

 

 

0 1 0 20 3O 40

ITERATIONS

.0
oo

.0
oz

Negative Loading Penalty
_0 o
N J:-

O

OPERATING POINT 2

Weight: 1800

 

 

 

 

0 1 O 20 30 4O

t—A ANN
o——o NS

 

 

 

 

 

 

 

 

ITERATIONS

TOTAL NEGATIVE LOADING PENALTY

60

.5
O

Negative Loading Penalty
N
O

O

 

I

I

 

 

 

HANN
O o—oNS

 

 

 

 

 

 

 

0 1 0 20 30 40

ITERATIONS

Figure 5.6 Convergence History. Neg. Loading Penalty at the three operating points.

146

 

5.2.3 Analysis of Optimized Geometries

If a designer is to select the most important criteria by which to make a selection
from a large amount of geometries, it would obviously be the efﬁciency. In this section
we will compare a few geometries that were produced during the optimization process
with the aid of Figure 5.7 and Tables 5.6, 5.7, and 5.8.

Figure 5.7 shows the value of the total-total and total-static efﬁciency versus the
number of iterations. The negative iterations represent the NS efﬁciencies of the
database, while the positive values represent the efﬁciencies obtained during the
optimization process. Indeed, this ﬁgure resembles the convergence history, but in a way
that has more physical meaning to the designer. That is, it is easier to understand
efﬁciencies, rather than penalties.

Looking at the total-total efﬁciency, one can see that in Figures 5.7a,b that the single
point optimum (Iteration -34) has the highest total-static efﬁciency when compared to the
rest of the iterations. This is because the exit width is wider than the rest of the Iterations
produced during the optimization. However, at 60 degrees pre—rotation, there is the
problem of ﬂow separation at the main blade hub (see ﬁg. 4.26c or 5. 101) which explains
why in Figure 5.7c that the total-static efﬁciency is closer to the efﬁciencies of the
optimized geometries. These optimized geometries all exhibit narrow exit width
channels. The reason for the separation could be that the wider exit is causing too much
diffusion at 60 degrees pre-rotation. In any event, the single point optimum is clearly not

the best candidate for multi-point operation.

147

Looking at both efﬁciencies in all three Figures, one can see that Iterations 3, 11, and
33 are clearly set apart from the rest of the geometries. Thus the author will attempt to
compare these geometries. However, before doing this, one must say that the contours in
Figure 5.8a,c,e are also compared to the baseline and single point optimum contours,
where the wide exit contour is the single point optimum. These contours are just to show
that the optimized geometry tends to resemble the baseline. In Figure 5.8a,c,e, there is not
much variation in the meridional contour to be seen.

Starting with Iteration 3, this geometry has a longer main blade, but a short splitter
blade, as can be seen in Figures 5.8a and 5.9a when compared to Iterations 11 and 33
which have the opposite traits. Apart from this, the only real difference between the
meridional contours is that the impeller inlet width of Iteration number 33 (Fig 5.8e) is a
little smaller than the other two. Since the values of the total-total and total-static
efﬁciencies listed in Tables 5.6, 5.7, and 5.8 are all the around the same value for each of
the Iterations, one will try to choose the optimum by means of the Mach number
distribution. Of course, the efﬁciencies change a little from operating point to operating

point. This comparison of Mach number distributions takes place just after Figure 5.8.

148

(a) Efficiences vs. Iterations
OPERATING POINT 1

 

 

 

 

 

 

 

 

 

- 1
Zafguilt; >—> DB_NS —ETA_TS
v—mv ITEFLNS — ETA_TS
<——< DB_NS —ETA__‘lT
O 9 t—a ITER__NS — ET A_‘l'T

 

 

 

 

 

 

, . .1
fl (ill l‘lr‘iTM PM

0.7 i

 

 

Efficiencies

 

 

06 W . "WWW"!
y l l y-

0.5
-40 —3O —20 -1 O O 1 0 20 30 4O

ITERATIONS

 

F":

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.7 Plots of the efﬁciencies at Various IGV Setting angles. (3) 0° (b) 40° (c) 60°

149

 

 

 

 

 

Efficiences vs. Iterations
OPERATING POINT 2

 

 

 

>——> DB_NS -ETA_TS

v——v ITER_NS - ETA_‘I'

<——< DB_NS -ETA_TT

t——A ITER__NS - ETA_‘l'T

 

 

b

 

 

 

 

‘lety‘w

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
V» l

 

 

 

 

 

 

 

 

 

 

D I»

 

 

 

 

 

 

 

 

 

 

(b)
1
Single _
opt at -34
iter \
0.9
a, 0.8
.9
0
C
.‘2
.9
E
0.7
0.6
0.5
-40

—20 —1 0

0 1 0 20 3O

ITERATIONS

Figure 5.7 (cont’d)

150

40

(C)

Efficiencies

0.5

-30

Efficiences vs. Iterations
OPERATING POINT 3

>——v DB_NS -ETA_TS
V———v ITER_NS —- ETA_
<——< DB_NS -ETA_TT
A——A ITER_NS — ETA_

 

—20 -10 O 10 20 3O
ITERATIONS

Figure 5.7 cont’d

151

40

 

Table 5.6 NS Results of the Optimizations at Pre-rotation = 0 Degrees

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lIteration # Mass Flow ETA_TT ' ETA_TS P02/P01 P2/P01 Inlet Lean (021) Outlet Lean (Beau
' 1 1.256 0.841 0.646 1.557 1.412 -6.298 45.000
2 1.238 0.849 0.645 1.551 1.402 -7.415 45.000
3 1.328 0.878 0.662 1.558 1.404 —4.984 45.000
4 1.272 0.856 0.649 1.560 1.408 0.000 42.679
5 1.259 0.852 0.640 1.554 1.399 -4.494 45.000
6 1.274 0.855 0.643 1.554 1.399 -5.984 45.000
7 1.255 0.862 0.648 1.549 1.396 -7.603 45.000
8 1.278 0.842 0.627 1.551 1.393 0.000 31.509
9 1.204 0.843 0.637 1.542 1.394 0.000 42.455
10 1.239 0.857 0.645 1.546 1.395 -i818 45.000
11 1.289 0.874 0.656 1.551 1.397 -7.058 45.000
12 1.091 0.822 0.636 1.529 1.395 0.000 44.050
13 1.236 0.858 0.647 1.547 1.396 -3.270 45.000
14 1.233 0.869 0.659 1.543 1.396 -1.868 45.000
15 1.271 0.864 0.646 1.549 1.394 0.000 41.029
16 1.372 0.860 0.631 1.561 1.394 0.000 17.202
17 1.351 0.851 0.631 1.558 1.397 0.000 18.523
18 1.279 0.870 0.652 1.552 1.397 -2.430 45.000
19 1.324 0.858 0.632 1.552 1.390 0.000 14.359
20 1.355 0.828 0.621 1.563 1.406 0.000 -33.080
21 1.245 0.858 0.660 1.554 1.411 -6.329 45.000
22 1.168 0.846 0.649 1.540 1.399 -1.060 45.000
23 1.305 0.842 0.620 1.549 1.387 0.000 -18.049
24 1.296 0.842 0.620 1.543 1.383 0.000 ~37.108
25 1.219 0.860 0.654 1.541 1.396 -2.040 45.000
26 1.243 0.850 0.639 1.549 1.397 -3.752 45.000
27 1.259 0.858 0.642 1.550 1.395 .4859 45.000
28 1.231 0.866 0.656 1.542 1.395 -7.494 45.000
29 1.303 0.868 0.657 1.555 1.403 -2.562 45.000
30 1.335 0.864 0.652 1.562 1.408 -5.211 45.000
31 1.281 0.864 0.650 1.550 1.398 0.000 42.861
32 1.283 0.863 0.646 1.552 1.397 -3.157 45.000
" 33 1.348 . 0.884 0.671 1.566 1.413 -7.914 45.000
_Sjﬂgle OPT ' 1.451 0.867 0.705 1.597 1.470 0.000 43.539
[gaseline 1.257 0.824 0.635 1.562 1.417 -4.336 45.000

 

 

 

 

 

 

 

 

152

 

Table 5.7 NS Results of the Optimizations at Pre-Rotation = 40 Degrees

 

153

Table 5.8 NS Results of the Optimizations at Pre-Rotation = 60 Degrees

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Iteration # Mass Flow Eﬁ_TT ETA_TS P02IP01 P2/P01 Inlet Lean (Deg) Outlet Lean (DegL
1 0.672 0.868 0.678 1.587 1.441 -6.298 45.000
2 0.665 0.869 0.679 1.581 1.437 -7.415 45.000
3 0.692 0.891 0.693 1.578 1.433 4.984 45.000
4 0.678 0.872 0.680 1.588 1.441 0.000 42.679
5 0.672 0.875 0.682 1.579 1.435 -4.494 45.000
6] 0.674 0.880 0.686 1.577 1.433 -5.984 45.000
71 0.670 0.885 0.690 1.572 1.430 -7.603 45.000
8] 0.698 0.882 0.681 1.573 1.426 0.000 31.509
9 0.650 0.875 0.684 1.570 1.429 0.000 42.455
10 0.659 0.88L 0.688 1.571 1.430 -6.818 45.000
11 0.676 0.887 0.692 1.574 1.431 -7.058 45.000
12 0.604 0.860 0.680 1.566 1.432 0.000 44.050
13 0.661 0.881 0.689 1.573 1.432 -3.270 45.000
14 0.650 0.882 0.694 1.567 1.431 -1.868 45.000
151 0.686 0.890 0.691 1.572 1.427 0.000 41.029
16} 0.731 0.887 0.681 1.575 1.425 0.000 17.202
17 0.724 0.889 0.684 1.578 1.428 0.000 18.523
18 0.670 0.886 0.692 1.573 1.431 -2.430 45.000
19 0.710 0.889 0.687 1.572 1.425 0.000 14.359
r 20 0.733 0.885 0.686 1.580 1.432 0.000 -33.080
21 0.654 0.864 0.678 1.581 1.440 -6.329 45.000
22 0.629 0.862 0.680 1.572 1.435 -1.060 45.000
2 0.723 0.888 0.683 1.571 1.423 0.000 -18.049
24 0.710 0.892 0.688 1.564 1.419 0.000 -37.108
25 0.648 0.879 0.690 1.568 1.431 -2.040 45.000
26 0.677 0.895 0.695 1.575 1.430 -3.752 45.000
271 0.689 0.883 0.683 1.575 1.428 -4.859 45.000
28 0.651 0.882 0.692 1.569 1.430 -7.494 45.000
29 0.686 0.887 0.691 1.576 1.433 -2.562 45.000
30 0.694 0.890 0.692 1.582 1.435 -5.211 45.000
31 0.678 0.884 0.688 1.573 1.431 0.000 42.861
32 0.682 0.885 0.688 1.576 1.431 -§.157 45.000
33 0.690 0.891 0.693 1.574 1.430 -7.914 45.000
Si le OPT 0.701 0.890 0.704 1.610 1.465 0.000 43.539
Iine . 0.657 0.892 0.699 1.596 1.450 -4.336 45.000

 

154

 

 

 

Meridional Contours

(a) 250

— ITERATION 3, 7
— Baseline
225 Single Point Optimum

200

175

150

125

Radius (mm)

100

75

50

25

 

0 25 50 75 100 125 150 175 200 225 250
Axial Distance(mm)

Figure 5.8 Meridional Contours (left) and blade proﬁles (right) of the Iterations.

155

 

ITERATION 3, 7 Blades

Blade Profiles

(b)

— Main blade hub
—— Splitter blade hub
-————-— Main blade shroud
—— Splitter blade shroud

Theta

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
m/R

Figure 5.8 cont’d

156

(C)

Radius (mm)

250

225

200

175

150

125

100

75

50

25

0

 

Meridional Contours

—— ITERATION11,7
—— Baseline
— Single Point Optimum

25 50 75 100 125 150 175 200 225 250

Axial Distance(mm)

Figure 5.8 cont’d

157

(d)

Theta

ITERATION 11, 7 Blades

Blade Profiles

— Main blade hub
— Splitter blade hub
—— Main blade shroud
—— Splitter blade shroud

 

1 1.5 2 2.5 3 3.5 4 4.5
m/R

Figure 5.8 cont’d

158

(e)

Radius (mm)

250

225

200

175

150

125

100

75

50

25

O

Meridional Contours

— ITERATION 33, 7
Baseline
Single Point Optimum

25

50

 

75 100 125 150
Axial Distance(mm)

Figure 5.8 cont’d

159

175 200 225 250

 

<0 ITERATION 33, 7 Blades

Blade Profiles

— Main blade hub
—— Splitter blade hub
—— Main blade shroud
———- Splitter blade shroud

Theta

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
rn/R

Figure 5.8 cont’d

Looking at Iteration 3, one can see in Figure 5.9(a-c) that the Mach number
distributions look quite good at all three operating points. That is, extremely high velocity
peaks are not present, there is no loading unbalance. There seems to be a little bit of
negative loading at the leading edge of the splitter blade hub of Figure 5.9a. In Figure
5.9c there is negative loading near the leading edge of the main blade hub. The negative
loading at 60 degrees pre-rotation was present in all the Iterations that were observed.

There must be some problem associated with the highly non-uniform pressure

160

 

 

distribution that is the observed at 60 degrees pre-rotation. On than, these few
observations, the Mach number distributions look much better than the baseline (Figure
5.10a-c)

Iteration 11 can be observed in Figure 5.9(d-t). In Figure 5.9d negative loading is
present near LE of the splitter blade at the hub. However the rest of the blade
distributions look acceptable. That is, no loading unbalance, no extremely high velocity
peaks. In Figure 5.9e, the distribution is quite good, perhaps one of the best that has been
observed yet. In Figure 5.9f, the distribution looks alright. However, the ever bothersome
negative loading is present at full blade hub leading edge

Iteration 33 can be observed in Figure 5.9g-i. In Figure 5.9g the high velocity peaks are
not present, which is good. There is no loading unbalance and no negative loading.
Continuing on with Figure 5.9h one can see a good distribution. Perhaps, there is not
enough diffusion however. Finally in Figure 5.9i, there is still negative loading that is
always present, plus a little amount of return ﬂow at the splitter blade hub.

Finally, it is obvious that the Mach number distributions in Figure 5.10 (a-t) are clearly
worse than the ones in Figure 5.9. Well, that is if one requires them to be suitable for
multiple point operation. So based on Mach number distribution, it looks like Iteration 3

is the best.

161

 

ITERATION 3, 7 Blades
P2/P01=1.39999998 ; Pre—Rotation = 0 Degrees

 

 

 

 

 

 

2 e I T
_ Full blade hub
1 8 - — Splitter blade hub j
. . — - — Full blade shroud
1 5 — - —— Splitter blade shroud
1.4 — "

--L

N

I V
l

 

lsentropic Mach Number

 

 

 

08 k
! ‘\‘\-‘ I“ ’;r—/
%I_ ~+~mm_~$:s/- .
f ‘ ----- L_ if‘x.
0.4 , ir-~-‘-:--:..:‘ _"/ _
[I
II _
0.2 '
0 i l r L r l . l .
0 0.2 0.4 0.6 0.8 1

S/C

Figure 5.9 Mach number distributions at various IGV settings of Iterations

 

162

0’) ITERATION 3, 7 Blades
P2/P01=1.41999996 ; Pre—Rotation = 40 Degrees

 

 

 

 

 

2 i i
Full blade hub
1 8 P —— Splitter blade hub -
' - - — Full blade shroud
1 6 ' - — — Splitter blade shroud

.8

I

A
l
1

_L
.

N

I t
1

.o
m
1
l

lsentropic Mach Number

.0
c»

 

.0
4:.

 

 

 

 

.0
o to

S/C

Figure 5.9 Cont’d

163

 

(C)

lsentropic Mach Number
.0 .0 7* r" r"
0') oo -& «b O) 00

.o
a

.0
m

ITERATION 3, 7 Blades
P2/P01=1.42499995 ; Pre-Rotation = 60 Degrees

 

_L
N

 

F I

—— Full blade hub
—— Splitter blade hub

— - — Full blade shroud
—-— Splitter blade shroud

 

 

 

 

 

 

 

I\. '\ W‘
'- \ ......... 1_‘_‘_F A-
I f _______ l / 4
} “"~1,l-£.‘_-\_ I,
' _____/" ‘ xx-‘ ‘47 _
r \-L—.;':/
r 1 'ml 4— l

02 04 0.6 08 1

S/C

Figure 5.9 Cont’d

164

 

(d) ITERATION 11, 7 Blades

P2/P01=1.39999998 ; Pre-Rotation = 0 Degrees

 

 

 

 

 

 

 

 

 

 

 

 

2 . a r
r Full blade hub
1 8 _ ————-— Splitter blade hub g
- , —-- Full blade shroud
1 5 ——— Splitter blade shroud
. 1.4 — -
(D
D .
=51 12 ~
2 ' I
.C
8 _I
2 1 ”
.2 I
Q.
g 0.8
C
<1)
fl)
" 0.6
0.4 '
0.2 I
l
0 ‘ ‘ L I 1
0 02 0.4 06 08 1

S/C

Figure 5.9 Cont’d

165

 

 

(e)

—L .5 _..L
P a) m

.0
m

lsentropic Mach Number

ITERATION 11, 7 Blades
P2/P01 =1 .41999996 ; Pre-Rotation = 40 Degrees

 

-L
N

1’

I I

Full blade hub

—-— Splitter blade hub

— - - Full blade shroud

— - — Splitter blade shroud

 

 

 

 

 

 

 

 

Figure 5.9

166

 

S/C

Cont’d

_i

 

(0

lsentropic Mach Number

 

ITERATION 11 , 7 Blades
P2/P01=1.42499995 ; Pre—Rotation = 60 Degrees

2 ‘ I I
Full blade hub T
1 8 -——-- Splitter blade hub
' — - -— Full blade shroud
— - - Splitter blade shroud

 

 

 

 

 

 
 
 
 

 

 

.4

 

 

 

 

 

 

 

S/C

Figure 5.9 Cont’d

167

 

 

 

ITERATION 33, 7 Blades

 

 

 

 

 

(g) P2/P01 =1 .39999998 ; Pre—Flotation = 0 Degrees
2 ' I ‘ I
Full blade hub
1 8 l_ —— Splitter blade hub _
° — - — Full blade shroud
1 6 b - - - Splitter blade shroud
1.4 ~ -

_t
C
N
I '
1

 

.0
00

lsentropic Mach Number

.0
m

 

 

 

 

 

0 0.2 0.4 0.6 0.8 1
S/C

Figure 5.9 Cont’d

168

 

(h)

lsentropic Mach Number
.0 .0 .0 .0 .-‘ .-‘ .-* .-‘
N -> O) 00 -& N J:- O) (D N

O

ITERATION 33, 7 Blades
P2/P01=1.41999996 ; Pre—Rotation = 40 Degrees

 

I I

 

 

Full blade hub

— Splitter blade hub

— - — Full blade shroud

— - — Splitter blade shroud

 

 

 

 

 

‘- l"-~~
-~-~-—-—-—--'—-~-:-.:-—-———-_-——-.-=-_-.~—~--~
Ir ....... ‘ .

I' ‘lr‘~\._-

' H \r-~

1 . ‘-~‘.:-r____=:/

I l I l

O 0.2 04 06 0.8

 

 

S/C

Figure 5.9 Cont’d

169

(i)

lsentropic Mach Number

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

ITERATION 33, 7 Blades
P2/P01=1.42499995 ; Pre-Flotation = 60 Degrees

 

I I

 

Full blade hub

—- Splitter blade hub
—-— Full blade shroud
--—- Splitter blade shroud

 

 

 

 

 

 

S/C

Figure 5.9 Cont’d

170

Baseline Mach number Distribution

 

 

 

 

 

 

 

a
( ) Pre-Rotation = 0 Degrees
2 r I
. Full blade hub
1 8 - —— Splitter blade hub -
- . —-- Full blade shroud
--— Splitter blade shroud

1.6 ~ I
- 1.4 ~ I
8 .
§ 12
z ' I —
.C I
8
2 1 ’ .
.2
D.
g 0.8
C
(D
_‘2

.O
or

_o
3:.

0.2

 

 

 

S/C

Figure 5.10 Mach number distributions of the Baseline and Single Point Optimum
geometries.

171

 

 

 

 

 

(b) Baseline Mach number Distribution
Pre—Rotation = 40 Degrees
2 i . i
, Full blade hub

1 8 _ —— Splitter blade hub g
' —-- Full blade shroud

1 6 ’ --— Splitter blade shroud

1.4 — «

_.L
I
N
I
l
1

P
m

I .
I

lsentropic Mach Number

.0
O)
,

 

9
.rs

 

 

 

.0
N

 

 

 

O
O
or-
N
nop-
.h
o-
O)
o.—
00
—L

Figure 5.10 cont’d

172

(C)

lsentropic Mach Number

1.8

1.6

1.4

1.2

0.8

0.6

 

Baseline Mach number Distribution

Pre—Rotation = 60 Degrees

 

 

I ' I

 

Full blade hub

—— Splitter blade hub

-- - — Full blade shroud

— ~ - Splitter blade shroud

 

 

 

 

 

 

 

 

S/C

Figure 5.10 cont’d

173

(d)

-L
m N

.L .a.
N a)

lsentropic Mach Number
.0 .0 9 .0
N b O) m .5

0

Single Point Optimum 7 Blades
P2/P01=1.4000 ; Pre—Flotation = 0 Degrees

 

_.L
A

 

r

I I

 

 

Full blade hub

—— Splitter blade hub

— - — Full blade shroud

— - - Splitter blade shroud

 

 

 

 

 

3"”x _________ I __‘_—_._::_:.~_.T._‘ ------
I, ~ ....... \ ﬂ‘\‘ _----.‘\
L' §

I. \_\

l "\-
1. l’
i I
l

0.2 0.4 0.6 0.8
S/C

Figure 5.10 cont’d

174

 

 

 

 

(e)

lsentropic Mach Number
0 o _. _. 1.. 74
b: be _. to 'A o> 00 N

.0
4:.

0.2

Single Point Optimum, 7 Blades

P2/P01=1.420000 ; Pre—Rotation = 40 Degrees

 

 

l r r

 

 

Full blade hub

— Splitter blade hub

— — — Full blade shroud
-~-- Splitter blade shroud

 

 

 

 

 

 

S/C

Figure 5.10 cont’d

175

. “3.1:: Wyn-pl.

 

(0

—L
m

—L -L
ix: 0:

lsentropic Mach Number
.0 .0 .0
b CD 00 —*

.0
N

Single Point Optimum, 7 Blades
P2/P01=1.425000 ; Pre—Rotation = 60 Degrees

 

—L
h

I I

Full blade hub

—— Splitter blade hub

— - — Full blade shroud
--- Splitter blade shroud

 

 

 

 

 

 

 

 

[\‘\ ......... FTJI‘JrJr;mm.—_—.—m\
/ ————————— ‘4
lr‘h‘Va-gT- /
I k V - \T ‘ T;‘:;—:;f7/
0 02 0.4 06 03

Figure 5.10 cont’d

176

 

 

 

5.3 Performance of Optimum Geometry and Conclusion

All of the geometries that have been analyzed are rather good, when compared to the
baseline. So the question is, how does one select the best one? It is quite difﬁcult to say
that there is the ultimate optimum geometry that is by far better than any other one. In
fact, there are many suitable candidates for the operation of an air conditioner and other
applications. Since the efﬁciencies are all around the same values, the designer then looks
at the Mach number distributions for some insight. Doing this, it is evident Iteration 3 is
the best even though Iteration 33 has a slightly higher efﬁciency. The reason why
Iteration 33 was not chosen was because there is a slight local return ﬂow at the splitter
blade hub as illustrated above in Figure 5.9i. Finally some characteristics of Iteration 3

and its performance map can be seen Table 5.12 and Figure 5.11.

Table 5.9 Characteristics of the Multi-Point Optimum Geometry

 

Geometric Characteristics

 

 

 

 

 

 

 

 

 

 

Beta-le-hub 24.98 Degrees
Beta-le-hub-sp 21.31 Degrees
Beta-le-shr 64.56 Degrees
Beta-le-shr-sp 57.38 Degrees
Beta-te 30.00 Degrees
Shroud clearance 0.575 mm
Max Hub Thickness 4 mm

Min Hub Thickness 3 mm

Max Shroud Thickness 3 mm

Min Shroud Thickness 2.5 mm

 

Percent blade chord length where ﬁnishes 25 %
the transition from Max to Min thickness.
Thickness remains constant from this
point all the way the trailing edge.

 

 

Impeller Tip Radius 100 mm

 

 

 

177

P02/ P01

1.8

1.7

1.6'

1.5

1.4

1.3

1.2

1.1

Pressure Ratio vs. Mass Flow Rate
Rotational Speed: 28, 000; Multi- PointOptimum Geometry

 

 

 

 

 

 

 

 

; f f T I fﬁr j—' I V T r 1 r r v I J , ‘
V-——-' Pre-Rotation= 0 Degree 3
5 0——0 Pre-Rotation: 40 Degree 3
F H Pre—Rotation = 60 Degree 3
E 2
i
: a A 4 gm . l . A . a l a . . . l . 1 :
0 0.5 1 1.5 2 2.5
Mass Flow Rate (Kg/s)

Figure 5.11 Performance Map: (a) Pressure ratio (b) ms (c) 71::

178

(b)

0.9
0.8
0.7

0.6

0.5

ETA T8

0.4

0.3

0.2

0.1

Total to Static Efficiency vs. Mass flow rate

Rotational Speed = 28000 RPM ; MultiPoint Optimum Geometry

 

 

h I

; v——-v Pre—Rotation = 0 Degree 3
:_ o—o Pre—Rotation = 40 Degree _:
5 H—e Pre-Rotation = 60 Degree 3
E 3
p 4
r 1
E p‘ a
E 3
E 3

l . . I r l m I l L

 

 

 

 

 

0.5 1 1.5 2 2.5

Mass Flow Rate (Kg/s)

Figure 5.11 cont’d

179

V

 

 

(C)

0.9
0.8
0.7

0.6

0.5

ETA TT

0.4

0.3

0.2

0.1

Total to Total Efficiency vs. Mass flow rate

Rotatio

nal Speed = 28000 RPM ; MultiPoint Optimum Geo

metry

 

'VIVYY'fYﬁV'va'Y'V"

 

1’ f I 7 I T ﬁ 1 fl V V r 1' Tﬁ

1""

 

F—v Pre-Rotation = 0 Degree
o—o Pre-Rotation = 40 Degree
.——. Pre-Rotation = 60 Degree

 

 

 

l 1 A A l A L A l

 

 

0.5

1 1 .5 2
Mass Flow Rate (Kg/s)

Figure 5.11 cont’d

180

 

Thus the multi-point optimum has an efﬁciency improvement over the baseline at the

various operating points. These improvements are:

OPl:

o A 5 point increase in total-total efﬁciency and almost a 3 point increase in total-

static efﬁciency

0P2:

o A 3 point increase in total-total efﬁciency and almost a 2 point increase in total-

static efﬁciency

0P3:

o A 0 point increase in either efﬁciencies. However, the Mach number distribution
for multi-point optimum is better than the baseline distributions. Perhaps there is
no improvement because of the fact that the losses coming from the IGV are too
high. Another alternative would be that perhaps not enough emphasis or weight
was placed at the operating point corresponding to the spring time (Pre-Rotation =

60 Degrees).

181

 

CHAPTER 6

OVERALL CONCLUSIONS AND FUTURE WORK

The following conclusions are supported by the results of the study:

Single Point:

/ The total to static efﬁciency of the impeller was improved by 7 points for the
single point case when compared to the baseline. The reason is that the exit width
is larger than the baseline, which naturally gives rise to increased total-static
efﬁciency. Another contributor to the increased efﬁciency is simply that the
Mach number distributions are much better than the ones corresponding to the
baseline.

\/ The total-total efﬁciency of the impeller was improved by 4 points. Again, the
Mach number distributions are much better. That is, no high velocity peaks are
observed and thus there should not be any corresponding separation or incidences
losses.

/ At 60 degrees Pre-Rotation the single point optimum exhibits separation at the
splitter blade hub, thus making it an unsuitable candidate for multi-point
operation. A possible reason for the separation can be attributed to the fact the

exit width is too wide, which results in too much diffusion.

182

Multiple Points:

V The total to static efﬁciency increased by 3, 2, and 0 points for the pre-rotations of
O, 40, and 60 degrees respectively. That is, relative to the baseline of course.
These improvements are less than the single point optimum because of the fact
that the geometry has to accommodate for the very non-uniform inlet proﬁle.
Even though there is really no improvement in efﬁciency at the 60 pre-rotation
case, the Mach number distribution is improved for all three setting angles.

V The total to total efﬁciency increased by 5, 3 and 0 points for the pre-rotations of
0, 40, and 60 degrees. That is, relative to the baseline of course. The improvement
is actually greater than for the single point optimum. Of course, if one were to
choose a different single point optimum, then it is possible to achieve one more

point. This gives rise to the ﬁnal conclusion

Overall:
V It is difﬁcult to determine the ultimate optimum, especially for the single point
case. Thus is can be said that there exist many feasible solutions in the design
space. Perhaps in the future it will be possible to ﬁnd the absolute, without a

doubt, optimum geometry.

183

Some suggestions for future work include:

> lrnplement for a variable vaned diffuser along with an IGV
> Implement for a Low solidity diffuser with the IGV
> Test even more operating points.

> Perform Multi-disciplinary optimization.

184

10.

11.

12.

13.

REFERENCES

Pierret, S. Designing T urbomachinery blades by means of the function
approximation concept based on Artiﬁcial Neural Network, Genetic Algorithm,
and the Navier-Stokes Equations.

Van den Braembussche, R. Fast Multidisciplinary Optimization of
Turbomachinery Components.

Doulgeris. Designing Nonsymmetrical IGVs by means of the WC] inverse design
method.

Alsalihi, Z. Optimization of Radial Impellers and Diﬂusers manual 58-b.
Alsalihi, Z. T RAF 3D SPLI TT ER-NOSPLI 77’ ER MANUAL

Sapiro, L. Centriﬁlgal Compressors.

Van den Braembussche, One dimensional design, VKI Course Note 134

Van den Braembussche, Radial Impeller Design Methodology, VKI Course Note
162

Van den Braembussche, Design and Optimization of Centrifugal Compressors,
VKI Course Note 141

Simon , H. et a1 , Improvements in Perﬁormance Characterigtics of Single-Stage
and Multistage Centrifugal Compressors by Simultpneous Adiustments of Inlet

Guide Vanes and Diﬁuser Vanes.

Cengel and Boles, Thermodynamics, 3rd Edition, 1998.

Van den Braembussche, Surge and Stall in Centrifugal Compressors, Pre-Print
1 984-1 5.

Van den Braembussche, Stability and Range in Centrifugal Compressors, Re-
Print 1996-51.

185

 

ililllililliliilliiilililiiii