MEANLINE PERFORMANCE ANALYSIS OF RADIAL COMPRESSORS

A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY

BY

EMRAH GÜLLÜ

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR
THE DEGREE OF MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING

SEPTEMBER 2015
 Approval of the thesis:

MEANLINE PERFORMANCE ANALYSIS OF RADIAL COMPRESSORS

submitted by EMRAH GÜLLÜ in partial fulfillment of the requirements for the

degree of Master of Science in Mechanical Engineering Department, Middle
East Technical University by,

Prof. Dr. M. Gülbin Dural Ünver

Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. Raif Tuna Balkan

Head of Department, Mechanical Engineering

Prof. Dr. Haluk Aksel

Supervisor, Mechanical Engineering Department, METU

Examining Committee Members:

Prof. Dr. Kahraman Albayrak

Mechanical Engineering Department, METU

Prof. Dr. M. Haluk Aksel

Mechanical Engineering Department, METU

Assist. Prof. Dr. Cüneyt Sert

Mechanical Engineering Department, METU

Assoc. Prof. Dr. M. Metin Yavuz

Mechanical Engineering Department, METU

Assist. Prof. Dr. Ö. Uğraş Baran

Department of Mechanical Engineering, TED University

Date:
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced
all material and results that are not original to this work.

Name, Last Name: EMRAH GÜLLÜ

Signature :

iv
 ABSTRACT

MEANLINE PERFORMANCE ANALYSIS OF RADIAL COMPRESSORS

Güllü, Emrah
M.S., Department of Mechanical Engineering
Supervisor : Prof. Dr. Haluk Aksel

September 2015, 106 pages

Due to the complex nature of flow within a compressor, two- and three-dimensional
CFD codes are commonly employed in compressor design today and they are
leading to levels of efficiency believed to be impossible a few years ago. However,
these sophisticated CFD codes are costly and time-consuming. Many improvements
can be done in a simple mean-line analysis during the preliminary design of a
compressor which leads to fewer hours of fine tuning in CFD codes. Despite its
inherent limitations, the one-dimensional flow analysis along a mean-line is a very
practical and powerful tool to inexpensively design and analyze a compressor.
Mean-line method employs fundamental flow equations along with some empirical
models to address the flow losses and some important flow phenomena. The
accuracy of the results directly depends on the validity of the empirical models
employed. The aim of this thesis is to develop a computer program that can be used
to predict the performance of a radial compressor using mean-line method.
Validation of the empirical models has been carried out by comparing the results
with the experimental data available in the literature.

Keywords: Compressor, Centrifugal Compressor, Radial Compressor, Mean-line

v
 ÖZ

RADYAL KOMPRESÖRLERİN ORTA ÇİZGİ PERFORMANS ANALİZİ

Güllü, Emrah
Yüksek Lisans, Makina Mühendisliği Bölümü
Tez Yöneticisi : Prof. Dr. Haluk Aksel

Eylül 2015 , 106 sayfa

Günümüzde iki ve üç boyutlu HAD analizleri sıklıkla kompresör tasarımı için

kullanılmaktadır ve bu analizler sayesinde bir kaç yıl önce imkansız gibi gözüken
verim değerleri yakalanmaktadır. Ancak bu karmaşık HAD analizleri pahalı ve
zaman alıcıdır. Ön tasarım sırasında bir çok iyileştirme basit bir orta çizgi analizi ile
yapılabilmektedir ve bu HAD analizlerindeki ince ayar süresini bir hayli
azaltmaktadır. Tabiatından dolayı kaynaklanan kısıtlamalara rağmen, orta çizgi
boyunca yapılan bir boyutlu analiz ucuz bir şekilde kompresör analizi ve tasarımı
yapmamızı sağlayan çok pratik ve güçlü bir araçtır. Bu yöntem temel akış
denklemlerini empirik modeller ile birlikte kullanır. Empirik modeller bazı önemli
akış olaylarını ve akış kayıplarını modellemek için gereklidir. Analizin sonuçları
tamamen empirik modellerin doğruluğu ile alakalıdır. Bu tezin amacı orta çizgi
analizi ile bir radyal kompresörün performansını tahmin edebilecek bir bilgisayar
programı yazmaktır. Kullanılan empirik modellerin doğruluğunun gösterilmesi için
literatürden alınan deneysel sonuçlar kullanılmıştır.

Anahtar Kelimeler: Kompresör, Santrifüj Kompresör, Radyal Kompresör, Orta Çizgi

vi
To my family

vii
 ACKNOWLEDGMENTS

First of all, I want to thank my family for their support and patience during this work.

Next, I wish to express my gratitude to my supervisor Dr. M. Haluk Aksel and to my

manager Taylan Ercan for their guidance.

Finally, I am thankful to my colleagues Gökhan Aran, Mert Erk and Mustafa Bilgiç
for their support, advice and friendship.

viii
 TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 BACKGROUND INFORMATION . . . . . . . . . . . . . . . . . . . 7

2.1 Classification of Compressors . . . . . . . . . . . . . . . . . 8

2.2 Flow in a Rotating Coordinate System . . . . . . . . . . . . 10

2.3 Performance Characteristics . . . . . . . . . . . . . . . . . . 13

2.4 Efficiency and Loss Coefficient . . . . . . . . . . . . . . . . 15

ix
 2.5 Dimensionless Parameters . . . . . . . . . . . . . . . . . . . 17

3 RADIAL COMPRESSOR MEANLINE THEORY . . . . . . . . . . 19

3.1 Impeller Performance . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 The Slip Factor . . . . . . . . . . . . . . . . . . . 23

3.1.2 Splitter Blades . . . . . . . . . . . . . . . . . . . 25

3.1.3 Impeller Tip Blockage . . . . . . . . . . . . . . . 25

3.1.4 Inducer Analysis and Optimization . . . . . . . . . 27

3.1.5 Throat Calculations . . . . . . . . . . . . . . . . . 27

3.1.6 Parasitic (External) Loss Models . . . . . . . . . . 32

3.1.7 Impeller Internal Losses . . . . . . . . . . . . . . 33

3.1.8 Exducer Analysis . . . . . . . . . . . . . . . . . . 37

3.1.8.1 One Zone Model . . . . . . . . . . . . 37

3.1.8.2 Two Zone Model . . . . . . . . . . . 41

3.1.8.3 Comparison of One and Two Zone

Models . . . . . . . . . . . . . . . . . 47

3.1.9 Impeller Gas Path Calculations . . . . . . . . . . . 48

3.1.10 Preliminary Impeller Sizing . . . . . . . . . . . . 54

3.1.11 Employing Impeller Analysis Procedure for Sizing 55

3.2 Vaneless Diffuser Performance . . . . . . . . . . . . . . . . 55

3.3 Vaned Diffuser Performance . . . . . . . . . . . . . . . . . . 60

3.4 Preliminary Diffuser Sizing . . . . . . . . . . . . . . . . . . 66

3.5 Return System Performance . . . . . . . . . . . . . . . . . . 69

x
 3.6 Inlet Guide Vane (IGV) Performance . . . . . . . . . . . . . 73

4 VALIDATION CASES . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Eckardt O-rotor Stage . . . . . . . . . . . . . . . . . . . . . 77

4.2 Radiver Stage . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 NASA CC3 Stage . . . . . . . . . . . . . . . . . . . . . . . 83

4.4 Jones Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 A Case with a Return Channel . . . . . . . . . . . . . . . . . 87

5 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . 95

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

APPENDICES

A THERMODYNAMIC MODELS . . . . . . . . . . . . . . . . . . . . 101

A.1 Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 Inlet Blockage . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.3 Skin Friction Coefficient . . . . . . . . . . . . . . . . . . . . 102

A.4 Hydraulic Diameter . . . . . . . . . . . . . . . . . . . . . . 103

B NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . . . . . 105

B.1 Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . 105

B.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . 105

xi
 LIST OF TABLES

TABLES

Table 3.1 Computing Station Numbers . . . . . . . . . . . . . . . . . . . . . 20

Table 3.2 Daily and Nece Torque Coefficients [13] . . . . . . . . . . . . . . . 34

Table 3.3 Suggested Values for Primitive TEIS [4] . . . . . . . . . . . . . . . 43

Table 3.4 Shroud geometry functions [12] . . . . . . . . . . . . . . . . . . . 51

Table 3.5 Hub geometry functions [12] . . . . . . . . . . . . . . . . . . . . . 53

Table 4.1 O-rotor Stage Geometry [18] . . . . . . . . . . . . . . . . . . . . . 78

Table 4.2 O-rotor Impeller Exit Comparison . . . . . . . . . . . . . . . . . . 81

Table 4.3 Radiver Stage Geometry [18] . . . . . . . . . . . . . . . . . . . . . 82

Table 4.4 NASA CC3 Stage Geometry [18] . . . . . . . . . . . . . . . . . . . 86

Table 4.5 Jones Stage Geometry [18] . . . . . . . . . . . . . . . . . . . . . . 90

Table A.1 NASA polynomial coefficients [48] . . . . . . . . . . . . . . . . . . 101

xii
 LIST OF FIGURES

FIGURES

Figure 1.1 Aerodynamic Design Methodology [3] . . . . . . . . . . . . . . . 6

Figure 2.1 A Typical Gas Turbine [26] . . . . . . . . . . . . . . . . . . . . . 7

Figure 2.2 Flow Field within an Axial Compressor [27] . . . . . . . . . . . . 9

Figure 2.3 Axial Flow Compressor Configuration [28] . . . . . . . . . . . . . 10

Figure 2.4 Radial Compressor Configuration [13] . . . . . . . . . . . . . . . 11

Figure 2.5 Velocity Triangles for the Rotor Section of a Compressor [29] . . . 12

Figure 2.6 Typical Compressor Characteristics Map [25] . . . . . . . . . . . . 15

Figure 2.7 Enthalpy-Entropy Diagram . . . . . . . . . . . . . . . . . . . . . 16

Figure 3.1 Computing Station Nomenclature [13] . . . . . . . . . . . . . . . 19

Figure 3.2 Impeller Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 3.3 Impeller Velocity Triangles . . . . . . . . . . . . . . . . . . . . . 22

Figure 3.4 Tip Velocity Triangle with Slip . . . . . . . . . . . . . . . . . . . 23

Figure 3.5 Relative eddy concept [31] . . . . . . . . . . . . . . . . . . . . . . 24

Figure 3.6 An Impeller with Splitter Blades [33] . . . . . . . . . . . . . . . . 26

Figure 3.7 Inlet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 3.8 Inducer Optimization Algorithm . . . . . . . . . . . . . . . . . . . 29

Figure 3.9 Inducer Optimization, Cm1 vs. W1s . . . . . . . . . . . . . . . . . 30

Figure 3.10 Throat velocity triangle [19] . . . . . . . . . . . . . . . . . . . . . 30

Figure 3.11 Throat Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 31

xiii
Figure 3.12 One Zone Model Procedure . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.13 Impeller Tip Conditions Estimation Procedure . . . . . . . . . . . 40

Figure 3.14 Impeller h − s diagram . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 3.15 Jet and Wake Model [15] . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 3.16 Two Zone Model Procedure . . . . . . . . . . . . . . . . . . . . . 43

Figure 3.17 TEIS Conceptual Model [4] . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.18 Jet Zone Calculations . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.19 Wake Zone Calculations . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 3.20 Mixed-out State Calculations [39] . . . . . . . . . . . . . . . . . . 47

Figure 3.21 Comparison of One and Two Zone Models on Came Impeller
[42], Total-to-total Pressure Ratio (P Rtt ) versus corrected mass flow rate
(ṁcorr ), 100% = 40000rpm . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 3.22 Assumed impeller geometry [12] . . . . . . . . . . . . . . . . . . 52

Figure 3.23 Assumed blade profile [12] . . . . . . . . . . . . . . . . . . . . . 52

Figure 3.24 Throat Line [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 3.25 Impeller Meridional View After Impeller Sizing . . . . . . . . . . 55

Figure 3.26 Preliminary Impeller Sizing Procedure [17] . . . . . . . . . . . . . 56

Figure 3.27 Impeller Sizing with Analysis Procedure [4] . . . . . . . . . . . . 57

Figure 3.28 Vaneless Diffuser Geometry, from left to right: constant width,
constant area, linear shroud . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 3.29 Vaneless Passage Analysis Procedure . . . . . . . . . . . . . . . . 61

Figure 3.30 Vaned diffuser geometry [13] . . . . . . . . . . . . . . . . . . . . 66

Figure 3.31 Vaned diffuser analysis procedure . . . . . . . . . . . . . . . . . . 67

Figure 3.32 Wedge Diffuser Geometry [17] . . . . . . . . . . . . . . . . . . . 70

Figure 3.33 Return Channel Geometry [13] . . . . . . . . . . . . . . . . . . . 73

Figure 3.34 Alternative Return Bend Analysis Procedure . . . . . . . . . . . . 74

Figure 3.35 IGV Geometry [12] . . . . . . . . . . . . . . . . . . . . . . . . . 75

xiv
Figure 3.36 IGV Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 4.1 O-rotor Performance Map . . . . . . . . . . . . . . . . . . . . . . 79

Figure 4.2 O-rotor, Distributions along Vaneless Diffuser . . . . . . . . . . . 80

Figure 4.3 Loss Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 4.4 Radiver Performance Map with Vaned Diffuser, P Rtt vs. ṁcorr . . 83

Figure 4.5 Radiver Performance Map with Vaned Diffuser, Vaned Diffuser
Throat Blockage Removed, P Rtt vs. ṁcorr . . . . . . . . . . . . . . . . . 84

Figure 4.6 Radiver Performance Map with Vaneless Diffuser, P Rtt vs. ṁcorr . 84

Figure 4.7 Radiver, Vane Diffuser Exit . . . . . . . . . . . . . . . . . . . . . 85

Figure 4.8 NASA CC3 with Vaned Diffuser Performance Map . . . . . . . . . 88

Figure 4.9 NASA CC3 with Vaneless Diffuser Performance Map . . . . . . . 89

Figure 4.10 Jones Stage with Vaned Diffuser Performance Map . . . . . . . . . 91

Figure 4.11 The Stage with a Return System . . . . . . . . . . . . . . . . . . . 92

Figure 4.12 Efficiency Map for the Stage with a Return System . . . . . . . . . 93

Figure 4.13 Pressure Ratio Map for the Stage with a Return System . . . . . . . 93

Figure A.1 Assumed Inlet Blockage Values [4] . . . . . . . . . . . . . . . . . 102

Figure B.1 Simpson’s Rule [49] . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure B.2 Newton Raphson Method with Numerical Differentiation . . . . . 106

xv
 NOMENCLATURE

Symbols

a m/s Speed of Sound

m Location of Point of Maximum Camber
A m2 Area
b m Passage Width
B - Blockage
c m Chord
C m/s Absolute Velocity
d m Diameter
D m Divergence Parameter
DF - Diffusion Factor
DR - Diffusion Ratio
E - Diffusion Efficiency
h j/kg Specific Enthalpy
i deg incidence
I - Work Input Coefficient
L m Meanline Length
m m Meridional Position
M - Mach Number
nCV - Number of Control Volumes
N rad/s Angular Velocity
p N/m2 Pressure
P W Power
peri m Perimeter
r m Radius
R j/kg Rothalpy
j/kg/K Gas Constant of Air
Re - Reynolds number
React - Degree of Reaction
s j/kg/K Specific Entropy
T K Temperature
U m/s Peripheral Speed
X - Choke Trigger
W m/s Relative Velocity
z - Number of Blades

xvi
α deg Flow Angle
β deg Blade Angle
γ - Specific Heat Ratio
θ - Normalized Total Temperature
deg Camber Angle
δ - Normalized Total Pressure
deg Deviation Angle
m Boundary Layer Thickness
η - Efficiency
- Effectiveness
µ - Stage Work Coefficient
kg/m/s Dynamic Viscosity
φ - Flow Coefficient
deg Wedge Divergence Angle
λ - Tip Distortion Factor
- Swirl Coefficient
σ - Slip Factor
- Solidity
 - Meanline Radius Ratio
- Secondary Flow Area Fraction
ρ kg/m3 Density
ω rad/s Angular Velocity
χ - Secondary Flow Mass Fraction
AR - Area Ratio
cf - Skin Friction Coefficient
cp j/kg/K Specific Heat at Constant Pressure
Cp - Pressure Recovery Coefficient
Cr - Contraction Ratio
CM - Disk Torque Coefficient
dH m Hydraulic Diameter
Deq - Equivalent Diffusion Factor
fc - Head Loss Correction
Fu - Tangential Velocity Factor
hth m Throat Line Length
LB m Blade Mean Camberline Length
ṁ kg/s Mass Flow Rate
sCL m Clearance Gap Width
tb m Blade Thickness
αc deg Streamline Slope Angle
θC deg Divergence Angle
κm rad/m Streamline Curvature
ω̄ - Loss Coefficient

xvii
Acronyms

CFD Computational Fluid Dynamics

IGV Inlet Guide Vane
ISC Inducer Stall Criterion
LC Loss Coefficient
NS Navier-Stokes
PR Pressure Ratio
SC Swirl Coefficient
TEIS Two-Element-in-Series

Subscripts

-1 IGV Inlet
0 IGV Exit
1 Impeller Inlet
2 Impeller Tip
Vaneless Diffuser Inlet
3 Vaneless Diffuser Exit
Vaned Diffuser Inlet
4 Vaned Diffuser Exit
Vaneless Space After Vaned Diffuser Inlet
5 Vaneless Space After Vaned Diffuser Exit
Return Bend Inlet
6 Return Bend Exit
Return Channel Inlet
7 Return Channel Exit
Exit Duct Inlet
8 Exit Duct Exit
B Blade
BL Blade loading
C Curvature
CH Choke
CL Clearance Gap
COR Corrected
CR Critical
D Diffusion
DF Disk Friction
DIF Diffusion
e Value at Boundary Layer Edge
FB Full Blade
h Hub

xviii
 HS Hub to Shroud
id Ideal
imp Impeller
INC Incidence
j Jet
jc Junction
L Leakage
LIM Limiting
m Meridional Component
p Polytropic
Primary
par Parasitic
r Radial Component
R Recirculation
s Isentropic
Shroud
Secondary
SB Splitter Blade
SEP Separation
SF Skin Friction
SH Shock
u Tangential Component
z Axial Component
t Total Condition
v Velocity
VD Vaned Diffuser
VLD Vaneless Diffuser
tar Target
th Throat
tt Total-to-Total
w Wake
∞ Perfect Guidance
? Throat Parameter

Superscripts

’ Relative Condition
?
Sonic Condition
Minimum
- Mean

xix
 CHAPTER 1

INTRODUCTION

There are a number of different approaches used for aerodynamic design and
analysis of turbo-machinery throughout the industry. They all utilize the basic
principles of thermodynamics along with the experimental data, but in different
combinations and in different complexity. Methods available in the literature vary
from being completely non-dimensional to three-dimensional. A simple scaling of
an existing design may be sufficient for some applications while a multistage
Navier-Stokes (NS) solver is required for the others. However, a proper combination
of methods of different complexities should be employed for an efficient design
procedure.

Design of a modern compressor is not possible without the help of Computational

Fluid Dynamics (CFD) and this dependency is increasing day by day. [1] Fully 3D,
multistage NS solvers are commonly employed in the compressor design today.
However, these solvers are still too expensive to be a design tool that all the design
process can rely on. Therefore, they must be supported by simpler and less
expensive methods and the design must be refined successively by the methods of
increasing complexity.

Figure 1.1 gives an overview of the standard design methodology employed for a
state-of-the-art compressor. Aerodynamic design is linked with many other
disciplines such as aeromechanical, structural and thermal analysis. Aerodynamic
design begins with cycle design where the specification of pressure ratio, mass flow
rate, shaft speed, geometrical constraints, target efficiency and target surge margin
are determined. After this, a meanline code is employed to determine the number of

1
stages required, thermodynamic conditions and velocities after each stage
component, preliminary sizes and off-design performance. Next, inviscid quasi 3D
solvers are used where the flow domain is divided into hub-to-tip (S2) and
blade-to-blade (S1) surfaces. [2] These methods are supported by empirical inputs to
account for blockage and flow losses. Streamline curvature method is generally
employed for hub-to-tip analysis, also referred as through-flow analysis, where
annulus geometry is optimized and radial distributions of all flow quantities are
checked. Blade profiles are optimized by a blade-to-blade solver, which generally
utilizes a 2D Euler code coupled with a boundary layer solver. Finally, 3-D Euler
and N-S codes are employed for detailed annulus and 3D blade geometry
optimization, stage matching, analysis of cavity and bleed flows and off-design map
generation.

1.1 Objective

In this thesis, it is aimed to write a meanline code that can be used to design and
analyze radial compressors. This code will be a part of design system mentioned
above in the near future. Meanline analysis is a very useful tool in the early stages of
compressor design. It allows the designer to make preliminary sizing and compressor
map predictions with the knowledge of only overall dimensions and blade angles.
This method makes use of fundamental flow equations along with the empirical input.
Accuracy of the obtained results is determined by the accuracy of the empirical input.

1.2 Literature Survey

There are a variety of approaches to the meanline performance prediction of

centrifugal compressors within the literature. Japikse [4] classifies them as Level 1,
Level 2 and Level 3 design.

Level 1 design employs similitude techniques to scale an existing, successful design.

Application of these techniques to radial compressors is explained in almost every
introductory level book about compressors, such as Whitfield and Baines [5], Dixon

2
[6], Cumpsty [7] and Japikse and Baines [8]. It is not always possible to employ
Level 1 design since, for example, the scaling may require too high rotational speed
which exceeds the stress limitations of the impeller. Another problem is that complete
similitude is hard to achieve due to the scaling of the Reynolds number. Despite these
limitations of the Level 1 design, it is employed very frequently in industry since it is
fast and accurate.

Level 2 design employs empirical correlations derived from prior test experience for
the overall component performance parameters, such as impeller efficiency, along
with one-dimensional calculations. These correlations are generally derived for a
certain type of machine and their application range is very limited. Japikse and Baines
[4, 8] explain how such a design system works if these component correlations are
available. It is not possible to find such correlations in the open literature since they
are usually a company property.

Level 3 design employs a detailed set of models to represent the physics of the flow
more accurately. This level of design requires the identification of different flow
phenomena within each stage component and suitable models to estimate their effect
on overall compressor performance. There are a variety of models to be used in the
open literature. Oh et al. [9] compares the accuracy of different loss models
presented in the open literature and suggest a set of loss models. Roberts [10]
employs these loss models in his master’s thesis and shows that they are indeed
capable of modeling the impeller performance accurately. Whitfield and Baines [5]
also summarizes the different models used in the open literature. Galvas [11]
provides a Fortran program to predict the off-design performance of centrifugal
compressors with channel diffusers. He employs a unique set of loss models taken
from the literature and a surge and choke criteria to predict the operating range.
Herbert [12] has an interesting study where he tries to model every stage component
by employing variety of geometrical features. He couples the calculations with a
boundary layer solver to estimate the blockage throughout the compressor stage. He
defines a preliminary impeller shape by means of prescribed functions which
enables a more detailed analysis to predict boundary layer growth, separation, flow
blockage and energy dissipation. [5] Aungier [13, 14] and Japikse [4] provide a
comprehensive set of models for every stage component of a radial compressor in a

3
well-documented and detailed way. Aungier employs one zone model to analyze the
impeller flow while Japikse employs two zone model. Both methods have their
advantages and disadvantages, which will be explained later in Chapter 3. One zone
model analyses the impeller flow as a whole while two zone model divides it into
two separate regions of different characteristics. Two zone model of Japikse is an
improved version of Jet-Wake model of Dean and Senoo [15], which is based on the
observations of Eckardt [16]. Eckardt observed and proved that the flow exiting
from the impeller blades of a centrifugal turbomachine is divided into two regions,
referred to as jet and wake. This idea is later improved by Japikse [4] and renamed
as the two-zone model. It is experimentally verified that jet flow is almost isentropic
while wake flow has all the flow losses. [4] Schiff [17], in his master’s thesis,
presents a detailed preliminary design and analysis procedure for the impeller,
vaneless diffuser and vaned diffuser. He combines these procedures with some
detailed geometrical calculations to create a preliminary impeller and diffuser
geometry.

1.3 Scope

Radial compressor meanline theory used in this thesis mainly follows the work of
Aungier [13], Japikse [4] and Herbert [12]. Aungier [13] provides very useful
correlations and methodologies for the preliminary design and performance analysis
of all stage components. Two zone model of Japikse [4] is a very popular technique
to analyze impeller flow due to its physical base. It requires fewer empirical
relations compared to traditional methods which makes it very practical in the
preliminary design phase. Impeller gas path geometry calculation is done using
equations provided by Herbert [12].

Validation of the developed code is done by the experimental test cases commonly
used in the literature. Eckardt O-rotor [18, 19], Radiver [18, 20, 21], NASA CC3
[18, 22, 23] and 6.5 pressure ratio centrifugal compressor of Jones [18, 24] are used
for this purpose.

The methodologies and correlations used in this thesis are widely used in the literature

4
and quite well documented. Validation of these methods has been done by many
researchers. Therefore, they present a good starting point for a mean-line code that
will be modified and corrected over the years by the experience of the author.

This thesis consists of 5 chapters. Chapter 2 gives some background information

regarding fundamental concepts in radial compressor aerodynamics and Chapter 3
explain the meanline methodology used. The results are compared with some test
cases in Chapter 4 and some concluding remarks are made in Chapter 5. Finally, a set
of appendices are added to include some bulky and extensive theories used.

5
 Cycle Design

Meanline (1D)

Throughflow (S2) Lifing

Blade to Blade (S1)

3D (multistage) Euler Aeroelastic analysis

Structural and
3D (multistage) NS
Thermal analysis

Rig and Engine Tests

Figure 1.1: Aerodynamic Design Methodology [3]

6
 CHAPTER 2

BACKGROUND INFORMATION

Gas turbine is one of the most popular means of producing mechanical power due to
its high reliability and high power-to-weight ratio. It consists of three main
components; the compressor, the combustion chamber and the turbine as shown in
Figure 2.1.

First step in the cycle of a gas turbine is the compression of the working fluid in the
compressor so that it can be expanded through a turbine. If there were no losses in
either component, the work provided by the turbine would be just sufficient to drive
the compressor. However, power developed by the turbine can be increased by the
addition of energy to the working fluid prior to this expansion. If the working fluid is
air, this energy addition can be done by the combustion of fuel in the compressed air.
[25]

Figure 2.1: A Typical Gas Turbine [26]

Compressor is probably the most challenging component to design from an

aerodynamic point of view. The flow through a compressor stage is diffusing, i.e.,
the flow is subject to adverse pressure gradients. Therefore, there is a risk of flow
separation which brings the need for a control over the diffusion process in every

7
stage component. Flow separation can cause dramatic performance drops and
stability problems.

Weight and cost are also other important parameters to consider besides the
compressor performance and stability. It is important to keep the number of stages
as low as possible in order to reduce weight and cost. This leads to more
aerodynamically loaded blades (higher diffusion) which makes the compressor
design even more challenging.

As it can be seen from Figure 2.2, the flow field within a compressor is quite
complicated, three-dimensional and unsteady. Boundary layers develop both on
blade surfaces and end walls. Vorticies form at the tip and hub corners. There are
flows through the tip clearances due to the pressure difference between suction and
pressure surfaces of a blade. There are also three-dimensional flow fields, named as
secondary flow, generated by the interactions between the annulus boundary layers
and the blade rows. If the maximum velocity on suction surface becomes supersonic,
shocks can form and these shocks can cause boundary layer separations if they are
strong enough. The flow field becomes even more complicated for a centrifugal
machine where curvature effects of the annulus and strong Coriolis forces as a result
of radial motion also comes into picture.

Basic information regarding compressor theory relevant to this thesis will be given in
the following sections.

2.1 Classification of Compressors

Compressors are commonly classified as positive displacement and dynamic

compressors. The positive displacement compressors achieve its pressure rise by
forcing fluid into a confined space whose volume is decreased to compress the fluid
while the dynamic compressors develop their pressure rise by the mechanical action
of rotating blades that impart velocity and pressure to the continuously flowing fluid.
Two basic types of dynamic compressors are centrifugal and axial flow compressors.
Axial flow compressors have streamlines through their rotating blade rows which
have almost constant radius while they undergo a substantial radius increase in

8
 Figure 2.2: Flow Field within an Axial Compressor [27]

centrifugal compressors. For this reason, it is possible to obtain a higher pressure

rise per stage in a radial compressor while an axial flow compressor can handle
greater mass flow rate per unit frontal area. Radial compressors are more robust and
of lower cost while axial flow compressors have the potential for better efficiency.
[13]

An axial flow compressor consists of series of stages, where a stage refers to a rotating
row (rotor) in combination with a stationary row (stator) as shown in Figure 2.3. The
rotor rows impart kinetic energy to working fluid by increasing the swirl (tangential)
velocity and stator rows remove the swirl developed to convert kinetic energy to static
pressure. In addition, an inlet guide vane (IGV) and an exit guide vane (EGV) can be
used to adjust the swirl velocity entering into and exiting from the stages for which the
stages designed. Often the IGV is adjustable so that it can be restaggered to broaden
the application range of compressors. [28] The working fluid passes through rotors
and stators in this manner to increase the total pressure to the degree required in the
engine cycle.

The basic operating principle of a centrifugal compressor is as follows. The fluid

enters to the impeller through the eye. An inlet guide vane (IGV) can be employed

9
 Figure 2.3: Axial Flow Compressor Configuration [28]

to introduce prewhirl to this flow and extend the operating range. The fluid then
moves into the inducer, which is the axial portion of the impeller. The inducer
transfers the fluid smoothly into the radial portion of the impeller, the exducer,
where the rotating blades energize the fluid. Some of this energy is recovered as
static pressure in the vaneless diffuser. A vaned diffuser can also be employed for
further recovery of the remaining kinetic energy. Finally, a return system, which
consists of a return bend and a return channel, directs the flow into the inlet of the
next stage or into a combustion chamber. Vanes in the return channel also reduce the
swirl developed by the preceding stage elements to the design value of the next stage
element. Alternatively, for single-stage machines, a volute can be employed to
smoothly collect the flow from the diffuser to the discharge pipe. Figures 2.4a and
2.4b show typical single and multistage centrifugal compressor configurations,
respectively. [13]

2.2 Flow in a Rotating Coordinate System

The analysis of the flow in a rotating blade row is best accomplished in a coordinate
system that rotates with the blades. The flow conditions in this coordinate system are
called as relative conditions. If a blade row is rotating with an angular velocity, ω,
blade speed at any radial position can be calculated as the product of rotational speed
with the radial position, U = ωr. Combining the blade speed vectorially with the
relative velocity results in the absolute velocity. Both relative and absolute velocities
have components in axial, radial and tangential directions. Combination of axial

10
 (a) Single Stage (b) Multistage

Figure 2.4: Radial Compressor Configuration [13]

and radial components are termed as meridional component. Meridional components

represent the mass flow through the stage while tangential components determine the
energy transfer of the stage. That is,

~ +U
W ~ =C
~ (2.1)

C~r + C~z = C~m (2.2)

C~m + C~u = C
~ (2.3)

where W is the relative velocity and C is the absolute velocity while subscripts r, z,
m and u represent radial, axial, meridional and tangential components respectively.
Sketching these velocity vectors entering and leaving the blade rows in a form of
so called velocity triangles is a very common and useful practice during compressor
design. It makes it easier to visualize velocity and flow angle changes in a compressor
stage. Velocity triangles for the rotor section of a compressor are shown in Figure 2.5.

If conservation of angular momentum is written between inlet (station 1) and outlet

(station 2) of the rotating blade row, the power input supplied by the blade row is

P = ṁ(r2 Cu2 − r1 Cu1 )ω (2.4)

where ṁ is the mass flow rate and subscripts 1 and 2 represent the inlet and exit
of the rotating blade row respectively. Combining above result with the first law of

11
 Figure 2.5: Velocity Triangles for the Rotor Section of a Compressor [29]

thermodynamics, the well-known Euler turbine equation is obtained as

ht2 − ht1 = ω(r2 Cu2 − r1 Cu1 ) (2.5)

where h is the enthalpy and subscript t represents the total condition. The above
equation can also be rewritten in the following form,

ht2 − ωr2 Cu2 = ht1 − ωr1 Cu1 (2.6)

Therefore, it is convenient to define a parameter, R, as

R = ht − ωrCu (2.7)

which is constant along a stream sheet. This parameter is known as rothalpy.

Another useful form the Equation (2.6) can be obtained using the cosine theorem in
the velocity triangle as,
W12 − W22 U22 − U12 C22 − C12
ht2 − ht1 = + + (2.8)
2 2 2

Therefore, total enthalpy rise is due to the decrease in the relative velocity, increase in
the absolute velocity and change of radius. Making use of the relative total condition

12
definition, h0t = h + W 2 /2, and Equation (2.8), rothalpy can also be defined as,

U2
R = h0t − (2.9)
2

2.3 Performance Characteristics

Two compressor stages are completely similar if the ratios of all corresponding
length dimensions, velocity components and forces are equal. Two completely
similar machines have also similar performance which can be expressed by the
dimensionless performance parameters. Although complete similarity is rarely
achieved in practice, it can be approximated to yield good enough results.

The dimensionless performance parameters can be obtained using the so called

Buckingham Pi theorem. Considering the various quantities that can influence the
compressor behavior, it can be stated that

pt2 , RTt2 = f (d, N, ṁ, pt1 , RTt1 ) (2.10)

where p is the pressure, T is the temperature, R is the gas constant of the working
fluid, d is a reference diameter of the compressor and N is the angular velocity.
Therefore, the function given by the Equation (2.10) can be reduced to a different
function of 7 − 3 = 4 non-dimensional groups. These non-dimensional groups in
their most useful form are
√
pt2 Tt2 ṁ RTt1 N d
, , ,√ (2.11)
pt1 Tt1 d2 pt1 RTt1

Viscosity is also an important physical property that can affect the compressor
behaviour. The presence of this variable would add another dimensional group
having the character of Reynolds number. It is found from experience that the effect
of this group is quite small in the normal operating range of most turbomachines. If
the effect is not negligible, empirical correlations exist in the literature to add its
influence. [25]

When dealing with the performance of a fixed size machine, R and d may be omitted
from the groups and the inlet total conditions may be normalized by the reference

13
ambient state. (pref = 1.013 bar, Tref = 288 K) Therefore,
√
pt2 Tt2 ṁ θ N
f( , , ,√ )=0 (2.12)
pt1 Tt1 δ θ

where θ = Tt1 /Tref and δ = pt1 /pref . The function as given in Equation (2.12)
states that any of the one group can be plotted against another while keeping a third
constant. Experience has shown that the most useful plots are pt2 /pt1 and Tt2 /Tt1
√ √
against ṁ θ/δ while keeping N/ θ constant. The total temperature ratio can be
replaced by isentropic efficiency, which is a function of total pressure and total
temperature ratio. The typical compressor characteristics plots are shown in Figure
2.6. [25]

Each blade row has a specific inlet flow angle for which losses are minimum and it
should operate close to this inlet flow angle at a specific operating condition,
commonly referred to as compressor’s design point. At flow rates less than the
design flow rate, losses increase to a point where pressure-mass flow characteristics
has a maximum. Lowering the flow rate even further, the characteristics have a
positive slope which is theoretically unstable and onset of this severe unstable
operation is called surge. Theoretical instability of the positive slope region can be
explained as follows. Pressure rise developed by the compressor is less than the exit
pressure of the compressor if the flow rate is reduced in this region. This causes
forward flow through the compressor to stop and reverse its direction. After some
point, pressure in the compressor exit drops below the pressure developed by the
compressor and forward flow starts again. This causes a high frequency cycle of
pressure build up and decay within the compressor which can cause severe
mechanical vibrations. Surge is a very complex phenomenon that is highly
dependent on the complete system, not just on the compressor. Although associating
surge only with a positive slope on pressure-mass flow characteristics is an
oversimplification, it is a useful one. [28]

Similarly, at flow rates greater than design flow rate, the increase in losses eventually
reduces the stage pressure rise to zero. This is commonly referred to as choking.
Large losses due to off-design operation can cause such an effect rather than a true
aerodynamic choking condition, where the velocity of the flow reaches sonic speed
somewhere within the compressor. [28]

14
 Figure 2.6: Typical Compressor Characteristics Map [25]

2.4 Efficiency and Loss Coefficient

Efficiency is a measure of the aerodynamic quality of a compressor and it can be

defined as the actual performance relative to an ideal performance. One measure of
efficiency is to compare the actual process to a reversible process. Since
turbomachines are essentially adiabatic, the reversible process is also an isentropic
one. Referring to Figure 2.7a, total-to-total isentropic efficiency of a compressor is
defined as
ht2s − ht1
ηs = (2.13)
ht2 − ht1

where subscript s represents the state after an isentropic process. Static conditions
can be preferred instead of total conditions at the exit in the isentropic efficiency
definition if the exit kinetic energy is wasted. This will yield the total-to-static
isentropic efficiency.

Although adiabatic efficiency is the most common efficiency definition used for
compressors, it cannot truly represent the aerodynamic quality of a compressor. Two
compressor stages having same aerodynamic quality but different pressure ratios can

15
 (a) Stage (b) Stationary Component

Figure 2.7: Enthalpy-Entropy Diagram

have different isentropic efficiencies since constant pressure lines diverge on an

h − s diagram. Therefore, another efficiency definition, named as polytropic
efficiency, is usually preferred over isentroic efficiency to evaluate the aerodynamic
quality. Instead of using a constant entropy path, polytropic efficiency makes use of
a path of constant efficiency. Assuming that a compressor consists of infinitesimal
successive stages, polytropic efficiency is given by the isentropic efficiency of an
infinitesimal stage such that it is constant throughout the whole process. [25]
Therefore, it can be expressed as

dhs
ηp = (2.14)
dh

which leads to the following formula for an ideal gas with constant specific heat,

R ln(p2 /p1 )
ηp = (2.15)
cp ln(T2 /T1 )

The above efficiency definitions cannot be used to evaluate the aerodynamic quality
of a stationary compressor component since, in the absence of heat transfer, the total
enthalpy is constant. However, it is still useful to compare the performance of a
stationary component to an ideal process. Pressure recovery coefficient, Cp , is one
popular parameter used for stationary diffusing components. It is defined as the
fraction of inlet dynamic pressure, pt1 − p1 , which is recovered as static pressure

16
rise. [13] Referring to Figure 2.7b,

p2 − p1
Cp = (2.16)
pt1 − p1

Another measure of the irreversibility within a compressor is the loss coefficient.

Since loss is usually proportional to kinetic energy, it is defined as

∆pt
ω̄ = (2.17)
pt1 − p1

where ∆pt is the total pressure loss.

2.5 Dimensionless Parameters

There are a number of useful dimensionless parameters that are useful during the
radial compressor design. They define the performance objectives the stage should
achieve and the type of design that will be most effective in a form general to any
stage design problem. [13]

Stage work coefficient and stage flow coefficient are defined as

∆ht,id
µ= (2.18)
U22

ṁ
φ= (2.19)
ρ1t πr22 U2

where ∆ht,id is the total enthalpy rise required to produce the total pressure rise of the
compressor via an ideal process and ρ1t is the inlet total gas density. The stage flow
coefficient determines the achievable efficiency levels and type of design which will
be most effective while the stage work coefficient represents how heavily loaded the
stage is.

The rotational Mach number (also named as machine Mach number), MU , is a

popular dimensionless parameter used to characterize a centrifugal machine. It is
given by
U2
MU = (2.20)
a1t

17
where U2 is the impeller tip speed and a1t is the speed of sound based on inlet total
conditions. It is a measure of the Mach number levels within the impeller and directly
affects the stage pressure and temperature ratios.

Diffusion factor and diffusion ratio are common parameters that are used to limit the
diffusion process. These parameters relate the peak velocity on the suction surface to
the outlet velocity. They are defined as,
Wmax
DR = (2.21)
W2
Wmax − W2
DF = (2.22)
W1

where Wmax is the maximum relative velocity on the suction surface. Distribution of
loading among the stage components is defined by the degree of reaction, React. It is
defined as the ratio of static enthaply rise along the impeller to that of the stage. That
is,
∆himpeller
React = (2.23)
∆hstage

Due to the influence of wall friction on performance, Reynolds number

(Re = ρW d/µ), which is the ratio of inertia forces to viscous forces, is also another
important characteristic parameter.

These parameters can be choosen or evaluated based on the design goals and previous
experience.

18
 CHAPTER 3

RADIAL COMPRESSOR MEANLINE THEORY

Radial compressor performance model used in this thesis has a modular structure.
The main components of the stage are modeled separately. The stage consists of an
impeller, a vaneless diffuser, a vaned diffuser and a return system. An inlet guide vane
system (IGV) can also be included to broaden the operating range. Volute modeling is
outside the scope of this thesis. Figure 3.1 and Table 3.1 show the station numbers and
stage components used in this thesis. Velocity components and basic thermodynamic
data are obtained at each computing station.

Figure 3.1: Computing Station Nomenclature [13]

19
 Table 3.1: Computing Station Numbers

(-1)-(0) IGV
(1)-(2) Impeller
(2)-(3) Vaneless Diffuser
(3)-(4) Vaned Diffuser
(4)-(5) Vaneless Space After Vaned Diffuser
(5)-(6) Return Bend
(6)-(7) Return Channel
(7)-(8) Vaneless Space into Next Stage

Mathematical description of the each stage component and the related physics are
explained in the following sections. Gas model used during these calculations is
presented in Appendix A. The working fluid is air, which can be assumed ideal.

Each stage component requires input from its upstream stage components as inlet
boundary condition. However, operation of the downstream components does not
affect the operation of upstream components.

Most of the empirical correlations used in this chapter are taken from Aungier [13].
He mentions in his book that the validation studies for these correlations include stage
flow coefficients ranging from 0.009 to 0.16 and pressure ratios up to 3.5, with actual
application range is limited to pressure ratios up to 4.2. [13]

Most of the procedures presented below require some kind of iteration and simple
guess and update procedure is used. These procedures are presented by means of
flowcharts to show how the iterations are performed.

3.1 Impeller Performance

The impeller is the rotating component of the centrifugal compressor stage which is
responsible for the energy transfer to the working fluid. It consists of an inducer,
where the working fluid enters to the impeller smoothly, and an exducer, where the
impeller energizes the working fluid. Figure 3.2 shows the impeller geometry with

20
 (a) Meridional Plane [19] (b) Mean Stream Surface Plane [13]

Figure 3.2: Impeller Geometry

some key geometric parameters in the meridional and mean stream surface planes.

The inlet and outlet flow areas are calculated by,

2 2
A1 = π(r1s − r1h ) − b1 zF B tb1 (3.1)

A2 = b2 [2πr2 − (zF B + zSB )tb2 ] (3.2)

where zF B is the number of full blades, zSB is the number of splitter blades, tb is the
blade thickness and b is the passage width. The impeller work input, I, is the total
enthalpy rise imparted to the working fluid by the impeller in the dimensionless form.
It is defined as,
∆ht
I= (3.3)
U22

The portion of this enthalpy rise supplied by the impeller blades is called blade work
input coefficient or useful work, IB . Rest of the impeller work input is termed as
parasitic losses, which do not contribute to the pressure rise. The main parasitic losses
are disk friction loss, recirculation loss and leakage loss. Disk friction loss, IDF , is
due to the adhesive interaction between the rotating disk and surrounding fluid in the
clearance gaps. The recirculation loss, IR , is generally seen in highly loaded impellers
and it is caused by the fluid reversing its direction at the impeller tip. Finally, some of
the leakage flow in the clearance gaps reenters into the impeller, creating the leakage

21
 (a) Inlet (b) Outlet

Figure 3.3: Impeller Velocity Triangles

loss, IL [30, 31]. Therefore, the total work input coefficient is written as

I = IB + IDF + IL + IR (3.4)

Using the Euler turbine equation given by Equation (2.5), the blade work input
coefficient is,
Cu2 Cu1 U1
IB = − (3.5)
U2 U22

Velocity triangles for the inlet and outlet of the impeller are shown in Figure 3.3.
Flow and blade angles are measured from tangential direction to be consistent with
the work of Aungier [13].

If the flow is assumed to receive perfect guidance from the impeller blades, α2 0 = β2 ,
Equation (3.5) can be rewritten as,
Cu1 U1
IB = 1 − φ2 λ cot β2 − (3.6)
U22

where λ is the tip distortion factor, φ2 is the tip flow coefficient and β2 is the blade
angle at the impeller tip. They are defined as,
1
λ= (3.7)
1 − B2

22
 ṁ
φ2 = (3.8)
ρ2 A2 U2

where B2 is area blockage at the impeller tip.

3.1.1 The Slip Factor

Perfect guidance of the flow by the impeller blades is only possible if there is infinite
number of blades with infinitesimal thickness. In the real case, the relative flow angle
always deviates from the blade angle for some appreciable amount. This is usually
termed as slip. Referring to Figure 3.4, the slip velocity is defined as,

Cslip = Cu2∞ − Cu2 (3.9)

where Cu2∞ is the tangential component of the absolute velocity if the flow follows
the impeller blades perfectly. Based on slip velocity, a slip factor definition can be
made as,
Cslip Cu2∞ − Cu2
σ =1− =1− (3.10)
U2 U2

Making use of this slip factor definition, velocity triangle given in Figure 3.4 and the
Equation (3.6), the blade work input coefficient can be written as,

Cu1 U1
IB = σ(1 − φ2 λ cot β2 ) − (3.11)
U22

Figure 3.4: Tip Velocity Triangle with Slip

23
Therefore, there is a linear relation between blade work input coefficient, IB , and
tip flow coefficient, φ2 . The slope of this curve depends on the value of β2 . For an
impeller with β2 < 90◦ , also referred to as back-swept impeller, the slope is negative
while for an impeller with β2 > 90◦ , also referred to as forward-swept impeller, it
is positive. Stability issues associated with positive slope was mentioned in Section
2.3. Therefore, a back-swept impeller is generally preferred. However, back-swept
angle decreases the blade work input for the same flow coefficient. To keep the blade
work input the same, impeller tip peripheral speed, U2 , should be increased with an
increase in stress levels. Therefore, back-swept blading extends the operating range
and improve the efficiency, but with the cost of increasing stresses. [4].

From Equation (3.11), it can be deduced that the slip factor is the ratio of actual blade
work input to the value that would be obtained with perfect flow guidance for the case
of zero inlet swirl velocity (Cu1 = 0). It is also the blade work input coefficient that
would be obtained for an impeller operating with zero mass flow for the same case.

The concept of slip factor can be explained by means of relative eddies forming
within the impeller as shown in Figure 3.5. It can be assumed that the flow entering
the impeller is irrotational. Hence, the flow must remain irrotational in the absolute
frame of reference throughout the impeller. Relative eddies rotating in a direction
opposite to the impeller are formed within the impeller to maintain this
irrotationality. This concept is accepted as the main source of slip in radial flow
turbomachines and people derived very useful correlations for slip factor just using
the kinematics of these relative eddies, for example the slip factor model by Stodola.
[32]

Figure 3.5: Relative eddy concept [31]

24
In this thesis, approximated Busemann slip factor by Wiesner is used to estimate the
slip factor [13, 14]. It is given as
√
sin αc2 sin β2
σ =1− (3.12)
z 0.7

where αc2 is the streamline slope angle at the impeller tip and z is the number of
impeller blades. Equation (3.13) is valid up to the limiting meanline radius ratio,
 = r1 /r2 , given by,
σ − σ?
LIM = (3.13)
1 − σ?

where σ? = 19◦ + 0.2β2 . If this limiting radius ratio is exceeded, a corrected slip
factor, σCOR , is used. It is given by,
 − LIM √β2 /10
σCOR = σ[1 − ( ) ] (3.14)
1 − LIM

3.1.2 Splitter Blades

Centrifugal compressors designed for high rotational Mach numbers usually employ
splitter blades, having partial length between adjacent full blades, as illustrated in
Figure 3.6. They are used to reduce the blockage at throat while maintaining an
acceptable blade solidity. Therefore, higher mass flow can be passed through the
impeller before choking occurs by employing splitter blades.

The effective number of blades, z, given by the Aungier [13] is used to include the
effect of splitter blades on the performance. It is given by,
LSB
z = zF B + zSB (3.15)
LF B

where LF B and LSB are the meanline lengths of the full and splitter blades.

3.1.3 Impeller Tip Blockage

High aerodynamic blockage is often encountered at the impeller exit due to the
diffusing flow within the impeller. Correct estimation of the impeller tip blockage,
B2 , or the impeller distortion factor, λ, is very important since it has a substantial

25
 Figure 3.6: An Impeller with Splitter Blades [33]

effect on blade work input coefficient. Aungier [13] made the following observations
during his experimental work:

(i) Blockage varies directly with the skin friction for very low flow coefficients

(ii) Blockage increases with the diffusion from impeller throat to impeller discharge

(iii) Blockage increases with blade aspect ratio, b2 /LB

(iv) Blockage increases with blade clearance

Based on above observations, Aungier [13] formulated the impeller tip blockage as,
s
pv1 W 1 dH b2 A2 ρ2 b2 sCL
B2 = ω̄SF + (0.3 + 22 ) R + (3.16)
pv2 W 2 b2 LB ρ 1 LB 2b2

where ω̄SF is the skin friction loss coefficient, dH is the hydraulic diameter, LB is
the blade mean camberline length, sCL is the clearance gap width, pv is the dynamic
pressure and AR is the ratio of impeller tip flow area to throat area. AR is defined by,

A2 sin β2
AR = (3.17)
A1 sin βth

where βth is the mean blade angle at the throat.

26
3.1.4 Inducer Analysis and Optimization

The aim of the inducer is to transfer working fluid from inlet to exducer with
minimum loss. To achieve this goal, the following parameters should be controlled:

(i) approach relative velocity, W1 ,

(ii) incidence, i1 = β1 − α10 ,

(iii) preswirl, Cu1 or α1 and

(iv) local acceleration effects with subsequent diffusion as the flow passes around
the blade leading edge

To establish the inducer velocity triangle shown in Figure 3.3a, the iterative procedure
shown in Figure 3.7 should be executed. For the inlet blockage, B1 , recommendations
given by Japikse [4] are included in Appendix A.

Meridional velocities at the hub and shroud of the inducer can be obtained using the
following relations given by Aungier [13],

Cmh1 = Cm1 (1 + 0.5κm1 b1 ) (3.18)

Cms1 = Cm1 (1 − 0.5κm1 b1 ) (3.19)

where κm1 is the streamline curvature at the inlet. If the inlet swirl angle distribution
and the meridional velocities are known, velocity triangles at the hub and shroud of
the inducer can also be obtained.

Given the inducer hub radius, r1h , or the inducer hub to tip ratio, r1h /r1s , as a
constraint, the inducer tip radius, r1s , which gives the minimum relative tip velocity,
W1s , can be found. This is referred to as inducer optimization. Flowchart given in
Figure 3.8 can be followed to plot W1s against Cm1 for a given range of Cm1 , as in
Figure 3.9, and optimum r1s can be spotted from this plot.

3.1.5 Throat Calculations

Throat of the impeller is the smallest area within the impeller passage where choking
is most likely to occur. It plays a vital role in performance calculations since it

27
 pt1 , Tt1 , ṁ, α1 , B1 , r1s , r1h , Nimp

2 2
A1 = π(r1s − r1h ) − b1 zF B tb1
2 2
r1 = [(r1h + r1s )/2]0.5
ρ1guess = pt1 /R/Tt1
cp1 = cp (Tt1 )

Cm1 = ṁ/[ρ1guess A1 (1 − B1 )]
C1 = Cm1 / sin(α1 )
ht1 = cp1 Tt1
h1 = ht1 − C12 /2
ρ1guess = ρ1
T1 = h1 /cp1
cp1 = cp (T1 ), γ1 = γ(T1 )
p1 = pt1 (T1 /Tt1 )γ1 /(γ1 −1)
ρ1 = p1 /R/T1

FALSE
abs(ρ1 /ρ1guess − 1) < 10−5

TRUE
Cu1 = Cm1 / tan(α1 )
U1 = Nimp r1
Wu1 = U1 − Cu1
2 2 0.5
W1 = (Cm1 + Wu1 )

Figure 3.7: Inlet Analysis

determines the maximum mass flow rate that can pass through the impeller. To
determine the onset of choking, total thermodynamic conditions (relative total
conditions for the impeller case) must be known.

Since the throat is usually placed just after the inlet, the process between inlet to throat
can be assumed isentropic, that is, total conditions at the inlet and throat are equal.
It can also be assumed that the throat meanline radius is equal to the inlet meanline
radius. These assumptions can be summarized as,

ptth = pt1 , Ttth = Tt1 , rth = r1 (3.20)

The aerodynamic blockage in the impeller throat due to the sudden change in area

28
 pt1 , Tt1 , ṁ, α1 , B1 , r1h or r1h /r1s , Nimp

cp1 = cp (Tt1 )
Define a Cm1 range

Repeat this block for each Cm1

Cu1 = Cm1 tan(α1 )
2 2 0.5
C1 = (Cm1 + Cu1 )
T1 = Tt1 − C12 /2/cp1
cp1 = cp (T1 ), γ1 = γ(T1 )
M1 = C1 /(γ1 RT1 )0.5
p1 = pt1 (T1 /Tt1 )γ1 /(γ1 −1)
ρ1 = p1 /R/T1
A1 = ṁ/ρ1 /Cm1 /(1 − B1 )
2 0.5
r1s = (A1 /π + rh1 )
or r1s = [A1 /π/(1 − (r1h /r1s )2 )]0.5
U1s = Nimp r1s
2 2 2
W1s = [Cm1 + (U1s − Cu1 )]0.5

Plot Cm1 vs. W1s

Figure 3.8: Inducer Optimization Algorithm

can be modeled by a contraction ratio as [13],

A1 sin β1 0.5 A1 sin β1
Cr = min[( ) ,1 − ( − 1)2 ] (3.21)
Ath Ath

where Ath is the geometric throat area. The area for which the given mass flow rate
and inlet total conditions yield a sonic velocity is given by,
ṁ
A?th = (3.22)
ρ?th Wth
?

2 1/(γ−1)
ρ?th = ρ0tth ( ) (3.23)
γ+1
0
? 2γRTtth
Wth =( )0.5 (3.24)
γ+1
?
where superscript represent sonic or critical condition. Choking will occur if
Cr Ath < A?th . Equations (3.22), (3.23) and (3.24) can be used for any vaned
component by changing the relative values with the absolute ones for non-rotating
components. The flowchart of the impeller throat calculations is given in Figure
3.11. Velocity triangle used during calculations is given in Figure 3.10.

29
Figure 3.9: Inducer Optimization, Cm1 vs. W1s

Figure 3.10: Throat velocity triangle [19]

30
 InletAnalysis, βth

ρthguess = ρ1
cpth = cp1
Uth = Nimp rth

Wth = ṁ/ρthguess /Ath /Cr

Wuth = Wth cos(βth )
Cmth = Wth sin(βth )
Cuth = Uth − Wuth
2 2
Cth = (Cuth + Cmth )0.5
2
hth = ht1 − Cth /2 ρthguess = ρth
Tth = hth /cpth
cpth = cp (Tth )
γth = γ(Tth )
pth = pt1 (Tth /Tt1 )γth /(γth −1)
ρth = pth /R/Tth

FALSE
abs(ρth /ρthguess − 1) < 10−5

TRUE
0 2
Ttth = Tth + Wth /2/cpth
p0tth = ptth (Ttth
0
/Ttth )γth /(γth −1)
ρ0tth = p0tth /R/Ttth
0

ρ?th = ρ0tth ((2/(γth + 1))1/(γth −1) )

? 0
Wth = (2γth RTtth /(γth + 1))0.5
A?th = ṁ/ρ?th /Wth
?

FALSE Exducer
Cr Ath <= A∗th
Analysis

TRUE
Stop Execution
Print ’Inducer Choke’

Figure 3.11: Throat Calculations

31
3.1.6 Parasitic (External) Loss Models

Power consumed by the compressor that is not doing any useful work is called
parasitic loss and the main components of parasitic loss are disk friction loss,
leakage loss and recirculation loss as mentioned in Section 3.1.

Leakage and disk friction losses are both dependent upon the flow in the clearance
gaps. Aungier [13] provides a clearance gap flow model which can be used to
calculate these parasitic losses for both open and closed impellers. Parts of the
analysis relevant to the closed impellers are omitted since closed impellers are
outside the scope of this thesis.

The flow through the clearance gap is created by the pressure difference between
the two sides of an impeller blade which must be balanced by the impeller torque.
Therefore, the average pressure difference across the clearance gap can be estimated
by,
ṁ(r2 Cu2 − r1 Cu1 )
∆pCL = (3.25)
zr̄b̄L

where r̄ = (r1 + r2 )/2 and b̄ = (b1 + b2 )/2. The leakage flow undergoes a sudden
contraction followed by a sudden expansion across the clearance gap and the velocity
of the clearance gap flow can be estimated by assuming a throttling coefficient of
0.816. Density is also assumed constant and equal to the value at the impeller tip
[13]. Therefore,
2∆pCL 0.5
UCL = 0.816( ) (3.26)
ρ2

Then, the mass flow through the clearance gap is given by,

ṁCL = ρ2 zsCL LF B UCL (3.27)

Using these clearance gap flow calculations, Aungier [13] provides the following
empirical formula for the leakage loss,
ṁCL UCL
IL = (3.28)
2ṁU2

Disk friction loss model presented in Aungier [13] follows the work of Daily and
Nece [34, 35]. They consider four different flow regimes with four different torque

32
coefficients as shown in Table 3.2. The Reynolds number, ReCM , used in these
coefficients is defined by,
ρ2 ωr22
ReCM = (3.29)
µ2

The true flow regime is represented by the torque coefficient having the largest value,
that is,
CM 0 = max(CM 1 , CM 2 , CM 3 , CM 4 ) (3.30)

Aungier [13] applies some empirical corrections to the Daily and Nece torque
coefficient, CM 0 , as,
0.75 CM 0 (1 − K)2
CM D = (3.31)
(1 − K0 )2

where K = Cu2 /U2 and K0 = 0.46/(1 + 2sD /d2 ). Therefore, the work input
coefficient associated with disk friction is,
CM D ρ2 U2 r22
IDF = (3.32)
2ṁ

Recirculation losses are dominant for the impellers with excessive blade loading. To
be able to evaluate the blade loading, Aungier [13] uses the diffusion ratio definition
given in Equation (2.21) as,
Wmax
Deq = (3.33)
W2

where Wmax = (W1 + W2 + ∆W )/2. The average blade velocity difference, ∆W , is

computed by,
2πd2 U2 IB
∆W = (3.34)
zLB

Lieblein [36, 37] accepted Deq > 2 as the blade stall limit for the axial machines and
Aungier [13] found Deq > 2 to be an appropriate limit for the radial impellers as well.
If this limit is exceeded, a recirculation work input coefficient is computed from,
Deq WU 2
IR = ( − 1)( − 2 cot β2 ) (3.35)
2 Cm2

3.1.7 Impeller Internal Losses

Besides the parasitic losses, blade work input is also compromised by the internal
losses. The internal loss models for the impeller used in thesis include:

33
 Table 3.2: Daily and Nece Torque Coefficients [13]

CM 1 Laminar, merged boundary layers 2π/[(sd /r2 )ReCM ]

CM 2 Laminar, separate boundary layers 3.7(sd /r2 )0.1 /Re0.5
CM
1/4
CM 3 Turbulent, merged boundary layers 0.08/[(sd /r2 )1/6 ReCM ]
CM 4 Turbulent, separate boundary layers 0.102(sd /r2 )0.1 /Re0.2
CM

(i) Inlet Bow Shock Loss

(ii) Incidence Loss

(iii) Entrance Diffusion Loss

(iv) Choking Loss

(v) Skin Friction Loss

(vi) Blade to Blade Loading Loss

(vii) Hub-to-Shroud Loading Loss

(viii) Abrupt Expansion Loss

(ix) Wake Mixing Loss

(x) Clearance Gap Leakage Loss

(xi) Super-critical Mach Loss

When the inlet velocity exceeds sonic conditions (M10 > 1), inlet bow shock loss is
calculated by a normal shock assumption at the inlet [38]. It is given by,
Wth 2 2 pth γ−1
ω̄SH = 1 − ( ) − [( ) γ − 1] (3.36)
W1 (γ − 1)M10 2 p1
The deviation between the relative flow angle and the blade inlet angle creates the
incidence loss [13] since the fluid has to change its direction suddenly. It can be
computed from,
Cm1 zF B tb1 2
ω̄IN C = 0.8(1 − )2 + ( ) (3.37)
W1 sin β1 2πr1 sin β1
Equation 3.37 can be applied at the hub, mean and shroud stream surfaces and a
weighted average of these values , where the mean value is weighted 10 times as
heavy as the hub and shroud values, is accepted as the incidence loss.

34
Entrance diffusion loss [13] is due to the flow diffusion from blade inlet to throat and
it can be computed as,
Wth 2
ω̄DIF = 0.8(1 − ) − ω̄IN C ; ω̄DIF ≥ 0.0 (3.38)
W1

The excessive flow diffusion between the inlet and throat can cause inducer stall and
the inducer stall criterion (ISC) [13] is given by,
W1s
ISC = ≥ 1.75 (3.39)
Wth

If this criterion is exceeded, the diffusion loss [13] is limited by,

W1s − 1.75Wth 2
ω̄DIF ≥ ( ) − ω̄IN C (3.40)
W1

As the throat Mach number approaches unity, choking loss [13] is calculated by,

ω̄CH = 0.5(0.05X + X 7 ); X > 0 (3.41)

where X is the choke trigger and it is calculated by,

10Cr Ath
X = 11 − (3.42)
A?

Skin friction loss [13] due to the adhesive forces between the channel surfaces and
fluid is calculated from,
W̄ 2 LB
ω̄SF = 4cf ( ) (3.43)
W1 dH

where W̄ 2 = max[(W12 +W22 )/2, (Wth

2
+W22 )/2]. Average of the throat and tip values
are employed to calculate the hydraulic diamater, dH , and the skin friction coefficient,
cf . Skin friction coefficient is obtained from generalized pipe friction data given in
Appendix A and it is correlated as a function of Reynolds number based on pipe
diameter, Re = ρV d/µ. Hydraulic diameter calculation is also included in Appendix
A.

Blade-to-blade and hub-to-shroud pressure gradients cause momentum losses due to

boundary layer build-up on the blade surfaces. The blade-to-blade loading loss [13]
is given by,
1 ∆W 2
ω̄BL = ( ) (3.44)
24 W1

35
where ∆W given by Equation (3.34). The hub-to-shroud loading loss [13] is given
by,
1 κ̄m b̄W̄ 2
ω̄HS = ( ) (3.45)
6 W1

where κ̄m = (αc2 − αc1 )/L, b̄ = (b1 + b2 )/2 and W̄ = (W1 + W2 )/2. High
aerodynamic blockage is often encountered at the impeller exit since the flow has
diffused substantially within the impeller. This results in a flow regime with two
separate zones; high momentum core flow (jet) and low momentum wake flow
(wake). This idea of two-zone flow is further explained in Section 3.1.8.2. Abrupt
expansion loss [13], or blockage loss, due to the impeller tip blockage is given by,
(λ − 1)Cm2 2
ω̄λ = [ ] (3.46)
W1

Loss associated with the mixing of the jet and wake flows is calculated by estimating
the velocity at which separation takes place, WSEP . It is given by,

WSEP = W2 for Deq ≤ 2 (3.47)

WSEP = 0.5W2 Deq for Deq > 2 (3.48)

where Deq is the equivalent diffusion factor given in Equation (3.33). Meridional
velocity before and after mixing are then calculated using WSEP and conservation of
mass as,
2
Cm,wake = (WSEP − WU2 )0.5 (3.49)
Cm2 A2
Cm,mix = (3.50)
πd2 b2

assuming constant gas density. Therefore, the wake mixing loss is given by,
Cm,wake − Cm,mix 2
ω̄M IX = ( ) (3.51)
W1

Due to the finite clearance gap between the impeller and the stationary casing, the
fluid will leak from the pressure side to the suction side of the blade. Clearance gap
model given in Equations (3.25), (3.26), (3.27) is used to calculate the clearance loss
as,
2ṁCL ∆pCL
ω̄CL = (3.52)
ṁρ1 W12

36
Using the average of relative thermodynamic conditions at the inlet and tip in
Equation (3.24), the local sonic velocity at the mid-passage, W ? , can be calculated.
Then, the inlet critical Mach number which corresponds to the onset of sonic
velocity at the mid-passage suction surface is estimated by,

0 M1 W ?
Mcr = (3.53)
Wmax

where Wmax is given in Equation 3.33. When this limit is exceeded, shock waves
are expected to form at the mid-passage and these shocks can induce boundary layer
separations. Therefore, the super-critical Mach number loss can be estimated by,
(M10 − MCR
0
)Wmax 2
ω̄CR = 0.4[ ] (3.54)
W1

3.1.8 Exducer Analysis

There are two widely used options to model the impeller for the meanline analysis:
one zone model and two zone model. One zone model analyses the impeller flow
as a whole while two zone model divides it into two separate regions of different
characteristics. One zone model completely relies on empirical models while two
zone model is more physical based. Both methods are explained and compared in the
following sections.

3.1.8.1 One Zone Model

The one zone model employs 1-D flow equations along with empirical flow and loss
models through a mean streamline. The mean streamline corresponds to an average
annulus area location. Typically, this corresponds to the radii defined by the rms of
the corresponding hub and shroud radii. Empirical correlations are used for the
estimation of slip factor, external losses and internal losses, given in the previous
sections. These correlations require velocity and thermodynamic property
information from the inlet, throat and tip of the impeller along with the geometrical
input. Since the tip information is unavailable at the beginning of the analysis, the
one zone model is an iterative procedure. The flow chart of this procedure is given in
Figure 3.12.

37
 pt1 , Tt1 , ṁ, Nimp
α1 , B1 , r1h , r1s , b1 , β1h , β1 , β1s , αc1 , κm1 , tb1
Ath , βth , perith , r2 , b2 , β2 , αc2 , tb2 Stop if Choked
zF B , zSB , Lb , LF B , LSB , sCL
Guess P Rtt , ηtt

Geometrical Calculations
Inducer and Throat Calculations
r1 , A1 , A2 , AR , z, dH

Estimate Tip Thermodynamic Calculate

Conditions and Velocity Components σ, ωSH , ωIN C , ωDIF , ωCH

Calculate
IB , IDF , IL , IR
ωSF , ωBL , ωHS , ωλ , ωM IX , ωCL , ωCR

Calculate Tip Thermodynamic

ρ2guess = ρ2
Conditions and Velocity Components

FALSE
abs(ρ2 /ρ2guess − 1) < 10−5

TRUE

Calculate
P Rtt , ηtt

Figure 3.12: One Zone Model Procedure

38
First, some geometrical parameters should be calculated. They are defined in the
previous sections. Second, inducer and throat is analyzed and the execution of the
program is terminated if choking occurs at the inducer throat. Third, estimation of
the slip factor and calculation of the losses which do not depend on the impeller tip
state is done. These losses include inlet bow shock loss, incidence loss, entrance
diffusion loss and choking loss. Next, estimation of the impeller tip conditions based
on the initial pressure ratio and efficiency assumption. The flowchart given in Figure
3.13 can be used for this purpose. Then, work input coefficients and loss coefficients
which depend on the impeller inlet, throat and tip states are computed. Employing
these coefficients, impeller tip state can be obtained and iterated until the impeller
tip density converges. Finally, necessary post-processing is done and the component
characteristics are obtained.

Impeller tip state is calculated as follows. Since rothalpy is conserved within the
impeller, ideal relative total enthalpy at the impeller tip is,

h0t2id = h0t1 + 0.5(U22 − U12 ) (3.55)

employing Equation (2.10). The deviation of the actual relative total enthalpy from
the ideal one is due to the parasitic loss components. Then, the ideal relative total
temperature and pressure are,
0 h0t2id
Tt2id = (3.56)
cp2
0
Tt2id
p0t2id = pt1 ( )γ2 /(γ2 −1) (3.57)
Tt1

These ideal conditions can be corrected to actual ones employing the total loss
coefficient as,
p0t2 = p0t2id − fc (p0t1 − p1 )ω̄tot (3.58)

where fc is a correction factor due to the fact that loss coefficients are based on inlet
dynamic pressure while they are applied at the impeller tip. It is defined as,
p0t2
fc = 0 (3.59)
pt1

At this point, it is useful to note that these calculations may require impeller tip
conditions and the values from the previous iteration are used if they are not updated

39
 pt1 , Tt1 , ṁ, Nimp
Estimate P Rtt and ηtt

pt2 = pt1 P Rtt

Tt2s = Tt1 (pt2 /pt1 )(γ1 −1)/γ1
Tt2 = Tt1 + (Tt2s − Tt1 )/ηtt
ρ2guess = pt2 /R/Tt2
U2 = Nimp r2
cp2 = cp1

Cm2 = ṁ/ρ2guess /A2

Wu2 = Cm2 / tan(β2 )
Cu2 = U2 − Wu2
2 2 0.5
C2 = (Cm2 + Cu2 )
ρ2guess = ρ2
T2 = Tt2 − C22 /2/cp2
cp2 = cp (T2 ), γ2 = γ(T2 )
p2 = pt2 (T2 /Tt2 )γ2 /(γ2 −1)
ρ2 = p2 /R/T2

FALSE
abs(ρ2 /ρ2guess − 1) < 10−5

TRUE
2 2 0.5
W2 = (Cm2
+ Wu2 )
0 2
Tt2 = Tt2 + W2 /2/cp2
p0t2 = pt2 (Tt2
0
/Tt2 )γ2 /(γ2 −1)

Figure 3.13: Impeller Tip Conditions Estimation Procedure

40
 Figure 3.14: Impeller h − s diagram

yet. Rest of the impeller tip conditions are calculated by employing total work input
coefficient, I, as,
ht2 = ht1 + IU22 (3.60)

along with the basic thermodynamic relations. Impeller h − s diagram is given in

Figure 3.14. Velocity components can be calculated by using the definition of blade
work input as,
U1 Cu1
Cu2 = IB U2 + (3.61)
U2

3.1.8.2 Two Zone Model

The two zone model of Japikse [4] is based on the jet and wake model of Eckardt [16],
which divides the flow within the impeller into two regions, namely jet (primary) and
wake (secondary) regions. It is experimentally verified that the jet flow is nearly
isentropic while wake flow contains all the flow losses. It is also verified that the
wake flow has a very low momentum compared to the jet flow as shown in Figure

41
 Figure 3.15: Jet and Wake Model [15]

3.15 [4]. The flowchart of two-zone modeling used in this thesis is given in Figure
3.16.

Two zone model employs Two-Element-in-Series (TEIS) model to estimate the

diffusion in the primary zone. The TEIS model considers the impeller as a rotating
diffuser, consists of two elements as shown in Figure 3.17. The first element,
element "a", is the the inlet portion from the impeller inlet to the impeller throat. The
second element, element "b", is the passage portion from the impeller throat to the
impeller exit. The diffusion within these elements can be modeled very easily by
treating them as a diffuser with constant effectiveness [4]. Therefore, the diffusion
ratio is calculated as,
W1s 2 1 1
DR2 = ( ) = (3.62)
W2j 1 − ηa Cpai 1 − ηb Cpbi

where Cpai and Cpbi are pressure recovery coefficients and they are defined as,
0
sin α1s
Cpai = 1 − ( )2 (3.63)
sin β1s
Ath 2
Cpbi = 1 − ( ) (3.64)
Ae

where Ae ≈ A2 sin β2 . Critical DR values that can cause impeller stall are given in
Table 3.3. The parameters, ηa and ηb , are the diffusion effectiveness values. They

42
 pt1 , Tt1 , ṁ, Nimp
α1 , B1 , r1h , r1s , β1s , Ath
r2 , b2 , β2 , tb2 , ZF B , ZSB , sCL
ηa , ηb , χ

Inducer and Throat Stop if

Calculations Choked

DR Estimation by TEIS

Jet Zone Calculation

Mixed
Parasitic Losses
Out State
Internal
Wake Zone Calculation
Losses

Figure 3.16: Two Zone Model Procedure

are taken as constants for the primitive TEIS model and suggested values of them are
given in Table 3.3. In reality, however, these parameters are not constants and depend
on impeller passage geometry and operating conditions. [4] Besides diffusion ratio,
two zone model also requires two other parameters to be specified. These parameters
are the secondary flow mass fraction, χ, and deviation of primary flow at the impeller
tip, δ2j . χ is generally assumed constant and suggested values by Japikse [4] are given
in Table 3.3. It can also be defined as a function of , the secondary flow area fraction.

Table 3.3: Suggested Values for Primitive TEIS [4]

Case ηa ηb DRstall χ
Large(>10"-12"D), well design 0.9-1.1 0.4-0.6 1.5-1.8 0.1-0.2
Medium size(4"-10"D), well design 0.8-0.9 0.3-0.5 1.3-1.6 0.15-0.25
Medium size(4"-10"D), ordinary design 0.6-0.8 0.0-0.4 1.2-1.5 0.2-0.3
Small(<4"D) or poor design 0.4-0.6 -0.3-0.3 1.1-1.4 0.25-0.35

43
 Figure 3.17: TEIS Conceptual Model [4]

They are defined as,

ṁw
χ= (3.65)
ṁ
Aw
= (3.66)
A2

The relation between these two parameters can be derived for a specific type of
impeller via experimental work or CFD investigations. Japikse [4] provides this
relation for Eckardt and CETI impellers respectively as,

χ = 2 (3.67)

χ = 0.52 + 0.05 (3.68)

Many correlations exist within the literature to predict the deviation of primary flow.
The slip factor model of Oh et al. [39] is used for this purpose. Equation (3.13) is not
used since it is derived for mixed out flow state, not for the primary flow. Combining
it with the definition of slip, relative flow angle at the impeller tip for the primary flow
is,
β2 U2 A2j
β2j = + 0.5 cos−1 [(11.46 ∗ 10−6 β22.5 + cotβ2 ) sin β2 ] (3.69)
2 W2j r2 b2 z

Since the relation between χ and  is not available in the preliminary design phase,
the iterative procedure shown in Figure 3.18 is instead used for the primary zone
calculations.

Deviation of the secondary flow, δ2s , is generally set to zero since it has a little effect
on the overall stage performance because the secondary flow is largely tangential [40].

44
 χ, r2 , b2 , β2 , z
DR from TEIS
Inlet Analysis

W2j = W1s /DR

U2 = Nimp r2
Guess guess = χ0.5 , cp2 = cp1

2
T2j = (ht1 − W2j /2 + U22 /2 − U1 Cu1 )/cp2
cp2 = cp (T2j ), γ2 = γ(T2j )
p2j = p1 (T2j /T1 )γ2 /(γ2 −1)
ρ2j = p2j /R/T2j guess = 
A2j = A2 (1 − guess )
β2j from Equation 3.69
 = 1 − (1 − χ)ṁ/ρ2j /A2 /W2j /cosβ2j

FALSE
abs(/guess − 1) < 10−5

TRUE
Cm2j = W2j cos β2j
Cu2j = U2 − W2j sin β2j
2 2
C2j = (Cm2j + Cu2j )0.5
2
Tt2j = T2j + C2j /2/cp2
pt2j = p2j (Tt2j /T2j )γ2 /(γ2 −1)

Figure 3.18: Jet Zone Calculations

45
 Fu
Primary Zone Calc.
Inlet Analysis

p2w = p2j
Cu2w = Cu2j Fu
T2wguess = T2j , cp2w = cp2j
R2w = ht1 − U1 Cu1

ρ2w = p2w /R/T2wguess

Cm2w = (ṁ/A2 − ρ2j Cm2j (1 − ))/ρ2w /
2
T2wguess =
W2w = (Cm2w + (U2 − Cu2w )2 )0.5
2
T2w
T2w = (R2w − W2w /2 + U22 /2)/cp2w
cp2w = cp (T2w ), γ2w = γ(T2w )

FALSE
abs(T2w /T2wguess − 1) < 10−5

TRUE
2 2
C2w = (Cu2w + Cm2w )0.5
2
Tt2w = T2w + C2w /2/cp2w
pt2w = p2w (Tt2w /T2w )γ2w /(γ2w −1)

Figure 3.19: Wake Zone Calculations

However, a tangential velocity factor is used in this thesis to account for δ2s [31]. It
is defined as,
Cu2w
Fu = (3.70)
Cu2j

The recommended range for this parameter is 0.92 to 0.94.

The secondary flow static pressure can be equated to that of primary flow to account
for the flow losses. This is often referred to as unloaded tip condition. A set of
loss correlations can also be employed instead. Usage of unloaded tip condition is
preferred in this thesis due to its simplicity. The iterative procedure shown in Figure
3.19 is used to calculate the wake zone conditions. Finally, mixing of primary and
secondary zones at the impeller tip is modeled by assuming that this process occurs
very rapidly after the flow leaves the impeller. One-dimensional energy, momentum
and continuity equations are solved at the impeller tip to calculate the mixed-out

46
 Primary Zone Calc.
Secondary Zone Calc.
Inlet Analysis

Cu2mix = (1 − χ)Cu2j + χCu2w

cp2mix = cp2j , γ2mix = γ2j
Tt2mixguess = (1 − χ)Tt2j + χTt2w

a = ṁ(γ2mix + 1)/(4πr2 b2 γ2mix )

b = [p2j A2 + ṁ((1 − χ)Cm2j + χCm2w )]/(2πr2 b2 )
2
c = ṁ[RTt2mixguess − (γ2mix − 1)Cu2mix /2/γ2mix ]/(2πr2 b2 )
Cm2mix = [b − (b2 − 4ac)0.5 ]/2/a
2 2
C2mix = (Cm2mix + Cu2mix )0.5 Tt2mixguess =
ρ2mix = ṁ/(2πr2 b2 Cm2mix ) Tt2mix
2
T2mix = Tt2mixguess − C2mix /2/cp2mix
cp2mix = cp (T2mix ), γ2mix = γ(T2mix )
Calculate External Losses, ∆hDF , ∆hR , ∆hL from Section 3.1.6
Tt2mix = (1 − χ)Tt2j + χTt2w + (∆hDF + ∆hR + ∆hL )/cp2mix

FALSE
abs(Tt2mix /Tt2mixguess − 1) < 10−5

TRUE

p2mix = ρ2mix RT2mix

pt2mix = p2mix (Tt2mix /T2mix )γ2mix /(γ2mix −1)

Figure 3.20: Mixed-out State Calculations [39]

state. Parasitic (external) losses are included in the energy equation to account for the
enthalpy rise they impart to the flow. The flow chart of the calculation procedure for
mixed-out state is given in Figure 3.20.

3.1.8.3 Comparison of One and Two Zone Models

Two zone model employs TEIS model to predict the diffusion ratio and primitive
TEIS models can predict the diffusion ratio around the design point with a good
accuracy. However, it fails for the off-design points, especially for high speeds [41].
This is due to the absence of accurate and broadly applicable models for the

47
modeling parameters, ηa and ηb , in the open literature. Japikse [4] states that several
years of additional work is required to develop an enhanced version of TEIS. One
zone model of Aungier, on the other hand, has a better off-design accuracy. A
comparison of one and two zone models applied to Came impeller [42] is presented
in Figure 3.21.

One advantage of two zone model is that it requires fewer geometrical information
than one zone model of Aungier, which makes it more practical in the preliminary
design phase.

Since one zone model gives better results for off-design points and it is very well
documented by Aungier [13], it is a superior option for now and it is used in this
thesis for the impeller analysis.

3.1.9 Impeller Gas Path Calculations

For the above mentioned impeller calculations, some impeller gas path parameters,
such as inducer throat area and meanline length, are required. These parameters are
diffucult to find or assume during the preliminary design and some rough calculations
are useful to estimate them. The procedure given by Herbert [12] is used for this
purpose based on the following assumptions:

(i) a rotor geometry with circular arc hub and double circular arc shroud as shown
in Figure 3.22

(ii) blades have constant thickness and circular arc leading edges as shown in Figure
3.23

(iii) a blade angle distribution for hub and shroud as given in Tables 3.4 and 3.5.

(iv) impeller exit is fully radial (αc2 = 90◦ )

The functions given in Tables 3.4 and 3.5 are derived by Herbert [12] to describe the
geometry of the rotor based on the hub and shroud profile assumptions mentioned
above. The following functions are also employed along with these functions,

x = (r2 − r1h )(sec αc1h − tan αc1h ) − b2 (3.71)

48
 (a) One Zone Model

(b) Two Zone Model

Figure 3.21: Comparison of One and Two Zone Models on Came Impeller [42],
Total-to-total Pressure Ratio (P Rtt ) versus corrected mass flow rate (ṁcorr ), 100% =
40000rpm

49
 y = r2 − r1s (3.72)
β2 β1h
βmin = (1 + ) (3.73)
6 15
√
mjc (1 − 2)y
=x+ √ (3.74)
m2s (2 − 2)(x + y)

where mjc denotes the meridional distance at the junction of two circular arcs at the
shroud of the rotor.

The blade angles are measured from the meridional direction in this section, not
from tangential direction as the rest of this chapter, to be consistent with the work of
Herbert [12]. A simple conversion of blade angles from one system to another can
be done at the beginning and end of these calculations.

Throat position at the hub and shroud are found from the blade geometry assumption
shown in Figure 3.24. Some geometrical calculations on this figure results as,
πr1 1 tb1
m? = + (3.75)
zF B cot βa + tan βb 2

where βa is the value of β at m = 2m? −tb1 and βb = 0.5(β1 +βa ). A simple iteration
is needed to calculate m? at the hub and tip since βa and βb are also function of m? .
It should be noted that all quantities differ between hub and tip. Knowing the throat
position at the hub and tip, throat position at any radius can be found by assuming
that throat line is linear . Referring to Figure 3.24, it can written as,
r − r?h
m? (r) = m?h + (m?s − m?h ) (3.76)
r?s − r?h

Blade angles along the throat line is also needed for the throat area calculations and
it can be calculated by assuming that tan β varies linearly with radius at any value of
m? . That is,
rs − r
tan β(r) = tan βs (r) − (tan βs (r) − tan βh (r)) (3.77)
rs − rh

Knowing the position of throat line and blade angle distribution along it, throat area
can be calculated as,
Z r?s
Ath = (2πr cos β(r) − zF B t? (r)) dr (3.78)
r?h

50
 Table 3.4: Shroud geometry functions [12]

m2 π(x + y)/4
m < mjc πm/4/mjc
αc
m > mjc π(1 + (m − mjc )/(m2s − mjc ))/4
√ √
m < mjc r1s + (1 − cos αc )(x + (1 − 2)y)/(2 − 2)
√ √
r r1s + (1 − cos(45◦ ))(x + (1 − 2)y)/(2 − 2) + ...
m > mjc √ √
(cos(45◦ ) − cos αc )((1 − 2)x + y)/(2 − 2)
β β1s (1 − 5(m/m2s )2 )

This integral can be calculated numerically using Simpson’s rule, as explained in

Appendix B. 10 equal intervals of r is sufficient for this calculation.

Throat radius, rth , mean blade angle at the throat, βth , and throat wetted perimeter,
perith , are also required in the impeller analysis procedure. Throat radius can be
assumed to be equal to the inlet radius, rth = r1 , since throat is usually placed very
close to the inlet. Knowing the blade angle distribution along the throat line, blade
angle at this radius can be calculated, βth = β(rth ). Throat wetted perimeter is
calculated using the following formula,

perith = 2πr?s cos β?s + 2πr?h cos β?h + 2zF B hth − 2zF B tb1 (3.79)

where hth is the throat line length and can be calculated as,

hth = [(r?s − r?h )2 + (m?s − m?h )2 ]0.5 (3.80)

Finally, the mean line meridional length, L, is calculated by the help of

quasi-normals. An equal number of discrete points can be distributed along the
shroud and hub curve and connecting the corresponding points with straight lines,
quasi-normals are obtained. RMS radius along each quasi-normal is found first and
these points are connected by linear lines along the impeller. Summing up the
lengths of these small linear lines, meanline meridional length can be estimated.
Accuracy of the calculation increases with the number of quasi-normal.

51
Figure 3.22: Assumed impeller geometry [12]

Figure 3.23: Assumed blade profile [12]

52
 Figure 3.24: Throat Line [12]

Table 3.5: Hub geometry functions [12]

m2 (π/2 − φ1h )(r2 − r1h )/ cos αc1h

αc αc1h + (π/2 − αc1h )m/m2h
r r1h + (r2 − r1h )(1 − cos φ/ cos αc1h )
β β1h − 4(β1h − βmin )(m/m2h )(1 − m/m2h )

53
3.1.10 Preliminary Impeller Sizing

Aungier [13] provides some useful correlations for the preliminary sizing of the
impeller as a function of stage flow coefficient, φ = ṁ/(ρ1t πr22 U2 ). Only the
relations given for open impellers are included here. They are given by,

φ 3 0.002
I = 0.68 − ( ) + (3.81)
0.37 φ

0.00025
µp,V D = 0.59 + 0.7φ − 7.5φ2 − (3.82)
φ
0.017
ηp,V LD = ηp,V D − 3
(3.83)
0.04 + 5φ + ηp,V D

0.002
Ipar = (3.84)
φ
α2,V LD = atan(0.26 + 3φ) (3.85)

α2,V D = 18 + 0.5lnφ + 585φ2 (3.86)

d2
∆zI = 0.014 + 0.023 + 1.58φ (3.87)
d0h

noting that µ = ηI. Therefore impeller blade work input coefficient is,

Cu2
IB = I − Ipar = (3.88)
U2

assuming that there is no inlet swirl.

Employing the inducer optimization algorithm and the gas path calculation
procedure mentioned above along with these empirical correlations enables us to
create a preliminary impeller geometry. The procedure given by Figure 3.26 is
followed in this thesis. During these calculations, a blade loading parameter is
defined and limited to a certain value as,

2∆W
≤ 0.9 (3.89)
W1 + W2

where ∆W is given by Equation (3.34). A meridional view of the obtained impeller

geometry is also plotted along with its meanline and throat line, as shown in Figure
3.25.

54
 Figure 3.25: Impeller Meridional View After Impeller Sizing

3.1.11 Employing Impeller Analysis Procedure for Sizing

It is possible to employ the above analysis procedures for sizing purposes. To

achieve this, the procedure given in Figure 3.27 can be employed. After specifying
target performance parameters, such as pressure ratio, P Rtar , and swirl parameter,
λtar , iterations can be carried out with the analysis procedure to obtain the sizing
parameters, r2 and b2 , which gives the desired performance. Since this problem is
actually a root-finding problem, Newton-Raphson method can be used due to its fast
convergence. This method is explained in Appendix B.

3.2 Vaneless Diffuser Performance

Both Aungier [13] and Stanitz [43] provide one-dimensional conservation of mass,
momentum and energy equations in a form suitable for vaneless passage analysis.
A boundary layer analysis along the vaneless passage is also needed to close these

55
 pt1 , Tt1 , ṁ, Nimp , α1
r1h , r2 , β1 , β1h , β1s , tb , BLmax
Type of diffuser to be used

φ = ṁ/ρt1 /π/r22 /U2

- Use empirical relations
given by Equations 3.81-3.87
- Optimize the inlet (Section 3.1.4)

- Impeller Tip Velocities and Properties

Cu2 = IB U2
Wu2 , Cm2 , C2 , W2 from velocity triangle
∆htP = µP U22
pt2 /pt1 = (1 + ∆htP /cp /T01 )γ/(γ−1)
Tt2 = (ht1 + IU22 )/cp
T2 = Tt2 − C22 /2/cp
p2 = pt2 (T2 /Tt2 )γ/(γ−1)
ρ2 = p2 /R/T2
A2 = ṁ/ρ2 /Cm2

Guess zguess and β2guess

b2 = A2 /(2πr2 − zguess tb / sin β2guess )

Calculate L from Section 3.1.9
Assume Lb ≈ L
∆W = BLmax (W1 + W2 )/2 zguess = z
z = 4πr2 U2 IB /Lb /∆W
Calculate λ and σ from Sections 3.1.1 and 3.1.3
Calculate β2 from Equation 3.11

FALSE
abs(z/zguess − 1) < 10−5

TRUE
P R, ηP , r1s , z, β2 , b2 , Ath
perith , βth , L, Lb , ∆zI

Figure 3.26: Preliminary Impeller Sizing Procedure [17]

56
 Inlet Geometry
Pt1 , Tt1 , B1 , α1
β2 , αc2 , tb2 , zf b , zsb , scl
ṁ, Nimp , P Rtar , λtar
Guess r2 , b2

Procedures of
Change ANALYSIS Section 3.1.9
r2 and b2 PROCEDURE for throat parameters
and meanline length

Check calculated P R, λ
Not OK against P Rtar , λtar

OK

END

Figure 3.27: Impeller Sizing with Analysis Procedure [4]

equations. The equation set given by Aungier is preferred since it includes more loss
models. He used more than 35 different compressor stage tests to developed these
models. [13] It is given by,

2πrρbCm (1 − B) = ṁ (3.90)

d(rCu )
bCm = −rCCu cf (3.91)
dm

1 dp C 2 sin αc dCm CCm cf dID

= u − Cm − − − IC (3.92)
ρ dm r dm b dm

ht = h + 0.5C 2 = constant (3.93)

Last three terms of Equation (3.92) and last term of Equation (3.91) are the source
terms which represents the loss contributions due to wall friction, flow diffusion and
passage curvature. Aungier [13] employs the classical diffuser analogy to estimate
the diffusion loss. A divergence parameter, D, is defined to identify the low loss
regime. It is given as,
b dC
D=− (3.94)
C dm

57
The critical value of divergence parameter below which the diffusion losses are low
is given by,
b1 0.35
Dm = 0.4( ) sin α (3.95)
L
Using this divergence parameter, the empirical diffusion efficiency is given as,

E = 1; D≤0 (3.96)

D 2
E = 1 − 0.2( ); 0 < D < Dm (3.97)
Dm
s
Dm
E = 0.8 ; D ≥ Dm (3.98)
D

The diffusion term in Equation (3.92) is given by,

dID 1 dC
= −2(pt − p)(1 − E) (3.99)
dm ρC dm

Besides this streamwise diffusion loss term, an excessive meridional gradient of the
vaneless passage can also lead to high losses. To check this, the maximum, stall free
local area is estimated by,

m
(rb)m = (rb)1 (1 + 0.16 ) (3.100)
b1

which corresponds to a diffuser divergence angle, 2θC of 9◦ . If the local area exceeds
this, a second estimate of the diffusion term is calculated by,

0.65(pt − p) (rb)m
ID = [1 − ] (3.101)
ρ (rb)

If this value exceeds the local value obtained by integrating Equation (3.99), it is used
instead.

The passage curvature term is omitted in this thesis since it has negligible effect on
vaneless diffuser performance. However, if one wishes to use these equations for
a vaneless passage with high curvatures, such as return bend, this terms should be
included. It is given by,
κm (pt − p)Cm
IC = (3.102)
13ρC

58
The area blockage and skin friction factor coefficient along the vaneless passage are
calculated using a boundary layer growth model, based on a 1/7th power law for the
boundary layer velocity profile as,

y y
Cm = Cme ( )1/7 , Cu = Cue ( )1/7 (3.103)
δ δ

where Cme and Cue are the values at the boundary layer edge. Using this assumption,
boundary layer area blockage is derived from the integration of mass flux across the
passage as,
2δ
B= (3.104)
8b
Similarly, integrating the angular momentum flux across the passage yields,

2δ
rCu = rCue (1 − ) (3.105)
4.5b

Therefore, if the inlet boundary layer thickness, δ2 , is known, rCue can be computed
from inlet rCu , which is the input coming from impeller analysis. Since the flow
domain outside the boundary layer is inviscid, rCue is conserved (rCue = constant)
until the boundary layer fills the passage, that is b = 2δ. Using this fact, boundary
layer thickness at any local position can be computed from the local predicted rCu .
Knowing the boundary layer thickness, blockage, B, at any local position is
calculated from Equation 3.104. The limit 2δ ≤ b should always be remembered. A
simple flat plate boundary layer thickness estimate of the impeller exit is sufficient to
start this analysis. It is given by [19],

LB
δ2 = 5.142 cf , 2δ ≤ dH (3.106)
2

where the computation of hydraulic diameter is given in Appendix A. The following

empirical equation can also be used instead if a vaneless passage analysis is conducted
by itself,
2δ b 0.15
= 1 − ( )in (3.107)
b r

The influence of boundary layer thickness, δ, is primarily on the local skin friction
coefficient, cf , along the vaneless passage which is calculated using the pipe friction
model described in Appendix A, using 2δ instead of pipe diameter.

59
Equations through (3.90) to (3.93) are solved by casting them in finite difference
form and employing a direct marching technique through the vaneless passage.
Vaneless passage is divided into small segments along the meridional direction and
each segment is solved by marching through the vaneless space. Exit condition of
one segment becomes the inlet condition of the next. Inlet conditions of the first
segment are taken from the impeller outlet. The procedure used for vaneless diffuser
analysis is given in Figure 3.29. Finite difference form of the equations are [19],

2πrj+1 ρj+1 bj+1 Cm,j+1 (1 − Bj+1 ) = ṁ (3.108)

∆(rCu )j
bj Cm,j = −rj Cj Cu,j cf,j (3.109)
∆mj
1 ∆pj C 2 sin αc,j ∆Cm,j Cj Cm,j cf,j ∆ID,j
= u,j − Cm,j − − − IC,j (3.110)
ρj ∆mj rj ∆mj bj ∆mj

where ∆()j = ()j+1 − ()j if the forward differencing is used and j indicates the
segment number. Segment number (or number of control volumes, nCV ) should
be increased until the results remain unchanged with its increase, that is, the mesh
independency is achieved.

The above analysis can be applied to any vaneless space including vaneless diffuser,
vaneless space after vaned diffuser and return bend. Three different vaneless diffuser
geometries are used in this thesis as shown in Figure 3.28.

Another important point is that rotating stall can form within the vaneless diffuser if
the impeller exit angle falls below a certain angle, especially at low mass flow rates.
This stall angle [18] is given by,
b2 3 b2 b2
α2,stall = 3821.5( ) − 1559.6( )2 + 238.24( ) − 0.0733 (3.111)
r2 r2 r2

3.3 Vaned Diffuser Performance

Vaned diffuser performance model given by Aungier [13] is similar to that of

impeller. Calculations are performed at the inlet, throat and discharge. Since some of
the loss coefficients also depend on the discharge conditions, an iterative procedure
is again required. This model is validated for conventional airfoil style vaned

60
Figure 3.28: Vaneless Diffuser Geometry, from left to right: constant width, constant
area, linear shroud

r3 , nCV
Diffuser shroud contour
Impeller exit conditions

δ2 from Equation 3.106

rCue from Equation 3.105

cf j from pipe friction model

(rCu )j+1 from Equation 3.109
δj+1 from Equation 3.105
Bj+1 from Equation 3.104

Guess ρj+1 = ρj

Cm,j+1 from Equation 3.108

Start calc. Pj+1 from Equation 3.110
ρj+1,guess =
for the next Tt,j+1 = Tt2
ρj+1
segment Knowing Cj+1 , Calculate Tj+1
ρj+1 = Pj+1 /R/Tj+1

FALSE
abs(ρj+1 /ρj+1,guess − 1) < 10−5

TRUE
Equate exit of one segment to
the inlet of the next
()j = ()j+1

Figure 3.29: Vaneless Passage Analysis Procedure

61
diffusers; however, it is also generalized to be able to use it for vaned diffusers with
nonparallel end walls and thick vanes. Aungier [13] states that this generalization
gives reasonable results although it is not validated with high quality test data.

The vaned diffuser geometry used in this thesis is given in Figure 3.30. The inlet and
discharge area of the vaned diffuser are given by,

A3 = 2πr3 b3 − zV D b3 tb3 , A4 = 2πr4 b4 − zV D b4 tb4 (3.112)

The procedure given in Figure 3.31 is employed for vaned diffuser analysis. The
analysis starts with the inputs from the upstream stage component, usually a vaneless
diffuser. Then, the choking and stall limits are estimated. Similar to impeller analysis,
the blockage at the vane throat is estimated by a throat contraction ratio given by,
s
A3 sin β3
Cr = (3.113)
Ath

Choking will occur if Cr Ath < A?th . Sonic flow area at the throat, A?th = ṁ/(ρ? C ? ),
can be calculated using the formulation given by Equations (3.22), (3.23), (3.24) with
absolute values instead of relative ones. Vaned diffuser stall estimation is based on
the following two parameters supplied by Auniger [13],
r3 cos α3
K= ( − 1) (3.114)
hth cos αth

M32 sin2 β3 cos β3

K0 = (3.115)
1 − M32 sin2 β3

After comparing with some experimental stall limits, Auniger [13] came up with the
following stall criterion for vaned diffusers,

K + K0 = 0.39 (3.116)

which can be solved for α3,stall . If the vaned diffuser inlet flow angle, α3 , falls below
this value, vaned diffuser stall is expected.

After checking for choking and stall, velocities and thermodynamic properties at the
throat can be calculated using a procedure similar to impeller throat calculation
procedure given in Section 3.1.5. Again throat and inlet total conditions can be taken
equal since they are usually very close.

62
Vane discharge flow angle, α4 , can be computed using the axial-flow compressor
correlations transformed to radial plane. The minimum loss deviation angle of Howell
[44] is given by,
θ[0.92(a/c)2 + 0.02(90 − β4 )]
δ? = √ (3.117)
σ − 0.02θ

where the location of point of maximum camber, a/c, solidity, σ, and camber angle,
θ, are given by,
a 1 β̄ − β3
= (2 − ) (3.118)
c 3 β4 − β3
z(r4 − r3)
σ= (3.119)
2πr3 sin β̄
θ = β4 − β3 (3.120)

where the vane angle at the midchord can be estimated as β̄ = (β3 + β4 )/2. Aungier
[13] models the variation of the deviation angle with incidence using the graphical
data presented by Johnsen and Bullock [45] as,
∂δ β3
= exp[((1.5 − )2 − 3.3)σ] (3.121)
∂i 60

Finally, the vaned diffuser discharge flow angle is calculated as,

∂δ
α4 = β4 − δ ? − (β3 − α3 ) (3.122)
∂i

Next, the choking and incidence loss coefficients, which are the loss coefficients
which do not depend on the discharge conditions, are calculated. They are the
choking loss coefficient, ωCH , and the incidence loss coefficient, ωIN C . The choking
loss calculation is identical to that of impeller. The minimum-loss incidence angle is
defined as,
q Cm3
sin α3? = sin β3 sin αth = (3.123)
C3?

The minimum incidence loss for this minimum-loss incidence is given by,
C3? − Cth 2 ztb3
ω̄IN C0 = 0.8( ) + (3.124)
C3 2πr3

The off-design incidence for incidence angles other than minimum-loss incidence is
given by,
C3 − C3? 2
ω̄IN C = 0.8( ) (3.125)
C3

63
for C3 < C3,stall , where C3,stall = Cm3 / sin α3,stall . If C3 > C3,stall ,
2
C3 Cth (C3,stall − C3? )2
ω̄IN C = 0.8[(( )2 − 1) + ] (3.126)
C3,stall C32 2
C3,stall

Similar to the impeller analysis, discharge conditions should be estimated to start the
iterations. This estimation can be performed assuming that there is no total pressure
loss within the vaned diffuser, that is, pt4 = pt3 .

After an initial estimation is done, the iterations are done for the discharge conditions.
During this iterations, the loss coefficients which depend on the discharge conditions
are calculated and updated with each iteration. They include the skin friction loss,
ω̄SF , abrupt expansion loss, ω̄λ , and the wake mixing loss, ω̄M IX . Aungier [13]
supplies the skin friction loss as,
4cf LB C̄
ω̄SF = 0.25
( )2 (3.127)
dH (2δ/dH ) C3

where the hydraulic diameter, dH , and skin friction coefficient, cf , are calculated
using the average of throat and discharge values as in the case of impeller.
Calculation of them are given in Appendix A. Similar to the vaneless diffuser
analysis, the boundary layer thickness at midpassage is estimated using a simple
flat-plate boundary layer approximation as,
LB dH
δ = 5.142cf , δ≤ (3.128)
2 2

The mean velocity, C̄, within the vaned diffuser is calculated by,

C̄ 2 = max[0.5(C32 + C42 ), 0.5(Cth

2
+ C42 )] (3.129)

For the remaining loss coefficients, the discharge area blockage, B4 , is needed and
the correlation provided by Aungier [13] employs the following two basic design
parameters,
b4 1
2θC = tan−1 [((w4 − tb4 ) − w3 + tb3 ) ] (3.130)
b3 2LB
∆C
L= (3.131)
C3 − C4

where the average blade-to-blade velocity difference is given as,

2π
∆C = (r3 Cu3 − r4 Cu4 ) (3.132)
zLB

64
from simple potential flow and w = 2πr sin β/z. It is observed that when L > 1/3
or 2θC > 11◦ , a sudden drop in vaned diffuser performance occurs. Based on this
observation, the following correction coefficients are defined,

1
K1 = 0.2(1 − ) (3.133)
CL Cθ

2θC 2θC
K2 = (1 − ) (3.134)
125Cθ 22Cθ

where Cθ = max(2θC /11, 1) and CL = max(3L, 1). The discharge area blockage is
defined as,
LB
B4 = [K1 + K2 (C̄R2 − 1)] (3.135)
w4

where C̄R = 0.5[Cm3 sin β4 /(Cm4 sin β3 ) + 1]. Similar to the impeller analysis, an
abrupt expansion loss can be defined as,

Cm4 2
ω̄λ = [(λ4 − 1) ] (3.136)
C3

where λ4 = 1/(1 − B4 ). Finally, a wake mixing loss is included to account for

excessive streamwise diffusion and vane discharge blade thickness. It is assumed that
the flow separates at a velocity given by,

C3
CSEP = , CSEP ≥ C4 (3.137)
1 + 2Cθ

The meridional velocities before and after the mixing are calculated from the
conservation of mass as,
q
2 2
Cm,wake = CSEP − Cu4 (3.138)
A4 Cm4
Cm,mix = (3.139)
2πr4 b4

Employing these velocities, wake mixing loss is defined by,

Cm,wake − Cm,mix 2
ω̄M IX = ( ) (3.140)
C3

After calculating all of the loss coefficients, the discharge total pressure can be
calculated from,
pt4 = pt3 − (pt3 − p3 )ω̄tot (3.141)

65
 Figure 3.30: Vaned diffuser geometry [13]

where ω̄tot = ω̄IN C + ω̄CH + ω̄SF + ω̄λ + ω̄M IX . Knowing α4 and using
conservation of mass to calculate Cm4 , all of the velocity components at the
discharge can be calculated. Since pt4 and Tt4 = Tt3 are also known, static properties
can be calculated using the definition of total temperature and isentropic relation.
This process continues until the convergence on discharge density, ρ4 , is achieved.

3.4 Preliminary Diffuser Sizing

Correlations and design limits given by Aungier [13] is used for preliminary sizing of
the vaneless and vaned diffuser. The vaned diffuser leading edge radius is estimated
by,
r3 α3 M2
=1+ + 2 (3.142)
r2 360 15

A simplified form of the angular momentum equation including wall friction effects
through the vaneless space is used to calculate Cu3 as,

r3 Cu3 −cf (r3 − r2 )

ln( )= (3.143)
r2 Cu2 b̄ sin ᾱ

66
 Vaneless Diffuser Analysis
r4 , b4 , Ath , hth , perith , β3
β4 , tb3 , tb4 , Zd , Lb

Check if choking occurs Throat

Stop if Choked
Check if vane diffuser stall occurs Calculations

Estimate Discharge Thermodynamic Calculate α4

Conditions and Velocity Components Calculate wch and winc

Calculate
wsf , wλ , wmix

Calculate Discharge Thermodynamic

Conditions and Velocity Components
pt4 = pt3 − (pt3 − p3 )ω̄tot ρ4guess = ρ4
....
ρ4 = p4 /R/T4

FALSE
abs(ρ4 /ρ4guess − 1) < 10−5

TRUE

Calculate
P Rtt , ηtt

Figure 3.31: Vaned diffuser analysis procedure

67
where b̄ = (b2 + b3 )/2 and ᾱ = (α2 + α3 )/2. Another constraint on passage width
comes from conservation of mass as,

r2 b2 Cm2 = r3 b3 Cm3 (3.144)

taking density as constant.

Sizing of the vaneless diffuser starts with the specification of α3 . Equation 3.142 is
employed to calculate r3 . Next, Equations (3.143) and (3.144) are used to calculate
b3 . However, these equations also contain b3 itself. Therefore, an iterative procedure
is required, for which b3 is guessed and updated until convergence. Another constraint
on b3 is that is should be less than b2 , that is, b3 ≤ b2 .

For the sizing of vaned diffuser, incidence angle, i3 , should specified so that β3 =
α3 + i3 is known. Selection of the vane number is based on both aerodynamic and
resonance considerations. It is preferred that zV D = zimp ± 1. It is also preferred that
10 ≤ zV D ≤ 20 due to improved stall incidence range. If zV D = zimp ± 1 do not fall
within this range, |zV D − zimp | ≤ 8 is required. Sizing of the discharge based on the
following three design parameters; divergence angle, 2θC , blade loading parameter,
L, and area ratio, AR . They are given as,
π(r4 sin β4 − r3 sin β3 )
tan θC = (3.145)
zV D LB
2π(r3 Cu3 − r4 Cu4 )
L= (3.146)
zV D LB (C3 − C4 )
r4 sin β4
AR = (3.147)
r3 sin β3

The preferred ranges for these parameters given by Aungier [13] are 7◦ < 2θC <
10.5◦ , L ≤ 1/3, and 1.4 < AR < 2.4. Besides these limits, it is also required to keep
the r4 below the values estimated by,

r4 = (1.55 + φ)r2 (3.148)

Discharge width, b4 can be taken equal to the inlet width, that is, b4 = b3 . Since no
detailed shape of the vaned diffuser is available at the preliminary design phase, the
following relation is used to estimate vane length,
2(r4 − r3 )
LB ≈ (3.149)
sin β3 + sin β4

68
Specifying r4 and θC , keeping the restrictions given above in mind, Equation (3.145)
can be used to calculate β4 . Next, AR and L are calculated and checked if they lie
within the specified ranges given above. If they don’t, specification of r4 and θC
should be revisited. The exit flow angle, α4 , can be computed using the minimum
loss deviation angle of Howell [44] described in Section 3.3.

Throat parameters of the vaned diffuser can be calculated by estimating the throat
width, hth . Using the estimated throat width, throat area and throat periphery can be
calculated as,
Ath = hth bth zV D (3.150)

perith = 2(hth + bth )zV D (3.151)

where it is usually assumed that bth = b3 .

Throat width of a wedge type diffuser can be calculated using the sizing procedure
given by Johannes [17]. He calculates the angles shown in Figure 3.32 as,
2π
γ= (3.152)
zV D
γ
φ= − θC (3.153)
2

where θC is the channel divergence angle, φ is the wedge divergence angle and γ is the
angle between two adjacent wedges. The channel divergence angle is calculated using
Equation (3.130) and wedge divergence angle is calculated using Equation (3.153).

After calculating these angles, he gives letters to the important points on the wedge
diffuser and employs a coordinate rotation of φcoord = −(β3 + φ) as shown in Figure
3.32. The new axes are shown with χ and η. Coordinate of each letter on this
coordinate system is calculated using basic geometrical relations. Finally, the throat
width is calculated as,
hth = bfχ (3.154)

3.5 Return System Performance

The return system geometry relevant to this thesis is given in Figure 3.33. As it can
be seen from this figure, it consists of a return bend, a return channel and an exit duct.

69
Figure 3.32: Wedge Diffuser Geometry [17]
70
Analysis of the return bend can be performed using the procedure explained in Section
3.2. However, since the return bend geometry is more complicated than a vaneless
diffuser geometry, it can be difficult to supply the geometrical information needed
by this procedure for a return bend. If this is the case and the designer can assume
a loss coefficient, LC, and a swirl coefficient, SC = Cu6 /Cu5 , from his previous
experience, the alternative approach given by Figure 3.34 can be employed. In this
procedure, a diffusion factor, DF , guess is made and iterated until convergence using
the given LC and SC along with conservation of mass.

Return channel analysis of Auniger [13] is similar to the analysis of vaned diffuser
mentioned in Section 3.3. Some differences occur due to distorted flow coming from
return bend and curvature of the return channel. Two estimates of the aerodynamic
blockage are made for the aerodynamic blockage at the vane entrance as,
(rb)m (κm b6 )2
B6 = 1 − , B6 = (3.155)
r6 b6 12 + (κm b6 )2

and the larger value of these two estimates is used in the analysis. The minimum loss
incidence angle is adjusted to account for this entrance blockage as,
Ath
tan α? = (1 − B6 ) tan[sin−1 ( )] (3.156)
A6

The incidence loss coefficient is given by,

Cm6 2
ω̄IN C = 0.8(1 − ) (3.157)
C6 sin α?

Skin friction loss coefficient is also modified to account for the loss associated with
the channel curvature as,
C̄ 2 LB |αc6 − αc7 |
ω̄SF = 4cf ( ) + Cm6 Cm7 (3.158)
C 6 dH 13C62

where C̄ = max((C6 +C7 )/2, (Cth +C7 )/2), dH is the average of throat and discharge
hyraulic diameters and cf is calculated from pipe friction model given in Appendix
A using the average values of throat and discharge. Blade loading loss coefficient is
given by,
1 ∆C 2
ω̄BL = ( ) (3.159)
6 C6

71
where the average blade-to-blade velocity difference is computed from
2π
∆C = (r6 Cu6 − r7 Cu7 ) (3.160)
zLB

Maximum velocity at the vane surface is estimated by,

Cmax = 0.5(C6 + C7 ) + ∆C (3.161)

assuming it occurs at the midpassage. Cmax > C6 is also required. From this velocity,
separation velocity is calculated as,

CSEP = 0.5Cmax if Cmax > 2C7 (3.162)

CSEP = C7 otherwise (3.163)

Therefore the meridional velocities before and after the wake mixing are,
q
2 2 Cm7 A7
Cm,wake = CSEP − Cu7 , Cm,mix = (3.164)
π(r7s + r7h )b7

Finally, the wake mixing loss is calculated as,

Cm,wake − Cm,mix 2
ω̄M IX = ( ) (3.165)
C6

The flow discharge angle is calculated from the minimum loss deviation angle of
Howell [44] and off-design incidence effect on flow deviation angle model of Johnsen
and Bullock [45] as explained in Section 3.3.

The losses due to the exit duct is given by,

1 Cm7 2
ω̄o = (4cf + )|αc7 − αc8 |( ) (3.166)
13 C6

which accounts for the friction and curvature losses. After calculating all loss
components, the total pressure at the discharge can be calculated from,

pt8 = pt6 − (pt6 − p6 )ω̄tot (3.167)

where ω̄tot = ω̄IN C + ω̄SF + ω̄BL + ω̄M IX + ω̄o .

The analysis procedure is the same as the vaned diffuser analysis procedure given in
Section 3.3.

72
 Figure 3.33: Return Channel Geometry [13]

3.6 Inlet Guide Vane (IGV) Performance

Procedure given by Herbert [12] is used for IGV performance analysis. He employs
the profile loss coefficient of Ainley and Mathieson [46] and secondary loss
coefficient of Dunham [47]. They are given by,
smean 2.64
ω̄p = 0.016 + 0.09(1 − ) (3.168)
cIGV
cIGV 1 (1 − B−1 )hmean 0.5
ω̄s = 0.022 sin2 α0 cos α0 (1 + tan2 α0 )0.5 [1 + 10( ) ]
hmean 4 cIGV
(3.169)
where the second term of equation 3.168 drops for smean /cIGV > 1. Total pressure
loss is calculated using these loss coefficients as [12],
pt−1 − pt0 Re0 −0.2
= (ω̄p + ω̄s )( ) (3.170)
pt0 − p0 2 ∗ 105
where Re0 = ρ0 C0 cIGV /µ0 .

The procedure given in Figure 3.36 can be followed to calculate the conditions after
the IGV. First some geometrical parameters are calculated which will be required by
the foregoing analysis. Second, profile and secondary loss calculations are carried
out using the calculated geometrical parameters. Next, the inlet is analyzed using a

73
 Exit of the previous
stage component
r6 , b6 , LC, SC

pt6 = pt5 − LC(pt5 − p5 )

Tt6 = Tt5
cp6 = cp5
Guess DFguess

C6 = C5 /DFguess
T6 = Tt6 − C62 /2/cp6
cp6 = cp (T6 ) and γ6 = γ(T6 )
p6 = pt6 (T6 /Tt6 )γ6 /(γ6 −1)
DFguess =
Cu6 = Cu5 SC
DF
ρ6 = p6 /R/T6
Cm6 = ṁ/ρ6 /A6
2 2
C6 = (Cu6 + Cm6 )0.5
DF = C5 /C6

FALSE
abs(DF/DFguess − 1) < 10−5

TRUE

Return Channel Analysis

Figure 3.34: Alternative Return Bend Analysis Procedure

74
 Figure 3.35: IGV Geometry [12]

procedure similar to Section 3.1.4. Finally, a guess of outlet total pressure is done as,

2
pt0guess = pt−1 − 0.5(ω̄p + ω̄s )ρ−1 C−1 (3.171)

and iterated with Equation (3.170) to converge to a final pt0 .

75
 ṁ, Pt−1 , Tt−1
α−1 , B−1 , zIGV , cIGV
r−1s , r−1h , r0s , r0h , α0s , α0h

Some geometrical calculations

2 2
A−1 = π(r−1s − r1h )
rmean = 0.25(r−1s + r−1h + r0s + r0h )
smean = 2πrmean /zIGV
hmean = 0.5(r−1s − r−1h + r0s − r0h )
α0 = 0.5(α0s + α0h )
2 2
A0 = π(r0s − r0h )

- Calculate Profile loss by Ainley, ω̄p ,

and Secondary loss by Dunham, ω̄s
- Inlet calculation similar to Section 3.1.4

B0 = (1 − B−1 )[(r−1s − r−1h )/(r0s − r0h )]

Tt0 = Tt−1
2
pt0guess = pt−1 − (ω̄p + ω̄s )ρ−1 C−1 /2

ρ0guess = pt0guess /R/Tt0

cp0 = cp (Tt0 )

Cm0 = ṁ/ρ0guess /A0 /(1 − B0 )

C0 = Cm0 / sin α0
T0 = Tt0 − C02 /2/cp0 ρ0guess pt0guess
cp0 = cp (T0 ) and γ0 = γ(T0 ) = ρ0 = pt0
γ0 /(γ0 −1)
p0 = Pt0guess (T0 /Tt0 )
ρ0 = P0 /R/T0

FALSE
abs(ρ0 /ρ0guess − 1) < 10−5
FALSE

TRUE
Re0 = ρ0 C0 cIGV /µ(T0 )
pt0 = (pt−1 + p0 (ω̄p + ω̄s )(Re0 /2/105 )−0.2 )/ ... abs(pt0 /pt0guess − 1) < 10−5
((ω̄p + ω̄s )(Re0 /2/105 )−0.2 + 1)

TRUE
END

Figure 3.36: IGV Analysis Procedure

76
 CHAPTER 4

VALIDATION CASES

4.1 Eckardt O-rotor Stage

The Eckardt O-rotor stage consists of a rotor with 20 radial blades with no backsweep
and a long vaneless diffuser with hyperbolic shroud. The detailed geometry of the
stage can be found in references [18], [19] and Table 4.1.

As it can be seen from Figure 4.1, slope of the performance curves over the stall
region is estimated poorly. Klausner and Gampe [18] explains this behavior as
follows. Since the impeller blades have no backsweep, Equation (3.11) reduces to
IB = σ for the no inlet swirl case. Therefore, the dependency of blade work input on
mass flow rate is removed for this case, which leads to an overestimation of blade
work input at high mass flow rates, causing early impeller stall. This overestimation
of blade work input also produces high Cu2 velocities and low α2 angles, which
contributes to the diffusion losses at the vaneless diffuser. It can also seen from
Table 4.2 that the prediction of Cu2 is poor, especially at high mass flow rates.
However, results are consistent with a reference solution which also uses a one zone
model. Figure 4.3 shows the contributions from different loss components. As it can
be seen from this figure, the incorrect gradient is caused by the parasitic losses and
vaneless diffuser losses. Despite this poor behavior, the error on pressure
characteristics reaches approximately 4% and the error on efficiency characteristics
reaches 12% at high speeds.

Variation of total pressure, static pressure and flow angle along the vaneless diffuser is
plotted in Figure 4.2 to assess the performance of vaneless diffuser analysis. Trends

77
 Table 4.1: O-rotor Stage Geometry [18]

Impeller
Inlet Throat Tip Overall
r1h [mm] 45 Ath [mm2 ] 32080.1 r2 [mm] 200 zF B 20
r1s [mm] 140 βth [◦ ] 39.76 b2 [mm] 26 zSB 0
β1h [◦ ] 57.12 perith [m] 4.422 β2 [◦ ] 90 Lb [mm] 202.26
β1 [◦ ] 33.83 αc2 [◦ ] 86.15 Lf b [mm] 171.26
β1s [◦ ] 26.75 tb2 [mm] 1.08 Lsb [mm] 0
αc1 [◦ ] 1.68 scl [mm] 0.372
κm1 [rad/m] -2.49 sd [mm] 0.372
tb1 [mm] 2.11
Vaneless diffuser
r3 [mm] 340
Type Hyperbolic shroud

are captured quite well although there is a slight shift due to the incorrect initial
condition given by the impeller exit.

4.2 Radiver Stage

The Radiver Stage consists of an impeller with 15 backswept blades and a diffuser of
different configurations. The detailed geometry can be found in references [18, 20,
21] and Table 4.3.

For one configuration, it has a short vaneless passage followed by a wedge type
vaned diffuser. Figure 4.4 gives the pressure ratio map for this configuration.
Unfortunately, no complete efficiency map is available in the literature. The pressure
ratio characteristics is estimated well except the choking regions at low speeds. A
much more abrupt choking than the real case is obtained. The error of the choke
estimation reaches 8% at low speeds. The reason for this behavior is probably due to
the incorrect estimation of blockage at the vaned diffuser throat. In fact, removing
this blockage and resolving the stage, Figure 4.5 is obtained. As it can be seen from

78
 (a) P Rtt vs. ṁcorr

(b) ηtt vs. ṁcorr

Figure 4.1: O-rotor Performance Map

79
 (a) Total and Static Pressure

(b) Flow angle

Figure 4.2: O-rotor, Distributions along Vaneless Diffuser

80
 Table 4.2: O-rotor Impeller Exit Comparison

Reference Error (%) by Error (%) by

Experiment Developed
Solution reference Developed
[18] Code
[18] [18] Code
ṁ [kg/s] 6.07 5.32 6.07 5.32 6.07 5.32 6.07 5.32 6.07 5.32
Tt2 [K] 364 363.5 363.3 363.3 362.6 362.7 0.192 0.06 0.38 0.23
ηs,imp 0.959 0.951 0.961 0.962 0.95 0.948 0.2 1.16 0.94 0.315
C2 /U2 0.97 0.981 0.967 0.945 0.967 0.945 0.3 3.7 0.3 3.7
Cu2 /U2 0.819 0.899 0.877 0.877 0.876 0.876 7.08 2.45 6.96 2.55
W2 /U2 0.502 0.407 0.424 0.371 0.43 0.376 15.5 8.85 14.3 7.62
α2 [◦ ] 29.8 23.7 24.83 21.78 25.13 22.08 16.7 8.1 15.67 6.84

Figure 4.3: Loss Contributions

81
 Table 4.3: Radiver Stage Geometry [18]

Impeller
Inlet Throat Tip Overall
r1h [mm] 30 Ath [mm2 ] 8668.28 r2 [mm] 135 zF B 15
r1s [mm] 72.9 βth [◦ ] 40.72 b2 [mm] 11.1 zSB 0
β1h [◦ ] 66.596 perith [m] 1.701 β2 [◦ ] 52.36 Lb [mm] 139.19
β1 [◦ ] 43.881 αc2 [◦ ] 88.28 Lf b [mm] 122.77
β1s [◦ ] 33.435 tb2 [mm] 1.68 Lsb [mm] 0
αc1 [◦ ] 7 scl [mm] 0.585
κm1 [rad/m] -5.22 sd [mm] 0.585
tb1 [mm] 0.92
Vaneless diffuser
r3 [mm] 139 (345 for the vaneless configuration)
Type Constant width
Vaned diffuser
Inlet Throat Tip Overall
β3 [◦ ] 19.97 Ath [mm2 ] 3919 r4 [mm] 277.62 Zd 23
tb3 [mm] 0.45 hth [mm] 17.25 b4 [mm] 11.1 Lb [mm] 183.69
perith [m] 1.304 β4 [◦ ] 58.35
tb4 [mm] 24.69

this figure, choking behavior is improved a lot.

Figure 4.6 gives the pressure ratio characteristics for another configuration, which
has a long vaneless diffuser. Again no efficiency data is available. A similar behavior
with O-rotor stage is obtained, which also has a long vaneless diffuser. Klausner and
Gampe [18] again explains this behavior with the small α2 at the impeller exit at low
mass flow rates which causes large diffusion losses at the long vaneless diffuser. The
errors on pressure ratio estimation reaches 7.5% at high speeds.

Figure 4.7 presents the total pressure and absolute Mach number curves against ṁ at
28541 rpm, compared with an experimental result and a meanline solution by the
reference [19] for the configuration with vaned diffuser. A good match with the

82
 Figure 4.4: Radiver Performance Map with Vaned Diffuser, P Rtt vs. ṁcorr

experimental case is obtained although deviations occur in the choking range due to
abrupt choking behavior.

4.3 NASA CC3 Stage

The NASA CC3 Stage consists of an impeller with 15 main and 15 splitter blades
having high backswept and a vaned diffuser of wedge type with 24 vanes. It has
also a configuration with a vaneless diffuser. The detailed geometry can be found in
references [18, 22, 23] and Table 4.4.

As it can be seen from Figure 4.8, a quite good estimation of both pressure ratio
and efficiency characteristics is obtained for the configuration with vaned diffuser.
Errors for the efficiency estimation are below 3% and errors for choking estimation
are below 1%.

Figure 4.9 shows the estimation of pressure ratio and efficiency characteristics for
the configuration with vaneless diffuser. The solution seems better compared to the
solutions obtained for the vaneless diffuser configurations of O-rotor stage and

83
Figure 4.5: Radiver Performance Map with Vaned Diffuser, Vaned Diffuser Throat
Blockage Removed, P Rtt vs. ṁcorr

Figure 4.6: Radiver Performance Map with Vaneless Diffuser, P Rtt vs. ṁcorr

84
 (a) Total Pressure

(b) Absolute Mach

Figure 4.7: Radiver, Vane Diffuser Exit

85
 Table 4.4: NASA CC3 Stage Geometry [18]

Impeller
Inlet Throat Tip Overall
r1h [mm] 41.37 Ath [mm2 ] 17499.5 r2 [mm] 215.2 zF B 15
r1s [mm] 104.95 βth [◦ ] 52.394 b2 [mm] 17.05 zSB 15
β1h [◦ ] 50.83 perith [m] 2.556 β2 [◦ ] 40 Lb [mm] 260.26
β1 [◦ ] 41.883 αc2 [◦ ] 86.15 Lf b [mm] 250.927
β1s [◦ ] 33.464 tb2 [mm] 3.0 Lsb [mm] 172.47
αc1 [◦ ] 1.68 scl [mm] 0.372
κm1 [rad/m] -2.49 sd [mm] 0.372
tb1 [mm] 3.0
Vaneless diffuser
r3 [mm] 232.41 (253.936 for the vaneless configuration)
Type Constant width
Vaned diffuser
Inlet Throat Tip Overall
β3 [◦ ] 12.23 Ath [mm2 ] 5980.798 r4 [mm] 362 Zd 24
tb3 [mm] 0.419 hth [mm] 14.09 b4 [mm] 17.05 Lb [mm] 242.05
perith [m] 1.49 β4 [◦ ] 56
tb4 [mm] 38.38

86
Radiver stage. This is due to the fact that vaneless diffuser of this stage is
comparatively short and impeller blades have high backsweep. Choking losses at the
highest speed are overestimated causing some large deviations in this region. The
deviations reach 17% for the pressure characteristics and 10% for the efficiency
characteristics. However, the deviations for the rest of the operating range are below
3%.

4.4 Jones Stage

The Stage of Jones consists of an impeller with 17 main and 17 splitter blades having
no backswept and an airfoil type vaned diffuser with 41 vanes. The detailed geometry
can be found in references [18, 24] and Table 4.5. The pressure ratio of the stage is
higher than the limits mentioned by Aungier [13]; however, it is included to see how
the analysis behave outside its limits.

As it can be seen from Figure 4.10, a good estimation of both characteristics

obtained besides the regions of choke for the 95% and 90% speeds, where some
overestimation of the choking flow rate is obtained. Maximum deviation of choking
mass flow is around 2.5% at these speeds, which causes some large deviations in
efficiency characteristics in the choking regions.

4.5 A Case with a Return Channel

Since there is no well defined test case for a stage with a return system in the
literature, a commercial meanline software named as Compal, which belongs to
Concepts NREC Inc., is used to create a stage with return channel and compare the
results with the developed code.

The stage shown in Figure 4.11 is created in Compal. It has an impeller with a
rotational speed of 42000 rpm and a backsweep angle of 60◦ (from tangential
direction). A wedge type diffuser is employed and a return system is included.

The obtained results are shown in Figures 4.12 and 4.13. The deviations are small

87
 (a) P Rtt vs. ṁcorr

(b) ηtt vs. ṁcorr

Figure 4.8: NASA CC3 with Vaned Diffuser Performance Map

88
 (a) P Rtt vs. ṁcorr

(b) ηtt vs. ṁcorr

Figure 4.9: NASA CC3 with Vaneless Diffuser Performance Map

89
 Table 4.5: Jones Stage Geometry [18]

Impeller
Inlet Throat Tip Overall
r1h [mm] 30.48 Ath [mm2 ] 5788.44 r2 [mm] 124.46 zF B 17
r1s [mm] 67.31 βth [◦ ] 39.87 b2 [mm] 5.08 zSB 17
β1h [◦ ] 51.9 perith [m] 1.701 β2 [◦ ] 90 Lb [mm] 114.175
β1 [◦ ] 36.525 αc2 [◦ ] 82 Lf b [mm] 102.801
β1s [◦ ] 30 tb2 [mm] 1.1 Lsb [mm] 59.3
αc1 [◦ ] 2.1 scl [mm] 0.55
κm1 [rad/m] -3.5 sd [mm] 0.55
tb1 [mm] 2
Vaneless diffuser
r3 [mm] 130.685
Type Constant width
Vaned diffuser
Inlet Throat Tip Overall
β3 [◦ ] 24.83 Ath [mm2 ] 1650 r4 [mm] 177 Zd 41
tb3 [mm] 0.48 hth [mm] 7.01 b4 [mm] 5.08 Lb [mm] 81.54
perith [m] 1.18 β4 [◦ ] 45.24
tb4 [mm] 0.9

90
 (a) P Rtt vs. ṁcorr

(b) ηtt vs. ṁcorr

Figure 4.10: Jones Stage with Vaned Diffuser Performance Map

91
 Figure 4.11: The Stage with a Return System

except in choking regions. It is not clear how the calculations are done for the throat
parameters or how the choking losses are calculated for the vaned diffuser within
Compal. The deviations within the choking region may be caused by the differences
in these calculations from the developed code.

92
 Figure 4.12: Efficiency Map for the Stage with a Return System

Figure 4.13: Pressure Ratio Map for the Stage with a Return System

93
94
 CHAPTER 5

CONCLUDING REMARKS

The aim of this thesis is to develop a computer program that can be used to estimate
the performance of radial compressors using mean-line method. For this purpose,
each stage component is analyzed individually using an extensive set of models for
flow losses, blockage and other important flow phenomena.

Two famous approaches to the impeller analysis, namely the one zone model of R.
Aungier [13] and two zone model of D. Japikse [4], are investigated and it is
observed that the one zone model performs better at off design points if the two zone
model is used together with the primitive TEIS model. Therefore, the one zone
model is preferred for the impeller analysis. The procedure given by Herbert [12]
and empirical correlations given by Aungier [13] are used to create the impeller
geometry and calculate some preliminary values for the impeller parameters needed
by the impeller analysis. One dimensional conservation equations are employed for
the vaneless diffuser analysis along with the loss models for diffusion, curvature,
wall friction and boundary layer growth. A simple marching technique is used to
solve these equations. Vaned diffuser and return channel are analyzed using the loss
models given by Aungier [13] along with minimum loss deviation model of Howell
[44] and off-design deviation model of Johnsen and Bullock [45]. The empirical
correlations and design limitations given by Aungier [13] and the procedure given by
Johannes [17] are used to size the vaned diffuser and calculate some preliminary
values for the vaned diffuser parameters needed by the vaned diffuser analysis. IGV
is analyzed using the profile loss coefficient of Ainley and Mathieson [46] and
secondary loss coefficient of Dunham [47] along with the procedure given by

95
Herbert [12].

Four different cases are investigated for the validation of the written code and it is
observed that the analysis procedure performs well for the stages with backswept
impellers and vaned diffusers. For the stages with radial blades and long vaneless
diffusers, an overestimation of the diffusion losses is observed at low mass flow rates.
This overestimation is due to the low α2 values at the impeller exit which reduces the
diffusion efficiency of the vaneless diffuser. An overestimation of the blockage at the
impeller and vaned diffuser throat is also observed for some cases.

As a future work, more test cases should be investigated, especially the ones with
a return channel. Vaneless diffuser diffusion loss correlation should be improved to
handle the cases with long vaneless diffuser and radially bladed impellers. Throat
blockage correlations should also be revisited to improve the accuracy of the analysis
within the choking region.

96
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Micro Gas Turbine Applications. PhD thesis, Dellft Technical University, 2012.

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losses and limiting blade loadings in axial-flow compressor blade elements.
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modeling. Master’s thesis, Brigham Young University, 2007.

[41] K. Im. Development of a Design Method for Centrifugal Comporessors. PhD

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[42] P.M. Came. The development, application and experimental evaluation of a

design procedure for centrifugal compressors. Proceedings of IMechE, 1978.

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[43] J.D. Stanitz. One-dimensional compressible flow in vaneless diffusers of radial
and mixed-flow centrifugal compressors, including effects of friction, heat
transfer and area change. NACA TN 2610, 1952.

[44] A.R. Howell. Development of british gas turbine unit. ASME, 1947.

[45] I.A. Johnsen and R.O. Bullock. Aerodynamic design of axial flow compressors.
NASA SP-36, 1965.

[46] D.G. Ainley and G.C.R. Mathieson. A method of performance estimation for
axial-flow turbines. ARC R and M 2974, 1957.

[47] J. Dunham. A review of cascade data on secondary losses in turbines. Journal

of Mechanical Engineering Science, 1970.

[48] E. Goos and A. Burcat. Third millenium ideal

gas and condensed phase thermochemical database.
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100
 APPENDIX A

THERMODYNAMIC MODELS

A.1 Gas Model

Fourth degree NASA coefficients [48] are used for the temperature dependent cp . It
is given as,
cp = R (a1 + a2 T + a3 T 2 + a4 T 3 + a5 T 4 ) (A.1)

where R is the gas constant of air (R = 287.062 j/kg/K) and coefficients of the
polynomial are given in Table A.1. The term in the parenthesis is dimensionless.
Knowing the gas constant, R, and the specific heat, cp , specific heat ratio is given by,

γ = cp /(cp − R) (A.2)

Sutherland’s formula is used to update the viscosity with temperature. It is given as,

T 383.55
µ = 1.716 ∗ 10−5 ( )1.5 (A.3)
273.15 T + 110.4

where T is in K and µ is in kg/m/s.

Table A.1: NASA polynomial coefficients [48]

T > 1000 T < 1000

a1 = 3.08792717E + 00 a1 = 3.56839620E + 00
a2 = 1.24597184E − 03 a2 = −6.78729429E − 04
a3 = −4.23718945E − 07 a3 = 1.55371476E − 06
a4 = 6.74774789E − 11 a4 = −3.29937060E − 12
a5 = −3.97076972E − 15 a5 = −4.66395387E − 13

101
 (a) Simple Axial Inlet (b) Curved Inlet (c) Axial IGV

(d) Radial IGV (e) 2D Impeller (f) Shrouded Impeller

Figure A.1: Assumed Inlet Blockage Values [4]

A.2 Inlet Blockage

Inlet blockage values which can be used during impeller inlet analysis are given in
Figure A.1

A.3 Skin Friction Coefficient

Wall friction is one of the loss mechanisms common to all stage components and a
general formulation of the skin friction coefficient is needed covering all flow
regimes. [13] The formulation used in this thesis is based on the generalized pipe
friction data, where the skin friction coefficients are correlated as a function of
Reynolds number based on pipe diameter. It is given as,

Red = ρV d/µ (A.4)

The usual practice is to replace d with dH for general passages. Skin friction
coefficients are also a function of wall surface finish; however, its effect is neglected
in this thesis.

For Red < 2000, the flow is laminar and the skin friction coefficient is calculated as
[13],
cf,l = 16/Red (A.5)

102
For Red > 4000, the flow is turbulent and the skin friction coefficient is calculated as
[19],
q
1/ 4cf,t = −1.8 log10 (6.9/Red ) (A.6)

The transition from laminar to turbulent is modeled as weighted averages of the above
calculated values. It is given by,

cf = cf,l + (cf,t − cf,l )(Red /2000 − 1) (A.7)

A.4 Hydraulic Diameter

Hydraulic diameter is given by,

CrossSectionalArea
dH = 4 (A.8)
W ettedP erimeter
From throat to tip, the mean hydraulic diameter of the impeller can be written as,
4Ath 1 4A2 sin β2 1
dH = + (A.9)
perith 2 2πd2 sin β2 + 2zb2 − 2ztb2 2
After some algebraic manipulations,
2Ath A2 sin β2
dH = + (A.10)
perith πd2 sin β2 + zb2 − ztb2
where perith ≈ 2πdth sin βth + 2zF B hth − 2zF B tb1 . Similarly for the vane diffuser,
2Ath A4 sin β4
dH = + (A.11)
perith πd4 sin β4 + zV D b4
The same formula can also be used for the return channel.

103
104
 APPENDIX B

NUMERICAL METHODS

B.1 Simpson’s Rule

Simpson’s rule is a numerical integration method that employs high order

polynomials (2nd order or higher) to estimate the value of a definite integral. If 2nd
order polynomials are used as shown in Figure B.1, the following Simpson’s formula
is obtained,
Z b
∆x
f (x) dx ≈ (y0 + 4y1 + 2y2 + 4y3 + 2y4 + ... + 4yn−1 + yn ) (B.1)
a 3

B.2 Newton-Raphson Method

If an initial guess of a root of the function, f (x), is xi , a tangent extending from the
point [xi , yi ] can be drawn and the point where this tangent crosses the x-axis usually

Figure B.1: Simpson’s Rule [49]

105
 y = f (x)
Calculate xcorr for ytar

Estimate xguess
Define a step ∆x

x1 = xguess
x2 = xguess + ∆x

y1 = f (x1 ) dE/dx = (y2 − y1 )/∆x

y2 = f (x2 ) x1 = x1 − E1 /(dE/dx)
E1 = y1 − ytar x2 = x1 + ∆x

FALSE
E1 < 

TRUE
xcorr = x1

Figure B.2: Newton Raphson Method with Numerical Differentiation

gives an improved estimate of the root, xi+1 . [50] That is,

f (xi )
xi+1 = xi − (B.2)
f 0 (xi )
If the derivative of the function, f 0 (xi ), can not be calculated directly, it can be
calculated numerically. The procedure given in Figure B.2 uses numerical
differentiation to find the root of y = f (x). The target y value is ytar and xcorr is the
corresponding root.

106