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DEVELOPMENT OF A COMPUTER-AIDED OPTIMIZATION
TOOL FOR CENTRIFUGAL COMPRESSOR IMPELLERS

presented by

YING MA

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DEVELOPMENT OF A COMPUTER-AIDED
OPTIMIZATION TOOL FOR CENTRIFUGAL

COMPRESSOR IMPELLERS
By

Ying Ma

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

MECHANICAL ENGINEERING

2009

ABSTRACT
DEVELOPMENT OF A COMPUTER-AIDED OPTIMIZATION TOOL FOR
CENTRIFUGAL COMPRESSOR IMPELLERS
By
Ying Ma
Development of a fast, automatic and effective computer-aided design and
optimization tool for centrifugal compressor impellers has attracted great attention
and interest both in industry and academia because centrifugal compressors are
widely used and more stringer criteria such as shorter design cycle time and higher
efficiency has been proposed by consumers.

In my study, a centrifugal compressor impellers optimization procedure is
established. A geometry generation tool is developed; a flow solver with streamline
curvature method is modified and linked to this geometry generation tool. This
geometry generation tool with the flow solver is used to generate the geometry cases
and calculate their corresponding performance to form a database. Two types of
Artificial Neural Networks (ANNs): F eed-forward Neural Network (FFNN) and
Radial Basis Function Network (RBFN) are used to create the performance map of
centrifugal compressor impellers based on this database. Genetic Algorithm (GA)
used as the optimization method to search the optimal geometry based on given
desired conditions.

Furthermore, Principle Component Analysis (PCA) or Independent Component
Analysis (ICA) is applied to improve optimization procedure by transforming training
database and make the creating of the performance map in a new coordinate system.
The aim of applications of PCA or ICA is to decrease the errors caused by
approximate performance map. In this dissertation, the accuracies of three different

trained ANNs: RBFN, RBFN with PCA, and RBFN with ICA. As well as total

performances of centrifugal compressor impeller optimization procedures using these
three different trained ANN s are compared.

An online flow solver is also developed to overcome the drawbacks of modeling tools,
in which the flow solver is used directly to evaluate the performances of centrifugal
compressor impellers. This optimization procedure is compared with offline flow
solver optimization procedure Furthermore; influences of GA operators, parameters
and local search algorithm on online and offline flow solver optimization procedure
are also investigated.

Finally, an industrial centrifugal compressor impeller designed by Solar Turbine Inc.
is optimized by using five different types of optimization procedures and new
impeller geometries are evaluated by AN SYS CF X.

Results show that GA has a good performance on this optimization problem and PCA
greatly increase the accuracy of created performance maps and following optimization
performances. It is indicated the developed optimization tool is capable of finding an
impeller geometry, which has the exact desired relative velocity distribution. Online
flow solver and offline flow solver with PCA optimization procedures have best
performance for achieving desired velocity distribution. However, results of CFX
suggest that all online flow solvers, offline flow solver with PCA and RBFN, offline
flow solver with FFNN optimization procedures are capable of reaching the desired

efficiencies.

ACKNOWLEDGMENTS

I would like to thanks to my advisor Dr. Abraham Engeda for his guidance, support and
also for recommending me to be an intern in Solar Turbine Inc., Caterpillar, in which I
learned a lot of industrial design and optimization experiences. His knowledge and

expertise in turbomachinery help me to build background and prepare for this project.

I would also like to thank to the guidance committee Dr. Clriu, Dr. Mueller and Dr.

Somerton for their helpfirl and constructive suggestions and comments.

I would like to express M. Cave and Dr. Di Liberti all my sincere gratitude for the
support and guidance which they gave me, for their knowledge and experience which

they shared with me during my academic years.

Next, I am also very thankful for all the students in turbomachine laboratory who help

and support me.

Finally, I would like to thank my parents for their loving support and education.

Ying Ma

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
NOMENCLATURE ........................................................................................................ xvii
CHAPTER 1 FUNDALMENTALS OF CENTRIFUGAL COMPRESSORS .................. 1
1.1 Introduction ....................................................................................................... 1

1.2 Centrifugal Compressors ................................................................................... 3
1.2.1 Inlet Casing ............................................................................................. 3

1.2.2 Impeller ................................................................................................... 4

1.2.3 Diffiiser ................................................................................................... 5

1.2.4 Volute ...................................................................................................... 7

1.3 Objectives of Research ...................................................................................... 8
CHAPTER 2 THEORY OF CENTRIFUGAL COMPRESSORS ................................... 11
2.1 Introduction ...................................................................................................... 11
2.1.1 Gas Properties ........................................................................................ 11

2.1.2 The First Law of Thermodynamics ....................................................... 12

2.1.3 The Second Law of Thermodynamics .................................................. 13

2.1.4 Compressible Gas Flow Relations ........................................................ 13

2.2 Basic Theories for Centrifugal Compressors... ............................................... 14
2.2.1 Velocity Triangle ................................................................................... 14

2.2.2 Mass Flow ............................................................................................. 15

2.2.3 Dimensionless Variables and Similitude ............................................... 15

2.3 Head and Efficiency ........................................................................................ 17
2.3.1 Rise of Stagnation Enthalpy .................................................................. 17

2.3.2 Specific Work and Head ....................................................................... 18

2.3.3 Conservation of Rothalpy ..................................................................... 20

2.3.4 Efficiency .............................................................................................. 21

2.3.5 Pressure Ratio ....................................................................................... 22

2.4 The Choking Mass Flow ................................................................................. 22

2.5 The Influences of Inlet Guidancing Vanes (IGV) ........................................... 23

2.6 The Influences of Inducer ................................................ - ................................ 25
2.6.1 Influences of Blade Blockage ............................................................... 25
CHAPTER 3 GEOMETRY GENERATION TOOL ....................................................... 28
3.1 Impeller Geometry Specification .................................................................... 28

3.2 Geometry Parameters ...................................................................................... 36
3.2.1 One-dimensional Geometry Parameters ............................................... 36

3.2.2 Two-dimensional Geometry Parameters ............................................... 37

3.3 Calculation of Lean Angle ............................................................................... 40

3.4 Calculation of Theta and Beta angle ............................................................... 47

3.5 Calculation of Leading and Trailing Edge ...................................................... 48
3.6 Comparison of BladeCAD and CCAD ........................................................... 54
3.7 Three-dimensional Design .............................................................................. 63
CHAPTER 4 FLOW SOLVER ....................................................................................... 64
4.1 Introduction ..................................................................................................... 64
4.2 Influences of Meshing ..................................................................................... 65
4.2.1 Comparison of Five Different Meshing Methods ................................. 65
4.2.2 Comparison of Relative Velocity Distribution ...................................... 68
4.2.3 Comparison of Relative Flow Angle .................................................... 71
4.2.4 Comparison of Relative Mach Number ................................................ 74
4.2.5 Comparison of Static Pressure Distribution .......................................... 76
4.2.6 Discussion ............................................................................................. 78

4.3 Comparison between T ASCF low and NASA Codes ....................................... 79
4.3.1 Geometry Cases .................................................................................... 79
4.3.2 Comparison of geometries .................................................................... 80
4.3.3 Comparison of loadings ........................................................................ 83
4.3.4 Comparison of relative velocity distributions ....................................... 85
4.3.5 Comparison of static pressure distributions .......................................... 89
4.3.6 Discussions ........................................................................................... 93
CHAPTER 5 IMPELLER OPTIMIZATION PROCEDURE WITH ANN & GA .......... 94
5.1 Introduction ..................................................................................................... 94
5.2 Parameterization .............................................................................................. 96
5.2.1 Geometry Parameterization .................................................................. 96
5.2.2 Performance Parameterization .............................................................. 99

5.3 Objective Function ........................................................................................ 102
5.4 Optimization Algorithm ................................................................................ 109
5.4.1 Genetic Algorithm ............................................................................... 109
5.4.2 Genetic Algorithm Procedure .............................................................. 110
5.4.3 Local Search Algorithm ....................................................................... 113
5.4.4 Test on GA and Local Research Algorithm ......................................... 113

5.5 Performance Mapping .................................................................................... 118
5.6 Results and Discussions ................................................................................ 120
5.6.1 Accuracies of RBFN and FFNN ......................................................... 120
5.6.2 Performances of Optimization Procedures using RBFN and F FNN .. 121

5.7 Summary ....................................................................................................... 126
CHAPTER 6 IMPROVED IMPELLERS OPTIMIZATION PROCEDURE ............... 128
6.1 Introduction ................................................................................................... 128
6.2 Application of ICA and PCA ........................................................................ 129
6.3 Results and Discussion .................................................................................. 132
6.3.1 Accuracy ofRBFN 132
6.3.2 Performance of Optimization Procedure ............................................ 135
6.3.3 Sensitivity Analysis of GA Parameters ............................................... 141

6.4 Conclusions ................................................................................................... 144

CHAPTER 7 ONLINE IMPELLERS OPTIMIZATION PROCEDURE ..................... 145
7.1 Introduction ................................................................................................... 145

7.2 Optimization Procedure ................................................................................. 145

7.3 Flow Solver ................................................................................................... 149

7.4 Results and Discussion .................................................................................. 150
7.4.1 Comparison of Online and Offline Flow Solver Optimization
procedures ......................................................................................................... 150

7.4.2 Influences of Optimization Method .................................................... 157

7.5 Conclusion ..................................................................................................... 163
CHAPTER 8 APPLICATION OF IMPELLER OPTIMIZATION PROCEDURES ....165
8.1 Optimization Conditions ............................................................................... 165

8.2 Optimization Using Impeller Optimization Procedures ................................ 167

8.3 Evaluation Using ANSYS CFX .................................................................... 171

8.4 Discussion and Conclusions .......................................................................... 175
CHAPTER 9 CONCLUSIONS ..................................................................................... 177
BIBLIOGRAPHY ............................................................................................................ 179

vii

LIST OF TABLES

Table 3-1 One dimensional impeller geometry parameters .............................................. 37
Table 5-1 Comparison of polynomial coefficients p between two W distributions ....... 102
Table 5-2 Comparison of W points between two W distributions ................................. 102
Table 5-3 Comparison of deceleration ratios of W distribution among five cases ......... 104
Table 5-4 Terminology of GA applied on centrifugal compressors ............................... 110
Table 5-5 Cons and Pros of ANN ................................................................................... 118
Table 5-6 Terminology of GA applied on centrifugal compressors ............................... 120

Table 5-7 Comparison of average computational time for centrifugal compressor impeller
optimization procedures using RBFN and FFNN ........................................................... 121

Table 6-1 Comparison of computational time for training three different ANNs: RBFN,
RBFN with PCA and RBFN with ICA ........................................................................... 135

Table 6-2 Comparison of average computational time for centrifugal compressor impeller
optimization procedures using three different ANNs: RBFN, RBFN with PCA and RBFN

with ICA ......................................................................................................................... 136
Table 7-1 Running Conditions and Gas Properties ........................................................ 150
Table 7-2 Computational time with different GA parameters ........................................ 158
Table 8-1 Parameters of preliminary design ................................................................... 165
Table 8-2 Mesh statistics ................................................................................................ 171

viii

LIST OF FIGURES

Figure 1-1 Illustration of inlet and outlet flow directions of three types of compressors:

axial, mixed flow and centrifugal ones [1] ......................................................................... 2
Figure 1-2 Components of centrifugal compressors[2] ...................................................... 3
Figure 1-3 Three types of pre-rotation caused by inlet guiding vanes[2] ........................... 4
Figure 1-4 Impeller nomenclature[2] .................................................................................. 4
Figure 2-1 h-s diagram for the centrifugal compressor stage[5] ...................................... 17
Figure 2-2 Velocity triangle at inlet .................................................................................. 18
Figure 2-3 Velocity diagram at outlet ............................................................................... 19
Figure 2-4 Illustration of influence of rotation speed on Cm ............................................ 24
Figure 2-5 Illustration of influence of preswirl on Cm ...................................................... 24
Figure 2-6 Influence of leading blade angle on throat area .............................................. 25
Figure 2-7 Influence of blade blockage on velocity triangle ............................................ 26
Figure 3-1 Illustration of coordinate system (z,r,9) ..................................... I ..................... 28
Figure 3-2 Whole blade surface[10] ................................................................................. 29
Figure 3-3 several section curves[10] ............................................................................... 29
Figure 3-4 Illustration of coordinate system (u,v,w)[l 1] .................................................. 30
Figure 3-5 Illustration of camber line ............................................................................... 32
Figure 3-6 normal thickness distribution on shroud and hub ........................................... 32
Figure 3-7 Transformation of a three dimensional curve into two two-dimensional plan3es
.......................................................................................................................................... 3
Figure 3-8 Shroud and hub profiles .................................................................................. 34
Figure 3-9 Blade angle distributions on shroud and hub .................................................. 34
Figure 3-10 Illustration of blade angle ............................................................................. 36

ix

Figure 3-11 Illustration of one-order Bezier polynomial with only inlet and outlet control

node points ........................................................................................................................ 38
Figure 3-12 Illustration of adding and moving node points on hub profile ...................... 40
Figure 3-13 Illustration of lean angle ...................................................................... 40
Figure 3-14 First definition of quasi-normal lines ............................................................ 41
Figure 3-15 The second definition of quasi-normal lines ................................................. 42
Figure 3-16 Lean angle distribution based on first definition of quasi-normal lines ....... 42

Figure 3-17 Lean angle distribution based on second definition of quasi-normal lines... 43

Figure 3-18 Generations of quasi-normal lines of first method ........................................ 44
Figure 3-19 Generations of quasi-normal lines of second method ................................... 45
Figure 3-20 Illustration of generating quasi-normal line in second method ..................... 47
Figure 3-21 Illustration of calculation of leading edge ..................................................... 51
Figure 3-22 Illustration of starting point on leading edge ................................................ 53
Figure 3-23 Illustration of trailing edge ............................................................................ 54
Figure 3-24 Comparison of Contours ............................................................................... 55.
Figure 3-25 Comparison of beta angle distribution on shroud ......................................... 55
Figure 3-26 Comparison of beta angle distribution on hub .............................................. 56
Figure 3-27 Comparison of theta angle distribution on shroud ........................................ 56
Figure 3-28 Comparison of theta angle distribution on hub ............................................. 57
Figure 3-29 Comparison of normal thickness distribution on shroud .............................. 57
Figure 3-30 Comparison of normal thickness distribution on shroud .............................. 58

Figure 3-31 Comparison of lean angle distribution based on first definition of quasi-normal
lines ................................................................................................................................... 58

Figure 3-32 Comparison of lean angle distribution based on second definition of
quasi-normal lines between BladeCAD and CCAD ......................................................... 59

Figure 3-33 Comparison of lead edge on shroud (ellipse aspect ratio = 2) ...................... 60

Figure 3-34 Comparison of lead edge on shroud (ellipse aspect ratio = 2) ...................... 60

Figure 3-35 Comparison of lead edge on shroud (ellipse aspect ratio = 4) ...................... 61
Figure 3-36 Comparison of lead edge on shroud (ellipse aspect ratio = 4) ...................... 61
Figure 3-37 Comparison of leading and trailing surface on shroud (ellipse aspect ratio =642)
Figure 3-38 Comparison of leading and trailing surface on shroud (ellipse aspect ratio =642)
Figure 4-1 First meshing method ...................................................................................... 65
Figure 4-2 Second meshing method ................................................................................. 66
Figure 4-3 Third meshing method .................................................................................... 67
Figure 4-4 Fourth meshing method .................................................................................. 67
Figure 4-5 Fifth meshing method ..................................................................................... 68
Figure 4-6 Relative velocity distribution based on first meshing method ........................ 69
Figure 4-7 Relative velocity distribution based on second meshing method ................... 69 A
Figure 4-8 Relative velocity distribution based on third meshing method ....................... 70
Figure 4-9 Relative velocity distribution based on fourth meshing method ..................... 70
. Figure 4-10 Relative velocity distribution based on fifth meshing method ..................... 71
Figure 4,-11 Relative flow angle distribution based on first meshing method .................. 71
Figure 4-12 Relative flow angle distribution based on second meshing method ............. 72
Figure 4-13 Relative flow angle distribution based on third meshing method ................. 72
Figure 4-14 Relative flow angle distribution based on fourth meshing method .............. 73
Figure 4-15 Relative flow angle distribution based on fifth meshing method ................. 73
Figure 4-16 Relative Mach number distribution based on first meshing method ............ 74
Figure 4-17 Relative Mach number distribution based on second meshing method ........ 74
Figure 4-18 Relative Mach number distribution based on third meshing method ........... 75
Figure 4-19 Relative Mach number distribution based on fourth meshing method ......... 75

xi

Figure 4-20 Relative Mach number distribution based on fifth meshing method ............ 76

Figure 4-21 Static pressure distribution based on first meshing method .......................... 76
Figure 4-22 Static pressure distribution based on second meshing method ..................... 77
Figure 4-23 Static pressure distribution based on third meshing method ......................... 77
Figure 4-24 Static pressure distribution based on fourth meshing method ...................... 78
Figure 4-25 Static pressure distribution based on fifth meshing method ......................... 78
Figure 4-26 Contours of five different cases .................................................................... 81
Figure 4-27 Blade angle distribuitons of five different cases ........................................... 82
Figure 4-28 Loading on hub calculated by TASCFlow .................................................... 83
Figure 4-29 Loading on hub calculated by MERIDL and TSONIC ................................. 83
Figure 4-30 Loading on shroud calculated by TASCFlow ............................................... 84
Figure 4-31 Loading on shroud calculated by MERIDL and TSONIC ............................ 84

Figure 4-32 Relative velocity distribution on pressure side of hub calculated by TASCFlow
.......................................................................................................................................... 85

Figure 4-33 Relative velocity distribution on pressure side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 85

Figure 4-34 Relative velocity distribution on suction side of hub calculated by TASCflow
.......................................................................................................................................... 86

Figure 4-35 Relative velocity distribution on suction side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 86

Figure 4-36 Relative velocity distribution on pressure side of shroud calculated by
TASCflow ......................................................................................................................... 87

Figure 4-37 Relative velocity distribution on pressure side of shroud calculated by
MERIDL and TSONIC ..................................................................................................... 87

Figure 4-38 Relative velocity distribution on suction side of shroud calculated by
TASCflow ......................................................................................................................... 88

Figure 4-39 Relative velocity distribution on suction side of shroud calculated by MERIDL
and TSONIC ..................................................................................................................... 88

xii

Figure 4-20 Relative Mach number distribution based on fifth meshing method ............ 76

Figure 4-21 Static pressure distribution based on first meshing method .......................... 76
Figure 4-22 Static pressure distribution based on second meshing method ..................... 77
Figure 4-23 Static pressure distribution based on third meshing method ......................... 77
Figure 4-24 Static pressure distribution based on fourth meshing method ...................... 78
Figure 4-25 Static pressure distribution based on fifth meshing method ......................... 78
Figure 4-26 Contours of five different cases .................................................................... 81
Figure 4-27 Blade angle distribuitons of five different cases ........................................... 82
Figure 4-28 Loading on hub calculated by TASCFlow .................................................... 83
Figure 4-29 Loading on hub calculated by MERIDL and TSONIC ................................. 83
Figure 4-30 Loading on shroud calculated by TASCFlow ............................................... 84
Figure 4-31 Loading on shroud calculated by MERIDL and TSONIC ............................ 84

Figure 4-32 Relative velocity distribution on pressure side of hub calculated by TASCFlow
.......................................................................................................................................... 85

Figure 4-33 Relative velocity distribution on pressure side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 85

Figure 4-34 Relative velocity distribution on suction side of hub calculated by TASCflow
.......................................................................................................................................... 86

Figure 4-35 Relative velocity distribution on suction side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 86

Figure 4-36 Relative velocity distribution on pressure side of shroud calculated by
TASCflow ......................................................................................................................... 87

Figure 4-37 Relative velocity distribution on pressure side of shroud calculated by
MERIDL and TSONIC ..................................................................................................... 87

Figure 4-38 Relative velocity distribution on suction side of shroud calculated by
TASCflow ......................................................................................................................... 88

Figure 4-39 Relative velocity distribution on suction side of shroud calculated by MERIDL
and TSONIC ..................................................................................................................... 88

xii

Figure 4-40 Static pressure distribution on pressure side of hub calculated by TASCflow

.......................................................................................................................................... 89
Figure 44] Relative velocity distribution on pressure side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 89
Figure 4-42 Relative velocity distribution on suction side of hub calculated by TASCflow

.......................................................................................................................................... 90
Figure 4-43 Relative velocity distribution on suction side of hub calculated by MERIDL
and TSONIC ..................................................................................................................... 90
Figure 4-44 Relative velocity distribution on pressure side of shroud calculated by
T ASCflow ......................................................................................................................... 91
Figure 4-45 Relative velocity distribution on pressure side of shroud calculated by
MERIDL and TSONIC ..................................................................................................... 91
Figure 4-46 Relative velocity distribution on suction side of shroud calculated by
TASCflow ......................................................................................................................... 92
Figure 4-47 Relative velocity distribution on suction side of shroud calculated by MERIDL
and TSONIC ..................................................................................................................... 92
Figure 5-1 Geometry parameterization of contour ........................................................... 98
Figure‘5-2 Geometry parameterization of blade angle distribution .................................. 98
Figure 5-3 Curve fitting of relative velocity distribution ............................................... 100
Figure 5—4 Discretization of relative velocity distribution .............................................. 100
Figure 5-5 Illustration of two relative velocity distributions with small differences ..... 101
Figure 5-6 Comparison of slope of W in inducer region for 3“1 Criterion ...................... 105
Figure 5-7 Comparison of times of sign changes based on 5th Criterion ........................ 106
Figure 5-8 Comparison of integral of loading differences for 7th Criterion ................... 107
Figure 5-9 Comparison of minimum velocities for 8th Criterion .................................... 108

Figure 5-10 Comparison of integrals on relative velocity differences between suction

surface on hub and shroud ...................................................... 108
Figure 5-11 Procedure of Genetic Algorithm ................................................................. 11 1
Figure 5-12 Illustration of De Jong test function in two-dimensions ............................. 114

xiii

Figure 5-13 Convergence histories of optimization based on De Jong test firnction in

twenty dimensions .......................................................................................................... l 15
Figure 5-14 Illustration of Rosenbrock test function in two-dimensions ....................... 115
Figure 5-15 Convergence histories of optimization based on Rosenbrock Test Function
optimization in twenty dimensions ................................................................................. 1 16
Figure 5-16 Illustration of Rastrigin test function in two-dimensions ............................ 117

Figure 5-17 Convergence histories of optimization based on Rastrigin test function
optimization in twenty dimensions ................................................................................. 117

Figure 5-18 Comparisons of accuracies of RBFN and FFNN in training database and
testing database ............................................................................................................... 120

Figure 5-19 Comparison of optimal W distributions calculated by employing RBFN and
FFNN .............................................................................................................................. 122

Figure 5-20 Statistical results of Average Absolute Error (AAE) and Maximum Absolute
Error (MAE) between optimal W points & desired ones between using RBFN and FFNN
........................................................................................................................................ 123

Figure 5-21 Comparison of optimal contours between using RBFN and FFNN ........... 124

Figure 5-22 Comparison of optimal beta distributions between using RBFN and FFNN
......................................................................... ‘ 125

Figure 5-23 Average Absolute Error (AAE) between optimal geometry & desired ones
........................................................................................................................................ 125

Figure 6-1 Centrifugal compressor impeller optimization procedure with PCA or ICA 130

Figure 6-2 Comparisons of accuracies of trained RBF N, RBFN with PCA, and RBFN with
ICA on Average Absolute Error (AAE) between predicted W points & those in testing
database ........................................................................................................................... 133

Figure 6-3 Comparisons of accuracies of trained RBFN, RBFN with PCA, and RBFN with
ICA on Maximum Absolute Error (MAE) between predicted W points & those in testing
database ........................................................................................................................... 133

Figure 64 Comparison of optimal W distributions calculated by employing RBFN, RBFN
with PCA, RBFN with ICA ............................................................................................ 137

Figure 6-5 Comparison of Average Absolute Error (AAE) between optimal W points &.
desired ones among three different ANNs: RBFN, RBFN with PCA, RBFN with ICA 138

xiv

Figure 6-6 Comparison of Maximum Absolute Error (MAE) between optimal W points &
desired ones among centrifirgal compressor impeller optimization procedures using three
different ANNs: RBFN, RBFN with PCA, RBFN with ICA ......................................... 138

Figure 6-7 Comparison of optimal profiles calculated by employing RBFN, RBFN with
PCA, RBFN with ICA .................................................................................................... 139

Figure 6-8 Comparison of optimal beta distributions calculated by employing RBFN,
RBFN with PCA, RBFN with IC .................................................................................... 139

Figure 6-9 Average Absolute Error (AAE) between optimal geometry & desired ones 140

Figure 6-10 Maximum Absolute Error (MAE) between optimal geometry & desired one

........................................................................................................................................ 141
Figure 6-11 Influence of population size ........................................................................ 142
Figure 6-12 Influence of maximum generation number ................................................. 142
Figure 6-13 Influence of crossover rate .......................................................................... 143
Figure 6-14 Influence of mutation rate ........................................................................... 143
Figure 7-1 Online flow solver optimization procedure .................................................. 146
Figure 7-2 Offline flow solver optimization procedure .................................................. 148

Figure 7-3 Comparison of optima contour calculated using online flow solver and offline
flow solver optimization procedure ................................................................................ 151

Figure 7-4 Comparison of Blade angle distribution calculated using online flow solver and
offline flow solver optimization procedure .................................................................... 152

Figure 7-5 Comparison of relative velocity distribution (W) calculated using online flow

solver and offline flow solver optimization procedure ................................................... 153
Figure 7-6 Converge history of online flow solver optimization procedure .................. 155
Figure 7-7 Converge history of offline flow solver optimization procedure .................. 155

Figure 7-8 Comparison of statistical results on optima calculated between using online
flow solver optimization procedure and using offline flow solver optimization procedure

........................................................................................................................................ 156
Figure 7-9 Influences ofpopulation size in GA ................................ 157
Figure 7-10 Influences of maximum generation in GA .................................................. 158
Figure 7-11 Influences of crossover rate in GA ............................................................. 159

XV

Figure 7-12 Influences of mutation rate in GA ............................................................... 160

Figure 7-13 Comparison of optima with and without using local search algorithm ...... 161
Figure 7-14 Comparison of two selection operators: Stochastic Universal Sampling (SUS)
and Tournament selection in GA .................................................................................... 161
Figure 7-15 Comparison of performances of reproduction operators ............................ 163
Figure 8-1 Illustration of inlet casing and diffuser contours .......................................... 166
Figure 8-2 Illustration of desired relative velocity distribution ...................................... 166
Figure 8-3 Comparison of optimal relative velocity distributions .................................. 169
Figure 8-4 Comparison of optimal contours ................................................................... 169
Figure 8-5 Comparison of optimal blade angle distributions ......................................... 170
Figure 8-6 Meshing of one blade passage ...................................................................... 172
Figure 8-7 Meshing of shroud ........................................................................................ 172
Figure 8-8 Comparison of hub ........................................................................................ 172
Figure 8-9 Comparison of total pressure ratio ................................................................ 173
Figure 8-10 Comparison of isentropic efficiency ........................................................... 174
Figure 8-11 Comparison of head .................................................................................... 174
Figure 8-12 Comparison of work .................................................................................... 175

xvi

NOMENCLATURE

0: Absolute flow angle measured from meridional plane or Slope angle of meanline
in meridional plane

,6 Blade angle or Relative flow angle measured from normal direction factor for
two-order polynomials

7 Ratio of specific heat or Slope angle

a) Weight factor

8 Ratio of mass flow or relative blade blockage
q? Flow coefficient

77 Efficiency

p Density

0' Slip factor

9” Head coefficient

0 Circular angle

7: Circular constant
A Area

a Sound speed

b Blade width

C Absolute Velocity

Specific heat under constant pressure

0 Velocity

child Chromosomes of child individual

xvii

Qxfirb

%M

%m

Ns

order

Pr

Diameter or Width at some certain radius
Fitness fimction

Penalty function or objective function

performance map or model

Gravitational acceleration or Constraint function
Head

Enthalpy

Rothalpy

Incidence

iteration number

Blade blockage coefficient

Mach number or Meridional distance
Percentage of section to begin splitter blades
Mass or Meridional distance

Normalized meridional distance

Mass flow rate

Blade rotating speed or Number of individuals in GA
number of control point nodes

Specific Speed

Throat width

Order of Bezier polynomials

Reduced Pressure

Pressure

xviii

parent

Tr

Tn

Coefficients of polynomials
Chromosome of parent individual

Heat added to the system or Volume

Volume flow

Radius or Specific gas constant
A random number

Entropy or Curve length
Temperature or Thickness

Reduced temperature

Normal thickness

Torque

Blade Velocity

Normalized arc length along streamlines or normalized distance of Bezier
polynomial

Specific Volume or Normalized arc length along quasi-normal lines

Work done by the system or Relative Velocity

Work per unit time

Fluid fiiction on the stationary component

Specific work per unit time

Specific Volume or Normalized arc length along normal thickness lines

Variables or x coordinate

y coordinate or Performance

xix

Z Blade number or Compressibility factor
2 Height

SUBSCRIPT

0 Stagnation, or Entry of IGV
1 Impeller inlet (Inducer Inlet)
2 Impeller outlet

4 Diffuser inlet

5 Diffuser Outlet

actu Actual

c Head or Center

Cur Current

d diffuser

des Desired

H At hub

i Ideal or impeller, or the ith point
j Jet or Integer number

kb Due to blockage of blades
Lean Lean angle

LE Leading edge

Max Maximum

Min Minimum

ML Meridional plane

In meridional direction

XX

node

At node points

PS Pressure surface

pre Pressure side or Pressure surface
Rake rake angle

r In axis or radial direction

SS Suction surface

S At shroud

s Isentropic process or slip

spl Splitter blades

suc Suction side or Suction surface
TE Trailing edge

th Throat

u In tangential direction
SUPERSCRIPT

‘ Revised

Desired

xxi

CHAPTER 1

FUNDALMENTALS OF CENTRIFUGAL COMPRESSORS

1.1 Introduction

A turbomachine describes a device that transfers energy between a rotor and a fluid.
The turbomachinery are constituted of a large class of machines. Their functions and
application area varies a lot. However, each of these includes several certain elements
including a rotor and a casing. A rotor is the rotating part and the most important
component, through which energy transfers. A casing provides a boundary as guides
to direct the flow. The turbomachinery are used for a wide range and are found
virtually everywhere in this world. The application field of turbomachinery includes
aerospace, automotive, refiigeration and air conditioning, power generation as well as
marine. The design of turbomachinery covers a wide range of subjects including fluid
mechanics, thermodynamics, aerodynamics, solid mechanics and vibration. Generally,
two main categories of turbomachine are identified based on its purpose. Those,
which produce energy by expanding fluid to a lower pressure, are classified as
turbines. Inversely, those that absorb energy to increase the fluid pressure are
classified as compressors or pumps. A pump uses liquids for a working fluid and a
compressor uses gases. For a compressor, three different terms (a fan, a blower, and a
compressor) may be used depending on the pressure ratio or the pressure rise
achieved. Compressors can be classified as axial, mixed flow and centrifugal (or
radial) depending on the discharge flow direction. The inlet and outlet flow directions
of axial, mixed flow and centrifugal compressors are illustrated in Figure 1-1

respectively.

 

 

 

 

 

Figure 1-1 Illustration of inlet and outlet flow directions of three types of
compressors: axial, mixed flow and centrifugal ones [1]

The fluid flows parallel to the rotation to axial coordinate in axial compressors.
Compared to centrifugal compressors, axial compressors have the large mass flow
capacity and higher efficiency. Therefore they are widely used in gas turbines,
especially jet engines. However, they provide lower pressure rise per state than
centrifugal compressors.

The increase of centrifugal compressor efficiency during last decades has resulted in
the wider industrial application. The centrifugal compressors offer several advantages:
small weight, lower maintenance, higher reliability, simplicity of components and
ease of manufacturing.

Mixed flow centrifirgal compressors combine impeller blade features from both the
axial and radial to produce a diagonal unit. The exit mean radiusis greater than one at
the inlet, which is similar to centrifugal compressor. However, the flows exit in both
axial and radial direction. Therefore, it eliminates the requirement of the diffuser,
which is another important component in compressors and introduced in the next

section.

1.2 Centrifugal Compressors
A centrifugal compressor, sometimes referred as a radial compressor shown in Figure
1-2, is generally made up from four basic components: an inlet casing, a rotating

impeller, a stationary diffuser of the vaneless or vaned type and a volute (a collector).

 

 

 

Figure 1-2 Components of centrifugal compressors[2]
1.2.1 Inlet Casing
The main purpose of inlet casing is to provide the pre-rotation by using inlet guiding

vanes, which allows circumventing the incidence and extending the flow range. There
are three different pro-rotations, shown in Figure 1-3. The positive pre-rotation leads
to a reduction in mass flow and a slightly less enthalpy rise. On the opposite, the
negative pre-rotation leads to a higher mass flow and an increased pressure ratio. The

comprehensive effects of pre-rotation will be discussed in the chapter 2.

With Rotation W C '
l
< (Positive Prewhirl) 1 U1 X Cu] > 0 Lower head

Type “I” ‘ ,91 W1 Reduced Q1 Reduced
U1 2 ,
U1 ' .Cul
W CI
< (Zero Prewhirl) Cal = 0
U Type 2 Am U1 x Cu] = 0
l U] .
Against Rotation W ' :C .
(Negative Prewhirl) 1 U] x Cal < 0 Higher head
Type “3” m. W1 Increased Q1 Increased
U1 A 9r
Ul _,
Cal

Figure 1-3 Three types of pre-rotation caused by inlet guiding vanes[2]

1.2.2 Impeller
The purpose of an impeller (rotor) includes: deflecting the flow in axial and radial

direction, increasing the static pressure as well as the kinetic energy of the flow.[3]
The impeller is the most important and complex element in geometry in the

centrifugal compressor. The nomenclature of an impeller is as shown in Figure 1-4.

Trailing edge Pressure side

   
   
  

"\ Suction side

 

 

  

Back face (Disk) = [A
Splitter vane ‘ . ,mpeller blade
- ' Shroud
, nducer throat ' d
4 ,n ucer
Hub - .
Leadmg edge

Figure 1-4 Impeller nomenclature[2]

The hub is the curve surface of revolution of the impeller, forming the inner boundary
to the flow. The shroud is the curved surface, forming the outer boundary to the flow.
At the entry of the impeller, the relative flow has a velocity in radial direction. And
the relative flow is turned into the axial direction since the entry section, which is
defined as inducer section. The inducer generally starts at the eye of impeller and
finishes in the region where the flow is beginning to turn into radial direction.[l] The
side of an impeller with higher pressure is called pressure side or driving face. On the
opposite, the side with lower pressure is called suction face. The pressure side, suction
side, hub and shroud form the four sides of the boundary to the flow. The contours of
them greatly effect the deflection of the flow. The effects of leading edge and trailing
will be discussed in chapter 2. The less the number of the impeller is, the less
blockage effects is. However, decreasing the number of impeller leads to the lager
pressure load, which formed by the pressure gradient between pressure side and
suction side, and also results in mechanical problems. An alternative solution is that
splitter blades are added to avoid this problem. Inducer throat has the smallest area
in the channel of the flow in the impeller. The maximum impeller inlet mass flow
occurs when the fluid passes through the inducer throat section at sonic speed.
Therefore, the calculation of throat area is required for the calculation of the
maximum mass flow and flow range.

1.2.3 Diffuser
As mentioned before, the fluid is drawn in through the inlet casing into the eye of the

impeller parallel to the axis of rotation. In order to add angular momentum, the
impeller whirls the fluid outwards and turns it into a direction perpendicular to the
rotation axis. As a result, the energy level is increased, resulting in both pressure and
velocity. In centrifugal compressors, energy is transferred to the fluid by the impeller.

Even though centrifugal impellers are designed for good diffusion within the blade

5

passage, approximately half of the energy imparted to the fluid remains as kinetic
energy at the impeller exit. Therefore, for an efficient centrifugal stage, this kinetic
energy must be efficiently converted into the static pressure. Thus, a diffuser, which is
stationary and is located downstream of the impeller, is a very important element in a
centrifugal compressor.

Since over the years the demands on the centrifugal compressors increased for higher
pressure ratios and efficiency, different types of radial diffusers have been developed.
These different types of radial diffusers can be classified as the vaneless diffusers, the
vaned diffusers, and the low solidity vaned diffusers.

Vaneless diffusers consist of two radial walls that may be parallel, diverging, or
converging. The flow entering a vaneless diffuser has a large amount of swirl. Thus,
the tangential component of momentum at low flow rates can be more than twice the
radial component The radial component of the flow diffuses due to the area increase
(conservation of mass), and the tangential component diffuses inversely proportional
to the radius (conservation of angular momentum). The vaneless diffuser is widely
used in automotive turbochargers because of the broad operating range it offers. It is
also cheaper to manufacture and more tolerant to erosion and fouling than the vaned
diffusers. However, the vaneless diffuser needs a large diameter ratio because of its
low diffusion ratio. The flow in a vaneless diffuser follows an approximate
logarithmic spiral path. The flow in a vaneless diffuser with a radius ratio of 2 and an
inlet flow angle of .6 degrees makes a full revolution before leaving the diffuser. This
will result in high friction loss due to viscous drag on the walls and accordingly its
pressure recovery is significantly lower than is found with vaned diffusers.

Generally the vaneless diffuser demonstrates lower pressure recovery by as much as

20% and lower stage efficiency by 10% compared to a vaned diffuser.

The role of vanes in a vaned diffuser is to shorten the flow path by deswirling the
flow, allowing a smaller outlet diameter to be used. A vaneless space precedes the
vaned diffuser to help reduce flow unsteadiness and Mach number at the leading edge
of the vanes so as to avoid shock waves. Boundary layer develops and generates
appreciable blockage at the vane leading edge. In order to reduce this blockage, the
vaneless space should be minimized until it doesn’t give any unfavorable effects such
as increase in noise level or pressure fluctuations due to interaction of the impeller
and diffuser. The flow exiting the impeller follows an approximate logarithmic spiral
path to the vane leading edge and is guided by the diffuser channels. The
semi-vaneless space follows the vaneless space, ending in a passage throat, which
may limit the maximum flow rate in a compressor. The number of diffuser vanes
has a direct bearing on the efficiency. With large number of vanes, the angle of
divergence is smaller and the efficiency rises until fiiction and blockage overcomes
the advantage of more gradual diffusion.

Although the vaned diffirser typically exhibits higher pressure recovery, the flow
range is limited at low flow rate due to vane stall. At high flow rates, flow choking at
the throat may also limit flow range.

1.2.4 Volute

Outside the diffuser is a scroll or volute whose function is to collect the flow from the
diffuser and deliver it to the discharge pipe. It is possible to gain a further deceleration
and thereby additional pressures rise. Volute plays an important role in influencing
the overall performance of the centrifugal compressor. The flow leaving the impeller
has the logarithmic spiral path. Therefore the volute has to be designed to match with
the flow of the impeller. The volute affects the circumferential pressure distribution
downstream the impeller, and then influence the impeller efficiency, off-design

operation, static and dynamic pressure and flow range.[4]

7

1.3 Objectives of Research
The conventional design, which is based on trial and error and still greatly depends on

the expertise of designers and existed database of companies, is widely used in the
industrial compressor companies.

Due to the wide applications of centrifugal compressors, only a small improvement on
centrifugal compressor performances will result in the significant savings in
expenditure. Furthermore, more stringent criteria such as higher efficiency, wider
flow operating range and shorter design cycle are required by consumers. Fortunately,
as the increase of computing capacity and the application of Computational Fluid
Dynamics (CFD) software, simulations has been widely applied, become a useful
designing tool and substitute experiments to a large extent. This greatly decreases the
design cycle time and makes the computer-aided design become possible. Therefore,
developing of a design and optimization tool or methodology for centrifugal
compressor impellers has attracted great attention and interest.

The conventional design process widely used in industry is a very complex procedure
and can be broadly divided into three loops: One Dimensional (1D) Preliminary
Design and Analysis, Two Dimensional (2D) Design and Aerodynamic Analysis, and
Three Dimensional (3D) Design and Aerodynamic/Mechanical Analysis. Actually,
these three steps are also closely related each other. 1D design is essential and a good
1D design can fasten the following 2D and 3D design. Defective 2D design cannot be
expected to obtain the good 3D performance. If the performance of 2D or 3D design
is unsatisfied, designers probably need to make modifications not only on 2D or 3D
design but also on 1D design.

Even for experienced designers, it will still take several weeks or months to modify
and analyze geometry to achieve customers’ requirements. Therefore, it will not be

realistic to expect that one automatic numerical optimization method can substitute

8

designers and be applied on total design and optimization procedure. In this study, an
optimization tool, working as a fast assistant tool aimed at improving 2D design and
analysis of industrial centrifugal compressor impellers is developed using quasi-3D
flow solver and Genetic Algorithm (GA). The objectives of the present research are to
improve the conventional design method procedure for the centrifugal compressor
impellers and the project is accomplished systematically with the following steps:

1) Developing a geometry generation tool (BladeCAD) including the following
fimctions:

a) Creating a new centrifugal compressor design, including an inlet casing, an
impeller, and a diffuser. All the geometric variables can be edited.

b) Loading existed centrifirgal compressor design files and also geometry files, e. g.
geometry files in meridional plane or blade-to-blade plane.

c) Generating 3D model, which allow the designers to visually observe the impeller
modeling.

2) Revising and linking the codes of Quasi-three dimensional (3D) flow solvers
MERIDL and TSONIC to the geometry generation tool BladeCAD. Comparing the
calculating results between Quasi-3D flow solvers and commercial software ANSYS
CFX using Naiver-Stroker equations to evaluate the accuracies of MERIDL and
TSONIC.

3) Developing an optimization procedure for centrifugal compressor impellers.
Creating a performance map by using an Artificial Neural Network (ANN) and
employing a Genetic algorithm (GA) as the optimization method.

4) Combing a Principle Component Analysis (PCA) and an Independent Component

Analysis (ICA) with the ANN and studying their influences on the ANN. Presenting

an improved centrifugal compressor impeller optimization procedure using the PCA
and GA.

5) Presenting a new online flow solver optimization procedure, in which the flow
solvers are directly used to eliminate the errors caused by created performance map.
Comparing this new one with the traditional optimization procedure which is called
offline flow solver optimization procedure in this study.

6) Using developed fast optimization procedures to find the optimal and AN SYS

CFX to evaluate the optimum as well as these optimization procedures eventually.

CHAPTER 2

THEORY OF CENTRIFUGAL COMPRESSORS

2.1 Introduction

Before introducing centrifugal compressor optimization, some basic theories on
evaluating the centrifugalt compressors have to be introduced firstly. There are
hundreds of formulas have been developed and used during decades of years of work
for design and performance analysis of centrifugal compressors. Only the theory and
equations related to the present research were presented here. Because of the complex
process happened in compressor, these formulas remains relative accurate. To bring
better accuracy, complex equations and practical interpretations have to be applied.
Besides, the combination of gases and operating conditions are also required to
consider.[2]

2.1.1 Gas Properties

The ideal equation of state for the perfect gas is:
pv 2 RT (2'1)

If the fluid is perfect gas, the enthalpy can be expressed as a linear function of

temperature T :

h = CpT (2-2)

The relationship between specific heat at a constant pressure C p and. specific gas

constant R is:

>U

7' .
C =—— (2-3)
P 7-

_|

However, the real equation of state is preferred to use in the industry for better
accuracy in the industry in Eqn. (24). And the deviation fiom perfect gases counts on

the compressibility fact Z.
pv = ZRT (24)

One simple and approximate equation for calculating compressibility factor Z is:

0.188 _ 0.468 _ 0.887e—5Tr
Tr Tr2 Tr2

 

z z 1+ Pr [5] (2-5)

Besides Eqn. (2-5), there are many methods have been proposed to calculate the
compressibility factor. please see reference [6] for others formulas.

2.1.2 The First Law of Thermodynamics

The first law of thermodynamics is introduced in Eqn. (2-6).
(flaw -— dQ) = 0 (2-6)

dQ denotes the heat supplied by the system to the surrounding, while d W denotes the

work done to the system. For a centrifugal compressor, the first law of

thermodynamics can be rewritten into:

. . C2 C2
w-em (hz-h1)+ -,2—--,1— +(822-gzr) <24)

The fluid in the centrifugal compressor is gas; therefore the potential energy g: is
negligible. Most turbomachinery processes are or very close to adiabatic process,
therefore the heat transfer is zero. The Eqn. above can be rewritten into as a function

of stagnation enthalpy:

2 2
Wm}! [h2+E22—]—[h1+-C-2L] ='h(h02-h01) (2'3)

Work done in Eqn. (2-8) is from the surrounding to the fluid.

2.1.3 The Second Law of Thermodynamics

Tds = dh — £8 (2-9)
p

The definition of isentropic process is:

pv7 = constant (2-10)

Therefore, by combining Eqns. (2-1) and (2-10), the relationship among pressure,

temperature and density in the isentropic process, which mean ds = O , are give as in

the Eqns. (2-11) and (2-12):

7
fl: IL /7—1 (2-11)
T2

1’2

1/(7-1)
'0_1 = [fl] (2-12)
p2 T2 .

2.1.4 Compressible Gas Flow Relations

The stagnation enthalpy (total enthalpy) is defined by combined static enthalpy h and
2

kinetic energy 67 :

2

h0=h+67 (2-13)

If the fluid is a perfect gas, combining Eqns. (2-2), (2-3) and (2-13) gives the
relationship between stagnation temperature and static temperature:

13

2 2 _
E=1+ c =1+(y-1)—i—=1+(L---QM2 (214)
T ZCPT 2yRT 2

 

Where the Mach number M is defined by:

M=c/a=c/ 7RT (2-15)
If the flow rest adiabatically and isentropically, combining Eqns. (2-1), (2-2), (2-3)

and (2-9) gives Eqns. (2-16) and (2-17):

_7_ _7_
£9. =[Igjr-l =[1+9’_flM2]7‘1 (2-16)
p T 2
_1_ _1_
[fl {Ely-l =[1+£7_‘_1_)M2]7‘1 (217)
p T 2

2.2 Basic Theories for Centrifugal Compressors

2.2.] Velocity Triangle

Both inlet and outlet velocity triangles play an important role on the performance of
centrifugal compressors. Therefore, they are paid great attention and carefully
designed.

The blade velocity is calculated from:

U = NR (2-18)
Therefore the blade velocity at inducer tip is:

U15 = NRIS (2-19)
The relative velocity Wof the fluid is a very important factor in analyzing the

performance of the centrifugal compressor. The relationship between relative velocity

W , blade velocity U and absolute velocity C is expressed by:

C=U+W cam

C, U ,W are velocity vectors. Because inlet casings and diffusers are stationary,
U = 0. Therefore, relative velocity Wis equivalent to absolute velocityC in inlet
casing and diffusers.

2.2.2 Mass Flow

The mass flow can be calculated by using of the integral form:

m = ijmdA (2-21)
A

If the inlet mass flow is uniform with a constant pre-rotation, and the meridional flow

velocity is normal to the blade leading edge, the then the mass flow at the inducer

inlet is defined by:

rir=p17r(RIS+R1H)lZIS-ZIHIC1,,, (2-22)

The volume flow is defined by:

Q = L". (223)
p .

However, the equation above needs to be revised because of the effect of blade
blockage, which is introduced in Section 2.6.2.

2.2.3 Dimensionless Variables and Similitude

The dimensionless variables are very useful in the analysis of turbomachinery

performance, The important variables in turbomachine performance included volume
flow Q, angular speed N and rotor diameterD.

The flow coefficient is defined as:

99.3. (2.2.)
ND

The head coefficient is defined as:

gH
w = (225)
N202

 

The specific speed is defined as:

 

 

1 Q E 1
¢2 ND3 NQ 2
Ns - —§ _ 2 = 2 (2-26)
4 gH )4 4
V’ H
(NZDZ (g )

The equality of dimensionless groups resulting from Similitude plays an important
role in analysis of compressor performances. The similarity velocity triangle gives
equal flow coefficient:

Q 3 = Q2 3 (2-27)
NlD1 N2D2
While the similar force triangle gives equal head coefficient:

H1 H2

= (2-28)
2 2 2 2
N1 Dr N2 Dz

 

For same compressor with different running speed, Eqns. (2-27) and (2-28) can be

rewritten into:

.91. = 9.2. (229)
N1 N2
E1. = [2%. (230)
N12 N 2

For the trimmed diameter D2 from the original diameter Dl while keeping the

rotating speed, Eqns. (2-27) and (2-28) can be rewritten into:

9.1. = 22.. (2-31)
1 2
fl = 51; (2-32)

2 2
0102

2.3 Head and Efficiency

2.3.1 Rise of Stagnation Enthalpy

 

 

 

 

 

 

 

 

 

 

 

 

hi 02 fig 04 _ 50/5
(:52
l T
2.23
2
r
25
Ru
0 PL __
l 2 2
IS i g
Inlet Vaneless D'ff g
Casin Impeller . space a I user #7
S

Figure 2-1 ill-s diagram for the centrifugal compressor stage [5]

The contribution of each element of the compressor is as shown in Figure 2-1. In
Figure 2-1, in the inlet casing, the fluid is accelerated fiom velocity co to c1 while
the static pressure decreased from p0 to p1 . Since there is no shaft works in inlet
casing. The loss in the inlet casing is small and negligible compared to others

elements. ”Therefore the stagnation enthalpy is constant in adiabatic flow:

2 2
hoo='*o-*%°=h~‘%-=M (2-33)

In the impeller, the rise of stagnation enthalpy is equivalent to:

2 2
Ah=h02—h01 =[h2 +le]-[h1+%] (2-34)

 

The flow is decelerated adiabatically from C4 to CS in the diffuser. The static

pressure rises from p4 to p5 (Figure 2-1). The stagnation enthalpy in steady

adiabatically flows without shaft work is constant. However, in the real situation, the

stagnation enthalpy decreases because of the losses in the diffusers.

oi cs2
2.3.2 Specific Work and Head
The specific energy transfer can be derived fi'om the velocity triangle at inlet and

outlet from the impeller as shown in Figure 2-2 and Figure 2-3.

I __________________ A

W1 ,6] a, C1 C"
Cm1

 

 

 

U1 CU1

Figure 2-2 Velocity triangle at inlet

 

 

 

 

 

U2 CU2

Figure 2-3 Velocity diagram at outlet
The rate of change of angular momentum will equal the sum of the moments of the

external forces 7;. When applied angular momentum theorem to an impeller, the

torque 7; , is given by:

T, =m<r26u2 -nCu1) (2-36)

Multiplying compressor rotating angular velocity N on both sides of Eqn. (2-28) gives
work of rotor done on the fluid per unit of time is:

W = NTq = Nim(’2Cu2 _’lCul) = m(U2Cu2 "UlCul) (2'37)

Applying the law of trigonometry to the velocity triangles of exit and inlet of the
impeller yields

UZCuZ = Z (Ui2 + C22 — W22) (2-3 8)
UrCur = %(Ur2 + C12 — le) (2-39)
Then by combing Eqns. (2-28) and (2-39), Eqn. (2-39) can be rewritten into:

° m 2 2 2 2 2 2
W=‘2'[(U2"U1)+(C2‘C1)’(W2 “Wt )] (240)
The work done on the fluid per unit mass or specific work per unit time is:

19

w: z z "1“]ng ac...)

m .
m

-1 2 2 2 2 2 2
_E[(U2 —U1)+(C2 —C1 )—(W2 —W1 M (2-41)
The head is defined as:

 

= (UzCuz ‘UICuD

H: Aho =(U2C112 _U1Cul) : (U2 —UI )+(C2 _C1)_(W2 ‘Wl )
g g 2g
2.3.3 Conservation of Rothalpy

(242)

In the centrifugal compressor, the specific work done on the fluid per unit time equals

to the rise of the stagnation enthalpy. Therefore combining Eqns. (2-34) and (241)

gives:
2 2
, 62 c1
Aho = hoz —h01=[h2 +7]_[h1 +7] = (U2Cu2 -U1Cu1) (2-43)
The Eqn. (2-43) can be also rewritten into:
1'02 —U2Cu2 = h01 —U1Cu1 (244)
01‘:
U§ W22 U12 W2
h ___+__= ___+_1 2-45
2 2 2 hi 2 2 ( )

In the Eqns. (2-44) and (2-45), the sum of all the variables in the left side are at the
entry equals to that in the right side at the exit of impeller. Therefore, a new function
rothalpy] is introduced. And value of rothalpy] is unchanged between the entry and
the exit of the impeller. However, some researchers found that an increase in rothalpy
was possible for steady, rothalpy flow without heat transfer or body forces. And the
increased rothalpy is because of the fluid fiiction acting on the stationary wall, such as
the shroud of centrifugal compressors. Therefore, a revised equation for rothalpy has

been proposed [6]:

20

h02 —U2Cu2 = hm *UrCur + Wf / "'1 (2-46)

Where W, denotes the power loss due to the fluid fiiction on the stationary shroud.
2.3.4 Efficiency

The overall efficiency of an adiabatic compressor is defined as the ratio of minimum
adiabatic work input per unit time to actual adiabatic work input to rotor per unit time,

or one of the isentropic head to actual head:

: hOSs 'hOI : Hs (2.47)
’05 Th0] Hact

The Eqn. (24?) can be rewritten into:

’73

T
gal—793-1]
zhoss—hor Jess-hm : 01

 

 

 

 

r]
c hos-her hoz-hor U2Cuz-UrCul
L11
CPT01[h-1] ($1) 7 -1
_ T01 - P01 ‘ 2-48
77,- _ T _T ( .)
CpTOS --Cme 05 01
T01
The efficiency of an impeller is defined as the same overall efficiency of a
compressor“.
hoz-hor

The efficiency of a diffuser is defined as ratio of the actual enthalpy change to the
isentropic enthalpy change.

hs 414

For steady and adiabatic flow in stationary diffusers, the stagnation enthalpy remains

2 2
c c . .
constant I104 +—§-=h053 +%; therefore the efficrency of a drffuser can be also

rewritten into:

21

2 2

 

 

C -65
5 851)
2.3.5 Pressure Ratio
The overall pressure ratio is defined by:
_Z_
P05 = [T053 )7-1 (2 52)
P01 T 01
Combine Eqns. (2-48) and (2-52) and get the overall pressure ratio
P05 _ 1+ (7 -1)77c (UzCuz - UrCu1) 7'1
_ 2 (2-53)
P01 a01

2.4 The Choking Mass Flow

When the flow velocity in a passage reaches at the sonic speed at some cross-section,
the flow chokes. Once the flow chokes, the mass flow cannot be increased further
either by decreasing the backpressure or increasing the rotational speed. For
centrifugal compressors, the behavior of choking can happen in an impeller or a
diffuser. However, the theories are different for rotating component and stationary
component. In the rotating component, choking occurs when the relative velocity is
equivalent to sonic speed at some cross-section, while when absolute velocity reaches
to speed of sound in the stationary component. An Eqn. is used to calculate in choking

mass flow is proposed in [1]:
""th = PthWthArh (2'54)
And when Wth reaches the sound of speed, the choking occurs. This equation is only

approximate equation to calculate choking mass flow. Because the actual choking
may occur at lower mass flows because the whirl of the flow cannot provide the

uniform velocity for the throat section. Therefore the supersonic part of the throat may

22

result in a choking mass flow while the left part still remain subsonic, which leads to
the actual choking mass flow is below the theoretical maximum value. A revised

equation for calculating choking mass flow has been supposed [3]:

fifth = I22“; Pth (Rth ) Wth (Rth )°Dth (Rth )thh (2-55)
2.5 The Influences of Inlet Guidancing Vanes (IGV)

The IGV has several important functions. The first function is to modify the Mach
number. The supersonic Mach number will lead to strong shock losses. Moreover,
supersonic Mach number will also induce early flow separation as well as higher
losses. Reducing the rotating speed is one possible method for this problem, which is
as shown in Figure 2-4. However, a lot of variables were required to redesign if the
designed speed changed. Another possibility is to induce preswirl vanes, which is as
shown in Figure 2-5 . The increase of turning of the flow results in a lower relative
velocity W and also lower Mach number. Besides, the increase of turning of the

flow from zero to positive prerotation results in the gradual increase of

Cul obviously. The Eqns. (2-43) and (2-53) explained that this also decreases the

enthalpy rise as well as the specific work.

23

U1

 

C1(Cm1)

 

 

 

Figure 2-4 Illustration of influence of rotation speed on C,"

 

 

 

 

Figure 2-5 Illustration of influence of preswirl on C,"

The second important function of IGV is to modify the mass flow. Mass flow

variation is limited by choking losses, incidence. The Clm varies as the change of

preswirl shown in the figure 2.4, which can change mass flow based on Eqn. (2-22).

24

The third influence of IGV is on the pressure ratio. The increase of prerotation

Cul decreased the overall pressure ratio based on Eqn. (2-53).

2.6 The Influences of Inducer

2.6.1 Influences of Blade Blockage

The inducer plays a very important role in the impeller performance. The good design
of an inducer should minimize the inlet relative Mach number and keep it subsonic if

possible.

The effect of leading edge blade angle on throat area is as shown in Figure 2-6.

Smaller value of leading edge blade angle ,6] leads to the larger throat area and
larger operating flow range.

_ , R2 _ R2
When the flows enter the mducer, the real free area 1s not” IS 1 H , but

smaller than this value because of the blockage by the presence of the blades.

\\\\\\ { '
:61 “// “61

m /

 

Figure 2-6 Influence of leading blade angle on throat area

The calculation of the zero loading incidence i k b has been proposed: [7]

mo )_ gkb 5‘” 'Blkb
kb _1-£ sinfl -tan,6
kb lkb lkb

 

(2-56)

Where the relative blade blockage Skb is defined by:

ZoTn (2-57)
27rR

Rh of the inducer is much smaller thanRS. Therefore the large value of zero loading

 

8kb=

incidence at the hub occurs due to the larger value of the relative blade blockage 5kb-

It was supposed the lower value at the shroud because of the lower value of gkb at
shroud. However, ,31 at shroud is larger than ,5] at hub, which results in the large

zero loading incidence. The influence of ,6] , le , T1 and I] were discussed as
following:

The relationship between new blade angle due to blockage and the designed blade
angle is:

flrkb = 31 "1% (2-58)
Assume there is no work done on the leading edge, therefore the tangential velocity
remains the same, which means the tangential component of relative velocity is

unchanged which is as shown in Figure 2-7.

 

 

 

 

Figure 2-7 Influence of blade blockage on velocity triangle

26

Therefore the new relative velocity is:

=__"flsinfll (2-59)
srn fllkb

The new axial velocity component is:

_ Clm tan(,61)
lmkb - ——tan ( 511:1») (2-60)

The new absolute velocity is:

 

Clmkb =fififnkb+vf +2W1mkbU1 sin flue (2-61)

Supposing there is no work done during the contraction of the fluid, the new static

temperature due to the blockage of the blades is:

2 2
C1 ’Crkb
2Cp

Supposing an isentropic process, therefore the new static pressure can be calculated

Trkb = T1 + (2-62)

by:

7—1

are = H (Ii—ii] 7 (2-63)
1

CHAPTER 3

GEOMETRY GENERATION TOOL

3.1 Impeller Geometry Specification

The impeller geometry parameters are introduced in this chapter. The most common

coordinate system to describe impellers is (z, r, 0) which is as shown in Figure 3-1.

I]
I,

\\

”’/1
I
”’III/

’4’

//
xxx/2,
2

”ll/I

I
////
flag”

’l/
//,
105..

”If.

I

\c‘.
.Q.
§~
s‘

l//

ill]. . .

\\“
‘ E‘

III
[willie

I ’r
We,
rrrrrrrrlll
llII"'.llI
'Iri‘lii‘hv 1
limit
I I
"lain“

r
1’ p“
fir
x ! cl‘i' ‘
‘ I‘l-“‘|I“‘|

"i. ' l |
iii? "

5538731551557

- . l

"’lzl'l'l!
I
Ir

 

Figure 3-1 Illustration of coordinate system (z,r,0)
Theoretically, coordinates of all points on each blade surface are required to specify
the geometry shape of an impeller. However, the impeller is axial symmetrical and
information on one blade shape is sufficient if the blade number is known.
If it is assumed that blade surface from hub to shroud can be approximated in
polynomials, then only several section curves (Figure 3;3) are required to be specified

instead the whole blade surface (Figure 3-2).

28

 

Figure 3-2 Whole blade surface[10]

“'2’
¢

Figure 3-3 several section curvesllo]

To calculate the points among these section curves, another coordinate system have to

be introduced here. Besidesthe coordinate system(r,z,t9) , all the points on surface
blade can also be defined in a relative coordinate system (u,v,w) shown in Figure

3-4.

 

 

 

 

 

Figure 3-4 Illustration of coordinate system (u,v,w)[11]
Along the streamline from leading edge (AB) to trailing edge (CD), u increased
flour 0 to 1 while u =0 at the leading edge and u=1 at the trailing edge. Along the
quasi-normal lines, v increased from 0 to 1 while v=0 at the hub and v=1 at the
shroud. Along the pressure surface to suction surface, w increased from 0 to 1 while
w=0 at the pressure surface, w=1 at the suction surface and w=0.5 at the camber
surface. Physically, u can be viewed as normalized arc length of streamlines and v
as normalized arc length of quasi-normal lines. As we mentioned above, if blade
Smface from hub to shroud can be approximated in polynomials, then the calculation

0f coordinates of any points on blade surface can be expressed by:

. n . - . T
[z,r,0]T= ZCfilO—vyflv'lzbrgflil (3-1)
.=0

Where (zi,r;,6I,-)denotes the coordinates of points on section curves, i=0denotes

the points on the hub, i = n denotes the points on shroud. n is the order of Bezier

. l
polynomial and C}, = (n——’i_)Tz'—°

The number of section curves i is depended on the order of polynomial. To the
author’s knowledge, blade surface from hub to shroud are linear for most of
compressors. This is because this type of compressors is much easier to manufacture
and save a lot of production cost. Therefore, if two section curves on hub and shroud
respectively are known, then all the points between hub and shroud can be calculated
by interpolation. Therefore, only two section curves on hub and shroud are required to
be specified. And coordinates of other points between hub and shroud can be
interpolated by coordinates of these points on hub and shroud. Eqn. (3-1) can be also
simplified into:

[z,r,0]T = (l —v)[zH,rH,6H]T + v[zS,rS,t95]T (3-2)
The shape of one section curves (Figure 3-3) can be firrther divided into the camber
lines and thickness t which is normal to camber lines (Figure 3-5). The mean line of
the section curve is called as camber line. Then only the camber lines on shroud and
hub as well as their corresponding thiclcness distributions normal to camber lines,
which are called as normal thickness distributions (Figure 3-6), are required to be
defined instead of whole section curves. The definition of s in Figure 3-5 is

eXpressed by:

 

ds = \/er +alz2 +r2a76I2 (3'3)

31

section

camber

Figure 3-5 Illustration of camber line

Normal thickness (inch)

0.4
0.3
Hub
O
02 0 ~— I

2 \K

.
fee—«o \
0.1 / //. Shroud N I

-0.1

 

I I I I I I I I I I I %M
0 10 20 30 40 50 60 70 80 90100

Figure 3-6 normal thickness distribution on shroud and hub
The camber line is a three dimensional curve in coordinate system (r, 2, 0). If all
Points on the camber line are project onto the cross-sections in meridional plane,
which is shown in Figure 3-7, then there is no change of 6? in the projected plane.
Therefore, this three dimensional curve can be transformed into two curves in two

dimensional planes: one is (r,z) coordinate system while another is (r,t9) or

32

 

Figure 3-7 Transformation of a three dimensional curve into two

two-dimensional planes
(r,z) coordinate system is called as meridional plane, the projected curve of camber
line on hub is called as hub profile while one on shroud is called as shroud profile
(Figure 3-8). The impeller outline which includes the shroud profile, hub profile,

leading edge and trailing edge is called as contour in turbomachinery.

1 Shroud 0

5.; /HUb
4i .

3-3 /

i ............................ Z (inch)
0

 

 

Figure 3-8 Shroud and hub profiles

Another coordinate system can be represented by (r, 6) or (2,6). However,
coordinate system of normalized meridional distance and beta angle distribution
(%m, ,6) are commonly used, which is shown in Figure 3-9.

Beta (deg)
0 ..

-10-g

-20
-30 - 0 Hub

40 -j /—‘\
-50 1%“

-60 .5
-70 '1
-80 -f

 

 

0 10 20 30 40 50 60 70 80 90 100
Figure 3-9 Blade angle distributions on shroud and hub

The derivative from, the continuous and discrete integer forms of calculation on .

meridional distance are respectively expressed by:

 

dmlu)=ldr(u)2+dz<u)2 (3-4)

 

u
"'09: I\/r'(u1)2 + Z'(U1)2du1 (3-5)
0
2
(3‘6)

Normalized meridional distance is ratio between current meridional distance from

leading edge and whole meridional distance. The discrete normalized meridional

distance is expressed by:

- ’=‘ (3-7)

 

Normalized meridional distance %m or meridional distance m are monotonically
increasing variables. Therefore each coordinates z , r and %m or m is

correlated each other.

The wrap angle, also called theta angle 6 represented the blade angle in an absolute
coordinate system. And the definition of blade angle ([3) is the angle between the
Camber line and axial cross section, which is actually impeller blade angle in the

relative coordinate system and. illustrated in Figure 3-10.

35

Camber line

 

Surface of
Blade Profile resolution

Figure 3-10 Illustration of blade angle
The equation to calculate blade angle distribution from wrap angle distribution is

expressed by:
_ d0 _1 d9-
fli=tan1[r;——‘—J=tan q—'— (3-8)
dmi (quz +dzi2

The Eqns. to calculate wrap angle distribution from blade angle are expressed by:

49‘. = I]. m (3-9)

r
1:0 I

+6LE

We can see that once normal thickness distributions, blade angle distributions, hub

profile and shroud profile, which are called contour in this study, are specified, then

the whole impeller geometry can be fixed.

3.2 Geometry Parameters

3.2.1 One-dimensional Geometry Parameters

All inlet and exit variables of these three two-dimensional distributions are

defined as one—dimensional parameters, which are listed in

Table 3-1. Besides, blade number can be also considered as one-dimensional

parameter.

36

 

Table 3-1 One dimensional impeller geometry parameters

 

Area Symbols

 

Contours (R-Z coordinates system) R15 , R1 H , R25 , R2H , Z15, Z1 H , 225 ’22 H

 

 

 

Angle Distribution 515,31H,,325,,32H,915,61H,
Thickness Distribution T n1 S , T n1 H , T n2 S , T n2 H
Number of full blade Z

 

Design on these geometry parameters in

Table 3-1 is called as one preliminary design, which is basis of the design. There is no
good performance of compressor can be obtained without a good 1D design.

3.2.2 Two-dimensional Geometry Parameters

Design on contour, blade angle distribution or normal thickness distribution can be
considered as two-dimensional design. Bezier polynomials are used to describe the
profiles of contours and distributions because of the characteristics of Bezier
polynomials including flexibility, smoothes and continuity. The general formulation

of Bezier polynomial is expressed by:

n . . _.
x(u)= chu' (1—u)" 'xnodej (3-10)
.=0
- 1
Where C}, zr—zjfi; nis the order of Bezier polynomial and n=nnode—l,
n—z .1.

"node is the number of node points.

The equations of Bezier polynomial used to calculate contour are expressed by:

. n . . .
2(u) T 2 Cir“, (1‘ u)n—-r Znode,i
< i=0 (341)
n . . -
”(14) = 2 CW (1 ”In—l rnodej
L i=0

 

 

37

After the preliminary design, there are only two node points. The order of Bezier
polynomial is one and the Eqn. (3-11) for calculating R and Z can be rewritten

into Eqn. (3-12) and the contour is also as shown in Figure 3-11.

I
. . 1_.
z(u)= Z Cllu' (l—u) lznodej =(l—u)zl +1122
.=0

    

1 (3-12)
. . 1_.
r(u)= Z Cllul (l—u) lrnOdej =(1—u)r] +102
i=0
R (inch)
0.3{
Shroud /
0.2 -'
0.1 {
.j ...... .r..., ......... l ...... Z(inch)

 

 

Figure 3-11 Illustration of one-order Bezier polynomial with only inlet and outlet
control node points

To change contours to achieve the desired curves, more node points are required to
add to Bezier polynomials. Without changing the profiles of contours, the new

coordinates of node points are expressed by:

' _ i'ZnodeJ—l + (n + 1 "1.) ' znodej
sze’i — n+1 ._

_ _ z—O,1,---,n+l (3-13)
l"irode,i—1+(" +1")"'rrode,i

I
r - =
node,z n +1

38

Where (znodej’rnodej) is the coordinates of new node points while

(Znode,i”'node,i) is the coordinate of old node points. It is the original order of Bezier

polynomial before adding one node and n +1 is the new order.

From the Eqn. (3-14), it can be seen that when i = 0 , Znode,i—l = Znode,—l doesn’t
exist. However the weight factor in front of Znode,—l is zero. Therefore, the whole
term i'znode,,-_1 goes to zero and has no influence on the result. The similarity also

occurred when i = n +1 .

During the editing of contour, the order of Bezier polynomial may be required to
reduce if too many node points have been added; the coordinates of new node points
are expressed by without significant changes on the profile of Bezier curves:

' (n ‘1 '- i) ' znode,i—l + i ' z'nodej

z .:
node,r n-l .—
. . z-O,1,---,n-l (3-14)
' _ ("’1_1)'rn0de,i—l ‘H'rnodej .
rnodej —

 

 

n —1
When the coordinates of node points of Bezier curves are changed, the profile of
Bezier curves is also changed correspondingly. The adding and moving of node points

on hub profile is as shown in Figure 3-12.

39

 

 

0.1{

 

 

. 6 ........ OT ....... 0.2 ........ 0.3 ...... e Z (rnch)

Figure 3-12 Illustration of adding and moving node points on hub profile
3.3 Calculation of Lean Angle
The definition of lean angle is illustrated in Figure 3-13, which is the view fi'om axis

direction. The calculation of lean angle is expressed by:

9Lean,i = 6S,i ‘0H,i - (3'15)

T Lean angle

 

 

Figure 3-13 Illustration of lean angle

Because 495,1- and 6;“ belongs to two different Bezier polynomials and how to

40

decide a pair of 6i on shroud and hub is the difficulty of calculating ELM”. This

depends on the definition of quasi-normal lines in the meridional plane. Different
definitions lead to diflerent results of lean angle distribution. Two different definitions
of quasi-normal lines are discussed here.

The first definition is that quasi-normal lines have same values of u on shroud and
hub (Figure 3-14); the second definition is that quasi-normal lines are normal to mean
lines of blades (Figure 3-15). Lean angle distributions based on first definition and
second definition of quasi-normal lines are as shown in Figure 3-16 and Figure 3-17.

And the lean angle at exit is named as rake angle.

 

 

 

Figure 3-14 First definition of quasi-normal lines

41

       

N , , . .. ,
, ,..,./,. -

 

47/

Figure 3-15 The second definition of quasi-normal lines

 

 

Leanangle

60'... . . .. ., ., __ .,
E?) .
Q) .
3 .
2 .
g” 40 f»
5 :
G)
1—1 .

20 —-

0 l - r . l . . r r l . . . . l r . r . l r . . . |

0 20 40 6O 80 100
%M

Figure 3-16 Lean angle distribution based on first definition of quasi-normal

lines

42

Lean angle

00
O
V 1' l

O‘
O
'r'

 

A
O
. I .

 

Lean angle (deg)

N
O
r

 

..
0 J A 1‘4 1 l J L l A 1 L4 1 l A L L l l 1 l

0 20 I 40 60 80 ‘100
%M

Figure 3-17 Lean angle distribution based on second definition of quasi-normal
lines

In Figure 3-16 and Figure 3-17, X coordinate denotes normalized meridional distance.
The value changes from 0% to 100%. Y coordinate denotes lean angle. The unit is
degree. In these two figures, lean angle changes fiom 0 to 63 as %M changes from
0% to 100%.

The great difference between first method and second method is that the quasi-normal
lines are vertical to mean lines in the second method. Because this definition, we
can see that at the leading edge and trailing edge, lean angle distribution is not
continuous. Moreover, the values of u of the intersection point between quasi-normal
lines and shroud or hub are not equal. Generally we count the leading edge and
trailing edge as the first and the last Quasi-Normal lines in second method, although
these two lines are not vertical to the corresponding mean line.

As mentioned above, lean angle distribution greatly depends on quasi-normal lines
distribution. Therefore, the comparison of first method and second method to define

quasi-normal lines are introduced here, which is as shown in Figure 3-18 and Figure

43

3-19. SiHi are lines which are used to calculate lean angle distribution. In the first
method, lengths of arcs $132, S2S3, 8384 are equal while those of H1H2, H2H3,H3H4

are equal. If there are 11 points on each shroud and hub, then corresponding u of each

points on shroud and hub are 0, 0.1, 0.2, ...,0.9,l.0. The determination of lines in

second method is as shown in Figure 3-19. Each quasi-normal line is normal to the
mean line at the intersection.

 

Trailing Edge [rmv— -~ -,
//I
/
/ /
/,/ l/r
/ I ”
./ , ,
/ ,
li/ ’/
. // ' I,
/ /
x / Ii
Shroud , / /
/,_/’/j // x I
/ /
, .2
S 83/ ’ / / ,
S S 2 3//'\
1 “/T’h‘ “\ / /l
P i “a \‘t Mean
l “x ./ -
‘1‘ \\ \\ / Lll’le Ix,
' i '. ‘~ /
Leadm g t, y/
l .\ \“( // /
Edge \ i‘\ (,1” ‘x‘ // hUb
‘1, \I/’ \n /
l ,/ First Method
1’), I" 1: K , /
1"”74 I is \l ‘\\ . / --—/ = / = ,x//.
l \.
t. H Arc(Sl 32) Arc(SZS3) Arc(S3 S 4)
4 /
l‘ \I/ I I I / /.
R \‘ ,.»"l’// _ [1,1], = 1/
it /// H3 Jae/”T, ///. //
),/
//
/» H2 Arc(HlH2) Arc(H2H3) Arc(H3H4)
Hr
Z

Figure 3-18 Generations of quasi-normal lines of first method

44

Leading
Edge

 

 

 

Figure 3-19 Generations of quasi-normal lines of second method
Generally, we define the leading edge as the 1st Quasi-Normal line. But we need to

find 2Ind Quasi-Normal line. The difficulty in finding that is because the contour of
Shroud, hub and mean line are not standard equations. Therefore, we have to use

iteration method to search. At the first, we calculate angle of leading edge a, . Then we

Calculate the angle a, of normal line at the beginning of mean line. If a, <0:2 , then

We can see that the position of 2ud Quasi-Normal line depends on shroud. On the
Contrary, it would depend on hub. SlHl is the 2Ind Quasi-Normal as shown in Figure

3—20. The next step is to find the foot of perpendicular from one point on shroud to

45

mean line. However, the contour of mean line is not standard equation either.
Therefore, iteration method is applied. If we want to find the point M2 which is the
foot of perpendicular fiom point S2 to the mean line. M2Lef and M2Rig are points
which are left and right of M2, but very close to. The intersection angle between
82M2 and normal line at point M2 should be zero, which means S2M2 is normal to
mean line at point M2. However, one intersection angle of M2Lef and M2Rig should
be positive and another one should be negative. Which one is positive depends on the
definition of intersection angle. Hence, once we find M2Lef and M2Rig by iteration
method, it will be very easy to find M2. The accuracy of coordinates of M2 depends

on resolution. However, high resolution will higher computational time.

46

Trailing

   

Edge 7
.//
/ f
,/
,/ ,, /
/'/’ /
Normal Shroud// / /
l'ne ’11 //
/
./
/
Normal sI 'Mean
Line Line / Hub
a 2 //
_/
Tangential //
Lips. '
Leading
Edge Quasi-Normal

Line

 

Figure 3-20 Illustration of generating quasi-normal line in second method

The equation for the calculation of lean angle distribution in BladeCAD:

Ri,shr (6i,shr - 9i,hub)

 

 

 

9i,Lean = 2 2 (3-16)
\j(Ri,shr " Ri,hub) +(Zi,shr ’ZiJrub)
The calculation of mean line in meridional plane:
R- h +R-h b
Rt,ML = "s r 2 " “ (3-17)
2' h +Z° hub
Zora = "S r 2 " (3-18)

3.4 Calculation of Theta and Beta angle

If the rake angle is known, the equations of calculating Theta Angle are expressed as

following:

47

 

9LE,shr = 0 (3-19)
. tan p.

9i,shr = 6i—1,shr + —-(—l’S—hr—)dmi i: 1,2, ...... ,m (3-20)

i,shr
0 _ 6, ZTE,hub - ZTE,shr 21
Tarmb - tam( Rake) R +975,shr (3- )
T E ,shr

tan )8.

6mm], = 9i+1, hub - —(—ih-ub—)d m,- i = 1,2, ...... , m (3-22)

Ri,hub
If the inlet shroud theta angle and hub inlet theta angle are known, then the Eqns. to

calculate the theta angle distribution are:

tan .3- h
or“), =t9i_1,s,,, +——IE.—‘:—')-dm,- i=l,2, ...... ,m (3-23)
1,3 r
tan ,6-
Bime = gi—IJIub +——Rg.—;:-’hb—ul-)ldmi i= l,2, ...... ,m (3-24)
1, u

where

19L E, Sh, , 9L E, hub denote theta angles of leading edge on shroud and hub respectively
9733),, , 9TB, hub denote theta angles of trailing edge on shroud and hub respectively
9i,shr denotes theta angle at point i on shroud

9i, hub denotes theta angle at point i on hub

rBi,shr denotes beta angle at point i on shroud
i denotes the number of Quasi-Normal lines
3.5 Calculation of Leading and Trailing Edge

The developed geometry design tool has the function of output geometry files, which

Can be used to generate mesh and calculate impeller performance using flow solver.
Variables (R,Z,%m,t9, ,6,tn) of points are output in this geometry output files. R

andZ are calculated from Eqn. (3-11). %m is calculated based on Eqn. (3-7). The

B,tn are calculated from node points on blade angle distribution and normal

48

thickness distribution. Eqns. to calculate )Bi, tn,- are respectively expressed by:

n . . .
[3(a) = Z C;(u)'(1-u)”" anodej (3-25)
.=O
n . . .
Tn(u) = Z C;(u)'(1—u)"“tnnode,.- (3-26)
.=0

However, it should be emphasize that each variable (R,Z,%m,t9, ,B,tn) should be
correlated to one point, which also means R,Z ,%m,t9, ,6, tn should be related to the
same value of u. If R,Z are calculated based on node points in contour, )6} is
calculated based on node points in blade angle distribution, and tn, are calculated

based on node points in normal thickness distribution. Then the same distributions of
u have to choose. Otherwise, u are required to interpolate in blade angle distribution

and normal thickness distribution to obtain the same distribution as u distribution in
the contour. After (R,Z ,%m, ,6, tn) are calculated, then 0 could be calculated based
on Eqn. (3-23) and (3-24).

The surface file contains variable (x, y, z, r, 6) on pressure side (leading surface) and

suction side (trailing surface) of section curves. And these data are calculated based

on data of camber line. As mentioned before, every point can be denotes in coordinate

System( z, r, 9) and x, y can be calculated fiom r,t9 , which are expressed by:

x = rsin(6) (3-27)
y = rcos(t9) (3-28)

~ Pressure and suction side should be calculated in the (m, 0r) coordinate system as

Shown in Figure 3-21. The calculation of (zsuc,rsuc, Osuc) on the suction side based
on data of camber lines has three steps.

49

a

J

b

I}

l

but.

Ti

The first step is to calculate zsuc (u) and 65W (u)-Rsuc (u), which are expressed

by:
zsuc (u)=z(u)-O.5Tn(u)sin(a) (3'29)
Gsuc(u)-Rsuc(u)=z(u)+0.5Tn(u)cos(a) (3-30)

The second step is to calculate Rsuc (u) based on the value of Z suc (u). If
Zsuc(u)=Z(“)’then Rsuc(u)=R(“)°

The third step is to calculate Bsuc (u) after the Rsuc (u) has been obtained by

gsuc (u) : asuc;::c1(e:lic (u)

The method of calculating (zpre,rpre,6pre) on the pressure side is similar to the

Eqns. for calculating (zpre,rpre,9pre) , while calculation of zpre (u) and

R pre (u) - Hpre (u) are difl‘erent:
zpre(u)=z(u)+0.5Tn(u)sin(a) (3'31)
6pm (to-Rpm (u) = z(u)—O.5Tn(u)cos(a) (3'32)

50

 

 

 

A
R*Thetai .
(inch) Thickness“)
j’\ Z (inch)
< Angle(l) Thickness(2)
a
b
An le(2)
g Leading
Surface
Line with Mean Line
minimum Z
Trailing
Surface

Figure 3-21 Illustration of calculation of leading edge
To calculating all points on pressure side and suction side are not sufficient, because
the leading edge and trailing edge has to be designed to reduce blockage and
separation, especially for leading edge. Therefore, the semi-ellipse is used to
substitute the blunt leading edge, which is also as shown in Figure 3-21. The ellipse
ratio is specified by designers. Therefore the center of ellipse is the most important
factor to be determined. The calculation of center of ellipse is based on trail-and-error
method and under the assumption that the blade angle is a constant in the leading edge,

which has several steps as following:

The first step is to choose a point as center of ellipse on the camber and then the

ellipse firnction A22 + Bzr + Cr2 + D2 + Er + F = 0 can be determined.

5]

The second step is to calculate the number of intersection points between ellipse and
line with minimum Z. Because the leading edge is assumed to be tangential to the line
with minimum Z. To calculate the number of intersection points between ellipse and

line with minimum Z, Zmin is input into the ellipse function and the values of R are

calculated. If there is no real solutions for R, then there is no intersection points
between ellipse and line with minimum Z, and the center of ellipse should be moved
toward the line with minimum Z; if there are two different real solutions for R, then
the center of ellipse should be moved away from the line with minimum Z. This
iteration ends during the there are two same real solutions have been obtained.

In Figure 3-22 we can see that the leading and trailing surface begin at 3rd point
instead of lst point. In fact, this number is influenced by many factors including ratio
of ellipse and the number of points to output. As shown in Figure 3-22, all calculation
of points on leading edge or leading surface or trailing surface are depends on camber
line. When the number of points P to output has been decided. BladeCAD divided
mean line into P small parts, as shown by red points. The green points on leading
surface or trailing surface output after the output of coordinates of blue points on
leading edge. The beginning number represents the position where the leading surface
and trailing surface begin. If the number for output is 50 instead, the space between
two close points increased 100%, which is shown with square in Figure 3-22. Besides,

if the ratio increases, the beginning number may also increases.

52

 

 

4)

R*Theta
(inch)
Z (inch)
a:
i 2 Leading
Surface
a
A Mean Line

 

Trailing
Surface

Figure 3-22 Illustration of starting point on leading edge
The condition at trailing edge is as shown in Figure 3-23. It can be seen that Z

coordinate of last point on leading surface is already larger than 0 while that on the
trailing surface is less than 0 although that of camber line is equal to O . This is the
influences of thickness and exit blade angle. In BladeCAD, the shape of trailing edge
is defined as blunt. The last point the leading surface is move left to the point with the
same Z coordinate as the last point on the camber line. The same method is used for

the last point on the trailing edge but in the inverse direction (Figure 3—23).

53

R‘Theta Z
(lflCh) (ll'lCh)

4% Leading Surface

Mean Line

. . 3*
Trailing Surface

 

 

Z=O

Figure 3-23 Illustration of trailing edge
3.6 Comparison of BladeCAD and CCAD

To compare the developed geometry tool, contour, blade angle distribution, normal
thickness distribution, theta angle distribution, lean angle distribution, leading edge,
leading and trailing surface of BladeCAD are compared with those of CCAD,

commercial sofiware.

54

+ BladeCAD Shroud
Plot of R vs Z + CCAD ShFoud

+ BladeC—AD_Hub

-><- CCAD_Hub

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 {M
a: 1 /
O 8 /
0.6 ‘
0.4 l/ T j T T T
-1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1
2 (inch)
Figure 3-24 Comparison of Contours
B An 1 Shr d + BladeCAD
eta ge on ou + C C AD
0
-10
A -20
3
3 -30
Q)
'31)
g -40
a /
a -50 /
~60
-70 I l l l
0 20 4O 6O 80 100
%M

Figure 3-25 Comparison of beta angle distribution on shroud

55

Beta Angle (deg)

Theta Angle (Degree)

Beta Angle on Hub

+ BladeCAD

 

 

 

 

 

 

 

 

 

 

100

 

 

 

 

 

 

 

 

 

 

+ CCAD
O

-5
-10
-15
-20
-25
-3O
-35 a r I I

0 20 40 60 80
%M
Figure 3-26 Comparison of beta angle distribution on hub
Theta Angle on Shroud + BladeCAD
+ CCAD

O

-5 \
-10 \
-15 \\
-20 \
‘25 x
-30
-35 1 T T l

0 20 4O 60 80 100
%M

Figure 3-27 Comparison of theta angle distribution on shroud

56

Theta Angle (Degree)
u’u .L.
o

Theta Angle on Hub + BladeCAD

+ CCAD

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-30 \\
-35
\D
-40 T . . .
O 20 40 6O 80 1 00
%M
Figure 3-28 Comparison of theta angle distribution on hub
Normal Thickness on Shroud + BladeCAD
+ CCAD
0.05
3 0.04
5 E
7° 0.03
E :
O x
Z
0.02 l l l T . r l
O 20 40 60 80 100

%M

Figure 3-29 Comparison of normal thickness distribution on shroud

57

Normal Thickness (inch)

0.07

0.065

0.06

0.055

0.05

0.045

Normal Thickness on Hub + BladeCAD
4" CCAD

 

 

 

 

 

 

db...“ -........._ _

20 40 60 80 1 00
%M

Figure 3-30 Comparison of normal thickness distribution on shroud

90
8O
70
60
50
40
30

20
10
0
-10

Loan Angle (deg)

 

+ BladeCAD

Lean Angle +CCAD

 

 

 

 

20 40 60 80 1 00
M (%)

Figure 3-31 Comparison of lean angle distribution based on first definition of

quasi-normal lines

58

+ BladeCAD
+ CCAD

Lean Angle

 

LII
O

 

A
M

 

‘5

 

U’I

/
MN. ./
//
//
//
//
I . , , .
0

20 40 6O 80 100
%M

 

 

O

 

 

 

O

 

M

 

Lean Angle (Degree)
H H N u 0) OJ

C

 

 

 

 

OU’I

Figure 3-32 Comparison of lean angle distribution based on second definition of
quasi-normal lines between BladeCAD and CCAD

The differences of lean angle distributions between BladeCAD and CCAD (Figure
3-32) are resulted from the different calculation method on quasi-normal lines.

Before we compare Leading Edge (L.E.), there are three parameters needed to be set.
The type of LB. has been set to semi-ellipse. The number of LB. points has been set
as 9. The L.E. ellipse aspect ratio has been set as two different values: 2 and 4, and

the comparison results are shown as following:

59

Leading Edge on Shroud (Aspect Ratio =2) + BladeCAD

 

 

 

R*Thet (inch)

 

 

 

 

+ CCAD
-0.01
-0.02
-0.03 \\
-0.04 T I I
-0.01 0.00 0.01 0.02 0.03

m (inch)
Figure 3-33 Comparison of lead edge on shroud (ellipse aspect ratio = 2)

 

 

leading Edge on Hub (Aspect Ratio :2 ) + BladeCAD
+ CCAD
0.00 \

 

f

-0.02 \

-0.03 \

-0.04 \

-0.05 \\
-0.06 I I ' I t

-0.02 0.00 0.02 0.04 0.06 0.08
m (inch)

 

 

R*Theta (inch)

 

 

 

 

 

Figure 3-34 Comparison of lead edge on shroud (ellipse aspect ratio = 2)

6O

leading Edge on Shroud (Aspect Ratio =4)

+ BladeCAD

+ CCAD

 

0.00

 

-0.01

 

-0.02

 

-0.03

\

 

-0.04

R*Theta (inch)

 

-0.05

 

\

 

 

-0.06
-0.01

T

l l

l T

0.00 0.01 0.02 0.03 0.04 0.05

111 (inch)

Figure 3-35 Comparison of lead edge on shroud (ellipse aspect ratio = 4)

+ BladeCAD

leading Edge on Hub (Aspect Ratio =4)

+ CCAD

 

m

 

\

 

\

 

\

 

 

\

 

 

0.01
-0.01
E?
0
E -003
e
d)
#3
* -005
a:
-007
-009
-002

l

0.03
m (inch)

T

0.08

0.13

Figure 3-36 Comparison of lead edge on shroud (ellipse aspect ratio = 4)

61

Leading and Trailing Surface on Shroud
— BladeCAD_l.eadingSurface

 

0.0
-0.2 '-

Leading
------ CCAD_LeadingSurface

— B1adeCAD_TrailingEdge

 

-0.4

Trailing
surfm

\m CCAD_TrailingEdge

 

-0.6

\\

 

-0.8

R*Theta (inch)

 

-1.0

 

 

 

-1.2

0.0

I l l l l l

0.2 0.4 0.6 0.8 1.0 1.2 1.4

m (inch)

Figure 3-37 Comparison of leading and trailing surface on shroud (ellipse aspect

ratio = 4)

Leading and Trailing Surface on Hub

 

0.0

-0.2

-0.4

Leading
surface _
' ------- CCAD_leadngdge

N — B1adeCAD_TrailingEdge __

-—- B1adeCAD_leadingEdge

 

-0.6

Trailing ....... . .
surface \\ CCAD_Trarlngdge

 

 

-0.8

R*Theta (inch)

-
~
\
s
a
a
\
o
\

 

-1.0

\
\
\
\
\
\

 

-1.2

 

\

 

 

-1.4

0.0

I l T l

0.5 1.0 1.5 2.0 2.5
m (inch)

Figure 3-38 Comparison of leading and trailing surface on shroud (ellipse aspect

ratio = 4)

62

3.7 Three-dimensional Design
There are two ways to calculate all points on surface of blades. The first method is to
calculate all the points on the pressure sides and suction sides on section curves
mentioned above, and then calculate all points on pressure surface and suction surface
based on data of pressure sides and suction sides on section curves by Eqn.
(3-29)-(3-32).
The second method is to calculate all the points on camber surface firstly, and then
calculate corresponding points on pressure surface and suction surface. The method to
calculate camber surface is the combination of Eqns. (3-1) and (3-10).

n

n . . _. . _ . . T
[2,“ng = Z Z CirCrJr (1_“)n (“1 (l—V)n J VJ [znodejarnodejrgnodej:l (3'33)
i=0j=0

63

CHAPTER 4

FLOW SOLVER
4.1 Introduction 7
MERIDL and TSONIC, which are free CFD codes fi'om NASA, are applied to
calculate the performance of centrifugal compressors with cubic spline curve. The
MERIDL is used to calculate meridional plane while TSONIC is applied to calculate
the blade-to-blade plane. The basic theory used in MERIDL/TOSNIC is streamline
curvature method and the basic idea for meshing is the use of arbitrary
quasi-orthogonals. MERIDL and TSONIC are two-dimensional codes and very low
time cost. The total time cost has been reduced to approximately 0.5 second per case
using CPU Pentium 3.206Hz. The combination of MERIDL and TSONIC can
provide quasi-three dimensional results. The detail introduction of MERIDL and
TSONIC are in References [12-14] respectively. To combine MERIDL and TSONIC
with the geometry generation tool BladeCAD, several changes are made to allow the
MERIDL and TSONIC can be directly called by BladeCAD. ’
1) Modifying Katsainis’s TSONIC to calculate geometry with Bezier polynomials
instead of original spline curves.
2) Combing the MERIDL and TSONIC with BladeCAD
3) Generating a file which included geometry file and running condition fi'om
BladeCAD as the input for MERIDL and TSONIC.
To further improve the accuracy of results, five different meshing methods are
compared and their corresponding results are analyzed.
Furthermore, to evaluate the accuracy of MERIDL and TSONIC, the calculation

results from MERIDL and TSONIC are compared with results calculated by

64

TASCFlow. The calculation results of different flow solvers are solved under the same
compressor geometry and running condition.
4.2 Influences of Meshing

4.2.1 Comparison of Five Different Meshing Methods
Five types of meshing are compared in this report. The aim is to study of the

influences of mesh. Based on the last report, there are oscillations in the results. Based
on the deep exploration on the codes, it has been found it occurs because of the
undesired meshing. Therefore, five different types of meshing are changed and their

corresponding influences are studied.

0.22

I

0.2

0.18

R (m)

0.1 -
J=21

BIB F

 

 

 

0. l l l I l l l
$12 -0.1 008 0.05 43.04 0.02 0 0.02
Z (m)

Figure 4-1 First meshing method
ln'the first meshing method, there are 50 grids from upstream line to downstream line

along meridional direction (I=50), which includes inlet casing, impeller and diffuser.
And there are 21 grids from hub to shroud along vertical direction (J =21). There are
10 grids in the inlet casing along meridional direction; while 30 grids in the impeller
and 10 grids in the diffuser. In the middle line (J=11), the meridional distances in the
inlet casing, impeller and diffuser are uniformly divided. Quasi—Normal lines are

65

created from middle line to shroud and hub to mesh. Therefore, the meridional
distances are not equal to each other on shroud and hub. Please refer to [12, 14] for

detail information on creating the arbitrary quai-orthogonals.

l

0.22

0’2 - Nil :1 Viv;

mm M
HHJIIHlIHIIIE III
‘1 1 8 I'll'll il‘II‘lt'lllllfili
:fiHI'IHlIIIIII :
0 15 _ 'Iri'flllllllhiIIIII

R (m)

0-1 * J=21

0.08

I

 

 

 

I1%i12 41.1 008 {LEE {1.04 43.02 [I 0.02
Figure 4-2 Second meshing method
In the second meshing method, I=50 and J=21. There are 10 grids in the inlet casing
along meridional direction; while 30 grids in the impeller and 10 grids in the difi'user.
The grids are divided based on the same normalized meridional distances on shroud
and hub. However, meridional distances on the inlet casing, impeller and diffuser are

not equal to each other.

66

0.22
0.2

0.18

I

f

0.16
0.14 I

R(m)

T

0.12

0.1

0.08 -

0.06 J l l l I
-0.12 -0.1 -0.08 -0.06 -0.04 -002 0 0.02
Z(m)

Figure 4-3 Third meshing method
In the third meshing method, I=50 and J=21. The method is similar to the first one.

 

 

However, not only middle line of each component is uniformly divided, but also the
whole middle line of compressor, which includes the inlet casing, impeller and

diffuser, is uniformly divided.

0.22 -
0.2 -
0.18 -

0.18 -

R (m)

0.14 -

0.12 -

0.1 - J=21

0.08

 

 

0. I l l l l l
93.12 -01 008 .005 43.04 -002 0 0.02
I (m)

Figure 4-4 Fourth meshing method
In the 4th case of meshing, I=50 and J=21. The method is similar to the second method.

67

However, the whole shroud or hub of compressor instead of each component is

uniformly divided.

0.22 P

0'2 3111111111111 |.
”IHIIImlII

“I'mlllll

O. 18 " EIIllllll‘Iit‘lill'lllllllllllit!‘l

0.1 - J=21

 

0.08

“98.12

 

 

 

 

Figure 4-5 Fifth meshing method
In the 5th case of meshing, I=50 and J=21. There are 10 grids in the inlet casing along

meridional direction; while 30 grids in the impeller and 10 grids in the diffuser. The
grids are divided based on the same normalized meridional distances on shroud and
hub in impeller. However, they are not uniform in the diffuser and inlet casing. The
meridional distance on shroud and hub of inlet casing gradually decrease while the
meridional distance on shroud and hub gradually increase.

4.2.2 Comparison of Relative Velocity Distribution
Results of five different meshing methods are introduced and their corresponding

results are compared on the following compressor performance parameters: relative

meridional velocity, relative flow angle, relative Mach number and static pressure.

68

 

_ ,. —°—PS onHUB —~— 53 onHUB "
100 _._ PS on SHD _._ 83 on SHD
O 1 1 1 1 1

0 20 40 60 80 100
Percentage Meridional Distance

 

 

Figure 4-6 Relative velocity distribution based on first meshing method

 

 

 

200 I —°—PS onHUB _... ss onHUB

100 ‘ _._ PS on SHD —-— ss on SHD

O I I I I I
0 20 40 60 80 100

Percentage Meridional Distance

Figure 4-7 Relative velocity distribution based on second meshing method

69

 

—°—PSonHUB —:— SSonHUB'
100 " -‘- PS on SHD —",' SS onSHD ‘

O I I I I I

0 20 40 ' 60 80 100
Percentage Meridional Distance

 

 

Figure 4-8 Relative velocity distribution based on third meshing method

 

—°—PS onHUB -*- SS onHUB

 

 

103‘ '—o— PSonS —~— SSonSHD
0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-9 Relative velocity distribution based on fourth meshing method

70

    

 

A
.‘
AAA--A-AAA
v' v
" v-

 

 

200 q . . fi—vvvv ... .. .
100 _ —‘-PS onHUB -*— SS onHUB -
0 —.—1 PS on SHD :4- SS on SI-ID j

0 20 4O 60 80 100
Percentage Meridional Distance

Figure 4-10 Relative velocity distribution based on fifth meshing method
4.2.3 Comparison of Relative Flow Angle

 

 

 

~30 _
A
or)
a)
'3
in 40 -
a) ’ ‘ 1
g d—PSonHUB --— SSonHUB
% 60 -°- PSonSHD_-‘- SSonSHD
Dd ' ‘ ' ' ‘ '

0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-11 Relative flow angle distribution based on first meshing method

7]

Relative Flow Angle (deg)

  

-°—PS onHUB —4— ss onHUB

 

 

-°- PS on SHD -*— SS on SHD

I I

0 20 40 60 80 100

Percentage Meridional Distance

Figure 4-12 Relative flow angle distribution based on second meshing method

Relative Flow Angle (deg)

-30 i

-35 _,

-40 —

    

—°—PS onHUB -*- SS onHUB

 

 

0

—-— PS on SHD —~— ss on SHD
20 40 60 80 100
Percentage Meridional Distance

Figure 4-13 Relative flow angle distribution based on third meshing method

72

Relative Flow Angle (deg)

   

~-°-PS onHUB —t- SS onHUB

 

 

I

20

._°— PS on SHD -‘— SS on SHD

40 60 80 100

Percentage Meridional Distance

Figure 4-14 Relative flow angle distribution based on fourth meshing method

Relative Flow Angle (deg)

-30 —

 

   

—‘-PS onHUB -‘- SS onHUB
j -°- PS on SI-ID -t- SS on SHD

 

I
I

20 40 60 80 100

Percentage Meridional Distance

Figure 4-15 Relative flow angle distribution based on fifth meshing method

73

4.2.4 Comparison of Relative Mach Number

0.60 — —‘—PS onHUB—*- SSonHUB
0 55 -'- PS on SHD —.— SS on SHD

050 p gggggg

 

p
S

0.35
0.30
0.25
0.20
0. 15 I I I I r
0 20 40 60 80 100
Percentage Meridional Distance

Relative Mach Number

 

 

Figure 4-16 Relative Mach number distribution based on first meshing method

0.60 — —°—PSonHUB-*- SSonHUB

 

 

5 0.55 _ m 1 -°- PS on SHD -*— SS on SHD

:4 , , ,,,,, “x

z 0.45 ‘ ‘

g 0.35 -

o 0.30 ‘

g 0.25 —

T) 0.20 ‘

m 0.15 I I I I 7
0 20 40 60 80 100

Percentage Meridional Distance

Figure 4-17 Relative Mach number distribution based on second meshing
method

74

060 _ —'—PSonHUB-t- SSonHUB
0'55 _ -°- PS on SHD -*- SS on SHD

0.50 - I A k
0.40 , h ‘
0.35 - ~+ +~ - ,_. ,
0.25 —
0.20 - e
0.15 . . . , .

0 20 40 60 80 100
Percentage Meridional Distance

Relative Mach Number

 

 

Figure 4-18 Relative Mach number distribution based on third meshing method

—‘—PS onHUB -*— SS onHUB

' —°- PS on SHD -‘- SS on SHD

0.35 r
0.25 ‘
0.2 - * *
0.15 I w I I I
0 20 40 60 80 100
Percentage Meridional Distance

0
O\
1

Relative Mach Number

 

 

Figure 4-19 Relative Mach number distribution based on fourth meshing method

75

   

 

 

 

._. 06— I—'—PSonHUB -*— SSonHUB

.2

g; (3.4 —‘

03 - i

o 0.2 ‘ I 5

m 0.15 I l I I I
0 20 40 60 80 100

Percentage Meridional Distance

Figure 4-20 Relative Mach number distribution based on fifth meshing method

4.2.5 Comparison of Static Pressure Distribution

 

 

 

600 s —°—PS onHUB -+- SS onHUB
A J -°- PS on SHD -t- SS on SHD
a 575
f? 550 —
-§
§ 525 s
13.1
.g 500 ‘
S
m 475 s .
450 1 . 1 a 1
0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-21 Static pressure distribution based on first meshing method

76

 

 

 

600 s -‘—PS onHUB —*— SS onHUB
-'- PS on SHD -*— SS on SHD
0.1
E 550 s
g 525 ~ + ~
3 500 ~
1% 475 -
450 1 1 1 T ‘
0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-22 Static pressure distribution based on second meshing method

 

 

 

600 ' -°—PS onHUB -*— SS onHUB
A -'- PS on SHD -*- SS on SHD
t? 550 -
i=2 _1
g 525
ft, 500
3 475 ~
450 I I I I I
0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-23 Static pressure distribution based on third meshing method

77

600 - -°-PS onHUB -‘- SS onHUB
-°- PS on SHD -‘- SS on SHD

)

575 s

l

M

V!

O
1

525 --~w

Static Pressure (ps
4:. UI
d 8

 

 

 

450 I I I I I
0 20 4O 60 80 100

Percentage Meridional Distance
Figure 4-24 Static pressure distribution based on fourth meshing method

600-—°—PSonI-IUB -*— SSonHUB
575 -°- PS on SHD -*— SS on SHD

)

1

LI!

M

O
1

525 s

Static Pressure (ps

J§ £11
d 8
a 1

 

 

 

450 1 1 1 1 1

0 20 40 60 80 100
Percentage Meridional Distance

Figure 4-25 Static pressure distribution based on fifth meshing method

4.2.6 Discussion

All results calculated based on these five different types of meshing are similar.
However, some of them are smooth while others have waves, which will result in
difficulties and bring large errors for the following steps in the optimization procedure.

Based on the results of relative meridional velocity, five cases of meshing were sorted

fi'om worst to best: Case1=Case5<Case2=Case3 <Case 4.

78

Based on the results of relative flow angle, five cases of meshing of were sorted fi'om
worst to best: Casel<Case2=Case5=Case3<Case 4.

Based on the results of relative Mach number, five cases of meshing were sorted from
worst to best: Casel<Case5<Case 2=Case 3<Case 4.

Based on the results of static pressure, five cases of meshing were sorted from worst
to best: Casel<Case5<Case 3<Case 2<Case 4.

Based on results of those four variables, the fourth meshing method (Figure 4-4) is the
best while the first meshing method (Figure 4-1) is the worst.

It should be mentioned that the influences on only those four variables in the blade to
blade direction are considered. The influences on the meridional direction are not
considered because almost no waves occur along meridional direction.

4.3 Comparison between TASCFlow and NASA Codes

To evaluate the accuracy of codes of MERIDL and TSONIC, a widely used
commercial software TASCflow is applied here. Five cases: Baseline, Casel, Case 2,,
Case 2b and Case 2c with different geometry are generated. TASCflow and NASA
codes are applied to calculate the performances of five compressor impellers. The
comparisons of loadings, relative velocity distributions and static pressure
distributions are compared.

4.3.1 Geometry Cases

Five cases of compressors with different geometries are compared using TASCflow
and NASA codes. The geometries of these inlet casing and diffuser are same. The one
dimensional geometries of impellers are same. And the normal thiclmess distributions
of five impellers are the same while blade angle distributions and contours are
difierent. Contours and blade angle distributions of five different cases are shown in
Figure 4-26 and Figure 4-27.The meshing used in TASCflow generated by FASTTasc

automatically, which is the three dimensional meshing tool. The meshing is

79

(I,J,K)=(115,25,25). In the NASA codes, there are two parts: MERIDL and TSONIC.
Both of them are 2D codes. The meshing in the MERIDL is (I,K)=(50,25), while in
the TSONIC, I=50. However, calculation in blade-to-blade surface is based on each
streamline, which means the calculation for each streamline is independent. And the
value of J is determined by codes automatically based on geometry and varied with
the location of streamline. In last section, the fourth meshing method has been proved
as the best one. Therefore, that meshing method is used here. For NASA codes, the

efficiency is an input instead of output. The input of NASA codes included: 1)

upstream Total Temperature: To]; 2) upstream Total Pressure: P01 ; 3) upstream
Absolute whirl: UOICul 4) downstream total pressure: P02 5) downstream absolute
whirl: UzCuz 6) gas specific heat ratio: 7 7) gas specific heat measured under

constant pressure: Cp. The comparison of results between TASCflow and NASA

codes are shown from Figure 4-28 to Figure 4-47.

4.3.2 Comparison of geometries

80

6.5

3.5

 

 

 

 

—O— Baseline_Hub

—'— Casel_Hub

 

—*— Case2_Hub
—><- Case2b_Hub

  

 

+ Case20_Hub
-'- Baseline_Shroud
-+— Casel_Shroud

 
 
 

 

 

-- Case2_Shroud
-"- Case2b_Shroud
—°- Case2c_Shroud -

 
 

 

 

 

 

 

 

 

 

 

 

 

 

-2 -1.5 -1 .05
2 (inch)

Figure 4-26 Contours of five different cases

81

 

Blade Angle (Deg)

 

 

 

 

 

 

 

 

 

 

 

—*— Baseline_Hub
—'— Casel_Hub

“fir- Case2_Hub

 

-'><— Case2b_Hub
+ Case2 c_Hub

-l- Baseline_Shroud

 

I

Casel_Shroud
Case2_Shroud

 

 

l l

Case2b_Shroud

 

 

 

—*- Case2c_Shroud

 

0 20 40

60 80 100

Percentage Meridional Distance

Figure 4-27 Blade angle distributions of five different cases

82

4.3.3

Loading

Loading

Comparison of loadings

 

 

 

 

 

 

 

0.8
0.7 ~
0.6 1'
0.5
0.4 s
0.3 s .
0.2 % mm +Baselme_HUB
+Case1_HUB
0-1 A ‘ +Case2__HUB
0 ‘ ' ”1...- Case2b_HUB
-0.l - , +Case2c_HUB
-02 I I I I fl
0 20 40 60 80 100
%M

Figure 4-28 Loading on hub calculated by TASCFlow

 

 

 

 

 

 

 

 

0.8 - .............
0.7 _
0.6 _
0.5 s ..........................
0.4 _
0.3 s‘ -°-Base1ine_HUB
0-2 -—Case1_HUB
0.1 s +Case2_HUB
0 4! +Case2b_HUB
-0.1 s" -- , +Case2c_HUB
-0.2 1 1 , 1 fl
0 20 40 60 80 100
%M

Figure 4-29 Loading on hub calculated by MERIDL and TSONIC

Loading

Loading

   

 

   

-*— Baseline_SHR I]
+ Casel _SHR

*- Case2_SHR

“r!- Case2b_SHR

+ Case2 c_SHR

0 20 40 60 80 100
%M

Figure 4-30 Loading on shroud calculated by TASCFlow

 
 

 
   
 

 

    

 

 

 

 

 

 

()2 —~-~~-\ ~ . -°- Baseline_SI-IR=- - ~
0.1 _ -+- Casel_SHR
0 W , 1, -— Case2_SHR _ ,_ u
Case2b_SHR .
'0'1 _ -°- CaseZc SHR "
-02 I I I - I 7
0 20 40 60 80 100
%M

Figure 4-31 Loading on shroud calculated by MERIDL and TSONIC

4.3.4 Comparison of relative velocity distributions

 

     
 

 

 

 

 

 
 
  

 

 

 

   
   
 

 

 

 

  

 

 

 

 

 

650 -
+ Baseline_P S_HUB
g 550 - -'- Casel_PS_HUB ' __
E1: -*- Case2_PS_HUB
"a 450 — — ~><~ Case2b_PS_HUB
O
:3 + Case2c_P S_HUB
E
a)
M 250 —
1 50 u I I I T
0 20 40 60 80 100
%M
Figure 4-32 Relative velocity distribution on pressure side of hub calculated by
TASCFlow
650 _ , -
_._ Baseline_P S_HUB
g —*— Case2_P S_HUB
2; '95- Case2 b_P S_HUB
§ 450 __ +Case2c_Ps_HUB W
T) -
>
g 350 - ----------
E
0
ad
250
1 50 1 1 1 1 1
0 20 40 60 80 100

%M

Figure 4-33 Relative velocity distribution on pressure side of hub calculated by
MERIDL and TSONIC

85

 

 
 
    
    

 

 

 

  

 

 

 

   
 

 

 
   
 

 

 

 

  

 

 

600 + Baseline_SS_HUB
4" Casel _SS_HUB
”1,? + Case2_SS_HUB
«3:; 550 - +— Case2b_SS_HUB
g + Case2c_SS_HUB
Lo) ,
g 500 s‘ s s
O
.2
E
a“; 450 s
400 1 ‘ “’ ‘1‘ 1 1 1 1
0 20 40 60 80 100
%M
Figure 4-34 Relative velocity distribution on suction side of hub calculated by
TASCflow
650 1
+ Baseline_SS_HUB
-'- Casel _SS_HUB
g 600 -+— Case2_SS_HUB
E "r" Case2b_SS_HUB
'g 550 ' " -— Case2c_SS_HUB
E
_g 500 r
E
32
450 _
400 fi 1 1 1 1
0 20 40 60 80 100
%M

Figure 4-35 Relative velocity distribution on suction side of hub calculated by
MERIDL and TSONIC

86

 

800 s .
-°- Baseline_P S_SHR

-'-Case1_PS_SHR
700 _m '3 if ' +Case2_PS_SI-IR
—><— Case2b_P S_SHR
.1... Case20_PS_SHR

 
 
 

 

 

 

600 4

 

500

Relative Velocity (ft/s)

 

400 I'm .. 1:4. _ w
I w w
300 I I H I I
0 20 40 6O 80 100
%M

Figure 4-36 Relative velocity distribution on pressure side of shroud calculated
by TASCflow

   
  
 
 

 

 

 

  

 

 

 

 

-+- Baseline_P S_SHR

,6 700 4» " +Casel_PS_SHR
‘33 +Case2_PS_SHR
i? 600 J-.- W + Case2b_PS_SHR
O
73 + Case2 c_P S_SHR
>
.“Z’ 500 r
E
32

400 is

300 I I I I r

0 20 40 60 80 100

%M

Figure 4-37 Relative velocity distribution on pressure side of shroud calculated
by MERIDL and TSONIC

87

  
  
 
 

 

-°- Baseline_SS_SHR
-'- Casel _SS_SHR

 

 

 

  
 
 
 

 

 

 

g 800 +Case2_SS_SHR

.3» ”s“; Case2b_SS_SHR

_‘9’ 750 I +Case2c_ss__SHR

§

f; 700 —

E

O

0‘ 650 ~

600 a 1 1 1 " 1
0 20 40 6O 80 100

%M

Figure 4-38 Relative velocity distribution on suction side of shroud calculated by
TASCflow

 

+ Baseline_SS_SHR
-'- Casel_SS_SHR
+ Case2_SS_SHR
—«+— Case2b_SS_SHR

 

    

 

 

 

Relative Velocity (ft/s)

 

 

 

 

l

0 20 40 60 80 1 00
%M

Figure 4-39 Relative velocity distribution on suction side of shroud calculated by
MERIDL and TSONIC

4.3.5 Comparison of static pressure distributions

 

 

 

 

 

 

 

 

 
 
   

 

 

 

 

 

 

80 .. + Baseline_PS_HUB H ,, _

5 + Casel_PS_HU B

A +Case2_PS_HUB \ 1

g 560 _- + CaSCZb_PS_HUB 1w

j? + Case20_PS_HUB 3} 2 _ f"

a .
§ 540 —

9.1

.2

3% 520 r

500 1 1 1 1 1 1
0 20 40 60 80 1 00
%M
Figure 4-40 Static pressure distribution on pressure side of hub calculated by
TASCflow
-*- Baseline_P S_HUB

A “*s' Case2_PS_HUB

-§ 560 _ —><— Case2b_PS_HUB g

8' .

q, - + Case20_PS_HUB , 1.

O-1

.2

33’

m 520 s

500 '" I I f I I
0 I 20 40 60 80 100
%M

Figure 4-41 Relative velocity distribution on pressure side of hub calculated by
MERIDL and TSONIC

89

580 s‘

 

..;11...111.;ss_111 "111' ,,
560 —- + Casel_SS_HUB
+ Case2_SS_HUB

  
    

 

 

 

540 _1 +Case2h_ss_HUB i . "
+Case2c_ss_HUB .1" "
520 —~ ~ .

Static Pressure (psia)

 

 

 

     
 
   
  

 

 

 

 

 

0 20 4° 60 80 100
%M
Figure 4-42 Relative velocity distribution on suction side of hub calculated by
TASCflow
+ Baseline_SS_HUB
560 — +Casel_SS_HUB
3 + Case2_SS_HUB
g 540 — + Case2b_SS_HUB
g + Case2c_SS_HUB
§ 520 _ , I .
O-I ‘ . ‘ . . .
.2 A ............
E 500
CD
480 s
460 F I I I I j
0 2° 40 60 80 100
%M

Figure 4-43 Relative velocity distribution on suction side of hub calculated by
MERIDL and TSONIC

90

M
00
O

 

     
 
   

 

 

 

 

A 560

.2

é

8 540

g 520 . -°- Baseline_PS_SHR ‘

9.1 + Casel _PS_Sl-IR

0

g 500 —~— Case2_PS_SHR

m
480 ,,,,,,,,,,, _ Case2b_PS_SHR

+ Case20_P S_SHR
460 1 I I I I I
0 20 40 60 80 100
%M

Figure 4-44 Relative velocity distribution on pressure side of shroud calculated
by TASCflow

 

-°- Baseline_PS_SHR

-'- Casel _P S_SHR

-*- Case2_PS_SI-IR

— _._,__._ CaseZb_P S_SHR -~——

1 + Case2c_PS_SHR

460 f I I r 1 .
0 20 40 60 80 100

%M

   

Static Pressure (psia)

 

 

 

 

 

Figure 4-45 Relative velocity distribution on pressure side of shroud calculated
by MERIDL and TSONIC

 

 

 

 

   
  

 

   

 

560 ssssssssssss
-'- Baseline_SS_SHR

540 ~ +Case1_SS_SHR
g -*-Case2_SS_SHR
{51 520 —‘ s~*£‘~s*sCaseZb_SS_SHR ,1» “ ‘
‘5 +Case20_SS_SHR .. , 1
o‘: y 2’”
.2 _ .
E 480 .
m “5.1-5 ’ ‘

460 __..- ‘

440 I I 1

 

20 40 60 80 100
%M

Figure 4-46 Relative velocity distribution on suction side of shroud calculated by

560 “s"

Ur U]

N h

C O
1 I

Static Pressure (psia)
A M
00 O
O O
I l

4;

as

O
l

 

TASCflow

+ Baseline_SS_SHR
+ Casel_SS_SHR
+ Case2_SS_SHR
"*1" Case2b_SS_SHR

  

 

 

 

 

 

  

‘ +Case2c_SS_SHR

 

440

20 40 60 80 100
%M

Figure 4-47 Relative velocity distribution on suction side of shroud calculated by

MERIDL and TSONIC

4.3.6 Discussions

The differences between TASCflow and NASA codes at the leading and trailing edge
are quite different. This is probably because interface between the stators and rotors.
Most of comparison results indicate that there are no significant differences of relative
velocity distributions and static pressure distributions between TASCflow and NASA
codes. NASA codes have the fairly sufficiently accurate for evaluating the compressor
performance and therefore are used to evaluate compressor performances in the

following optimization procedure.

93

CHAPTER 5

IMPELLER OPTINIIZATION PROCEDURE WITH ANN & GA

5.1 Introduction

Compared to other components in compressors, such as diffusers[15], return
vanes[16], the optimization of impellers is more important and challenging because of
its dominant role in compressing flow and relatively complex geometry. Design and
optimization techniques can be broadly divided into two categories: inverse
method[16, l7] and direct methods[18, 19]. Direct method, which is less efficient but
more effective than inverse method, is studied and applied in this study. The general
procedure of direct method included five steps and these five optimization steps are
interdependent

1) Parameterization

Parameters x are extracted from complex geometry for optimization. There are dozens
of parameters required to be considered during the design of centrifugal compressor
impeller. However, only parameters with significant effects called as optimization
parameters are considered in the optimization because the increase of optimization
parameters will result in “the curse of dimensionality”, which is introduced by
Bellman[20]. Theoretically, the less the number of optimization parameters is, the
easier the optimal solution, especially the global one, can be found and the lower the
computational cost is. On the other hand, optimization parameters should be sufficient,
effective and accurate to represent the geometry of impellers.

2) Proposing objective function

Optimization problem can be subdivided into single-objective[21, 22] and

94

multiple-objective[23] optimization. The objective function f can be written in the
form: f0)

Where f should be a linear or nonlinear function and y is generally the performance of
impellers, such as efliciency, entropy, design-point loss, flow range, pressure
distribution, or velocity distribution. The chosen objective function greatly depends
on the available information from flow solver.

3) Using flow solver to calculate compressor performance

The flow solver used in optimization can be broadly divided into two categories: 2D
(Quasi-3D) and 3D flow solver. k — a) turbulence model is widely used in 3D flow
solver for calculating compressor performance.[15, 16] Theoretically, 3D flow solver
is much more accurate than 2D codes and can provide much more information such as
secondary flow, convex, etc. 2D codes are much lower computational time cost than
3D codes. Performance y can be calculated fi'om a geometry case, which is
represented by a set of optimization parameters x. Once we have a set of optimization
parameters x, we can calculate performance y by using flow solver. Combined with
objective function fly), we can calculate the value of objective firnction f and make an
evaluation on corresponding geometry.

4) Forming metamodel between optimization parameters and performance.
Considering the high computational cost of flow solver, especially 3D flow solver, a
metamodel function between optimization parameters x and performance y is required
to form: y=G(x). Once metamodel has been formed, the objective function f(v) can be
rewritten in the form with only optimization parameters f(G(x)), then the value of
objective function can be calculated without using flow solver . The general used
mapping methods are Response Surface Methodology (RSM)[24] and an Artificial

Neural Network (ANN) [25] while the kriging model is actually an improved

95

RSM[21]. It seems that RSM is more applicable in forming mapping for axial
compressor impeller and centrifugal compressors with low number of optimization
parameters because of the relative simplicity in the relationship between geometry
and performance. However, low number of optimization seems not to meet the
industrial requirements among most of cases. Therefore, many papers used the ANN
to form performance map, also called as metamodel.

5) Applying optimization algorithm to find optimal optimization parameters x*.
Optimization algorithm can be also divided into two categories: Gradient-Based
Method (GBM) and Evolutionary Algorithm (EA). In GBM classified as local
optimization method and known for its efiiciency, steepest descent, conjugate gradient
or quasi-Newton techniques can be applied. This method is widely used on
optimization of stators, such as inlet vanes, difiuser vanes or airfoil shape, of which
the optimization problem is lower dimension, convex or lowly nonlinear. For the
optimization of centrifugal compressor impeller, which is a highly nonlinear problem,
GBM is not effective and easy to converge into a local minimum. BA, in which
biology evolutionary ideas are used to deal with highly dimensional nonlinear
problem and find the global optimal solution, is applied in recent years.

5.2 Parameterization

5.2.1 Geometry Parameterization

There is an extensive list of factors, which must be considered in order to reach the
most suitable design for a given application in industry.[26] Therefore, it seems it is
impossible to optimize all the variables simultaneously limited by the existed
optimization algorithm as well as the computation cost. Therefore, the variables are
required to be chosen and pararneterized carefully to match other components in the

optimization algorithms.

96

Fan[27] use 12 discrete values of blade angle distribution as parametric variables
under the assumption that a centrifugal compressor impeller with a smaller exit width
and two-dimensional blades of constant thickness.

Oyama[27] parameterized the mean camber line and a thickness distribution by the
three order B-spline curves.

Fan[28] assumed that the blades of diffusers have the constant height and thickness,
and parameterized the blade profile of diffusers by fourth-order Bezier curve, the
blade suction and pressure surfaces with fifth-order polynomials.

Geometry parameters are parameterized from the impeller shape. Chosen geometry
parameters should be sufficient to represent impeller geometry. Contour and blade
angle distribution of centrifugal impeller are represented by Bezier polynomials.
Therefore profiles of contour and blade angle distribution are completely determined
by control point nodes and the locations of these control point nodes are chosen as
geometry parameters and used as input layer to train ANN s. The geometry
parameterization is shown in Figure 5-1 and Figure 5-2. Twelve geometry parameters
are chosen fi'om contour shown in Figure 5-1 while eight from blade angle
distribution shown in Figure 5-2. Inlet and outlet parameters are not considered in the
parameterization because these parameters have already been calculated in one
dimensional design and there are extensive studies on one dimensional design of
radial gas compressor. Normal thickness distributions have more effects on
mechanical performance and lifetime of impellers, and therefore remain conservative
in compressor industy. With this consideration, locations of control nodes points of

normal thickness distribution are not chosen as optimization parameters.

97

 

 

 

 

”5:? .
0
.e ;
M I
4 '.
3 -
...,.... ........... ,..Z(inch)
-2 -1 0
Figure 5-1 Geometry parameterization of contour
Beta (deg)
0 -
-10 3‘13 x14
-20 H 3617 II X123
-30 ‘:I x16 :flx
x15 / Egg
-40 //£)C149fl
I!

 

._,.,..,._%M
O 10 20 30 40 50 60 70 80 90100

 

figure 5-2 Geometry parameterization of blade angle distribution

5.2.2 Performance Parameterization

Performance parameters are parameterized from evaluation results of a flow solver,
and should be easily evaluated by designers. The chosen performance parameters can
be efliciency[21, 25], total pressure ratio[29], losses[23, 29], velocity distribution[30,
31], pressure distribution[22] and loading[21, 25]. Three dimensional (3D)
Computational Fluid Dynamics (CFD) flow solver could be applicable for calculating
the compressor performance. However, the application of 3D CFD flow solvers to
generate training data cases and test data cases still requires long calculating time.
Therefore, Quasi-3D flow solver MERIDL [14] and TSONIC [12], which employ
streamline curvature method, are applied instead. The total computational time is
approximately one second per case using a CPU Pentium 3.20GHz. Parameters
related to relative velocity distribution are chosen as performance parameters because
of its importance on analyzing the performance of compressors as well as the limits of
streamline curvature method. There are two methods of parameterization for relative
velocity distribution: curve fitting and discretization, which are shown in Figure 5-3

and Figure 5-4 respectively.

99

 

1701

 

° original data
curve fitting

 

 

 

 

I
L

160

w (m/s)

130-

120- -

 

 

l l l

60 70 80

 

110 1 1 l

10 20 30 40 50 90
%M

Figure 5-3 Curve fitting of relative velocity distribution

J’I

 

 

 

 

 

 

 

 

 

0% 20% 40% 60% 80% 100%

%M

Figure 5-4 Discretization of relative velocity distribution

However, it is found that very small changes on the curve shape can result in a

large change on these polynomial coefficients p. Two velocity distributions with very

small differences are compared in Figure 5-5. Their corresponding polynomial

coefficients p are list in

IOO

Table 5-1 by using curve fitting method for relative velocity distribution. The
comparison of change of W points between two relative velocity distributions is
shown in Table 5-2. We can see that the average change of coefficient polynomials is
approximately 6% while the maximum change is 11.67% for curve fitting method.
The average change of W points is 0.4% while the maximum change is 1.56%.
Therefore the results suggest that the curving fitting brings nonlinear characteristics to
the optimization problem and increases the difficulties of the whole optimization
problem. Therefore, the discretization is used for parameterization. Nine W points on
relative velocity distribution are used as the performance parameters shown in Figure
5-4. The corresponding normalized meridional distance of these nine W points ranges

from 10% to 90%.

 

 

 

 

 

 

 

 

1 80 1 1 1 1 1 1 1
_— first relative velocity distribution
170? ° second relative velocity distribution ,
(I.
K.
160 - ‘1‘ 1
I“.
r

a 150 “' 3‘ i
E In
v R
3 140 1 N, 1

130 \ ,r.‘

.\ /4. I
‘0‘ .6.
120 _ .‘N'W. d
1 1 0 1 L L 1 1— 1 W
10. 20 30 40 50 60 70 80 90

°/oM

Figure 5-5 Illustration of two relative velocity distributions with small

differences

lOl

Table 5-1 Comparison of polynomial coefficients p between two W distributions

 

 

 

 

P0 P1 P2 P3 P4
First relative
velocity distribution 2.761 e-006 -0.0004815 0.046806 -3.0436 195.68
Second relative 2.761e-006 -0.0005179 0.052267 —3.3039 199.6
velocity distribution
Percentage 0f 0% 7.56% 11.67% 8.55% 2.0%
change

 

Table 5-2 Comparison of W points between two W distributions

 

W0 W1 W2 W3 W4 W5 W6 W7 W3

 

First relative

velocity 169.5 150.1 135.7 125.1 117.6 113.4 113.1 118.3 131.1
distribution
Second

"313“?” 171.5 150.6 135.8 125.1 117.6 113.4 113.1 117.9 129.0
velocrty

distribution

”2:23;" °f 1.21% 0.32% 0.05% 0.00% 0.00% 0.00% 0.06% 0.41% 1.56%

5.3 Objective Function

The object fimctions defined in papers are quite different, because it is related to a lot

of factors such as parametric variables, flow solver, optimization algorithms, and
requirements of product.
Generally, optimization problems can be grouped into two categories: constrained

optimization problem and unconstrained problems.

In the unconstrained problems, there are only object functions, such as f (x). The aim

is to maximize or minimize f (x). On the other hand, in the constrained problems,
both object functions and constrained conditions existed. For example, the constrained

conditions in [32] are expressed by:

mdes " mactu

mactu

g 0.005 (5-1)

 

 

102

Idees - Cpactu l
l Cpactu

 

s 0.01 (5-2)

The objective function and constraints can be transformed into unconstrained type
incorporating the exterior penalty function and the mathematical expression can be
expressed by Sun[33]:

"c
min P(x,q,) = min f(x)+r7cZi|min(gi (x),0)’ (5-3)

1:
Where '7‘ = max (1 / (1, 17,4 ) , (1 denotes the mean values of distance from polyhedron
centroid to each vertex, while r denotes the penalty factor and k denotes the times of
the iteration.
Generally, the object functions can be grouped into two categories: the single and
multi object fimctions[34]. The multi objective functions can be combined into single
objective fimctions by using weight factors:

11 V
F=anfn (5-4)

1' =1

71
where Z wn =1

i =1
Jang[34] used the adiabatic efficiency as objective function. The efficiency is widely
used in objective function, because a small improvement in efficiency can result in
significant saving in annual cost[29]. Oyama[32] used the entropy production. as the
objective function to be minimized. Brian[35] used maximum principal stress at each
node as the objective function within the blade for thermoelastic optimization.

Wang[2l] used the total pressure losses as the objective function.

103

For a multi-objective function, Oyama[29] considered one, which involved
maximization of efficiency, mass flow rate, total pressure ratio and durability as well
as minimization of weight.

Loading and velocity distribution are considered to have influences on the efficiency
and flow range. Therefore ten criteria are considered to use in the objective function
in this study.

1’"t Criterion: The deceleration ratio should be larger than 0.65. And deceleration
ratio on pressure side should be larger than 0.5. The definition of deceleration ratio is
that of outlet velocity at the impeller to maximum velocity.

Table 5-3 Comparison of deceleration ratios of W distribution among five cases

 

 

 

 

 

Deceleration Ratio Base Casel Case2 CaseZb Case2c
Pressure side on Hub 0.467 0.444 0.459 0.324 0.469
Suction side on Hub 0.700 0.692 0.694 0.646 0.742
Pressure Side on 0 475 o 589 0 514 0 667 o 655
Shroud ' ' ' ' '
Pressure “‘16 on 0.706 0.705 0.753 0.706 0.712
Shroud

 

Based on the evaluation, we already know that the Base case is the best case while the
case 2b is the worst. The deceleration ratio on pressure side on hub shows that the
Base and case 2c has the minimum penalty while the case 2b the maximum. It
matched our evaluation results. There are no penalty on suction side both on hub and
shroud because all the deceleration ratios are larger than 0.65. However, results in
Figure 4-37 show that maximum relative velocity at inducer of Base Case is much
larger than others, which results the higher deceleration ratio and bring the penalty to
Base case. This penalty is not reasonable for Base case because there is no such high

inducer relative velocity for Base case.

2"d Criterion: The maximum loading should be reached shortly after the inlet. Based
on experiences, if the maximum loading is within the 6% normalized meridional
distance, then maximum loading has been reached shortly after the inlet. And all the
results meet this criterion. However, it is sensitive to give a criterion range. For
instance, the maximum loading of Base case on shroud occurs at 5.5% normalized
meridional distance, those of Casel on hub and of Case 2 on shroud at 5.4% and 5.6%
respectively. If a case with maximum loading at 6%-7% normalized meridional
distance, it is still quite possible this case is a good design. Therefore, this criterion is
not effective to make the judgment, especially for computer-aided optimization.

3rd Criterion: Rapid Deceleration is preferred in the inducer region. The slope of the
velocity distribution in the inducer region is calculated. Because the velocity
decelerates, the slope should be less than zero. The higher the absolute value of the
slope is, the more quickly the deceleration is. Firstly, it is very difficult to give an
exact numerical definition of inducer region. Secondly, the slope of the velocity
distribution greatly depends on the given inducer region. The slopes of five velocity
distribution are compared in Figure 5-6. If the uniform weight factors are given, then

Case 2 is considered as the best one based on this criterion.

Baseline Case1 CaseZ Caanb CaeeZc

 

    
 

 

 

 

o _

.1000 -
g :35 me
8. usssna
% ' ups SHR

 

 

 

 

 

1:33;:

 

Figure 5-6 Comparison of slope of W in inducer region for 3rd Criterion

105

4th Criterion: High loading of the inducer compared to the outlet of the impeller. All
five cases meet this criterion.

5th Criterion: All unnecessary acceleration or deceleration must be avoided in the
overall process. The derivation of velocity is calculated. Times of sign changing are
accounted. If sign is positive, it means that the velocity accelerate, while the sign
negative, the velocity decelerate. The higher times of sign changes indicate more
unnecessary acceleration and deceleration. The times of sign changes for each case

are shown in Figure 5-7.

 

 

 

 

 

 

 

 

 

 

8
8 7
go
5 5_ .ssnua
§4_ IPSHUB
: ISSSHR
o 3-
. nPssI-IR
O
E 2-
1:

1 _

04

 

 

 

 

Baseline Case1 Case2!) CaseZc

Figure 5-7 Comparison of times of sign changes based on 5“I Criterion
6th Criterion: Relative velocity must be kept positive everywhere to avoid reverse flow.
No negative relative velocity occurs in these five cases.
7th Criterion: Blade loading should be distributed as unifomily as possible. The

' numerical integration of loading difference along meridional distance is calculated

based on Eqn. ZabS(L0ading(l )— L0ading(i -— 1))Am. While Loading(i)is
i=1

the loading at point i , and Loading(i—l) is the loading at point i—l ,

Loading(i)— Loading(i — l) is loading difference, Am = m(i)— m(i — 1). If loading

l06

is perfectly uniformly distributed, the loading difl'erence at any point is zero.
Therefore the integral is zero. The results (Figure 5-8) indicate that Case 1 is the worst

case based on this criterion while case 2b is the best.

9
8

 

P

 

9999.0
883

PP
88

 

 

Integral of Loading Difference

 

p
8

Baseline Caee1 Case2 Case2!) Case2:

Figure 5-8 Comparison of integral of loading differences for 7‘“ Criterion
8m Criterion: Large deceleration in regions, which will cause thick boundary layers as
well as separation, must be avoided. It seems that this criterion is contradictory with
Criterion 3. However, they are different. For the Criterion 3, the rapid deceleration is
preferred only in the inducer. However, the too much deceleration is not preferred
because of the separation. Therefore, the highest minimum velocity is preferred.
Therefore, the minimum velocity at the end of deceleration process is used as standard.
However, no significant difi'erences can be observed among these five cases (Figure

5-9).

 

 

 

 

 

 

700
600-
? 500-
E—L ussnua
%' 400- IPSHUB
i 300- assault
5 DPSSHR
2 m0-
100-
o-

 

 

 

Baseline Caee1 Case2 CaeeZD CaseZc
Figure 5-9 Comparison of minimum velocities for 8"I Criterion
9m Criterion: The difference between Mach number on suction surface of hub and

shroud should be minimized to avoid secondary flow.

 

200

'3‘

 

U!
9
1

 

Integral of W Difference
between $8 HUB and SS SHR
A
8

 

 

 

 

O

Baseline Case1 CaseZ CaseZb Case2c

Figure 5-10 Comparison of integrals on relative velocity differences between
suction surface on hub and shroud

10th Criterion: Blade loading should have a limit, which is 0.81 based on mechanical
stress limit nowadays. However, numerical results at the inducer is very sensitive to
the interface between impeller and inlet casing and is not sufficient accurate the make
such a judgment.

Cassey[18] also suggests using suction surface peak Mach number, suction surface

108

. t . .
average Mach number and etc. However, based on the analysrs from 18 Criterion to

th . . . . . .
10 criterion, we found that some of crrterra are not efl‘ectrve to make the judgments

such as 2“, 4th, 6th, 10th. There are some wrong judgment are made because of

numerical errors. Moreover, the optimum greatly depends on the weight factors and
different values of these will result in different optimums. Therefore, Root mean
square error (RMSE) between the predicted relative velocity points of each case and
the desired ones is calculated and also defined as objective firnction in this study. The
difference between calculated and target relative velocity distribution is used for
objective function for the following reasons:

1) If the designed impeller can reach the relative velocity which results in the least
separation and friction loss, the minimum loss or maximum efficiency can be
obtained.[36]

2) The loading can be determined directly from relative velocity distribution.
Unfortunately, an exact optimal relative velocity distribution, which leads to the
optimal performance, cannot be defined accurately. However, designing expertise as
well as general design rules can help designers to identify‘the optimal relative velocity
distribution introduced in [3 6]. Once the optimal relative velocity distribution called
as aerodynamic design criteria is defined, the whole procedure will be degraded into a
standard optimization problem.

5.4 Optimization Algorithm

5.4.1 Genetic Algorithm

Genetic algorithm (GA) is categorized as global search heuristics. 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

109

of the fittest”. GA used reproduction, mutation, and genetic recombination to "evolve"
a solution to a problem. The terminology of GA applied in optimization methods for
centrifirgal compressors is presented in Table 5-4. h

The main advantages of GA are as Robustness, Intrinsic parallelism, Globality.[28]
However, the application of GA is very time consuming.

The goal of GA in this study is to find the optimal set of optimization parameters,
which is corresponding to the maximum fitness. There are twenty genes of each
individual, which are corresponding to twenty optimization parameters. The
optimization parameters have been normalized into range (0,1) based on their given
range. The correlation between GA terminology and the parameters used in

centrifirgal compressor impellers optimization are listed in Table 5-4.

Table 5-4 Terminology of GA applied on centrifugal compressors

 

 

 

 

 

Gene One design Variable
A Chromosome _ Impeller geometry parameters
An individual A geometry case
Population A group of cases
A Fitness The objective fimction value or the evaluation of

centrifugal compressor impeller performance

 

5.4.2 Genetic Algorithm Procedure
The procedure of GA used here is as shown in Figure 5-11. First of all, a new

population has been initialized randomly. The fitness of all individuals was calculated

by flow solver and the individual with best fitness was saved.

110

 

 

Initialization

 

 

 

 

 

Fitness Calculation

 

 

 

 

 

 

Best individual Saving

 

 

 

 

Selection

 

 

 

 

 

 

Reproduction (Crossover and Mutation)

 

 

 

 

Local Search Optimization

c 3

Figure 5-11 Procedure of Genetic Algorithm
In the selection step, a group of current population called as parents was selected to

 

 

 

   

 

 

breed next generation. In this study, tournament selection[3 7] and Stochastic
Universal Sampling (SUS)[3 8] are applied respectively and results were compared.
Tournament selection is a selection operator in which a few individuals are chosen
randomly from current generation and the winner is selected based on their fitness. In
SUS, the probability of being selected from current generation is based on order of
fitness. And the results showed that the found optimal solution based on these two
selection operators were very similar while the converging was faster based on
Tournament Selection and Stochastic Universal Sampling showed better performance
on keeping genetic diversity.

In the reproduction step, the next generation of population was generated through
genetic operator: crossover (also called as recombination) and mutation.

There are three different crossover operators are applied in sequence: Interpolation

lll

Crossover, Extrapolation Crossover and Two Points Crossover.[39]

The Interpolation Crossover operator is expressed in the equation 5.
childl = md x parentl +(1— md ) x parentz (5-5)
childz = (l — md) x parent] + rnd x parentz (5-6)
The extrapolation crossover operator is expressed in the equation 6.

childl = (1+ rnd) x parentl — rnd x parentz (5-7)
childz = —rnd x parent] + (1 + rnd ) x parentz (5-8)

where md is random value between 0 and 1.

Parent] and Parent; are two parent individuals while child] and chile are children
individuals. Values of children individuals should in ranges of those of their parent
individuals based on Interpolation Crossover while outside of range based on
Extrapolation Crossover. Chromosomes of children individuals don’t change but
switch between their parents based on two-points crossover operator. Two-Points
crossover operator exchanges genes between two randomly generated points from
parent individuals.

In mutation operator, analogous to biological mutation, chromosomes changes from
their original states. One of the most important function of mutation is to remain
genetic diversity and avoid the results converge to local optimum. Four different
mutation operators used in this study was given in sequence: Boundary Mutation,
Multi-NonUniform Mutation, NonUniform Mutation, Uniform Mutation.[3 9] In the
Boundary Mutation, one of genes is chosen randomly and changed to its upper or
bottom boundary. In the Multi-NonUniform Mutation, all genes move toward

boundary with a dreasing damping factor based on the equation 7.

112

ite order
child = parent + (l — parent) - [I —.—Cur] , rnd 6 [0,05]

lt
emax (5-9)

0

zte order
child = parent— parent- 1-_—£‘L ,rnd E (0.5,1]
Itemax

 

\

In the NonUniform Mutation, only one chromosome is randomly chosen for mutation
based on equation 7. In the Uniform Mutation, one gene is chosen randomly and
substituted by a random value. For the details on selection, crossover and mutation
operators, please refer to [39].

5.4.3 Local Search Algorithm

Local search algorithm is combined with GA in this study. Because derivative of
objective function is not available and the optimization problem is high dimensional,
the procedure of local search algorithm is:

1) One or several genes of best individual are chosen randomly.

2) A random value closed to the original value of gene is given and new individual is
generated.

3) If the fitness of new individual is better than the original one, then original one is
substituted by the new one.

Because of the high computational cost for local research algorithm as well as the
importance of best individual, this algorithm was only applied to update the best
individual.

There are two termination conditions, which are the global optimum has been found
or the maximum iteration number reached.

5.4.4 Test on GA and Local Research Algorithm

Before applying this combined optimization method for centrifugal compressor
impellers optimization problem, three test equations are used to test this optimization
method as following. In the testing, all testing dimension is 20, which is the same as

113

that of optimizing centrifugal impellers. However, all equations, which are plotted

here, are shown in two-dimensions.

5.4.4.1 De Jong test function

The Eqn of De J ong test fimction is expressed by:

7'
f(x) = x Z xiz -5.12< x, <5.12 (5-10)
i=1

 

 

Figure 5-12 Illustration of De Jong test function in two-dimensions
We can see that finding the minimum of De Jong function is a standard convex

problem (Figure 5-12). Because there are some random factors in the optimization
method, therefore the test rims five times and their corresponding convergence

histories are shown in Figure 5-13.

114

A
Q.

 

—l
I I I lliLlF;

 

 

a:

l IIIIIIII

 

3..

r I rIIlIIl

Objective function f

9..

I I Illllll

 

 

l l l I l l l l I I l
200 400 600
Epoch
Figure 5-13 Convergence histories of optimization based on De Jong test function
in twenty dimensions

1

14 .14
104 800

5.4.4.2 Rosenbrock Test Function

n—l
f(x) = % Z (xi+1 — x,- )2 + (1 — x,- )2 -2.048< x,- <2.048 (5-11)
.=1

 

 

 

-4 .4

Figure 5-14 Illustration of Rosenbrock test function in two-dimensions

115

 

101

5;

Objective function f
5‘2.
1 1 111N111.

5?.

I I IIIIII'

I IIIIIII

 

1 st
————— 2nd

 

 

 

I I IIIIII'

 

 

103

r l 4 1 1 1 1 1 l 1 1 1 l 1 1 1
200 400 600 800 1000
Epoch

Figure 5-15 Convergence histories of optimization based on Rosenbrock Test

Function optimization in twenty dimensions

5.4.4.3 Rastrigin Test Function

n—l
f(X) =% 201-)2 “Oil—003271351) -5.12< xi<5.12
i=1

116

 

 

-10 ~10

Figure 5-16 Illustration of Rastrigin test function in two-dimensions

 

10' 1'_ _— 1st
' — ix.

 

 

 

Objective function f

 

 

101 r 1 r I 1 r 1 L I 1 1-1 1 r r _L J—L 4;]
Epoch

Figure 5-17 Convergence histories of optimization based on Rastrigin test
function optimization in twenty dimensions

We can see that the performances of optimization method for Rosenbrock and De

117

Jong test function are good. However, Rastrigin test function is a high nonlinear and
finding a global optimal based on this function is very difiicult (Figure 5-16). The
results of five times running indicated that the final error is around 0.1 and it is
obvious that no global optimal or even close solutions have been found. To the
author’s knowledge, there is no quite effective optimization method for such as highly
nonlinear problem.

5.5 Performance Mapping

An ANN is one type of nonlinear mathematical models, which is used to form a
complex relationship between input and output data. The training of the ANN is an
admtive process in which parameters such as weights and bias are changed as internal
or external information that flow through the ANN. Moraal et a1 [40] indicate that
ANN can produce better performance compared to other curve fitting techniques if
ANN is sufficiently trained. The advantages and drawbacks of ANN is listed in Table
5-5.Feed-Forward Neural Networks (FFNNs) [23, 30, 41]with back-propagation
learning algorithm and Radial Basis Function Network (RBFN) [42, 43] are most

widely used ANN to train performance map for turbomachinery.

Table 5-5 Cons and Pros of ANN

 

 

 

Pros Cons
Relative low number of calculation Searching and applying approximate
functions
Efficient if the parameter space if unimodal, Has no ability for extrapolating

convex and continuous

 

No need for the knowledge of physics

 

High mapping capacity

 

Suitable for frmctions with multiple inputs
and outputs.

 

FFNN, which is one type of ANN, also called as multi-layer perceptrons, consists of

one input layer, one or several hidden layers, and one output layer. Information flow

118

 

moves only in one direction from the input layer to the output layer without loops in it.
Each neuron in one layer has directed connections to all neurons in the subsequent
layer. Each neuron performs a weighted summation of the inputs fi'om the previous
layer, which then passes an activation function such as sigmoid function. RBFN,
another type of ANN, has a less flexible structure compared to FFNN.

RBFN typically has only three layers: an input layer, one hidden layer with nonlinear
radial basis function (RBF) used as activation function, and one linear output layer.
The most widely used RBF is a Gaussian RBF in the hidden layer while the output is
a weighted summation of neurons in the hidden layer. Both F FNN and RBFN are used
to create performance maps for centrifugal compressor impellers in this work. Results
show that accuracy of FFNN is similar to that of RBFN. However, FFNN provides
higher robustness while RBFN provides much lower computational time. The average
training time of F FNN varies from 1 to 2 hours depending on its detail structure while
that of RBFN takes only 3 to 15 minutes. In this study, a large number of cases
including geometry and performance parameters are provided to train neural network
because of applications of fast Quasi-3D flow solvers, which help make the trained
RBFN more robust and overcome drawbacks of RBFN to some extent. Therefore
RBFN is chosen as mapping tool because it has much lower computational time.
There are five cases are generated to train the FFNN or RBFN while other five
hundreds cases are generated to evaluate the performance.

The performances of two types of ANN s: F FNN and RBFN are compared. The results
show that RBFN can achieve the higher accuracy on the training database than FFNN.
However, FFNN shows the better performance on testing database. Therefore, it
seems compared with FFNN, RBFN is over-train to some extent. Considering the

computation time, that of RBFN is approximately one tenth of that of FF NN.

119

Table 5-6 Terminology of GA applied on centrifugal compressors

 

 

 

 

 

 

Performance mapping Method RBFN FFNN
Average error of training database (ft/s) 3.3007e-014 3.9261
Maximum error of training database (ft/s) 5.1159e-013 27.601
Average error of testing database (ft/s) 19.504 5.607
Maximum error of testing database (ft/s) 106.62 53.512
Computational time (s) 159.54 1446.6

 

5.6 RESULTS AND DISCUSSIONS

5.6.1 Accuracies of RBFN and FFNN

Five hundred cases in training database are generated randomly to create the ANN
while other five hundred cases in testing database to evaluate the created ANN. The

number of cases in training database are indicated by X coordinate values in Figure

5—18.

 

Conparison of RBFN and FFNN I RBFN I FFNN

 

 

35
30
25
20
15
10

Error of W (m/s)

e

 

AAEofWin MAEofWin AAEofWin MAEofWin

training training testing
database database database
(m/s) (In/s) (m/s)

and testing database

In Figure 5-18, Average Absolute Error (AAE) and Maximum Absolute Error (MAE)
between W points in the training database and e W points calculated using RBFN is

much smaller than those using F FNN . However, Average Absolute Error (AAE) and

120

testing
database

(In/S)
Figure 5-18 Comparisons of accuracies of RBFN and FFNN in training database

 

Maximum Absolute Error (MAE) between W points in the testing database and W
points calculated using FFNN is larger. Therefore, it seems that compared to FFNN,
RBFN is over-trained. However, the training of FFNN is 1446.6 seconds while
training of RBFN is 159.54 seconds, approximately one tenth of training time of
FFNN.

5.6.2 Performances of Optimization Procedures using RBFN and FFNN

The trained RBFN and FFNN are used in centrifugal compressor impeller
optimization procedures. Therefore, the total performances of optimization procedures
using RBFN and FFNN are compared. The setting parameters for GA in optimization
procedures are the same and the only difference is that two different types of trained
ANN s: RBF N and FFNN. In the application of the optimization procedure, desired
geometry should be unknown. The desired W points or desired W distribution are
given directly by designers. Eventually, optimal geometry can be found. Optimal W
points, which are calculated based on the optimal geometry, can then be compared
with desired ones. In this study, the desired geometry and its corresponding W
distribution are given. Therefore both optimal W points and optimal geometry
parameters can be compared with desired ones. This is more effective for comparison
between these two optimization procedures. Because of the influences of random
factors, each optimization procedure runs for five times under the same setting.

Table 5-7 Comparison of average computational time for centrifugal compressor
impeller optimization procedures using RBFN and FFN N

 

15t 2nd 3'“ 4th 5th

' - - . . average
trme trme trme trme trme

 

Computational time of
Optimization using RBFN (s) 499.6 498.6 473.2 457.5 471.9 480.2
Computational time of

OptimizationusingFFNN(s) 408.5 535.1 542.1 545.8 546.7 515.6

 

121

 

In

Table 5-7, results show each computational time and average computational time. It
can be seen that there is almost no differences between using RBFN and FFNN. This
is because FFNN or RBFN is applied in the training procedure, and there is no extra
computational time for using them to evaluate geometries in optimization procedures.
The computational time differences are because of the different structures between
RBFN and FFNN, which results in the different application time. The results show
that it takes more time to call performance map trained by FFNN than that by RBFN.
Centrifugal compressor impeller optimization procedures, which employ RBFN and
FFNN, are used to calculate optimal geometries based on a given desired W
distribution. Their corresponding optimal W distributions calculated from optimal

geometries are compared with the desired W distribution in Figure 5-19.

180 1-

160 -

 

120 -

 

o 20 4o 60 80 100
%M 1

Figure 5-19 Comparison of optimal W distributions calculated by employing
RBFN and FFNN

It is observed that the optimization procedure can find an optimal W distribution

122

"I

 

much closer to the desired one by using RBFN than FFNN. In order to
comprehensively evaluate RBFN, statistical results: AAE and MAE between optimal
W points and desired ones are shown in Figure 5-20, respectively. In GA, the initial
populations are required to give randomly, which leads that difl‘erent optimal
geometries are found even under the exactly same given desired W points,
optimization algorithm and RBFN. Therefore, optimization procedure has been run
five times under the same algorithm specification. All AAE and MAE in Figure 5-20

are illustrated in the form of meaniSD (Standard Derivative).

Comparison of performances of optimintion procedures

16

 

 

l4

 

 

12 IRBFN IFFNN

 

 

 

 

 

l0

 

 

 

 

Error of W (mls)

 

 

 

 

 

AAEofW MAEofW AAEofW MAEofW
evaluated by the evaluated by the evaluated by the evaluated by the
performance map performance Imp flow solver flow solver

Figure 5-20 Statistical results of Average Absolute Error (AAE) and Maximum
Absolute Error (MAE) between optimal W points & desired ones between using
RBFN and FFNN

All results show that AAE and MAE between optimal W points and desired ones
based on performance maps using FFNN has higher values compared to FFNN.
However, AAE and MAE of W evaluated by flow solver show that RBFN has lower
values. This demonstrates that the optimization procedure using RBFN is able to find

optimal W points much closer to the desired ones compared to FFNN although the

123

accuracy of RBFN is lower than that of FFNN (Figure 5-18). Therefore, the RBFN is

used for the following chapters because of its much lower computational time and

slight better performance than those of F FNN.

6.5

5.5

R(inch)
.e
01
IIII'IIII'IIII'IIFTIIIII'IIIIIIIIIIIIII'

 

2.5

 

-2.5

 

Figure 5-21 Comparison of optimal contours between using RBFN and FFNN

-30

 

 

I Desired
- — — — RBFN Hub
-35 - ....................... FFNN .1. __ _ ~
- '/ / I“ x.
A ‘40 f .1/
g - _./'
v b /
g - .
g -45 " ./ z
a : /
5 - *“/
a - -- /
: v/
_ '. /
-55 .. /
_ g 1 1 I 1 r 1 I r 1 1 I I 1 I 1 I
600 20 40 60 80 100

124

Figure 5-22 Comparison of optimal beta distributions between using RBFN and
In Figure 5-21 and Figure 5-22, opthflzoljrtom and blade angle distribution, also
called beta distribution, are compared with desired ones, respectively. The
optimization procedure, which employs RBFN, finds closer hub profiles, beta angle
distribution on shroud than that using F FNN. However, the optimization procedure
using F FNN finds the closer beta angle distribution on hub. There is no guarantee that
all geometry parameters can be found closer to the desired one because this
optimization problem is highly nonlinear problem. Figure 5—23 show statistical results:
AAE and MAE between optimal geometry parameters and desired ones. Units of
geometry parameters are diverse, e.g. inch for contour, degree for blade angle and

dimensionless for normalized meridional distance. Therefore, all geometry parameters

are normalized to dimensionless quantity in the range (0, 1).

Comparison of performances of optinization procedures

 

0.7

 

 

 

0.6 I RBFN FFNN

 

 

 

 

0.5

 

 

0.4

 

0.3

 

 

 

 

Error of normalized geometry parameter

 

 

   

AAE of geometry parameters MAE of geometry parameters

Figure 5-23 Average Absolute Error (AAE) between optimal geometry & desired
ones
Results in Figure 5-23 suggest that the optimization procedure employing RBFN is

able to find optimal geometry closer to the desired ones compared to FFNN in

125

 

average.

5.7 Summary

In this chapter, the centrifugal impeller optimization procedure using artificial neural

network and genetic algorithm is established, which concludes the following steps:

1) Impeller geometry contour and blade angle distribution are used for
parameterization, from which twenty geometry parameters are chosen. Two methods:
curve fitting and discretization are compared. The discretization is used to
parameterize the relative velocity distribution because the results show that

parameters from discretization are more stable as change of curve.

2) The limits of Quasi-3D flow solver MERIDL and TSONIC are discussed. The

reasons of the use of relative velocity distribution are also discussed.

3) The possible criteria in objective function are proposed and applied on existed
cases. The calculation results of objective functions are compared with judgments of
engineering designer. It is found that these criteria is not effective because either these
are too loose for design cases or too dificult to express in exact numerical equation.
Eventually, the diflerences between relative velocity distribution and desired the
calculated and tried to apply on existed cases. Finally, root mean square error (RMSE)
between the predicted relative velocity points of each case and the desired ones is

defined as objective function.

4) Feed-forward Neural Network (FFNN) and Radial Basis Function Network
(RBFN) are used to create performance maps respectively. These two performance
maps are further used in the optimization procedure in which the Genetic Algorithm
(GA) is used as optimization method. Created performance maps as well as the total

optimization procedures between using RBFN and FFNN are evaluated and

126

 

compared.

5) The results show that the RBFN has higher accuracy on training database while
lower accuracy on the testing database than FFNN, which indicates the RBFN is
over-trained. However, the optimal results show that optimization procedure using
RBFN is able to find the better optimal based on the evaluating results of flow

solvers.

6) Although the application of modeling tools greatly decreases the computational
time, it also brings errors to the optimization procedure due to the application of
approximate performance map. Because the dimension of created performance map
equals to the number of geometry parameter, which is a high number. To the authors’
knowledge, it is very difficult to create an exact performance map for such a high
nonlinear problem. Based on our calculating results, the errors of the approximate
map is much larger than those caused by an optimization method. The errors of
approximate performance maps diminish the effects of high fidelity flow solvers and
highlight that the increase on modeling is more important than searching a better and
more effective optimization method for this centrifugal compressor impeller
optimization problem. Therefore, an improved offline flow solver optimization
procedure and an online flow solver optimization procedure are presented in Chapter

6 and Chapter7 respectively.

127

 

CHAPTER 6

IMPROVED IMPELLERS OPTIMIZATION PROCEDURE

6.1 Introduction

In last chapter, Artificial Neural Networks (ANNs) is used to create an approximate
performance map. However, it is found that although the introduction of ANN greatly
decreases the computational time, it also brings errors to the optimization procedure
because of the application of approximate performance map. Wang et al. [21] consider
only blade angle distribution for parameterization, which helps reduce geometry
parameters, the dimension of input layer as well as the whole dimensions of this
optimization problem. This makes training of ANN and searching of global optimum
become much easier. Verstraete et a1. [25] propose to use the online-trained ANN,
which uses the new calculated results to update the existed ANN, and increases the
accuracy of performance gradually as the iteration proceeds. Ghorbanian .et a1. [42]
use rotated general regression neural network (RBRNN). Its basic theory is that
rotation can reduce the nonlinear characteristics of relationship between geometry
parameters and performance parameters in new rotated coordinate system. In our
research it is found that the chosen geometry parameters and performance parameters
are of importance because ANN s are trained to represent the relationship between
geometry parameters and performance parameters. The simpler the relationship
between geometry parameters and performance ones is, the more easily ANNs can be
trained and the higher accuracies of ANNs will be. In our study, RBFN, a type of
ANN, is applied to create a performance map for centrifugal compressor impellers.
Instead of applying training database to train RBFN directly, Principle Component
Analysis (PCA) or Independent Component Analysis (ICA) is applied to transform

128

training database into new transformed coordinate system, in which RBFN is trained.
Accuracies of three different trained ANN s: RBFN, RBFN with PCA and RBFN with
ICA are compared. Performances of centrifugal compressor impeller optimization
procedures employing three different trained Artificial Neural Networks (ANN s):
RBFN, RBFN with PCA, and RBFN with ICA are also compared.

6.2 Application of ICA and PCA

We use PCA and ICA to improve the accuracy of RBFN. Generally, PCA is a
mathematical technique to transform a high-dimensional space into a few orthogonal
axes called principle components, along which the variances of data are maximized.
In our study, PCA is used as an orthogonal transformation, which preserves the
dimension of data and only transforms data into a new coordinate system on which
maximum variances of data are projected. ICA is superficially related to principle
component analysis. ICA involves a computational procedure for separating
multivariate data, which are assumed to be linear mixtures of unknown latent
variables, into non-gaussian and mutually independent components called the

independent component of the observed data.

129

 

 

 

 

Centrifugalcompreesorimpeileroptimizationtool

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(quasi-3D flow solver)
//i\-
Testing database Training Database DesireTiTJatabase
(Geometry. W Distribution' ) (Geometry. W Distributin) (Geometry, W Distribution' )
1 l | 1 i i
Parametenzatro‘ ' n Discrehza‘ tion Parameterization Discretizatro' ' n Parametenzatron‘ ‘ D‘Iscretrzatron' ‘
Geometry Performance Geometry Perfomrance Geometry Performance
Parameters Parameters Parameters Parameters Parameters Parameters
11111.11
1 1‘
Transionned Transformed Transformed
Gm, Geometry Performance
pmm Parameters Parameters
i
/\ 0W MN. Optimal Geometry and
§ FUMO" Perionnance Parameters
PPfiiétE g Trained t (T ransiormed egrordinate
erl'ormance S RBFN '- 8113191“
Parameters § PM“
(Transformed . Transoirmed
coordinate stern g Perionnance [ Inverse PCA or ICA ]
1 Paraemters ] ]
cases Performance Parameters
Pr—erficted (Original coordinate
Perionnance 1 system)
Parameters
(Original coordinate Gm 111 11
system) Wm Evaluate differences between Optimal
Geometry. Perionnance Parameters and
Desired ones

 

Evaluating Accuracy of Trained Artificial
Neural Network

 

Figure 6-1 Centrifugal compressor impeller optimization procedure with PCA or
ICA

The flowchart of centrifugal compressor impeller optimization procedure using PCA
or ICA is shown in Figure 6-1. It consists of three steps: generation of three databases,
training and evaluation of RBFN, searching and evaluation of optimum based on
Genetic algorithm (GA) and desired W points. Firstly, geometry generation tool with
a quasi-three dimensional flow solver is used to generate three groups of database:
training, testing and desired database. Training and testing database are generated
randomly while desired database is given by designers. Each database can be divided

into two parts: geometry parameters and performance parameters. As for each case,

130

there are twenty geometry parameters and nine performance parameters. Secondly, if
PCA or ICA is not used, geometry parameters and performance parameters in training
database are used respectively as input layer X and output layer Y to train RBFN.
Otherwise, PCA or ICA is used to transform training database to those in a new
coordinate system which is used to train RBFN in this new coordinate system. Then
twenty geometry parameters in testing database are also transformed into those in a
new coordinate system by PCA or ICA, and then transformed geometry parameters in
the new coordinate system are input into trained RBFN, followed by the prediction of
corresponding performance parameters using RBFN. These predicted performance
parameters are represented in the new coordinate system because RBFN is trained in
new one. Therefore, by using inverse PCA or ICA, these predicted performance
parameters in the new coordinate system are transformed back to those in the original
coordinate system, which are used to compare with performance parameters in testing
database. The differences between predicted performance parameters in the original
coordinate system and performance parameters in the testing database are used to
evaluate the accuracy of trained RBFN. Because there is no transformation using
RBFN without PCA or ICA, therefore accuracies of RBFN between with and without
using PCA or ICA can only be compared in the original coordinate. This is the reason
why inverse PCA or ICA is used to transform predicted performance parameters back
to the original coordinates. The smaller errors between predicted performance
parameters and performance parameters in the testing database means higher accuracy
of trained ANN. Thirdly, nine W points calculated from desired geometry is used as
the desired W points or desired performance parameters. GA is used to generate the
first generation: a group of geometry cases and their corresponding performance

parameters are predicted using RBFN. Root mean square error (RMSE) between the

13]

 

predicted performance parameters of each case and the desired ones is calculated and
also defined as objective fimction in this study. GA is then used to generate the next
generation based on genes and finesses of first generation using selection, crossover
and mutation. This process continues until it meets these requirements: the exact
desired performance parameters are found or maximum generation number reaches.
Optimal geometry and performance parameters are transformed into the original
coordinates and compared with desired ones to evaluate the total performance of
optimization procedure.

6.3 Results and Discussion

6.3.1 Accuracy of RBFN

In this optimization procedure, cases in training and testing databases are generated
randomly. It seems that one group of results is not suficiently strong to evaluate the
effects of PCA or ICA on the accuracies of trained RBFN and the performance of
centrifugal compressor impeller optimization procedure due to influences of random
factors. Therefore, four independent groups with different numbers of cases in
training database are generated to better evaluate the application of PCA and ICA.
The number of cases in training database are indicated by X coordinate values in

Figure 6-2 and Figure 6-3.

I32

 

Fr: RBFN I RBFN+PCA

 

 

 

 

AAEofW(m/s)
O H N w 4:. Ln Ox

 

 

500 625 780 936
Number of Cases m Training (Testing) Database
Figure 6-2 Comparisons of accuracies of trained RBFN, RBFN with PCA, and

RBFN with ICA on Average Absolute Error (AAE) between predicted W points
& those in testing database

 

I D RBFN I RBFN+PCA I RBFN+ICA

 

 

U)
M

 

O

 

I-II-INNU)
UIOUr

MAE ofW (In/s)
O U! C

 

 

500 625 780 936
Number of Cases in Training (Testing) Database

Figure 6—3 Comparisons of accuracies of trained RBFN, RBFN with PCA, and
RBFN with ICA on Maximum Absolute Error (MAE) between predicted W
points & those in testing database

As mentioned above, each training database and testing database in these four groups
is used to train and evaluate three different ANNs: RBFN, RBFN with PCA and
RBFN with ICA. Accuracies of RBFN, RBFN with PCA, and RBFN with ICA are
compared and results are shown in Figure 6-2 and Figure 6-3. Results in Figure 6-2
and Figure 6-3 show Average Absolute Error (AAE) and Maximum Absolute Error
(MAE) between W points in the testing database and predicted W points based on
three different ANNs: RBFN, RBFN with PCA, and RBFN with ICA for four

different testing databases. The unit of AAE and MAE is m/s. In our study, numbers

133

 

of cases in the training database are equal to those of the testing database. The
comparison of three different ANNs: RBFN, RBFN with PCA, RBFN with ICA on
both AAE and MAE are shown in different colors. In Figure 6-2 and Figure 6-3,
results of these four groups uniformly show that AAE and MAE of RBFN with PCA
is approximately 50% of those of RBFN with ICA or RBFN. AAE and MAE of
RBFN with ICA are slightly lower than those of RBFN. As the increase of case
numbers in the training database, the AAE of RBFN or RBFN+ICA gradually
decrease while AAE of RBFN+PCA remain the same. This is probably because
RBFN or RBFN+ICA can create more accurate ANN as the increase of training cases
while RBFN+PCA has already achieve the highest accuracy, which is limited by its
characteristics. However, there is no general rules can be concluded fiom MAE. The
MAE with 936 cases in training database and testing database is larger than others.
This is because the possibility of generating a geometry case, on which the created
performance map does not has good prediction, is increased as the increase of testing
database. However, the increase of the number of cases results in unfavorable
exponential increase on the training time for ANN shown in Table 6-1. Therefore,
increase of the number of cases is not reasonable when it is sufficient to create a fairly
accurate RBFN. The comparison of computational time for training RBFN, RBFN
with PCA and RBFN with ICA are shown in in Table 6-1. Results of four groups of
database show that there is a slight increase, approximately 5-6%, on the training
RBFN caused by the introduction of PCA or ICA. In summary, trained RBFN with
PCA has a much higher accuracy than RBFN or RBFN with ICA because it shows

lower value of AAE and MAE with only a slight increase on the training time.

I34

Table 6-1 Comparison of computational time for training three different ANN s:
RBFN, RBFN with PCA and RBFN with ICA

Number of Cases in a training Database
' 500 625 780 936
Computational Time of RBFN (s) 110.73 199.46 480.97 875.38
C°mputan°nal T‘m" 0f RBFN “b 115.87 231.22 474.69 881.08
PCA(s)
Computational Time of RBFN with
ICA(s)

6.3.2 Performance of Optimization Procedure
It has been approved that RBFN with PCA has better performance than RBFN and

 

 

111.05 239.06 483.96 887.30

 

RBFN with ICA. However, this does not guarantee that RBFN with PCA can bring
benefits to total optimization procedure. Because training of ANN is only one part of
optimization procedure. It is possible that RBFN with the increased accuracy does not
play an important role in the total optimization procedure. Moreover, introduction of
PCA may lead to some drawbacks and diminish the effects of higher accuracy of
RBFN with PCA. Therefore, the total performances of optimization procedures using
three different ANNs should be compared. The setting parameters for GA in
optimization procedures are the same and the only difference is that three different
trained ANNs: RBFN, RBFN with PCA, and RBFN with ICA are applied in these
optimization procedures.

In the application of the optimization procedure, desired geometry should be unknown.
The desired W points or desired W distribution are given directly by designers.
Eventually, optimal geometry can be found. Optimal W points, which are calculated
based on the optimal geometry, can then be compared with desired ones. In this study,
the desired geometry and its corresponding W distribution are given. Therefore both
optimal W points and optimal geometry parameters can be compared with desired
ones. This is more effective for comparison among three ANNs as well as the

evaluation of application of PCA or ICA.

135

 

Table 6-2 Comparison of average computational time for centrifugal compressor
impeller optimization procedures using three different ANNs: RBFN, RBFN with
PCA and RBFN with [CA

Number of Cases in a training
Database
500 625 780 936

56.95 67.11 75.45 80.69

 

 

Computational time of optimization using

RBFN (s)
Computational time of optimization using
RBFN with PC A(S) 56.32 70.33 75.34 78.86
Computational time of optimization using
RBFN with ICA(s) 56.72 71.86 74.81 79.54

 

In Table 6-2, results show average computational time of three optimization
procedures. It can be seen that there is almost no difference among using RBFN,
RBFN with PCA, or RBFN with ICA in the optimization procedure. This is because
PCA or ICA is applied in the training procedure, and there is no extra computational
time for using them to evaluate geometries in optimization procedures. Also,
structures of RBFN, RBFN with PCA and RBFN with ICA trained by same databases
are identical although they are trained in different coordinate systems. However,
results in Table 6-2 reveal that it takes more time to use RBFN trained with higher
number of cases because there are more neurons in the hidden layer and therefore
prediction by using such a RBFN requires more calculations. Centrifugal compressor
impeller optimization procedures, which employ RBFN, RBFN with PCA and RBF N
with ICA, are used to calculate optimal geometries based on a given desired W
distribution. Their corresponding optimal W distributions calculated from optimal

geometries are compared with the desired W distribution in Figure 6-4.

136

 

180

 

 

 

 

170?-
160; ————— RBFN
; . -- -.- RBFNHI'IICA
_ .\‘ — — — Rmmpca
— \ nested
70.150— .
E -
3140:—
130:—
120;
' 1 1 l L 4 4
1100

l l l l l I l l l 1 l I
20 4O 60 80 100

Figure 6-4 Comparison of optimal W distributions calculated by employing
RBFN, RBFN with PCA, RBFN with ICA

It is observed that the optimization procedure can find an optimal W distribution
much closer to the desired one by using RBFN with PCA. In order to
comprehensively evaluate RBFN with PCA, the optimization procedure uses RBFNs
trained with four different numbers of cases indicated by X coordinate and statistical
results: AAE and MAE between optimal W points and desired ones are shown in
Figure 6-5 and Figure 6-6, respectively. In GA, the initial populations are required to
give randomly, which leads that different optimal geometries are found even under the
exactly same given desired W points, optimization algorithm and RBFN. Therefore,
optimization procedure has been run five times under the same algorithm
specification. All AAE and MAE in Figure 6-5 and Figure 6-6 are illustrated in the

form of meaniSD (Standard Derivative).

137

 

D RBFN I RBFN+PCA I RBFN+ICA

 

 

 

 

 

1.5

 

 

 

AAE of W (tn/s)
'—-|

 

 

 

 

 

 

500 625 780 936
Number of Cases in Training/Testing Database

Figure 6-5 Comparison of Average Absolute Error (AAE) between optimal W
points & desired ones among three different ANNs: RBFN, RBFN with PCA,
RBFN with [CA

 

D RBFN I RBFN+PCA I RBFN+ICA

 

 

 

 

 

 

 

MAE ofW (m/s)

 

 

 

 

 

 

 

 

 

500 625 780 936
Number of Cases in Training/Testing Database

Figure 6-6 Comparison of Maximum Absolute Error (MAE) between optimal W
points & desired ones among centrifugal compressor impeller optimization
procedures using three different ANNs: RBFN, RBFN with PCA, RBFN with
ICA
All results of four different groups of databases show that AAE and MAE between
optimal W points and desired ones using RBFN with PCA has lower values compared
to RBFN or RBFN with ICA. This demonstrate that the optimization procedure using

RBFN with PCA is able to find optimal W points much closer to the desired ones

138

 

 

compared to other two different ANN s. However, optimization procedure which

employs RBFN with ICA does not show better performance although RBFN with ICA
has a slightly higher accuracy.

0.16 -
0.14 -

0.12 -

R(m)

0.1 -

0.08 -

 

 

1erlrrirlrriJlrrrmlrrrmlrrrrlr
0050.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

2 (m)

Figure 6—7 Comparison of optimal profiles calculated by employing RBFN,
RBFN with PCA, RBFN with ICA

 

 

 

r
-351— //.:':" .... 3' .m-
- was,“
”N.
L Hub “#:5.
-40 h— .4“ . "(r .4. . .-"' r
7 / K.
g / / Shroud
e /‘
3-45- / — ''''' RBF"
.2 / ------------- RBFNthICA
g / — — - RBFNthPCA
< ,4 Dashed
o /,I
6-501-
S 1 ./'./
1n _ //'
-55—
_htrrrmr.1.r.rrtirrtrr
600 20 40 60 80 100
%M

Figure 6-8 Comparison of optimal beta distributions calculated by employing
RBFN, RBFN with PCA, RBFN with IC

139

9. swam—r a

 

F

In Figure 6-7 and Figure 6-8, optimal contour and blade angle distribution, also called
beta distribution, are compared with desired ones, respectively. The optimization
procedure, which employs RBFN with PCA, finds closer profiles on the second half
part of impeller and beta distribution on hub. However, there is no guarantee that all
geometry parameters can be found closer to the desired one because this optimization
problem is 20 dimensions, which is also the number of geometry parameters. It is
possible that the application of PCA can improve performances on some of
dimensions while it may decrease those on other dimensions. This problem is also
highly nonlinear and has many local optimums, which makes a group of geometry
parameters provide a high value of objective function although these geometry
parameters are not close to the desired ones. Figure 6-9 and Figure 6-10 show
statistical results: AAE and MAE between optimal geometry parameters and desired
ones. Units of geometry parameters are diverse, e. g. inch for contour, degree for blade
angle and dimenSionless for normalized meridional distance. Therefore, all geometry

parameters are normalized to dimensionless quantity in the range (-1, 1).

 

 

 

C] RBFN I RBFN+PCA I RBFN+ICA

 

 

.o
co

 

9
as

AAE of Geometry Parameters
p o
N h
I I

 

 

 

 

 

 

 

     

O

500 625 780 ' 936
Number of Cases in Training/Testing Database

Figure 6-9 Average Absolute Error (AAE) between optimal geometry & desired
ones

140

 

 

U RBFN I RBFN+PCA I RBFN+ICA

 

 

 

 

 

 

 

 

MAE of Geometry Parameters

 

 

 

 

 

 

 

 

 

 

500 625 780 936
Number of Cases in Training/'1‘ esting Database

Figure 6-10 Maximum Absolute Error (MAE) between optimal geometry &
desired one

The form of Figure 6-9 and Figure 6-10 is exactly the same as that of Figure 6-5 and
Figure 6-6 because W points in Figure 6-4 are correlated to the geometry parameters
in Figure 6-7 and Figure 6-8. Results of four different groups of databases in Figure
6-7 and Figure 6-8 suggest that the optimization procedure employing RBFN with
PCA is able to find optimal geometry closer to the desired ones compared to other two
ANNs in average; AAE and MAE between RBFN with ICA and RBFN can hardly be
discriminated.

6.3.3 Sensitivity Analysis of GA Parameters

In addition to trained ANN s and initial populations, there are some other parameters,
e.g. GA parameters which also influence optimums. Population size is set as 100,
°maximum generation number 100, crossover rate is 6 and mutation rate is 18 for all
the above calculations shown above. Crossover rate represents the number of pairs of
chromosomes for crossover while mutation rate means the number of chromosomes
for mutation in each generation. In this section, one of these four GA parameters

varies while three others remain the same to study the influences of the variation of

141

 

these parameters. The influences of population size, maximum generation number,
crossover rate and mutation rate on whole optimization procedure are shown in Figure

6-11, Figure 6-12, Figure 6-13 and Figure 6-14 respectively.

I AAE of geometry parameters EJMAE of geometry parameters

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I AAE of W (m/s) DMAE of W (m/s)
1 l ‘
(18 l in 1
*5 O 6 I 5E . 3 '
s ' 2
0.4
0.2
0
Population Size
Figure 6-11 Influence of population size
I AAE of geometry parameters In MAE of geometry parameters
I AAE ofW (m/s) DMAE ofW (m/s)
1.2
1 l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A60 100 200 300 400 800
Maximum generation number

Figure 6-12 Influence of maximum generation number

142

I AAE of geometry parameters EIMAE of geometry parameters
I AAEofW(m/s) UMAEofW (m/s)

 

1.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
  

 

 

 

6
Crossover rate

Figure 6-13 Influence of crossover rate

I AAE of geometry parameters El MAE of geometry parameters
I AAE ofW (m/s) [:1 MAE ofW (m/s)

L4
L2 4
1
08
06
04
02

O

 

 

 

 

 

 

 

Enor

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

24
meflmume

Figure 6-14 Influence of mutation rate
For each parameter, changes of AAE and MAE between desired W points and optimal

W points as well as AAE and MAE between desired geometry parameters and optimal
geometry parameters are compared. AAE between optimal W points and desired W
points is considered as the most important factor because its value is directly related
to that of objective function here. In Figure 6-11, Figure 6-12, Figure 6-13, the results
show the possibility of finding a better optimum, which is represented by a lower
value of AAE between optimal W points and desired W points, is increased as the
value of population size, maximum generation number or crossover rate increase.

I43

 

However, the results in Figure 6-14 show that the mutation rate at 24 provides better
performance on finding optimum for this optimization problem compared with higher
values.

6.4 Conclusions

ANN, a nonlinear statistical data modeling tools, is widely used to create a
performance map to substitute the direct application of flow solvers during
optimization procedure, especially the application of GA. Because the ANN is used to
create an approximate map, the accuracy of the trained ANN is of critical importance
and greatly influences its applications and final optimal results.

PCA and ICA are also applied to transformed training database and make RBFN
trained in a new coordinate system. The accuracies of these three trained ANN s:
RBFN, RBFN with PCA, RBFN with ICA have been compared. Then these different
ANNs are used in the optimization procedure. The influences of PCA or ICA on total
performances of optimization procedure are also studied .by comparing the
performances of optimization procedures employing diflerent ANNs. Also, the
influences of other GA parameters on the performances of centrifugal compressor
optimization procedure have also been studied.

These results suggest that PCA can significantly decrease evaluation error and
improve the accuracy of trained RBFN with slightly increased computational time.
Using RBFN with PCA in the optimization procedure can increase total performance
and help find better optimum without increasing computational time. ICA can slightly
improve the accuracy of the trained ANN while no subsequent benefits on the total
optimization procedure. Large number of population size, maximum generation
number and crossover rate as well as mutation rate at 24 result in a better optimum

based on statistical results.

CHAPTER 7

ONLINE IMPELLERS OPTIMIZATION PROCEDURE

7.1 Introduction

In last chapter, Principle Component Analysis (PCA) is applied to improve the
performance of existed centrifugal compressor impeller optimization procedure.
Many papers on this optimization problem have been published and different
optimization procedures have been proposed A flow solver is directly used as
performance evaluation tool in optimization method, which is called as an online flow
solver optimization here. If the flow solver is only used to create a performance map
in the optimization procedure, then this optimization procedure is called as offline
flow solver optimization procedure in this study. This is because the flow solver in
this optimization procedure is not directly used for evaluating geometry in the
optimization method. In order to evaluate the performances of online flow solver
optimization procedure, firstly, the optimal results calculated by optimization
procedure using online flow solver and Genetic algorithm (GA) are compared to those
calculated by offline flow solver optimization procedure with the same GA parameter
setting. These comparisons both on performances and geometry are used to estimate
the influences of difl'erent applications of the flow solver. Furthermore, the
influences of GA parameters and local researchalgorithm combined with GA on the
performance of online flow solver optimization procedure are also investigated.

7.2 Optimization Procedure

Comparison of flowcharts between online flow solver optimization procedure and

offline flow solver optimization procedure is as shown in Figure 7-1 and Figure 7-2.

145

 

 

 

Desired Database
(Geometry, W Distribution)
1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
Discretization Parameterization
v v
Performance Geometry
initialization Parameters Parameters
of GA
Generation ~+ Quasi-3D flow solver Fitness _> Objective
(Geometry cases) Function

 

 

 

 

 

 

 

 

 

   

 

 

 

Crossover 8r

Mutation 89W"

 

 

 

 

 

 

Y

v
Optimal Geometry and
Performance Parameters

 

 

 

 

 

 

 

 

 

Evaluate differences between
Optimal Geometry.
Performance Parameters and
Desired ones

 

Figure 7-1 Online flow solver optimization procedure
In online flow solver optimization procedure (Figure 7-1), GA parameters are firstly
set in the initialization. A group of geometry cases are generated randomly and
represented by the first generation of individuals in GA. Each individual consists of
twenty chromosomes, which indicate twenty geometry parameters of each geometry
case. Mathematically, each individual is a vector and each chromosome is a float
number. This generation of individuals, or this group of geometry cases is input into
quasi-three dimensional (3D) flow solver MERIDL and TSONIC, and their
corresponding performances, especially relative velocity distributions are calculated.
Nine W points are discretized from each relative velocity distribution. RMSE between
these nine W points and desired ones, which is the objective function and also called
the fitness in GA, is calculated. If the global optimum is found or maximum
generation reaches, which are termination criteria in this study, then this generational

process and the total procedure stops. Otherwise, selection operator is applied to the

146

 

first generation of individuals. Two different selection operators: tournament
selection[44] and Stochastic Universal Sampling (SUS)[45] are applied respectively
and comparison results are shown in next section. Tournament selection is a selection
operator in which a few individuals are chosen randomly fi'om current generation and
the winner with best fitness among these individuals is selected. In SUS, all the
individuals are arranged in a order of fitness fiom high to low, and then are mapped to
contiguous segments of a line, such that each individual's segment on the line
decrease exponentially and represents probability of being selected. Followed by
selection step, these selected individuals are used to generate new individuals for next
generation via genetic operators: crossover and mutation. In the crossover step, three
different crossover operators are applied in sequence: arithmetic crossover, heuristic
crossover and two points crossover operators[46, 47]. In the mutation step, four
different mutation operators: boundary mutation, multi-nonuniform mutation,
nonuniform Mutation, Uniform Mutation are used in sequence [46, 47] in this study.
After new generation of individuals are created, quasi-3D flow solvers are used again
to calculate their performances and their corresponding finesses. Genetic algorithm

continues until termination criteria mentioned above reach.

147

 

 

Centrifugal compressor impeller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

optimization tool
(quasi-SD flow solver) Desired Database
(Geometry. W Distribution)
Training Database Dischtizaticn Parameferization
(Geometry. W Distributin) v t
T 1 Performance Geometry
Parameterization Discretization
Initial' 1 ' Parameters Parameters
of GA Geometry Performance
Parameters Parameters ’
l l
Generation . . Objective
([3 | ) —> Trained RBFN —-> Fltness ~+ Function
Cmover & . . . .
Mm saw..."
Y
Y
Optimal Geometry and
Perionnance Parameters

 

 

 

 

 

 

Geometry. Performance Parameters and

Evaluate differences batman Optimal
Desired ones

 

Figure 7-2 Offline flow solver optimization procedure
The oflline flow solver optimization procedure is very similar to the online flow
solver optimization procedure with the exception of evaluation tool for impeller
performance, which is shown in Figure 7-2. A group of geometry cases are generated
and their corresponding performances, especially relative velocity distributions, are
calculated by the quasi-3D flow solver MERIDL and TSONIC. Geometry parameters
are parameterized from their contours and blade angle distributions. Performance
parameters W points are discretized from their relative velocity distributions. Both
geometry parameters and performance parameters are used as input and output layers
respectively to train Artificial Neural Network (ANN) in this procedure. Actually, the
most widely used Artificial Neural Networks (ANN s) include: Radial Basis Function
Network (RBFN) and F cad-forward Neural Network (F FNN ). Both FFNN and RBF N,
two types of ANN s are used to create performance maps for centrifugal compressor

impellers in this work. It is found that the accuracy of FFNN is similar to that of

148

 

RBFN. However, FFNN provides higher robustness while RBFN provides much
lower computational time. The average training time of FFNN varies fi'om 1 to 2
hours depending on its specific structure while that of RBFN takes only 3 to 15
minutes. In this study, a large number of cases are provided to train neural network
because of applications of fast quasi-3D flow solvers, which helps to make the trained
RBFN more robust and overcome drawbacks of RBFN to some extent. Therefore
RBFN is chosen as mapping tool in offline solver flow optimization procedure. The
performance map is trained by RBFN and used to calculate W points and evaluate
fitnesses of individuals in GA.

7.3 How Solver

Flow solvers are used to calculate impeller performance based on its geometry and
can be generally categorized into three groups: flow solvers using Navier—Stokes
equations, e.g. Fluent[48], EURANUS/TURBO[21], TRAF code[17] and
CFX-Tascflow[16], those using two-dimensional Euler equations without considering . I
viscosity[49], and those with streamline curvature throughflow method[l3, 50, 51],
e.g. MERIDL[14] and TSONIC[12]. Flow solvers using Navier—Strokes equations are
the most accurate but very time consuming while those using streamline curvature
methods are most efficient. As mentioned above, flow solvers are directly used in
online flow solver optimization procedure to evaluate impeller performances.
Obviously, three dimensional flow solvers using Naiver-Strokes equations are not
feasible due to its high computational time and high times of its application in
optimization procedure. Hence MERIDL and TSONIC using streamline curvature
method are used in optimization procedure here, which is the same as flow solvers
used in references [28] and [18]. MERIDL is used to calculate the flow conditions in

meridional plane while TSONIC in blade-to-blade plane. In the original codes of

149

MERIDL and TSONIC, arbitrary quasi-orthogonals are used for meshing. However, it
is found that there are some waves in the results, which are caused by numerical
errors due to the application of the arbitrary quasi-orthogonals. Therefore, uniform
quasi-orthogonals are used to substitute of arbitrary quasi-orthogonals for meshingn,
which has been mentioned in Chapter 5. New relative velocity distribution calculated
based on uniform-orthogonals is very close to the original one based on
arbitrary-orthogonals but most of waves of original results are eliminated.

7.4 Results and Discussion

7.4.1 Comparison of Online and Offline Flow Solver Optimization procedures
In the application of the optimization procedure, desired geometry is unknown. The
desired W points or desired W distribution are given directly by designers and optimal
geometry can be found eventually. Optimal W points, which are calculated based on

the optimal geometry, can be compared with desired ones.

Table 7-1 Running Conditions and Gas Properties

 

 

 

 

 

 

 

 

Mass flow rate 37 (kg/s)
Rotating speed 1685.15 (rad/s)
Inlet Total Temperature 287.78 (K)
Inlet Total Pressure 344737865 (Pa)
Blade Number 17
Specific Heat Capacity Cp 1969.53 (J/kg.K)
Specific Heat Capacity Cv 1515.01 (J/kg.K)
Viscosity 1.006x10'5(kg/m.s)

 

However, in order to better evaluate the performance of online flow solver
optimization procedure, both the desired geometry and its corresponding relative
velocity (W) distribution are given in this study. Hence both optimal W points and
optimal geometry parameters can be compared with desired ones. The running

150

 

conditions of centrifugal compressor and gas property are given in Table 7-1. Optimal
contour, blade angle distribution and relative velocity distribution calculated by online
flow optimization procedure are compared with optimal ones by offline flow solver

optimization procedures in Figure 7-3, Figure 7-4 and Figure 7-5.

 

 

 

r — Optimal cortou’ uslng onllne flow solver optbnlzatlon procedue
0.16 - — - - - Optimal contou' uslng ofllhe solver tlowoptl'nlzdlon procedue
. -------------------------- Deslred contort
0.14 - _
” f
A * Shroud /.-"
E 0.12 -
ll! -
0.1 -
0.08 -
l 1 J L I J 1 J 1 J J I J J I J l l l J l l l l I I 1 l l I J
0'0-(006 -0.05 -0.04 -0.03 -0.02 -0.01 0
2 (m)

Figure 7-3 Comparison of optima contour calculated using online flow solver and
offline flow solver optimization procedure

151

 

— -— - Optimal beta dstrlbutlon calculated using onlhe flowsolver optimization procedure
............ Optimal beta astrlbutlon calculated using oflllne flow solver optimization procedure
Desired beta distribution

 

-35

I
ab
0

Beta (degree)
a a.
l I I I I I I I I I l I l I I I I I l l I 1

 

Jig

 

Figure 74 Comparison of Blade angle distribution calculated using online flow
solver and offline flow solver optimization procedure

Typically 100 generations containing 50 individuals are used in GA. Average
computational time of online flow solver optimization procedure is 1400s while that
of flow solver flow varies from 800 to 2000s, which depends on the number of cases
in the training database and the specific structure of RBFN. Results in Figure 7-3
show that there are no significant differences between two optimal shroud profiles and
both to them match desire shroud profiles very well. Also, there are no significant
differences between these two optimal shroud profiles and desired shroud profile.
However, the optimal hub profile calculated by online flow solver optimization

procedure is closer to the desired one compared with that by offline flow solver

optimization procedure.

152

onlmalW dstrlbwon ushgorihefloweolveroflimzaion procedue
Optimal W datrlbrllon ushg ol'l‘lne low solver opttnuation procedue
Desired W Distributor:

1&3E

EH>O

17o _-

130;

120:-

 

110

 

Figure 7-5 Comparison of relative velocity distribution (W) calculated using
online flow solver and offline flow solver optimization procedure

Results in Figure 5-3 show the similar phenomena as those in Figure 7-4. Optimal
blade angle distributions on shroud calculated by these two optimization procedures
are very similar while optimal blade angle distribution on hub calculated using online
flow solver optimization procedure is closer to desired one. In Figure 7-5, differences
between desired relative velocity distribution and optimal one calculated by online
flow solver optimization procedure are much smaller. Due to the effects of random
factors in GA, e.g. randomly generated initial individuals, this centrifugal compressor
impeller optimization problem is calculated for five times using online and offline
flow solver optimization procedures respectively. Five converge histories of online
flow solver and offline flow solver optimization procedures are shown in Figure 7-6
and Figure 7-7. It can be seen the significant differences among five converge
histories due to the influences of random factors in optimization method (Genetic

Algorithm and local research algorithm). It should be reminded that the best fitness is

l53

 

calculated by flow solvers in online flow solver optimization procedure while that
calculated by approximate performance map in offline flow solver optimization
procedure. Statistical results of these five optimal solutions are represented in the
form of meaniSD (Standard Derivative) shown in Figure 7-8. These results are
reevaluated using MERIDL and TSONIC eventually. It can be seen that mean value
of root mean square error (RMSE) between optimal W points and desired ones using
online flow solver optimization procedure is only 50% of that using offline flow
solver optimization procedure. As for RMSE between optimal geometry parameters
and desired ones, there are no significant differences. RMSE between optimal W
points and desired ones is much more important than RMSE between optimal
geometry parameters and desired ones in evaluating performance of optimization
procedure because RMSE between optimal W points and desired ones is the definition
of fitness in GA and smaller RSME reveal that the better solution is found by
optimization procedure. However, optimal geometry parameters closer to desired ones
do not guarantee a better fitness because this optimization problem is high nonlinear

and there are many local optima.

154

 

 

 

Best fitness
to

i

l

t

3 —— 131th”

_\ — — — - 2nd tine

. ‘‘‘‘‘ $ — ------ 3rd time
‘ ........................ 4th um.

: 1‘ — -------- - srn time

: l

 

 

 

Figure 7-6 Converge history of online flow solver optimization procedure

0.3

0.25

P
to

Best fitness
9
a

0.1

0.05

 

 

 

 

.4
‘
_‘.
-
)-
l-
-
)-
l 1 l l A 1 l J l 41 4L 4 l L J 1 1 I
20 60 so 100
Generations

Figure 7-7 Converge history of offline flow solver optimization procedure

155

 

Ionl i no flow solver opti rrizati on procedure
Uoffl i ne flow solver opti nization procedure
1.8

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

RSME of W points RSME of geometry
parameters

Figure 7-8 Comparison of statistical results on optima calculated between using
online flow solver optimization procedure and using offline flow solver _
optimization procedure

Although the values in objective function (RSME of relative velocity points) is less
than 0.1 m/s (Figure 7-8), these values are calculated based on approximate
- performance map. RMSE of relative velocity points using flow solver is about 1.17
m/s (Figure 7-8). This is proved that errors of approximate performance map play an
important factor on the optimal results: these errors can diminish the effects of high
fidelity flow solvers. Resolution of optimization method is less than 0.1 m/s while the
average errors of approximate performance map, however, is larger than 1 m/s.
Therefore, the increase on modeling is much more important than searching a better
and more effective optimization method. This is the reason why. online flow solver
method is employed in this study. All results fi'om Figure 5-3 to Figure 7-8 suggest the
better performance of online flow solver optimization procedure compared to oflline

flow solver optimization procedure.

156

 

 

7.4.2 Influences of Optimization Method
Influences of GA parameters, GA operators and local research algorithm on the

optimization procedure performances are studied and results are shown in Figure 7-9

to Figure 7-15.

 

 

RSME of W points (m/s)

 

 

 

 

100 300 500
Population size

Figure 7-9 Influences of population size in GA
Centrifugal impeller online flow solver optimization procedure runs five times under
each set of parameters due to the influences of random factors. Mean and standard
derivatives (SD) of these five objective function values, which are RMSE between
optimal W points and desired ones, are calculated. No relationships between
population size and optimization performance can be concluded from results in Figure
7-9. Results in Figure 7-10 suggest 'that possibility of finding a better optimum
represented by a lower objective function value increase as maximum generation
increases. However, computational time also linearly increase with maximum

generation shown in Table 7-2.

157

 

 

0.5

 

RSME of W points (m/s)

 

 

     

 

300 500 . .
Maximum generation

Figure 7-10 Influences of maximum generation in GA

Table 7-2 Computational time with different GA parameters

 

 

 

 

 

 

Population size 3 6 9
Average computational time (s) 1342 1437 1533
Maximum generation 8 18 26
Average computational time (s) 1342 3910 6475
Crossover rate 3 6 9
Average computational time (s) 1114 1342 1574
Mutation rate 8 18 26
Average computational time (s) 957 1342 1714
Using Local search algorithm Yes No

Average computational time (s) 1431 1342
Selection operators Tournament SUS
Average computational time (s) 13 3 1 1342

 

 

Crossover rate (Figure 7-11) presents the number of pairs used for crossover and

158

 

generating new individuals in each generation. Statistical results reveal that crossover

rate at 6 leads to better found optimum compared to other values.

 

0.9
0.7
0.6
0.5
0.4
0.3
0.2
0. 1

 

 

 

RSME of W points (m/s)

 

 

 

 

 

 

Crossover rate

Figure 7-11 Influences of crossover rate in GA
In Figure 7-12, mutation rate indicated by X coordinate denotes the number of
individuals used for mutation in GA in each generation. Results reveal that mean
value of RMSE of W points decrease gradually as mutation rate increases. However,
computational time of whole optimization procedure almost become double when

mutation rate increases from 10 to 26 shown in Table 2.

159

 

 

 

0.6

 

 

0.4 - . T

 

 

 

0.3 M

 

0,2 —-~~----- -- Mm .. MM _

RSME of W points (m/s)

 

 

 

 

 

 

 

Mutation rate

Figure 7-12 Influences of mutation rate in GA
Optima calculated using only GA as optimization method and those using combined
GA and local search algorithm are compared in Figure 7-13. The mean of RMSE of W
points greatly decreases approximately 40% by using local search algorithm. The use
of local research algorithm averagely increases the computational time from 1342 to
1431 seconds, approximately 6.5%. Results in Figure 7-13 and comparison of

computation time in Table 2 suggest that with slight increase in computational time.

160

 

 

0.5

0.4 — ~

RSME of W points (m/s)

0.1 —_n

 

 

   

without local research with local research
algorithm algorithm

Figure 7-13 Comparison of optima with and without using local search algorithm
The performance of two selection operators: Tournament selection and stochastic
universal sampling (SUS) are compared and results (Figure 7-14) indicate SUS has

better performance than Tournament selection on this centrifugal impeller

optimization problem.

 

0.7

0.6 - — —- r 4774444 . , _. .._ -IJ

0.5 _ -W . . ,_-_-- _._____w_ WWW—"J, .,

0.4

0.3

0.2

RSME of W points (m/s)

0.1

 

 

   

Stochastic Uriversal tournament selection
Sampling

Figure 7-14 Comparison of two selection operators: Stochastic Universal
Sampling (SUS) and Tournament selection in GA

161

 

 

 

During the reproduction step, seven reproduction operators (three crossover operators
and four mutation operators) are used in sequence. In order to evaluate the
performance of each operator, effective times and effective ratio of each reproduction
operator are shown (Figure 7-15). The definition of effective ratio of each operator is
the ratio between its effective times and sum of all effective times of these seven
operators. Therefore, effective times and effective ratio are in accordance with each
other. Statistical results in Figure 7-15 _is based on the 55 times of impeller
optimizations. The average crossover rate is 2 while average mutation crossover rate
is 4. However, each crossover operator can generate two individuals while one for
mutation operator. Therefore, each operator generates the same amount of new
individuals in these 55 times of optimizations totally. Theoretically, the higher
effective times or ratio an operator obtains, the better performance the operator has.
ReSults (Figure 7-15) show that the mutation operator averagely works better for this
centrifugal compressor impeller optimization problem. Nonuniform mutation seems
the most effective mutation operator while arithmetic crossover has best performance

among these three crossover operators.

162

 

 

 

 

30%
25%
20%
15%
10%

5%

0%

 

 

 

 

Effective ratio
Effective times

 

 

 

 

 

Figure 7-15 Comparison of performances of reproduction operators

7.5 Conclusion

In this chapter, the online flow solver optimization procedure method are presented
and optimal parameters of optimization method are analyzed in order to diminish the
influences of the errors of created performance maps and improve the performance of
centrifugal compressor impeller optimization procedure. Quasi—three dimensional
flow solvers MERIDL and TSONIC, which use streamline curvature method and have
low computational time, are directly used to evaluate impeller performance in Genetic
Algorithm (GA). This is called as online flow solver optimization procedure here.
Optima calculated by online flow solver optimization procedure are compared with
those by offline flow solver optimization procedure under same GA parameters. In
offline flow solver optimization procedure, same flow solvers are only used to
calculate impeller performances in the training database, which are then used to create
a performance map. This performance map is used to evaluate impeller performances

in GA and substitute the direct application of the flow solvers. Statistical results of

163

 

 

RSME between optimal relative velocity (W) points and desired ones, which is the
definition of objective firnction, reveal that online flow solver Optimization procedure
can find better the impeller geometry with closer relative velocity distribution to
desired one comparing to offline flow solver optimization procedure.

Influences of optimization method on the performances of optimization procedure are
also evaluated. Results suggests that increases of mutation rate and maximum
generation in Genetic Algorithm can result in the higher possibilities of finding a
better solution besides the use of online flow solver. However, the increases of these
two parameters greatly increase the computational time. The alternative method is to
increase the mutation rate and decrease crossover rate simultaneously, especially the
increase of the mutation rate on uniform mutation and the decrease of the crossover
rates on two points crossover and heuristic crossover. Moreover, the use of local
search algorithm combined with GA as optimization method seems the most effective
. way to increase optimization procedure performance with only slight increase on

computational time.

CHAPTER 8

APPLICATION OF IMPELLER OPTIMIZATION PROCEDURES
8.1 Optimization Conditions
In this chapter, the different types of developed centrifugal compressor impeller
optimization procedures, which are introduced in Chapter 5, 6 and 7 respectively, are
applied on an industrial gas centrifugal compressor impeller made by Solar Turbine
Inc, Caterpillar Company. The running condition and gas properties for this

compressor have been introduced in the preliminary design parameters are listed in

 

 

 

 

 

 

Table 8-1.
Table 8-1 Parameters of preliminary design

Rrs (inch) 4 Rm (inch) 2.78
er (inch) -2.13 Zm (inch) -1.64
R25 (inch) 6.04 R23 (inch) 6.04
Z25 (inch) -0.57 Z2” (inch) 0
firs (deg) -57.62 ,6,” (deg) -57
,st (deg) —40 .3211 (deg) -40

 

Z 17

 

There are no vanes in the inlet casing and diflirser. The contour profiles of the inlet

casing and difl’user as shown in Figure 8-1.

165

 

 

 

 

 

 

 

.4 ‘ -3 _2 _1 0 l Z(inch)

Figure 8-1 Illustration of inlet casing and diffuser contours

The desired relative velocity distribution is given and shown in Figure 8-2.

18" 170.64
170

160
150
140
130
120
110

W(m/s)

 

%M

Figure 8-2 Illustration of desired relative velocity distribution

166

 

8.2 Optimization Using Impeller Optimization Procedures

Five different types of centrifugal compressor impeller optimization procedures are
applied here and represented by RBFN+GA, FFNN+GA, RBFN+PCA+GA,
RBFN+ICA+GA, and Online-lGA. The similarities of these five optimization
procedures are introduced as following:

1) The parameterizations of impeller geometries are same. Twelve parameters are
chosen from contour between leading and trailing edge while eight fiom blade angle
distribution for each case.

2) The discretizations of relative velocity distributions are the same. The definitions
of objective functions are same and based on the RMSE between the calculated
relative velocity points and desired ones.

3) The flow solvers: Quasi-3D codes MERIDL and TSONIC, used in these five
optimization procedures are same and meshing methods in flow solver are under same
setting. The flow solvers are used to evaluate the compressor performance and also
create the naming and testing database.

4) The Genetic Algorithms used in these optimization procedures are same, which
means the selection, crossover, and mutation operators and parameters in GA are
same.

5) The training database and testing database for RBFN+GA, FFNN+GA,
RBFN+PCA+GA and RBFN+ICA+GA are same though the cases in these databases
are generated randomly. .
The diflerences between these five optimization procedures are mainly focus on the
application of performance map and the optimization methods:

1) RBFN+GA employ the Radial Basis Function Network (RBFN) to train the

performance map

167

 

2)

3)

4)

5)

FFNN+GA use the feed-forward Neural Network (FFNN) to train the
performance map
RBFN+ICA+GA use Independent Component Analysis (ICA) to transfer the

training database into the new coordinate system, in which the coordinates are
independent fi'om each other, and make the training finished in this new
coordinate system.

RBFN+PCA+GA use Principle Component Analysis (PCA) to transfer the
training database into the new coordinate system, in which the maximum
variances of data are projected in the coordinates and make the training done in
this new coordinate system.

Online+GA directly uses the flow solvers instead of performance maps to evaluate

the compressor performance.

The optimal impeller geometries (contours and blade angle distributions) found by

these five types of optimization procedures are shown and compared with desired

ones in Figure 8-4 and Figure 8-5. The corresponding optimal relative velocity

distributions are shown and compared with desired one in Figure 8-3.

168

 

 

W (mls)

 

 

 

r
160 - \ — — —— - RBFN+GA
\ — ------ FFNN+GA
- - 'r.\ RBFN+ICA+GA
.\ - ---- RBFN+PCA+GA
_ \ —--—--—--- Online+GA
~._f‘\ __-._.__..____ Desked
'._\
14o - \
,\\.\\
1- ‘“ \\ /
\\\ .\\\ \ ”N5;
' ‘ \‘\ ~ l 2;?
. “was
120 - ~ R.“ ”My
. . . 1 i r r 1 1 J r 1 r t i 1 . . . r
0 20 4O 60 80 100

Figure 8-3 Comparison of optimal relative velocity distributions

 

 

6 E f I
— .— — - RBFN+GA / .
5.5 r. ~ ------ FFNN+GA f ;
; RBFN+ICA+GA j 1‘
~ RBFN+PCA+GA /$ I”
5 L — -------- - Online+GA /-~’
: Desired {5’ 4?
_ Afi/ sf,
.. 4.5 - z 4’"
= I a” 42"“
S '- 4+?" /V
'5 '- Mfyfl‘ .' 6"
m 4 I: ./'.»V
.. ///
.. .' /
- ./,.”
- 'I/
3.5 P 1;}
L .
- /2‘/’
h . /;m’;"f
3 - ”.32.!
2.5 _-
1‘ 1 I 1 t r 1 I 1 J; 1 I 1 1 1 1 I J 1 1 1 I
-2 -1.5 -‘| -0.5 0
Z (itch)

Figure 8-4 Comparison of optimal contours

169

 

   

 

 

-35 L , «r: s; ,,_. a ;\;‘
- 5: Hub _ . _ — — - :§§
~ .31" \aq;
'40 ' // _,/_-I‘--/‘ 1“” \
- / 2‘; i /
' 7% I; fl-r/ shroud
' “3")" I, 2’"
A ' ”I ‘g/
a '45 - W; '1 7’; ‘
g _ .,-
3 M ....-,;1/
' i/ //
8 - :7/ 7 /
(I [7:
13 ' (1’ //-’.’
-50 - /} - — — - RBFN+GA
- :I 21/7 — ------ FFNN+GA
- //> -~---- » RBFN+ICA+GA
‘ / -— -—~ ~— - RBFN+PCA+GA
' / — -------- - Onllne+GA
'55 -/ I — DOSII'Dd
'r
-60 '- L l l I l I l I I; J I l l l 1 I l l l I
0 20 40 60 80 100
%M

Figure 8-5 Comparison of optimal blade angle distributions
Results in Figure 8-3 show that optimal relative velocity distribution calculated by

Online+GA is the closest to desired one. Besides, W distribution calculated by
RBFN+GA and RBFN+PCA+GA are also closed to the desired ones. However, there
are slightly differences between RBFN+GA and desired one at 60% to 90%
normalized meridional distance and between RBFN+PCA+GA and desired one at
10% to 40%. There are significant differences between FFNN+GA, or
RBFN+ICA+GA and desired ones at 30% to 80% or 40% -70% normalized
meridional distance respectively.

As for the contour (Figure 8-4), optimal shroud are similar while RBFN+GA,
RBFN+PCA+GA and Online+GA are slightly better than FFNN+GA and
RBFN+ICA+GA; RBFN+PCA+GA calculates the closest optimal hub to the desired
one while optimal hub profiles calculated from RBFN+GA and FFNN+GA are

slightly closer to the desired one compared to RBFN+ICA+GA and Online+GA.

170

 

 

The comparisons of optimal blade angle distributions are shown in Figure 8-5. For the
first half of blade angle distribution on hub, RBFN+ICA+GA is the closest while
FFNN+GA is the furthest compared to the desired one. For the second half,
Online+GA provides closest while RBFN+ICA+GA becomes the firrthest.
RBFN+PCA+GA provides the closet optimal blade angle distribution on shroud while
FFNN+ICA and RBFN+ICA+GA has significant differences from desired one, which
may be the reasons for slightly worse performance of optimization procedure.

8.3 Evaluation Using AN SYS CFX

AN SYS CFX is used to evaluate the optimal geometry eventually. The mesh of flow
passage of one blade is shown in Figure 8-6. The mesh of flow passage is mainly
determined based on mesh on shroud and hub. Therefore, the mesh on shroud and hub
are shown in Figure 8-7 and Figure 8-8 respectively. The mesh statistics of shroud,

hub and passage are as shown in Table 8-2.

Table 8-2 Mesh statistics

 

Minimum face angle Maximum face angle Maximum aspect ratio

 

 

 

 

 

Mesh on shroud 37.0 142.175 48.5826
Mesh on hub 26.5801 160.337 58.3881
Number of Nodes Number of Elements Hexahedra
27594 22680 22680
Mesh of passage . Max Edge Length
Volume. Ra .
tlo

 

0.000184265 [mA3] 80.045

 

I71

 

Figure 8-8 Comparison of hub

172

The fluid is used CH4 Ideal Gas and the inlet condition is Total Pressure and equal to
500 psi while the outlet condition is the static pressure, and equals to 615, 610, 600,
580, 540, 500, 460 psi for different mass flow rate. The fluid timescale control is Auto
Timescale and convergence criteria is Max residual is less than 1e-4. The calculation
results of optimal geometries calculated by using five diflerent types of optimization

procedures are compared in 1) Total pressure ratio, 2) isentropic efficiency 3) Head 4)

Work and shown in Figure 8-9 - Figure 8-12.

     
 

1.30 — __ _
:8
as
m L
g 1.20 1
m :
§ 1 ——+ —RBFN+GA
g - - . - - FFNN+GA
3 . W. RBFN+ICA+GA -.
L mun-Online+GA
——+—-Desired ‘ o
1.00 1W- W WWW—— -— .--.----- 3, .._ m _ __ 1
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
Flow coeflicient

Figure 8-9 Comparison of total pressure ratio

173

Isentropic efficiency

Head (m"2/ S"2)

 

9093;

7090 §

6005!

5096-§~w~

400/0 -

0.04

20000

18000 lm--

HS—lI—i

co

osoag

COCO

OOOOO
l

6000 9
4000 ~

0.04

   
 

-~—-——oW—IUBFNHK1A

- - - - a- - - RBFN+ICA+GA

  

 

--O--FFNN+GA

-—Gé— Bunnq+PthA+cux “ 2-1.
mun-Online+GA ‘.“.\ \

0.05 0.06 0.07 0.08 0.09 0.1 0.11
How coefficient

Figure 8-10 Comparison of isentropic efficiency

m... RBFN+ICA+GA
-—%&-IUBFNH4K1A+CHK
w —-I-~(DnfirH{LA

  

_...» .. .._._... - ..... _ .. ... .. .... _ __ _..

0.05 0.06 0.07 0.08 0.09 0.1 0.1 1
How coefficient

Figure 8-11 Comparison of head

174

20000 WWWW ---~-—~—~WW------ffigan f _
18000 "
16000 .-

 

  

——o —RBFN+GA
" --O--FFNN+GA
__ -----t RBFN+ICA+GA ,
—>é- RBFN+PCA+GA

  

Work (m"2/S"2)
oo '5 ES '3
8 8 8 8
o o o o

 

6000 - _W mun-Online+GA
' -——0——— Desired
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
How coefficient

Figure 8-12 Comparison of work

8.4 Discussion and Conclusions

In this chapter, AN SYS CFX is used to evaluate the comprehensive performances of
optimal impeller geometry found by five different types of optimization procedures:
RBFN+GA, FFNN+GA, RBFN+ICA+GA, RBFN+PCA+GA, and Online+GA, which
has been introduced in former chapters.

Total pressure ratio, isentropic efficiency, work, head are compared not only each
other, but also with desired one. The results in Figure 8-9 - Figure 8-12 indicate that
there are no significant differences between FFNN+GA, RBFN+PCA+GA or
Online+GA and desired ones. It means that FFNN+GA, RBFN+PCA+GA and
Online+GA provides the better performance and find optimal one closer to the global
optimal compared to RBFN+GA and RBFN+ICA+GA. Furthermore, performances
impellers found by FFNN+GA, RBFN+PCA+GA and Online+GA are not quite
different from each other though their relative velocity distributions are different.
Although impeller found by RBFN+GA is not as close as desire one, it has better

performance: higher isentropic efficiency, higher total pressure ratio. This suggests

175

that the given desired relative velocity is not correlated to the highest achieved
efficiency. There is still possibility existed to reach the geometry correlated to the

better performance, which is based on the numerical simulation.

176

CHAPTER 9

CONCLUSIONS

The background and purposes of developing a computer-aided optimization tool for

centrifugal compressor impellers have been proposed. Its aim is to assist designer to

reach a desired geometry within a more efficiency and effective way before impellers
are manufactured and tested.

First of all, a geometry generation tool, which is called BladeCAD is developed in this

study. In BladeCAD, centrifugal impellers as well as inlet casings and diffusers can be

created and modified. NASA Quasi-3D dimensional codes MERIDL and TOSNIC are
modified and linked to BladeCAD and calculate the compressor performance

' including velocity, pressure, flow angle distribution and etc. The reasons of

developing this geometry generation tool are to allow the optimization finished in this

tool and decrease the adesigning time.

Five different types of optimization procedures are developed as following:

1) In the first optimization procedures, Radial Basis Function Network is used to
create the performance map, and Genetic algorithm is used as optimization
method to search for the optimal geometry.

2) The second optimization procedure is very similar to the first one. However,
Feed-forward Neural Network (F FNN) is used to create the performance map
instead of Radial Basis Function Network.

3) In the third optimization procedure, Independent Component Analysis (ICA) is
applied to transform the coordinate system into new one, in which the
optimization procedure finished.

4) Principle Component Analysis (PCA) instead of ICA is used to transform the

177

original coordinate system in fourth optimization procedure. Although
transformations used in third or fourth optimization procedure, the new coordinate
systems are quite different. In ICA, the coordinates are independent fi'om each
other while maximum variances are projected on the coordinates in PCA.
5) The online flow solver is used in the fifth optimization procedure.
The calculation results of each optimization procedure are compared. Moreover,
AN SYS CFX is applied to evaluate optimal geometries. The advantages and
drawbacks of each method are discussed. During the comparison, Genetic Algorithm
(GA) is used in all five types of optimization procedures and settings are same. The
influences of GA operators and parameters on optimization procedures are also

studied and presented.

178

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