DEVELOPMENT OF A DESIGN METHOD
FOR CENTRIFUGAL COMPRESSORS
By
Kangsoo Im

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2012

ABSTRACT
Development of a Design Method for Centrifugal Compressors
by
Kangsoo Im

This dissertation is the development of a centrifugal compressor design and analysis code that
can be used to inexpensively design and analyze the performance of a centrifugal compressor. It
can also be used to match the components of centrifugal compressor and to integrate and
optimize system performance. The design system is being developed also with the intent to
reduce the time taken to experimentally match a centrifugal compressor with the operational
environment, a task that is key to process industry application. The design and analysis code will
use both one and two-dimensional thermo-fluid equations to analyze the compressor and its
components. For each compressor component, the tool calculates the velocities, pressures,
temperatures, pressure losses, energy transfer and transformation, and efficiencies for a specified
set of compressor geometry, atmospheric conditions, rotational speed, and fluid mass flow rate.
The design-tool will be based on and include established loss models found in literature and
extensive industrial experimental data. The other main feature of the design-tool will be its
ready and easy integration and interaction with other CFD, CAM, and Stress Analysis
commercial packages.

ACKNOWLEDGEMENTS

I wish to express my gratitude to my advisor Dr. Abraham Engeda, who has supported
and guided me in my study and personal life. I specially thank him for encouragement whenever
I was discouraged. Sincere thanks to Professors Craig Somerton, Norbert Mueller, Wei Liao for
their advice, discussion, and interest in this work.
I would like to express my sincere appreciation to The Solar Turbines Incorporated. I also
want to express my gratitude to Mike Cave for support and advice during internships and other
research. In addition I am also thankful to Min Ji and Russ Marechale for experiment data and
advice.
Finally, I am very thankful to my lovely wife, Hye-Young, and my family, parents,
parents in law, Su-Kyoung, Jung-Min and Yu-Kyoung for support and pray. And I would like to
express special thanks to my friend, Hyun-Jin and his family. Most of all, I greatly give thanks to
God for all I am. I am Yours.

iii

TABLE OF CONTENTS

LIST OF TABLES…………………………………………...………………………………….vi
LIST OF FIGURES…………………………………………………………………………….vii
NOMENCLATURE………………………………………………………………………….…x
1. FUNDAMENTALS OF CENTRIFUGAL COMPRESSORS..…………………………… 1
1.1 INTRODUCTION…………………………………………………………………………….1
1.2 THE IMPELLER……………………………………………………………………………...3
1.3 THE RADIAL DIFFUSSER………………………………………………………………….7
1.4 OBJECTIVE OF CURRENT WORKING……………………………………………………9
1.5 THE NEED FOR DESIGN METHOD IN CENTRIFUGAL COMPRESSORS…………....11
2. PROCEDURES AND MATHEMATICAL FORMULATIONS USED IN DEVELOPING
THE DESIGN METHOD FOR CENTRIFUGAL COMPRESSORS………………………13
2.1 TWO-ZONE MODELING…………………………………………………………………..13
2.1.1 TEIS MODEL……………………………………………………………………………17
2.2 EMPRICAL BASED METHOD…………………………………………………………….21
2.2.1 IMPELLER WORK INPUT……………………………………………………………….21
2.2.2 RELATIVE PRESSURE LOSSES WITHIN IMPELLER………………………………...26
2.2.3 VANELESS DIFFUSER ANALYSIS…………………………………………………….30
3. 1D ANALYSIS CODE VALIDATION WITH TESTED IMPELLER DATA………….32
3.1 COMPARISON RESULTS………………………………………………………………….33
4. CORRECTED TWO ZONE WITH TEIS MODEL………………………………………46
4.1 PROBEM DEFINITION IN DIFFUSION RATIO IN TEIS MODEL……………………..46
4.2 LOSS MODEL IN EMPIRICAL MODEL………………………………………………….53
4.3 CORRECTED DIFFUSION MODEL IN TWO-ZONE MODEL……………….………….68
4.4 CORRECTED DIFFUSION MODEL RESULTS…………………………………………..70
4.5 CONCLUSION…………………………………………………..…………………………..84
5. RE-DESIGN IMPELLER……………………………………………………….…..………86
5.1 OBJECTIVE………………………………………………………………………..………..86
5.2 ORIGINAL IMPELLER………………………………………………………………….….86
5.2.1 1D PERFORMANCE ANALYSIS ON ORIGINAL IMPELLER………..…….…………88
5.3 RE-DESIGN USING 1D ANALYSIS CODE…………………………………….…………91
5.3.1 DIFFUSION RATIO…………………………………………………………………..…..93
5.3.2 MERIDIONAL AND BLADE CURVE DESIGN………………………..………………95
5.4 1D PERFORMANCE ANALAYSIS ON THE 4 CASES……………………………….100
5.5 CFD RESULTS……………………………………………………………………..…….104
5.6 CONCLUSION……………………………………………………………………………108
iv

6. CONSIDERATION OF AN AXIAL LENGTH OF IMPELLER………...……………110
6.1 DESIGN CASES…………………………………………………………………………112
6.2 CFD ANALYSIS…………………………………………………………...…………….117
6.3 RESULTS………………………………………………………………………………..…119
7. CONCLUSIONS……………………………………………………….………………….138
7.1 1D PERFORMANCE ANALYSIS ON TESTED IMPELLER………………….….……..138
7.2 TWO-ZONE MODEL WITH CORRECTED DIFFUSION MODEL………….…….……139
7.3 RE-DESIGN IMPELLER…………………………………………………………..………138
7.4 CONSIDERATION OF AXAIL LENGTH FOR IMPELLER DESIGN……………….….141
7.5 CONTRIBUTION……………………….……………………………………….…...…...141
7.6 RECOMMENDATION…………………………………………………………...………..142
REFERENCES………………………………………………………………………………..144

v

LIST OF TABLES
Table 2-1 Suggested values for primitive TEIS model…………………………………....……..20
Table 3-1 Tested Impeller information…………………………...………………………...........32
Table 4-1 Pressure recovery effectiveness used in this study………………………………....…70
Table 5-1 Information of the original impeller……………………………...………….………..87
Table 5-2 Cases for impeller re-design designate diffusion ratio of 0.6……...……….………...95
Table 6-1 Mach number and absolute flow angle range for unstable flow condition...…..........112
Table 6-2 1D geometry conditions for the cases……......…….…………………...…………113
Table 6-3 Axial lengths along with respect to factor of a……………………...………..……114

vi

LIST OF FIGURES
Figure 1-1 Configuration of a single stage centrifugal compressor……..………………………..2
Figure 1-2 h-s diagram for the centrifugal compressor stage………….………………………….3
Figure 1-3 (a) Velocity triangles for an idealized impeller………………………………………..6
Figure 1-3 (b) Effect of the impeller exit blade angle………………….…………………………6
Figure 1-4 The idealized jet-wake model………………………………………………………11
Figure 2-1 TEIS conceptual model………………………..……………………………………..18
Figure 3-1 Meridional shape of two tested impellers……..……………………………………..33
Figure 3-2 Work factor and head coefficient result according to flow coefficient using Two Zone
model with TEIS and Ron Aungier’s model (Impeller #1 13k, 19k, 21k
rpm)……………………...….……………………………...………………………..34
Figure 3-3 Work factor and head coefficient result according to flow coefficient using Two Zone
model with TEIS and Ron Aungier’s model (Impeller #2 13k, 19k, 21k
rpm)……………………………………………..…………….……………………..37
Figure 3-4 Absolute flow angle according to flow coefficient along the various rotational speeds
(impeller #1, 13k, 19k, 21k rpm)……………………………………….……….…..40
Figure 3-5 Absolute flow angle according to flow coefficient along the various rotationsl speeds
(impeller #2, 13k, 19k, 21k rpm)………………………….………….……………..43
Figure 4-1 TEIS model characteristic for estimating diffusion ratio……………..…..…………46
Figure 4-2 Comparing head coefficient results using Two-Zone model and test results (Impeller
#1, 13k, 19k, 21k rpm)………………………………………………………………48
Figure 4-3 Comparing head coefficient results using Two-Zone model and test results (Impeller
#2, 13k, 19k, 21k rpm)………………………………………………...…………….51
Figure 4-4 Loss models in impeller #1 and impeller #2 with various speeds……………....……55
Figure 4-5 Effectiveness calculated by corrected diffusion ratio (Impeller #1, Impeller #2)…....71
Figure 4-6 Head coefficient and work factor calculated by corrected diffusion model (Impeller
#1, 13k, 19k, 21k rpm)……………………………………………………..………..73

vii

Figure 4-7 Head coefficient and work factor calculated by corrected diffusion model (Impeller
#2, 13k, 19k, 21k rpm)……………………………………………………...……….76
Figure 4-8 Absolute flow angle calculated by corrected diffusion model (Impeller #1, 13k, 19k,
21k rpm)………………………………………………………………………….….79
Figure 4-9 Absolute flow angle calculated by corrected diffusion model (Impeller #2, 13k, 19k,
21k rpm)………………………………………………………..……………………82
Figure 5-1 Meridional shape of the original limpeller with wide and smaller tip width……...…88
Figure 5-2 Comparing work factor, head coefficient and efficiency, the 1D analysis results with
test data (Original impeller)…………………………………………………...…….89
Figure 5-3 Comparing absolute flow angle, 1D analysis with test data (Original impeller)…...90
Figure 5-4 Comparing diffusion ratio, 1D analysis, with test data (Original impeller)……..…91
Figure 5-5 Impeller exit flow coefficient, impeller slip factor, work input factor and blade angle
(from tangent) as function of the inlet flow coefficient (empirically determined
values for the best efficient point)………….………………………….…………….92
Figure 5-6 The diffusion ratio according to the impeller tip width and blade exit angle, calculated
by 1D analysis code using Ron Aungier’s method………………………………….94
Figure 5-7 Meridional shape of all the cases…………………………………………...………..97
Figure 5-8 Blade angle distribution of all the cases according to m`%.........................................98
Figure 5-9 Blade theta angle distribution of all the cases according to m`%................................99
Figure 5-10 Area distribution (A/A1) of all the cases according to m`%....................................100
Figure 5-11 Efficiency and head coefficient according to flow rate (Case1, Case2, Case3,
Case4)…………………………………………………………………………...….102
Figure 5-12 Diffusion ratio and absolute flow angle at the diffuser inlet………………………103
Figure 5-13 Parasitic work input (recirculation, disk friction and leakage loss).………..…104
Figure 5-14 3D shape of re-designed impeller…………………………………………………105
Figure 5-15 Grid generated for re-designed impeller………………………………………..…105
Figure 5-16 Comparison of the head coefficient and efficiency of 1D analysis and CFD
result………………………………………………………………………………..107
viii

Figure 5-17 Comparison of the diffusion ratio and absolute flow angle, results of 1D analysis and
CFD…………………………………………………………………………...……108
Figure 6-1 Meridional curves of the all cases…………………….…………………….…….115
Figure 6-2 Camber-line vs. Meridional distance……………………….…………………….116
Figure 6-3 3D impeller shape………………………………………..…………..………….…117
Figure 6-4 Mesh elements at 50% span…………………………………………….………118
Figure 6-5 Domains used in the analysis………………………….…………………………118
Figure 6-6 Absolute flow angle contour at 0% from the impeller exit (a=1.0, a=1.3, a=1.5)….120
Figure 6-7 Absolute flow angle contour at 3% from the impeller exit (a=1.0, a=1.3, a=1.5)….121
Figure 6-8 Absolute flow angle contour at 6% from the impeller exit (a=1.0, a=1.3, a=1.5)….122
Figure 6-9 Absolute flow angle contour at 9% from the impeller exit (a=1.0, a=1.3, a=1.5)….123
Figure 6-10 Absolute Mach number contour at 0% from the impeller exit (a=1.0, a=1.3,
a=1.5)…………………………………………………………………………...….125
Figure 6-11 Absolute Mach number contour at 3% from the impeller exit (a=1.0, a=1.3,
a=1.5)………………………………………………………………………...…….126
Figure 6-12 Absolute Mach number contour at 6% from the impeller exit (a=1.0, a=1.3,
a=1.5)………………………………………………………………………………127
Figure 6-13 Absolute Mach number contour at 9% from the impeller exit (a=1.0, a=1.3,
a=1.5)………………………………………………………………………………128
Figure 6-14 Static pressure contour at 0% from the impeller exit (a=1.0, a=1.3, a=1.5)………130
Figure 6-15 Static pressure contour at 3% from the impeller exit (a=1.0, a=1.3, a=1.5)…....…131
Figure 6-16 Static pressure contour at 6% from the impeller exit (a=1.0, a=1.3, a=1.5)…..…..132
Figure 6-17 Static pressure contour at 9% from the impeller exit (a=1.0, a=1.3, a=1.5)…..…..133
Figure 6-18 Mass flow averaged absolute flow angle, Mach number and static pressure after
impeller exit……………………………………………………………….…….…134
Figure 6-19 Hub to Shroud mass averaged absolute flow angle distribution at the impeller
exit…………………………………………………………………………….……136
ix

Nomenclature
B

Blockage

BL

Blade loading coefficient

b

Blade height [in]

C

Absolute fluid velocity

CM

Disk torque coefficient

CP

Pressure recovery coefficient

Deq

Equivalent diffusion factor

d

Diameter

dH

Hydraulic diameter

DR

Diffusion ratio

H

Head

h

Enthalpy

I

Work input coefficient

K

Blade angle design parameter

L

blade mean streamline meridional length

LB

Length of blade mean camberline

M

Mach number

m

Meridional distance

m

Mass flow rate [lb/s]

N

Impeller rotational speed [rpm]

Pv

Velocity pressure
x

p

pressure

R

Gas constant

Re

Reynolds number

t

Seal fin thickness

Q

Volume flow rate [cfm]

r

Radius [in]

U

Blade speed [ft/s]

v

Fluid velocity [ft/s]

W

Relative fluid velocity [ft/s]

Z

Number of blades dimensionless

α

Absolute velocity angle, measured from radial direction [degrees]

β

Relative velocity angle, measured from tangential direction [degrees]

δ

Incidence angle [degrees]

φ

Inlet flow coefficient dimensionless

φ2

tip flow coefficient

κ

blade angle with respect to meridional angle

κm

streamline curvature

λ

impeller tip distortion factor

ξ

distance along the blade mean camberline

ρ

gas density

η

Isentropic efficiency dimensionless

ψ

Isentropic head coefficient dimensionless

ω

Rotational speed [rad/s]
xi

ω

total pressure loss coefficient

Subscripts
1

Impeller inlet

2

Impeller exit

3

Diffuser inlet

4

vaneless diffuser exit

5

diffuser exit

B

a blade parameter

C

cover parameter

D

disk parameter

h

Hub

imp

Impeller

isen

Isentropic

L

leakage parameter

LE

Blade leading edge

M

Meanline

m

meridional direction

p

primary flow

ps

Pressure side

r

radial direction

s

Shroud, secondary flow

ss

Suction side

sta

Stage

xii

TE

Blade trailing edge

t

Blade tip, total thermodynamic condition

z

axial direction

θ

theta direction

xiii

1. FUNDAMENTALS OF CENTRIFUGAL COMPRESSOR

1.1.

INTRODUCTION
Flows in the impellers of radial flow centrifugal compressor are among the most complex

in turbomachinery. These flows occur in twisted trapezoidal passages. They are often transonic,
with shocks in the inducer section that could be accompanied by significant viscous and
secondary flow effects with the possibility of regions of separation. The complexities, however,
did not deter many researchers in the last fifty years from tackling very important problem areas
such as flow visualization using the laser Doppler anemometry and the production of highly
sophisticated computerized techniques that can handle the analysis of both a viscous and nonviscous, three-dimensional flow picture.
In spite of the above mentioned advancements, however, the aero-thermodynamic design
methodologies for radial flow centrifugal impellers do not appear to have kept pace with these
important developments, open literatures dealing with the design methodologies particularly
from industrial sources are still very scarce. It can be generally stated, however, that the primary
objective of the centrifugal impeller designer is to produce a detailed impeller geometry that
operates at a given rotational speed and can handle the design mass flow as efficiently as
possible. Effective modeling procedures, however, must also take into account the physical
conditions imposed on the working fluid as it passes through the passage such as diffusion levels,
relative inlet flow angles, relative outlet flow angles, etc. All of these parameters and their
effects on other dependent design parameters should be taken into account in any comprehensive
design method.
A single stage centrifugal compressor consists of an impeller (the rotating part), a diffuser
(non-rotating), and a volute. Inlet guide vanes are sometimes used in front of the rotor to direct
xiv

1. FUNDAMENTALS OF CENTRIFUGAL COMPRESSOR

1.1.

INTRODUCTION
Flows in the impellers of radial flow centrifugal compressor are among the most complex

in turbomachinery. These flows occur in twisted trapezoidal passages. They are often transonic,
with shocks in the inducer section that could be accompanied by significant viscous and
secondary flow effects with the possibility of regions of separation. The complexities, however,
did not deter many researchers in the last fifty years from tackling very important problem areas
such as flow visualization using the laser Doppler anemometry and the production of highly
sophisticated computerized techniques that can handle the analysis of both a viscous and nonviscous, three-dimensional flow picture.
In spite of the above mentioned advancements, however, the aero-thermodynamic design
methodologies for radial flow centrifugal impellers do not appear to have kept pace with these
important developments, open literatures dealing with the design methodologies particularly
from industrial sources are still very scarce. It can be generally stated, however, that the primary
objective of the centrifugal impeller designer is to produce a detailed impeller geometry that
operates at a given rotational speed and can handle the design mass flow as efficiently as
possible. Effective modeling procedures, however, must also take into account the physical
conditions imposed on the working fluid as it passes through the passage such as diffusion levels,
relative inlet flow angles, relative outlet flow angles, etc. All of these parameters and their
effects on other dependent design parameters should be taken into account in any comprehensive
design method.
A single stage centrifugal compressor consists of an impeller (the rotating part), a diffuser
(non-rotating), and a volute. Inlet guide vanes are sometimes used in front of the rotor to direct
1

the flow to the impeller inducer. The impeller is used to impart energy to the flow by increasing
the velocity and pressure of the fluid. The diffuser is used to convert the kinetic energy available
at the impeller exit into static pressure by decelerating the fluid. The diffuser is followed by a
volute, which collects the fluid from the periphery of the diffuser. Usually an exit cone is
connected to the volute exit to delivery the compressed fluid to the pipeline of desired
application. The exit cone is essentially a divergent device where the fluid is further diffused.
Figure below a typical configuration of a single stage centrifugal compressor.

throat

8
7

volute
5

Exit Cone

y



x



4
3

vaneless
diffuser

7
8

5

4
3
2







2

shroud
hub
1
impeller



1
Z

Fig. 1- 1 Configuration of a single stage centrifugal compressor

2

Z

P02

02

05

P05
P5

5
C

2/2
5

Enthalpy

C
P4

4

05s

2/2
2

P2

5s
2
4s
P01

2s
01

2/2
1

P1

C

Impeller

Vaneless
space

00

1
1s
Inlet
casing

Diffuser
channel

Entropy
Fig. 1- 2 h-s diagram for the centrifugal compressor stage

1.2. THE IMPELLER
The impeller is the rotating component of the centrifugal compressor stage, where energy
transfer of the compressor stage occurs.

h0c  h05  h01  h02  h01

(1-1)

3

The specific energy transfer can be derived from the velocity triangle at inlet and outlet
from the impeller. The rate of change of angular momentum will equal the sum of the moments
of the external forces, TQ. When the angular momentum theorem applied to an impeller, the
torque, TQ, is given by

TQ  m(r2Cu 2  r Cu1)
1

(1-2)

The energy transfer is given by the product of the angular velocity and the torque, given
by the Euler equation
mh0c  W  TQ  m(U 2Cu 2  U1Cu1)

(1-3)

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

2
2
2
U 2Cu 2  1 (U 2  C2  W2 )
2

(1-4)

2
2
U1Cu1  1 (U1  C1  W12 )
2

(1-5)

Combining (1-3), (1-4), and (1-5) results in the enthalpy rise in terms of velocity relations
2
2
2
2
2
h0c  1 (U 2  U1 )  (W12  W2 )  (C2  C1 ) 
2


(1-6)

The sum of the first and the second terms on the right hand side represents the increase in
static pressure; and the kinetic energy increase is shown in the last term. In an axial compressor
machine, there is no impeller tip speed variation (U2=U1=constant) explaining a higher enthalpy
rise in a radial compressor.

4

If we neglect inlet angular momentum, which is generally acceptable, the theoretical
enthalpy reduces to

h0c 

W
 U 2Cu 2
m

(1-7)

Then the effect of the impeller exit blade angle, β2b on the theoretical enthalpy rise
becomes

C
W
2
 U 2 (1  r 2 cot  2b )
m
U2

(1-8)

For a given value of impeller exit blade angle, β2b there is a linear relationship between
specific energy transfer and flow rate Theoretical enthalpy of impeller backward-curved blades
decreases as the flow rate increases, while that of impeller with the radial blades remains
constant The positive-slope condition can be unstable and cause surge. For this reason a
backward-curved blade impeller is generally preferred.

5

C2

U2=r2

Cr2
2

W2

2b

Cu2
C1

W1

1
R2

U1=r1

R1


Fig. 1-3 (a) Velocity triangles for an idealized impeller

.

W
.

2b>90o (Forward-curved)

mU 22

2b=90o (Radial)

1

2b<90o (Backward-curved)

Cr2/U 2
Fig. 1- 3 (b) Effect of the impeller exit blade angle
6

1.3.

THE RADIAL DIFFUSER
Diffusion occurs where velocity reduction occurs. Velocity is a vector quantity having

both magnitude and direction. In other words, diffusion can occur on an isolated surface or in a
duct.
In centrifugal compressors, energy is transferred to the fluid by the impeller. Even though
centrifugal impellers are designed for good diffusion within the blade passage, approximately
half of the energy imparted to the fluid remains as kinetic energy at the impeller exit. Therefore,
for an efficient centrifugal stage, this kinetic energy must be efficiently converted into the static
pressure. Thus, a diffuser, which is stationary and is located downstream of the impeller, is a
very important component in a centrifugal compressor.
Since over the years the demands on the centrifugal compressors increased for higher
pressure ratios and efficiency, different types of radial diffusers have been developed. These
different types of radial diffusers can be classified as the vaneless diffusers, the vaned diffusers,
and the low solidity vaned diffusers.

VANELESS DIFFUSERS
Vaneless diffusers consist of two radial walls that may be parallel, diverging, or
converging. The flow entering a vaneless diffuser has a large amount of swirl; the swirl angle

[  tan 1(Cr / Cu )] is typically between 10o - 30o. Thus, the tangential component of
momentum at low flow rates can be more than twice the radial component. The radial component
of the flow diffuses due to the area increase (conservation of mass), and the tangential
component diffuses inversely proportional to the radius (conservation of angular momentum).
However, the radial component has to surmount the radial pressure gradient for the tangential

7

component to diffuse continuously. When reverse flow of the radial boundary layer occurs, it is
not possible for the tangential component to continue diffusing, as this would imply a pressure
increase in one component and not in the other. Therefore, in such cases the breakdown of flow
occurs, and this can cause the diffuser to stall and produce other flow instabilities such as surge
or stall. However, backflow into the impeller is less frequent with vaneless diffusers than with
vaned diffusers where local pressure disturbs caused by vane pressure loading.
The vanelsess diffuser is widely used in automotive turbochargers because of the broad
operating range it offers. It is also cheaper to manufacture and more tolerant to erosion and
fouling than the vaned diffusers. However, the vaneless diffuser needs a large diameter ratio
because of its low diffusion ratio. The flow in a vaneless diffuser follows an approximate
logarithmic spiral path. The flow in a vaneless diffuser with a radius ratio of 2 and an inlet flow
angle of 6o makes a full revolution before leaving the diffuser. This will result in high friction
loss due to viscous drag on the walls and accordingly its pressure recovery is significantly lower
than is found with vaned diffusers.
Generally, the vaneless diffuser demonstrates lower pressure recovery by as much as 20%
and lower stage efficiency by 10% compared to a vaned diffuser.

VANED DIFFUSERS
The role of the vane in a vaned diffuser is to shorten the flow path by deswirling the flow,
allowing a smaller outlet diameter to be used. A vaneless space precedes the vaned diffuser to
help reduce flow unsteadiness and Mach number at the leading edge of the vanes so as to avoid
shock waves. A boundary layer develops and generates appreciable blockage at the vane leading
edge. In order to reduce this blockage, the vaneless space should be minimized until it doesn’t
give any unfavorable effects such as increase in noise level or pressure fluctuations due to
8

interaction of the impeller and diffuser. The flow exiting the impeller follows an approximate
logarithmic spiral path to the vane leading edge and is guided by the diffuser channels. The semivaneless space follows the vaneless space, ending in a passage throat, which may limit the
maximum flow rate in a compressor. The number of diffuser vanes has a direct bearing on the
efficiency. With a large number of vanes, the angle of divergence is smaller and the efficiency
rises until friction and blockage overcomes the advantage of more gradual diffusion.
Although the vaned diffuser typically exhibits higher pressure recovery, the flow range is
limited at a low flow rate due to vane stall. At high flow rates, flow choking at the throat may
also limit flow range.

1.4.

OBJECTIVE OF THE CURRENT WORKING
The objective of this study is to develop an integrated environment to design and analysis

centrifugal compressors. 1D sizing, 1D analysis, and 2D analysis of the centrifugal compressor
code, used to design centrifugal compressor, usually work independently. Generally, the design
and analysis procedure of centrifugal compressors includes 1D sizing, 1D analysis, determining
blade shape, 2D analysis, and 3D CFD. And this procedure is repeated until satisfying the
requirement of the customer or finding the optimized geometry. Many aero designers spend their
time to making centrifugal impellers since they have many variables dealing with. And they use
convenient commercial tools such as COMPAL or CENCOM to make design easy. The
programs use different 1d analysis code, Two Zone modeling [1], and Ron Aungier’s method [2].
Two Zone modeling is based on the jet and wake model, Dean and Senoo [3], as shown in Fig
1.4, which explains the impeller exit flow velocity profile. The Two Zone model is mostly based
on physics equations, although it uses loss models in friction and leakage. The Two Zone model

9

has high accuracy in the flow characteristics at the impeller exit. However, it requires variables
(deviation angle, mass fraction between jet, and wake and pressure recovery coefficients) to
calculate properties in impeller flow. And in the other method, Ron Aungier’s method, most of
the equations explain loss models and he uses empirical correlations for work coefficients. It
explains relative pressure loss models that are based on the entropy rise resulting in various loss
models. But this method must assume 3D path line for 1D analysis. The deviation angle is only
variable for the 1D analysis. Both of the two methods are mostly depend on slip factor
commonly. In order to design a high efficiency or maximum performance impeller, there are
variables the designer consider, impeller blade exit angle, impeller exit width and impeller exit
diameter, etc. And to get an impeller that work properly according to its goal, it is necessary to
determine these variables.
To determine 3D shape of an impeller, these 1D analysis and sizing has to be conducted
to reduce cost for 3D CFD works, which is taking long time or making prototype. It is efficient
to integrate design procedure, which are 1D sizing, 1D analysis, and 2D analysis. Although there
are already existing programs, such as COMPAL and CENCOM, house code is good for
accessibility to the other users.
For 1D analysis code, this study focuses on Two Zone modeling and Ron Aungier’s
method, used in COMPAL and CENCOM respectively. And both methods or selectively can be
used in 1D analysis code in this study.

10

Fig. 1- 4 The idealized jet-wake model (Dean and Senoo) [3]

1.5. THE NEED FOR DESIGN METHOD IN CENTRIFUGAL COMPRESSORS
When designing a centrifugal impeller, 1D analysis gives the approximate 1d sizing
results and a chance to reduce the cost for experiments. There are two 1d analysis methods that
are well known in the world, Two Zone modeling and Ron Aungier’s method. They are verified
through many experiments and other’s research about them. And they have their own
characteristics. There are basically two ways to do 1d analysis on an impeller: flow physics
based and empirical based. While the Two Zone model usually depends on flow physics, Ron
Aungier’s method uses both of them. Most 1D analysis usually depends on empirical
formulations because of the complicated impeller 3D shape. Although the Two Zone model uses
flow physics using 1D geometry condition, and effectiveness, it makes more accurate analysis
11

results. This can be explained in Two Elements In Series (TEIS) [1]. The TEIS model can make
more accurate performance predictions. It is composed of two elements: blade inlet to throat and
throat to blade exit. A first element may be used for the diffuser or nozzle according to high or
low flow and the second element is mostly a diffuser in character. According to the inlet portion
and the passage portion, the overall diffusion ratio can be made. And it is necessary to assume
the effectiveness for accurate performance prediction.
Ron Aungier’s 1D analysis method can be said to be more empirical then the Two Zone
model. Ron Aungier’s method starts with a fundamental thermodynamic relation for entropy.
Ron Aungier’s method calculates entropy rise in relative total pressure loss in flow within the
impeller. It uses various empirical equations about relative pressure loss for normal shock wave,
incidence loss at blade inlet, entrance diffusion loss, chocking loss at the impeller throat, blade
loading loss, hub to shroud loading loss, skin friction loss, distortion loss for mixing at the
impeller exit, clearance gap loss for open impeller and mixing loss at the impeller exit and,
supercritical Mach number loss. And it can calculate all thermodynamic conditions at the
impeller exit. The loss models used in the Ron Aungier’s method are mostly based on relative
velocity and geometry conditions. Although this method has high accuracy, some of the loss
models cannot be used in the case of approximate blade 3D geometry. In order to use Ron
Aungier’s model, a blade length and axial length should be assumed.

12

2. PROCEDURES AND MATHEMATICAL FORMULATIONS USED IN
DEVELOPING

THE

DESIGN

METHOD

FOR

CENTRIFUGAL

COMPRESSORS

2.1. TWO ZONE MODELING
The Jet and Wake model, the origin of the Two Zone model, is proposed first by Eckardt
[4]. Dean and Senoo[3] explain first that Jet flow is isentropic and wake flow contains all
flow losses and very low momentum as shown in Fig 1-4. After that Johnson and Moore[5]
show that while jet is almost isentropic, wake has a little mass flow. Japikse develops Two
Zone modeling supported by the Jet and Wake model [1]. Two Zone modeling has high
accuracy and contains equations about flow physics. Two Zone modeling refer to the Jet
flow as a primary isentropic core flow or primary flow and Wake flow as secondary flow.
Before introducing Two Zone modeling equations there are basic assumptions in the Two
Zone modeling [1].
1. The primary flow is nearly isentropic and deviates from the impeller blade.
2. The secondary flow contains all the impeller losses and is perfectly guided by the
impeller blade.
3. The impeller tip static pressure is the same for the primary and secondary zone, which
is the unloaded condition.
4. The primary flow and secondary flow do not mix in the impeller passage but mix
very rapidly after leaving the impeller exit.
5. The secondary mass flow fraction of the through-flow is in the range of about 0.15
and 0.25.

13

The following equations are written in ideal gas form for the primary flow.

T2 p  T1t 


p2 p 

p1t 

( 1)/ 

(2-1)

2
2
h2 p  W2 p 2  U 2 2  hT

(2-2)

2
hT  h00  U1C1t  c pT1t  C1t / 2  U1C1t

(2-3)

U 2  2 r2 N

(2-4)

h2 p  c pT2 p

(2-5)



2
Cm2 p  Wwp  U 2  C 2 p



2

(2-6)

2
2
C2 p  Cm2 p  C 2 p

(2-7)

2
T02 p  T2 p  C2 p 2C p

(2-8)

  1

 T02 p 
p02 p  p2 p 

 T2 p 



(2-9)

1    2 pW2 p Ageo cos 2 p  m(1   )

(2-10)

  ms m

(2-11)

  As / Ageo

(2-12)

2 p  p2 RT2 p

(2-13)

Ageo  2 r2b2  Zbb2t2b cos 2b

(2-14)

2 p  2b   2 p

(2-15)

14

C 2 p  U 2  Cm2 p tan 2 p



Or: 2 p   tan 1  U 2  C 2 p


(2-16)

 Cm2 p 


(2-17)

The secondary zone or wake flow equations are following:
p2 p  p2s  p2

(2-18)

2s RT2s  p2

(2-19)

Cm2s  m 2 r2b2  2Cm2 p 1    2s



(2-20)

W2s  Cm2s cos 2s

(2-21)

2s  2b   2s

(2-22)

C 2s  U 2  Cm2s tan 2s

(2-23)

s
2
C2s  Cm2s  C 2s

(2-24)

hT 2s  hT  W front cover

(2-25)





2
2
h2s  hT 2s  W2s 2  U 2 2



(2-26)

h2s  c pT2s

(2-27)

2
T02s  T2s  C2s 2c p

(2-28)

  1

T

p02s  p2s  02s 
 T2s 

(2-29)

Mixed out conditions are derived by following equations.

15

From radial momentum equation

m pCm2 p  msCm2s  mCm2m  ( p2m  p2 )2 r2b2

(2-30)

From tangential momentum equation

m pC 2 p  msC 2s  mC 2m

(2-31)

From energy equation

m pc pT02 p  msc pT20s  ( Pdisk friction  Precirc )  mc pT02m

(2-32)

And continuity equation becomes

m  2 r2b2Cm2m 2m

(2-33)

2
T2m  T02m  C2m 2c p

(2-34)

2m  p2m RT2m

(2-35)

2
2
2
C2m  C 2m  Cm2m

(2-36)

M 2m  C2m

 RT2m

(2-37)

  1

T

p02m  p2m  02m 
 T2m 

(2-38)

2m  C 2m Cm2m

(2-39)

 2m  2m  2m  tan 2b  2m

(2-40)

2m  C 2m U2

(2-41)

16

2.1.1. TEIS MODEL
In order to predict the relative velocity at the impeller exit, there is some convenient
parameter, such as the relative Mach number ratio for the primary flow. This ratio means the
ratio of relative Mach number the inlet to exit of the impeller (M1t,rel/M2p,rel). It is a good
parameter to get the relative velocity at the impeller exit. With sufficient kinetic energy within
the impeller, efficiency drop can always be found. Therefore, it is necessary to focus on diffusing
the impeller primary flow (core flow).
The TEIS model starts with considering the impeller passage as a rotating diffuser [1].
And the impeller inlet part, element “a”, can be considered a diffuser or nozzle according to its
flow rate as shown in Fig 2-1. The part of the impeller throat to exit, element “b”, is diffuser in
character. The second element is a constant geometry element regardless of flow rate since throat
and exit area does not change. The variables in TEIS model are defined as the following.

17

(a) Conceptual Model

(b) Actual Element “a”

(c) Actual Element “b”
Fig.2- 1 TEIS conceptual model [1]
18

Pressure recovery coefficient:

Cp 

p
q

(2-42)

Effectiveness: η
Inlet portion:

a 

C pa

(2-43)

C pa,i

 cos 1 
C pa,i  1 
 1 

2
 cos 1b 
ARa
1

2

(2-44)

Passage portion:

b 

C pb

(2-45)

C pb,i

C pbi,i  1 

A

 1   throat 
 A2 
AR 2
b
1

2

(2-46)

Overall:
DR2  W1t W2 p

(2-47)

1
1
2
DR2 
1  aC pa,i 1  bC pb,i

(2-48)

The constants ηa, ηb can determine the diffusion ratio. Table 2-1 shows that suggested
values for primitive TEIS model.

19

Table 2- 1 Suggested values for primitive TEIS model. (Japikse Centrifugal Compressor
Design and performance) [1]
Case
χ
U D
ηa
ηb
DRstall
Re  2 2
 00
Large ,welldesigned rotors

0.9 to 1.1

0.4 to 0.6

1.5 to 1.8

 1.2 106

0.1 to 0.2

(>10" to 12"D, or smaller if well-designed)
Medium size, well
designed

0.8 to 0.9

0.3 to 0.5

1.3 to 1.6

 0.5 106

0.15 to 0.25

0.6 to 0.8

0.0 to 0.4

1.2 to1.5

 0.5 106

0.20 to 0.30

0.4 to 0.6

-0.3 to +0.3

1.1 to 1.4

 0.5 106

0.25 to 0.35

(0.4" to 10"D)
Medium size,
ordinary design
(0.4" to 10"D)
Small or poorly
designed
(<4"D)

This study uses either a relative Mach number ratio or the TEIS model to determine the
diffusion ratio of the impeller.

20

2.2. EMPIRICAL BASED METHOD
The empirical-based method for the 1D analysis method used in this study is Ron
Aungier’s method. The model consists of work-input parameters and relative pressure loss
models. The work input parameters are used for calculating head increase within the impeller
and between the impeller exit and diffuser inlet. The relative pressure loss models are used to
estimate pressure drop within the impeller using the thermodynamic law for entropy.

2.2.1. IMPELLER WORK INPUT
BLADE WORK INPUT
Ron Aungier defines various work input parameters for impeller, blade parameter, disk
friction parameter, leakage parameter, and recirculation parameter [2].

2
I  ht U 2  I B  I DF  I L  I R

(2-49)

The blade work input coefficient can be determined from the Euler equation.

2
I B  C 2 U 2  U1C1 U 2

(2-50)

For the ideal case no slip at the impeller exit

C 2,i U 2  1   m cot 2  2 A2U 2   1  2 cot 2
where, the tip distortion factor   1 1  B2 

(2-51)
(2-52)

Therefore, the blade work input coefficient can be derived in terms of slip factor

2
I B    2 cot 2b  U1C1 U 2
Where, slip factor (U.S)   1 

(2-53)

Cslip

(2-54)

U2

Ron Aungier derives the empirical top blockage equation, impeller distortion factor [2]
21

2

b2 

pv1 W1 d H 

B2  sf
 0.3

pv 2 W2 b2  L2 
B


2
AR 2b2 CL

1LB
2b2

(2-55)

where pυ = velocity pressure (pt-p) and area ratio defined by

AR  A2 sin 2b

 A1 sin th 

(2-56)

DISK FRICTION PARAMETER
To get the disk friction work parameter, the clearance gap leakage flow has to be
determined. Ron Aungier derives empirical relations from the empirical relations [6] using the
forced vortex flow model.

p
 K 2 2  r
r

(2-57)

Where K  C r

(2-58)

 2s 
K0  0.46 1  
d 


(2-59)

K  K0  Cq (1.75K F  0.316) r2 s

(2-60)

Cq 

mL   r2U 2  1 5
2
2 r2 U 2

(2-61)

For seal leakage from the impeller tip K F  C 2 U 2 , but toward the tip, K F  0 .
For an open impeller, the clearance gap leakage flow velocity is given by

UCL  0.816 2pCL 2

(2-62)

The average pressure difference across the gap can be expressed by
pCL 

m  r2C 2  r C1 
1

(2-63)

zrbL

22

r   r  r2  2
1

(2-64)

b   b1  b2  2

(2-65)

Consequently, the blade gap leakage flow is given by

mCL  2 zsLUCL

(2-66)

Ron Aungier brings in windage and disk friction losses from Daily and Nece’s correlation [7, 8].
The disk torque coefficient is defined by
CM 

2
 2r 5

(2-67)

And in order to determine the disk torque coefficient, it is necessary to consider four
different flow conditions.

1. Laminar, merged boundary layers

CM 1 

2
 s r  Re

(2-68)

2. Laminar, separate boundary layers

CM 2 

3.7  s r 0.1

(2-69)

Re

3. Turbulent, merged boundary layers
CM 3 

0.08
 s r 1 6 Re1 4

(2-70)

4. Turbulent, separate boundary layers
CM 4 

0.102  s r 0.1
Re0.2

(2-71)

23

where Reynolds number is

 r 2
Re 


(2-72)

The torque coefficient can be taken to be the largest one among the four different values
from the above equations.
And the fully rough disk torque is given by

1
 3.8log10  r / e   2.4  s r 0.25
CMr

(2-73)

And for the smooth wall

Res CMs  1100  e r 0.4

(2-74)

Ron Aungier suggests the Reynolds number for the fully rough disk instead of the correlations of
Daily and Nece [7].

Rer  1100 r e  6 106

(2-75)

CM 0  CMs   CMr  CMs  log  Re Res  log  Rer Res 



(2-76)

Aungier develops the following torque coefficient using the above equation from Daily and Nece.

CM  CM 0 1  K 2

1  K0 2

(2-76)

For the disk

CMD  0.75CM

(2-77)

For the cover
5
CMC  0.75LCM 1   d1s d2  





 r2  r1 

Therefore, the disk friction work parameter is given by

24

(2-78)

2
I DF   CMD  CMC  2U 2r2 2m

(2-79)

LEAKAGE FLOW PARAMTER
Ron Aungier provides seal leakage mass flow from Egli’s formulation[9].

mL   d Ct CcCr  RT
Cr  1 

Ct 

(2-80)

1
 54.3 
3 
1  100 / t 


2.143 ln N  1.464
N  4.322

(2-81)

3.45

1  pR 0.375 pR

(2-82)

Cc  1  X1  p  X 2 ln(1   p) 1  X 2 

(2-83)

 p  X 2 1
0.50712 N  
 ;
X1  15.1  0.05255e

N≤12

(2-84)

X1  13.15  0.1625N ;

N>12

(2-85)

X 2  1.058  0.0218N ;

N≤12

(2-86)

X 2  1.32 ;

N>12

(2-87)

Therefore, the leakage flow work parameter is given by

I L  mL I B m

(2-88)

RECIRCULATION WORK
Lieblein suggests a diffusion factor, Deq, for radial and mixed flow blades [10].

Wmax  W1  W2  W  2

(2-89)
25

Deq  Wmax W2

(2-90)

The average blade velocity difference, ΔW, can be calculated using irrotational flow
relations.

W  2 d 2U 2 I B zLB



(2-91)



I R  Deq 2  1 W 2 Cm2  2cot 2b 

(2-92)

IR  0

2.2.2. RELATIVE PRESSURE LOSSES WITHIN IMPELLER
Ron Aungier suggests various loss mechanisms that result in a relative pressure drop
within the impeller [2]. From the fundamental thermodynamic relation, entropy rise equation, he
shows a total relative pressure equation at the impeller exit.


pt 2  p 2,i  fc ( pt1  p1) i
t
i

(2-93)

And Ron Aungier uses a correction factor.


 1
fc   t 2Tt2   t1Tt 

(2-94)

Total relative pressure loss models suggested by Aungier are the following:
1. Incidence loss due to the difference between the inlet blade angle and flow angle.

Cmh1  Cm1 1   m1b1 2

(2-95)

Cms1  Cm1 1   m1b1 2

(2-96)
2

inc  0.8 1  Cm1 W1 sin 1    zFBtb1  2 r sin 1 
1





2

(2-97)

Total incidence loss can be calculated by weighted average 10 times as heavy as
the hub and shroud values.

26

2. Entrance diffusion loss due to the diffusion inlet to throat.

DIF  0.81  Wth W12  inc

(2-98)

DIF  0
Inducer stall criterion is

W1s Wth  1.75

(2-99)

When occurring inducer stall

DIF  W1s  1.75Wth  W1   inc



(2-100)

3. Chocking loss due to the relative Mach number at the throat.
Contraction ratio is used to evaluate aerodynamic blockage.

Cr  A sin 1 Ath
1

(2-101)

Cr  1   A sin 1 Ath  1
1

2

(2-102)

X  11  10Cr Ath A*

(2-103)

CH  0 ;

CH 



X≤0



1
0.05 X  X 7 ;
2

(2-104)

X>0

(2-105)

4. Skin friction loss due to wall skin friction
Reynolds number based on pipe diameter;

Red  Vd 

(2-106)

For Red<2,000 and laminar flow, Skin friction coefficient is

C fl  16 Red

(2-107)

For Red>2,000, turbulent flow and smooth wall
27


1
2.51
 2 log10 
 Red 4C fts
4C fts







(2-108)

and for turbulent flow but fully rough surface

1
 e 
 2 log10 
4C ftr
 3.71d 


(2-109)

The surface roughness has to be considered when

Ree   Ree  2000 e d  60

(2-110)

When Red  4000 turbulent skin friction coefficient becomes

C ft  C fts ;

Ree<60





C ft  C fts  C ftr  C fts 1  60 Ree  ;

(2-111)

Ree≥60

(2-112)

When 2000  Red  4000 , the skin friction coefficient is given by



C f  C fl  C ft  C fl

  Red

2000  1

(2-113)

5. Blade loading loss due to the pressure difference between blade to blade.

BL   W W1 2 24

(2-114)

6. Hub to shroud loading loss is given by

HS   mbW W1 

2

(2-115)

6

 m  c2  c1  L

(2-116)

b   b1  b2  2

(2-117)

W  W1  W2  2

(2-118)

7. Abrupt expansion loss due to the distorted flow mixing with the free stream flow
28

     1 Cm2 W1 



2

(2-119)

8. Wake mixing loss due to mixing with jet flow right after impeller exit.
Separation velocity is given by

WSEP  W 2 ;

Deq≤2

(2-120)

WSEP  W2 Deq 2 ; Deq>2

(2-121)

2
2
Cm, wake  WSEP  W

(2-122)

Cm,mix  Cm2 A2  d2b2 

(2-123)

mix   Cm, wake  Cm,mix  W1 



2

(2-124)

9. Blade clearance loss due to clearance gap flow.

CL  2mCL pCL

 m1W12 

(2-125)

10. Super critical Mach number loss due to shock wave loss.


M cr  M1W * Wmax

(2-126)


cr  0.4  M1  M cr Wmax W1 
 


2

(2-127)

After a summation of all losses, relative pressure loss can be calculated. Since all
ideal conditions are known, all thermodynamic conditions at the impeller exit can
be computed using isentropic equations, the equation of state, and conservation of
rothalpy.

29

2.2.3. VANELESS DIFFUSER ANALYSIS
This study uses only Ron Aungier’s method to calculate the vaneless diffuser flow
because this method is well known for good prediction of the vaneless diffuser flow.
Aungier developed governing equations for 1-dimensional flow analysis from Johnston
and Dean’s equation[11] in vaneless diffuser.

2 r bCm 1  B   m
bCm

d  rC 
dm

(2-128)

 rCC c f

(2-129)

2
CCmc f dI D
dC
1 dp C sin c

 Cm m 

 Ic
 dm
r
dm
b
dm

(2-130)

1
ht  h  C 2
2

(2-131)

The divergence parameter is given by

D

b dC
c dm

(2-132)

Dm  0.4  b1 L 0.35 sin c

(2-133)

Empirical diffusion efficiency is the following:

E 1 ;

D≤0

(2-134)

2
E  1  0.2  D Dm  ;

0<D<Dm

(2-135)

E  0.8 Dm D ;

D≥Dm

(2-136)

The diffusion term is given by

dI D
1 dC
 2  pt  p  1  E 
dm
C dm

30

(2-137)

Local maximum area is

 rb m   rb 1 1  0.16m b1

(2-138)

If  rb    rb m ,

I D  0.65  pt  p  1   rb m  rb  



(2-139)

And curvature loss term is obtained by
IC   m  pt  p  Cm 13 c 

(2-140)

Blockage in the vaneless diffuser is given by

B  2 8b 

(2-141)

If the boundary layer at the diffuser inlet is known, Cθe can be calculated from following
equation.
rC  rC e 1  2  4.5b 



(2-142)

And Cθe must be conserved until the boundary layer fills the passage, i.e. 2δ=b.
If the boundary layer at the diffuser inlet is not given, the inlet boundary layer can be calculated
from
0.15
2 b  1   b r in

(2-143)

31

3. 1D ANALYSIS CODE VALIDATION WITH TESTED IMPELLER
DATA
Two tested impellers are used to validate two 1D analysis codes, table 3-1 shows
information of the two tested impellers. Impeller #1 and impeller #2 are tested in Solar Turbines,
Inc. The design rotational speed is 19240 rpm and design flow coefficients of the impeller #1 and
#2 are 0.026 and 0.045 respectively. And their exit angles are 60 and 40 degree respectively.
Fig.3-1 shows meridional shape of the impellers.

Impeller #

Design speed, N

Φ

b2/d2

2

1

19240

0.026

0.036

60

2

19240

0.54

0.044

40

Table 3- 1 Tested Impeller information

The tests are performed along the three different rotational speeds of 13k, 19k, 21k rpm.
Impeller #1 and #2 are tested at 544R and 90 psi, 544R and 44psi respectively. And both
impellers have vaneless diffuser with pinch. All tested results, work factor, head coefficient and
absolute flow angles are taken at the exit of vaneless diffuser.

32

(a) impeller #1

(b) impeller #2

Fig.3- 1 Meridional shape of two tested impellers (Impeller #1, Impeller #2)

3.1. COMPARISON RESULTS
Fig 3.2 and Fig 3.3 show the head coefficient and work factor of the tested impellers
according to flow coefficient,t comparing 1D analysis results and tested data with different
rotational speeds. As shown in the figures, Ron Aungier’s method predicts better than the Two
Zone model in the head coefficient map. While the Two-Zone model predicts well around the
design point, the Two-Zone model does not capture the head coefficient at the surge and choking

33

region. However, the results of Ron Aungier’s method predicts more accurately around the surge
and choking region.

Fig.3- 2 Work factor and Head Coefficient result according to flow coefficient using Two Zone
model with TEIS and Ron Aungier’s model. (Impeller #1,13k, 19k, 21k rpm)
For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this dissertation.

34

Fig.3-2 (cont’d)

35

Fig.3-2(cont’d)

As shown in fig. 3-2, Ron Aungier’s 1D analysis method, which is the empirical method, is
more accurate at 19k and 21k rpm around surge region.

36

Fig.3- 3 Work factor and Head Coefficient result according to flow coefficient using Two Zone
model with TEIS and Ron Aungier’s model. (Impeller #2, 13k, 19k, 21k rpm)

37

Fig.3-3 (cont’d)

38

Fig.3-3 (cont’d)

Fig. 3-3 clearly shows that the head coefficient result of Ron Aungier’s method is more
accurate than the results of the Two-Zone model around the choking region. At an increased
rotational speed, Ron Aungier’s model predicts the head coefficient and work factor better than
the results of Two-Zone model.
Fig. 3.4 and Fig 3.5 present the absolute flow angle at the exit of vaneless diffuser of the
impeller #1 and impeller #2 respectively. As shown in the figures, around the choking area Ron

39

Aungier’s method predicts flow angles better, especially in impeller #2 as the rotational speed
increases as well as the head coefficient prediction.

Fig.3- 4 Absolute flow angle according to flow coefficient along the various rotational speeds
(impeller #1, 13k, 19k, 21k rpm)

40

Fig.3-4 (cont’d)

41

Fig.3-4 (cont’d)

42

Fig.3- 5 Absolute flow angle according to the flow coefficient along the various rotational speeds
(impeller #1, 13k, 19k, 21k rpm)

43

Fig.3-5 (cont’d)

44

Fig.3-5 (cont’d)

45

4. CORRECTED TWO ZONE WITH TEIS MODEL

4.1. PROBLEM DEFINITION IN DIFFUSION RATIO IN TEIS MODEL
To predict the head coefficient of a centrifugal compressor, estimating diffusion ratio,
which determine pressure through the impeller, is important. As discussed in chapter 2, the TEIS
model is one of the popular methods for calculating the diffusion ratio in Two Zone model. Fig.
4-1 presents the diffusion ratio corresponding to pressure recovery effectiveness [1]. There are
two effectivenesses for the inlet portion and passage portion. These effectivenesses are used for
calculating the diffusion ratio and invariant for whole flow range. As shown in Fig. 4-1, the
results of the diffusion ratio calculated by TEIS model are predicted well around the design point.
The results, however, depart from the test data in the surge and choking region.

Fig.4- 1 TEIS model characteristic for estimating the diffusion ratio [1]
46

Fig. 4-2 and Fig. 4-3 show performance prediction results of the Two Zone model with
TEIS for the impeller #1 and impeller #2 along the three different rotational speeds. The inlet
and passage pressure recovery effectivenesses are fixed during calculating the head coefficients
for the whole range of flow. As shown in Fig 4-2 and Fig 4-3, the Two Zone model can predict
head coefficients around the design point. The Two-Zone model, on the other hand, cannot
capture the head coefficient in the surge and choking region. The errors in head coefficient
become larger as the impeller rotational speed increases for both impellers. The differences in
the choking region show that they become significant as rotational speeds increase.

47

Fig.4- 2 Comparing head coefficient results using the Two Zone model and test results. (Impeller
#1, 13k, 19k, 21k rpm)

48

Fig. 4-2 (cont’d)

49

Fig. 4-2 (cont’d)

50

Fig.4- 3 Comparing head coefficient results using the Two Zone model and test results. (Impeller
#2, 13k, 19k, 21k rpm)

51

Fig. 4-3 (cont’d)

52

Fig. 4-3 (cont’d)

4.2. LOSS MODEL IN EMPIRICAL MODEL
Head coefficient is mainly a function of the total pressure ratio. The 1D performance
analysis errors in head coefficient in the surge and choking region result from underestimating
the total pressure loss. The Two Zone model with TEIS uses fixed pressure recovery
effectiveness to calculate the diffusion ratio, which determines the total pressure ratio. For this
reason, the Two Zone model with TEIS can predict the performance of the impeller only in a
narrow range of flow.

53

In order to find the influence of loss model to total pressure ratio, the loss models, which are
defined in chapter 2, are applied to impeller #1 and impeller #2. The loss models applied to
impeller #1 and impeller #2 are presented in Fig.4-4 for three different speeds, 13k, 19k, 21k rpm.
As shown in the Fig. 4-4, the loss coefficients are independent on the impeller speeds except the
diffusion loss. The diffusion loss coefficient shows that there are differences in the surge and
choking region by the three speeds. Particularly in the choking region, there are substantial
differences compared to the surge region as well as the total loss coefficient. In terms of the total
pressure loss coefficient, there are differences in the choking region as much as the diffusion loss
coefficient. As the figure shows, the diffusion loss coefficient shows similar tendency as total
pressure loss coefficient and differences between the speeds are almost same. In this sense, the
differences in the total pressure loss coefficient resulted from the inlet diffusion loss coefficients.
In other words, influence of the inlet condition to the total pressure ratio is very significant in the
total pressure ratio or total head rise.

54

(a) Incidence loss coefficient according to flow rate along with 3 different speeds.(13k, 19k, 21k
rpm)
Fig.4- 4 Loss models in impeller #1 and impeller #2 with various speeds.

55

Fig. 4-4 (cont’d)

56

Fig. 4-4 (cont’d)

57

Fig. 4-4 (cont’d)

(b) Diffusion loss coefficient according to flow rate along with 3 different speeds.(13k, 19k, 21k
rpm)

58

Fig. 4-4 (cont’d)

59

Fig. 4-4 (cont’d)

(c) Skin friction loss coefficient according to flow rate along with 3 different speeds.(13k, 19k,
21k rpm)

60

Fig. 4-4 (cont’d)

61

Fig. 4-4 (cont’d)

(d) Blade loading loss coefficient according to flow rate along with 3 different speeds.(13k, 19k,
21k rpm)

62

Fig. 4-4 (cont’d)

63

Fig. 4-4 (cont’d)

(e) Hub to shroud loading loss coefficient according to flow rate along with 3 different
speeds(13k, 19k, 21k rpm)

64

Fig. 4-4 (cont’d)

65

Fig. 4-4 (cont’d)

(f) Abrupt expansion loss coefficient according to flow rate along with 3 different speeds.(13k,
19k, 21k rpm)

66

Fig. 4-4 (cont’d)

67

Fig. 4-4 (cont’d)

(g) Mixing loss coefficient according to flow rate along with 3 different speeds(13k, 19k, 21k
rpm)
4.3. CORRECTED DIFFUSION MODEL IN TWO ZONE MODEL
As discussed in previous section, to improve accuracy of total pressure ratio through the
impeller, it is important to predict the diffusion ratio through the impeller. The TEIS model,
which is used for predicting the diffusion ratio in the Two-Zone model, divides the impeller
passage into two elements as shown in Fig.2-1. Element “a” and element “b” mean impeller inlet
to throat and impeller throat to exit respectively. Each element estimates their diffusion ratio and
68

calculates the diffusion ratio through the impeller whole passage by multiplying. The TEIS
model uses pressure recovery effectiveness to estimate relative velocity losses. Using fixed
pressure recovery effectiveness through the whole range of flow rates, the TEIS model does not
estimate impeller performance in the surge and choking region where the losses are too large to
neglect compared to around the design point. To compensate these problems, the TEIS model
needs to consider internal losses through the passage.
The internal loss models can be divided into the inlet portion and passage portion. Incidence
loss and inlet diffusion loss are included in the inlet category and others are placed in the passage
portion. The inlet losses, incidence and inlet diffusion loss, can be applied to element “a”. The
passage losses can be applied to element “b”. To apply all loss models to the TEIS model, all
loss models should be calculated.
One of the important things in the Two Zone model and Ron Aungier’s method when
calculating impeller exit condition is that a different slip factor is applied. The reason for using a
different slip factor is that the Two-Zone model does not consider impeller exit aerodynamic
blockage, but Ron Aungier’s model calculates aerodynamic blockage using an iterative method
and applies to the impeller exit area. The blockage makes a different exit velocity diagram so
that two different 1D analysis models share the same slip factor. The inlet loss models, incidence
loss and inlet diffusion loss, are functions of the inlet condition and not affected by the slip factor.
In order to apply all loss models to TEIS, a different slip factor for Two-Zone and loss models
should be used. As shown in Fig. 4-4 (b), however, inlet diffusion loss is very useful to predict
total loss model and a new slip factor is not necessary to calculate it.

69

For these reasons, this study focuses on improving the accuracy of the Two-Zone model
with TEIS using the inlet loss models. Corrected diffusion models applying to Tow-Zone are the
following.



a,new  a 1  10dif





a,new  a 1  10dif

(Inlet is diffuser)



(4-1)

(Inlet is nozzle)

(4-2)

0  a,new  2

4.4. CORRECTED DIFFUSION MODEL RESULTS
The corrected diffusion model is applied to impeller #1 and impeller #2. Table 4-1 shows
the effectiveness used in this analysis. New effectiveness is calculated for every flow rate by
equations (4-1) and (4-2). The corrected effectivenesses are shown in Fig. 4-5.

ηa

ηb

Impeller #1

0.79

0.0

Impeller #2

1.0

0.4

Table 4- 1 pressure recovery effectiveness used in this study

Fig. 4-6 and Fig. 4-7 present the head coefficient and work factor calculated by corrected
diffusion loss. As shown in the figure, the corrected Two-Zone model can capture the head
coefficient and work factor better in the choking region in both impeller #1 and impeller #2. In
the surge region, the corrected model overestimates total pressure loss in impeller #1.

70

Fig.4- 5 effectiveness calculated by the corrected diffusion model (impeller #1, impeller #2)

71

Fig.4-5 (cont’d)

72

Fig.4- 6 head coefficient and work factor calculated by the corrected diffusion model (impeller
#1, 13k, 19k, 21k rpm)

73

Fig. 4-6 (cont’d)

74

Fig. 4-6 (cont’d)

75

Fig.4- 7 head coefficient and work factor calculated by the corrected diffusion model (impeller
#2, 13k, 19k, 21k rpm)

76

Fig. 4-7 (cont’d)

77

Fig.4-7 (cont’d)

78

Fig.4- 8 Absolute flow angle calculated by the corrected diffusion model (impeller #1, 13k, 19k,
21k rpm)

79

Fig.4-8 (cont’d)

80

Fig. 4-8 (cont’d)

81

Fig.4- 9 Absolute flow angle calculated by the corrected diffusion model (impeller #2, 13k, 19k,
21k rpm)

82

Fig. 4-9 (cont’d)

83

Fig.4-9 (cont’d)

4.5. CONCLUSION
This chapter discussed the diffusion model in 1D analysis and showed a corrected diffusion
model to improve the accuracy of the head coefficient prediction. The TEIS model, which is
used for the diffusion model in the Two-Zone model, has two pressure recovery effectivenesses
to predict relative velocity at the impeller exit. Generally, the pressure recovery effectivenesses
are invariant parameter during off design calculation. However, fixed pressure recovery

84

effectiveness cannot predict an accurate diffusion ratio through the impeller. In order to improve
the accuracy of the diffusion ratio, the study applies the impeller inlet loss model to element “a”,
inlet portion. By applying impeller inlet loss, the inlet diffusion loss, to the existing diffusion
model without other loss models, the accuracy of the Two-Zone model with TEIS is improved.
This is because the inlet diffusion loss is changed significantly among total loss models in surge
and choking losses.

85

5. RE-DESIGN IMPELLER

5.1. OBJECTIVE
1D performance analysis has been conducted for the previous chapters. This study showed
accurate 1D analysis results and improved accuracy through the corrected model. This chapter is
going to show how use 1D analysis when re-designing the existing impeller. The main purpose
of the re-design impeller is to make a scaled impeller so that it shows better performance and
efficiency than the existing impeller. This chapter shows an impeller re-design procedure using
1D performance analysis and a comparison of the results of 1D analysis and CFD results. Finally,
this study compares the CFD results of the original impeller and the re-designed impeller. The
requirements of re-design process maintain a machine Mach number of 0.7 and a design point
flow coefficient of 0.025.

5.2. ORINAL IMPELLER
The characteristics of the original impellers, two different tip widths, are presented in Table
5-1. As shown in Table 5-1, the impeller with a wider tip width gives an additional surge margin.
The head coefficient is decreased by 8.2% and diffusion ratio is increased by 24% because of a
wider tip width condition. The corrected machine Mach numbers of 0.75 are same. Fig. 5-1
shows the meridional shape of impeller.

86

Table 5- 1 Information of the original impeller
b2/d2

0.0393

0.0294

Flow Coefficient, Φ

0.0245

0.0245

Head Coefficient, Ψ

1.22

1.12

Rotor efficiency

0.84

0.84

Work factor, q

0.73

0.675

Surge Margin

20%

40%

Diffusion ratio W2/W1t

0.62

0.77

Flow angle, α2

79.5

79.5

Backsweep angle

45

45

Machine Mach #, Mn

0.75

0.75

87

Fig. 5- 1 Meridional shape of the original impeller with wider and smaller tip width.

5.2.1. 1D PERFORMANCE ANALYSIS OF ORIGINAL IMPELLER
Before conducting the re-design procedure, this study performed a 1D analysis on the
original impeller with a wider tip width. Fig. 5-2 shows the head coefficient, the work factor total
to total efficiency, the results of 1D performance analysis of original impeller using the Ron
Aungier’s method, and the corrected Two-Zone model. As shown in the figure, Ron Aungier’s
method predicts these properties better than the Two Zone model. Fig. 5-3 and Fig. 5-4 show an
absolute flow angle at the impeller exit and a diffusion ratio through the impeller passage, which
are the result of two 1D analysis methods. As the figure shows, the results of the absolute flow
angle at the impeller exit are fairly well predicted in both analysis methods. The results of the
diffusion ratio show that Ron Aungier’s method is more accurate in the diffusion ratio.

88

Fig. 5- 2 Comparing work factor, head coefficient and efficiency, the 1D analysis results, with
test data (original impeller)

89

Fig. 5- 3 Comparing absolute flow angle, 1D analysis, with test data (original impeller)

90

Fig. 5- 4 Comparing diffusion ratio, 1D analysis, with test data (original impeller)

5.3. INVERSE DESIGN USING 1D ANALYSIS CODE
This study shows the re-design procedure using the 1D analysis method. The centrifugal
compressor design starts with a 1D preliminary design. However, when the new impeller
characteristics were similar to the existing impeller or redesign, 1D performance analysis can be
used instead of the preliminary design. This is beneficial to reduce the range of the variables,
which is needed for the preliminary design procedure. A main issue of the redesign in this study
91

is finding trade-off points of the impeller tip width and the impeller blade exit angle. To make a
better head increase, the blade exit angle (from radial) must decrease keeping with impeller tip
width. However, as the blade exit angle decreases, impeller tip width needs to decrease to
maintain efficiency. To find the trade-off point, this study conducted 1d performance analysis
changing both blade exit angle, and impeller tip width keeping with other variables. Fig. 5-5
shows blade exit angle (from tangential) at the best efficiency point according to the flow
coefficient [12]. As shown in Fig. 5-5, the blade exit angle near the designated impeller design
point, at the best efficiency point is roughly from 40 to 50 degree.

92

Fig. 5- 5 Impeller exit flow coefficient, impeller slip factor, work input factor and blade angle
(from tangent) as functions of the inlet flow coefficient (empirically determined values for the
best efficiency point) [12]
5.3.1. DIFFUSION RATIO
The diffusion ratio affects impeller efficiency and head rise. The lower diffusion ratio
drops the impeller efficiency but brings on a head rise. Therefore, it is important to determine the
diffusion ratio through the impeller. Generally, the diffusion ratio (W2/W1t) lies on between 0.55
and 0.70. In order to make a high performance impeller, the diffusion ratio approaches 0.55; and
to make high efficiency impeller, the diffusion ratio becomes close to 0.70.
93

The diffusion ratio (W1,rms/W2) that resulted from the 1D performance analysis on the
original impeller is shown in Fig. 5-4. Based on the 1D analysis results of the diffusion ratio, the
diffusion ratio of the re-design impeller was determined. Fig. 5-6 shows the diffusion ratio
according to impeller tip width (b2) and impeller blade exit angle (β2). As shown in the figure,
the impeller tip width becomes larger as blade angle increases to keep the diffusion ratio. The
cases used in this study are determined by Fig. 5-6. Four cases are selected by the diffusion ratio
of 0.62, with considering both head rise and efficiency. Table 5-2 shows four cases used in this
chapter.

94

Fig. 5- 6 The diffusion ratio according to the impeller tip width and blade exit angle calculated
by 1D analysis code using Ron Aungier’s method.

95

Table 5- 2 Cases for impeller re-design designate diffusion ratio of 0.6
Blade exit angle, β2 (degree)
(from radial)

Impeller tip width, b2/d2

Case1

48

0.0403

Case2

46

0.0373

Case3

44

0.0350

Case4

42

0.0329

5.3.2. MERIDIONAL AND BLADE CURVE DESIGN
After the 1D calculation, the impeller design procedure is on the blade design including
meridional curve and blade curve design. This procedure determines the impeller 2D and 3D
shape. Flow separation and blade loading to get total pressure rise within the impeller are
determined by these 2D and 3D design procedures as well as the 1D design. Because of this, this
procedure should be done carefully.
To complete the re-design and be independent on the 2D and 3D design, meridional
shape changed as little as possible and blade angle distribution is determined by the same
equation. Ron Aungier suggested an equation to be used to calculate the blade angle distribution
[2].



 s  1s   2b  1t  3 2  2 3



h  1h  A  B 2  C 3
h  90K  1  K   2b  1h  2

96

A  4  2b  2h  1h 
B  112b  16h  51h
C  162b  8h  21h

(5-1)

The parameter K is for a blade angle at the midpassage so that it determine maximum
blade angle. Fig. 5-7 shows meridional curves for the cases used in this study. All cases have the
same impeller tip diameter and different impeller blade exit angles and tip widths. Fig 5-8 shows
blade angle distribution calculated by the equation Ron Aungier suggested, K=0.1. In order to be
independent on exit lean angle, the lean angles of all the impellers are in the range of 0 to 5
degrees. Theta angle distributions of the impellers are presented in Fig 5-9. Theta angle at the
impeller exit are around 55 degrees. Area distributions of the impeller passage are shown in Fig.
5-10. The area distributions have some difference because their areas are almost same until the
meridional distance of 0.3, but after that the impeller passage areas are different because of
impeller exit width.

97

Fig. 5- 7 Meridional shape of all the cases

98

Fig. 5- 8 Blade angle distribution of all the cases according to m`%

99

Fig. 5- 9 Blade theta angle distribution of all the cases according to m`%

100

Fig. 5- 10 Area distribution (A/A1) of all the cases according to m`%

5.4. 1D PERFORMANCE ANALYSIS ON THE 4 CASES
Before CFD analysis on the cases, this study conducted a 1D performance analysis on the
cases for predicting impeller performance. Fig 5-11 shows efficiency and head coefficient results
using a 1D analysis code according to the flow rate of all cases. As shown in the Figure 5-11, all
cases show almost the same results of head coefficient and efficiency at the impeller exit. At the

101

diffuser inlet, however, efficiencies are different between the cases and efficiency difference
become larger around the surge region.
Fig 5-12 presents the absolute flow angle and diffusion ratio at the diffuser inlet. The
figure shows that the diffusion ratio is almost the same along the whole flow rate and the
absolute flow angles have some difference. Absolute flow angle is a very important property at
the impeller exit because of high tangential flow results in backflow at the vaneless diffuser. As
shown in the Fig. 5-11, over 78 degrees of the absolute flow angle at the impeller exit lead to a
significant impeller efficiency drop. Among the cases, case 4 shows the best efficiency at the
diffuser inlet because the absolute flow angles are the least along the whole flow range.
Fig 5-13 shows the work input parameter calculated by 1D analysis code. As discussed in
chapter 2, the work input parameter consists of blade work, recirculation, leakage, and disk
friction parameter. Among them, recirculation, leakage and disk friction are parasitic loss
between the impeller exit and diffuser inlet and contribute to total loss. As shown in the figure,
the recirculation occupies most of the parasitic work input parameter and other parasitic losses
do not change largely as much as the recirculation parameter according to the flow rate.

102

Fig. 5- 11 Efficiency and head coefficient according to the flow rate (Case1, Case2, Case3,
Case4)

103

Fig. 5- 12 Diffusion ratio and Absolute flow angle at the diffuser inlet.

104

Fig. 5- 13 Parastic work input (Recirculation, Disk friction and Leakage loss)

5.5. CFD ANALYSIS AND COMPARE WITH 1D ANALYSIS
It is important for the impeller design process to do the CFD analysis that predicts
impeller performance. The impeller design program, Bladegen, for the CFD analysis is used in
the simulation. TurboGrid is used for the grid generation. Fig. 5-14 and Fig. 5-15 show 3D
impeller shape and grid generated for the simulation respectively. Total grid points are 350000 to
400000 according to the cases and all grids in the cases have almost the same grid points and
shape to reduce uncertainty from the mesh. Grid generation is conducted on one passage and a
periodic condition is used in the simulation. ANSYS CFX is used for CFD analysis. The
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domains for the simulation consist of rotating inlet domain, rotating main passage domain, and
stationary outlet domain. Inlet boundary conditions are set as total pressure, total temperature,
and velocity direction; outlet boundary conditions are set as mass flow rate.
interface was used between the stationary and rotating blocks.

Fig. 5- 14 3D shape of redesigned impeller

Fig. 5- 15 Grid generated for re-design impeller

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A frozen – rotor

At the design point, flow coefficient of 0.025, 1D analysis and CFD analysis are
conducted. Fig. 5-16 shows efficiency and head coefficient for all cases. As shown in the figure,
both efficiency and head coefficient of CFD results are bigger than the results of 1D analysis.
Efficiencies in CFD and 1D analysis are around 0.962 and 0.958 respectively. Head coefficients
in CFD and 1D analysis are around 1.35 and 1.25 respectively. Fig. 5-17 shows the diffusion
ratio (W2/W1t) and absolute flow angle results of 1D analysis and CFD of all cases. Absolute
flow angle and diffusion ratio are very close between 1D analysis and CFD results. As blade exit
angle increases, absolute flow angle increases. It means that blade exit angle affects the diffuser
inlet condition at the same diffusion ratio. The diffusion ratios are close to 0.6. All cases show
almost same result.

107

Fig. 5- 16 Comparison of Head coefficient and efficiency of 1D analysis and CFD result

108

Fig. 5- 17 Comparison of the diffusion ratio and absolute flow angle, results of 1D analysis and
CFD
5.6. CONCLUSION
Impeller re-design using 1D analysis code is conducted in this chapter. 1D analysis code
is used for the diffusion ratio, which is important to start the impeller design and affect its
performance and efficiency. The impeller re-design targeted the diffusion ratio of 0.6 and better
efficiency and head coefficient than those of original impeller. 1D analysis found a trade-off
point between the blade tip width (b2) and the blade exit angle (2). This study made four
different speeds and conducted CFD analysis.

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As results of the 1D analysis and CFD, the results of efficiency, diffusion ratio, and
absolute flow angle in 1D analysis are almost the same as the CFD result. However, 1D analysis
predicts head coefficient and efficiency less than the results of the CFD results. The results could
be compared with test results because wall condition in CFD analysis is a smooth condition that
affects skin friction loss.

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6. CONSIDERATION OF AXIAL LENGTH FOR IMPELLER DESIGN
In the definition of an initial meridional profile, building on the 1-D parameters and
design process, the first task is to select the axial length-diameter ratio, Lax/d2. The impeller
length affects a number of significant design issues, including the overall length of the machine,
the shaft dynamics, the impeller bore stresses, and the aerodynamic performance. The curvature
along the shroud is driven by the need to turn the flow from an axial direction at the leading edge
to a radially oriented flow at the exit. The length over which this turning takes place becomes a
critical factor. The most effective way to reduce the curvature is to increase the distance over
which the turning takes place; i.e. ,make the impeller longer. So wide is the impeller axial length,
Lax, impact, in fact, that Lax/d2 cannot sensibly be included as a varying parameter in the iterative
aerodynamic design procedure; it is better to select a value at the outset with guidance from
empirical rules or established custom and practice. From an aerodynamic standpoint, Birdi [13]
has suggested

 d d d 
Lax
  K1  M w1s  K 2  1  1m  1s 1h 
d2
d2 
d2




(6-1)

where, K1 = 0.28 and K2=2 0.8. Equation (6-1) yields Lax/d2 values of 0.32 - 0.37 for
inlet Mach numbers of 0.9 - 1.2 for typical ranges of diameter ratios in the expression. The
physical basis for equation (6-1) lies in the balance of reduced skin friction, prompting low
Lax/d2 especially at low d1m/d2, against the need to allow adequately gentle turning from axial to
radial, particularly at high Mw1t which prompts higher Lax/d2. For a single-stage compressor
there is usually freedom to choose an aerodynamically optimum axial length, in contrast to

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multistage designs in which the axial length of the rotor tends to be limited by overall shaft
length considerations.
Similarly Aungier[14] developed a correlation for optimum Lax/d2 based on his
experience and wide sources of data a function of diameter ratio (d2/d1t) and flow coefficient ( Φ)
as

Lax
d
 0.014  0.023 2  
d2
d1s

(6-2)

Sorokes et al.[15] carried out an extensive work to determine the impact of shroud
curvature on the performance of a centrifugal impeller or stage. They showed that the value of
Lax/d2 and related curvature geometry are especially important in the design of high inlet relative
Mach number centrifugal compressor impellers. They showed that if the impeller has rapid
curvature or tight turning in the leading edge region, the inlet relative Mach number will be
unacceptably high. Conversely, efforts to reduce the curvature typically result in longer impellers
that can lead to rotor-dynamics concerns in multi-stage compressors.
Brammert et al [16] investigated three centrifugal impellers having an elliptic blade shape
and identical blade geometry at inlet and exit but different shapes of the meridional contours
(Lax/d2 = 0.325 – 0.5). They showed that with increasing impeller length, the wake zone at the
suction side of the blades can be only partially influenced and friction losses become dominant.
A smooth curvature of the meridional cross section of the impeller channel leads to a better jetwake ratio at impeller exit.
Al-Zubaidy [17] developed a general direct-design method for radial flow impellers
(based on a prescribed relative velocity schedule). He varied the axial length, L ax, of each
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impeller systematically in order to assess its impact on the efficiency. The results showed that
there is a specific region of axial length band where the highest efficiency for all designs
occurred.
With the above background information, the design was carried out varying axial length
based on difference in length between radii, impeller exit and impeller inlet shroud (r2-r1). The
study presents axial length ratio Lax to d2 of 0.30 to 0.45.

6.1. DESIGN CASES
In terms of flow stability at the impeller exit and diffuser inlet, high Mach number and
high swirl flow should be avoided. In such conditions, axial length could be one of the design
considerations. To make the instability condition, a high Mach number and high swirl condition
are defined in Table 6-1. High Mach number and high swirl defined between 0.8 and 1.1 and 78
and 85 degree respectively. With these requirements 1D design results are shown as table 6-2.

Table 6- 1 Mach number and absolute flow angle range for unstable flow condition
Value
Mach number

0.8~1.1

α2

78 ~85

o

113

o

Table 6- 2 1D geometry conditions for the cases
83.15 mm

Inlet shroud diameter (Ds1)
Inlet hub diameter (Dh1)

40 mm

Inlet shroud blade angle (βh1)

53.89

Inlet hub blade angle (βs1)

33.40

o
o

210.407 mm

Exit diameter (D2)

o

Exit blade angle (β2b)

60

Exit flow angle (α2b)

10

Blade number

16

o

Rotational speed

40000 rpm

Blade exit width

6.137 mm

Mass flow rate (ṁ)

0.5 kg/s

To make meridional curves, an elliptic equation was used and a 3D camber line was made
by Casey’s method[18]. The study considered six cases of varying axial length based on the
difference between r2 and r1 to determine proper axial length using equation (6-3). The cases are
shown in Table. 6-3. Meridional curves are presented in Fig. 1. And a is used as normalized value
in each case.

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Lax  a(r2  r )
1

(6-3)

Table 6- 3 Axial length along with respect to factor a
A

Lax

1.0

63.63mm

1.1

69.99mm

1.2

76.35mm

1.3

83.71mm

1.4

89.08mm

1.5

95.44mm

And hub and shroud camber lines are determined by cubic equation (6-4). Boundary
conditions of this equation are determined by blade angles, θ2 and angles between axial
coordinate and meridional curve so as to determine constants in the equation. Camber lines in the
study are shown in Fig. 6-2.
3

2

c=rθ=Am +Bm +Cm+D

(6-4)

After determine all curves to make 3D impeller geometry impeller shape is presented in
Figure 6-3.

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Fig. 6- 1 Meridional curves of the all cases

116

Fig. 6- 2 Camber line vs. Meridional distance

117

Fig. 6- 3 3D Impeller shape

6.2. CFD ANALYSIS
In the study, CFD analysis was conducted using ANSYS CFX. The impeller grid nodes
account is 300K-330K. Fig 6-4 shows mesh element shape. The domains for the study are
consisted of stationary inlet domain, rotating main passage domain, and stationary outlet domain.
Inlet boundary conditions are set as total pressure, total temperature, and velocity direction; and
outlet boundary conditions are set as mass flow rate.
A high resolution scheme has been used as an advection scheme. A k-ε model was
employed as a turbulent model. A frozen – rotor interface was used between the stationary and
rotating blocks. The domains in the study consisted of stationary inlet domain, rotating main
passage domain, and stationary diffuser domain.

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Fig. 6- 4 Mesh elements at 50% span

Fig. 6- 5 Domains used in the analysis

119

6.3. RESULTS
The performance of the impeller has been carried out for all cases. The Mach number,
absolute velocity flow angle, and static pressure at 0%, 3%, 6%, and 9% from the impeller exit
are given in Fig. 6-6 to Fig. 6-17 respectively.
As shown in the absolute velocity flow angle contour, the case of a=1.5 has the most
uniform flow at 0%; and all cases show that the absolute flow angle decreases as going to the
shroud from the hub. Also, it is observed that the absolute flow angle is higher at the pressure
side near the hub. Generally over the whole region, impeller, a=1.5, has a high tangential flow at
the impeller exit.

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Fig. 6- 6 Absolute flow angle contour at 0% from the impeller exit (a=1.0, a=1.3, a=1.5)

121

Fig. 6- 7 Absolute flow angle contour at 3% from the impeller exit (a=1.0, a=1.3, a=1.5)

122

Fig. 6- 8 Absolute flow angle contour at 6% from the impeller exit (a=1.0, a=1.3, a=1.5)

123

Fig. 6- 9 Absolute flow angle contour at 9% from the impeller exit (a=1.0, a=1.3, a=1.5)

124

Similar to the absolute flow angle, in Mach number contour, Mach number distribution
shows the most uniform flow at the case of a=1.5. In particular, the case of 1.0 and 1.3 at 9% in
Figure 13, there is still not uniform region near the hub. But throughout the check point, 0%, 3%,
6%, and 9%, the case of a=1.5 shows that the Mach number is smaller than the other cases.

125

Fig. 6- 10 Absolute Mach number contour at 0% from impeller exit (a=1.0, a=1.3, a=1.5)

126

Fig. 6- 11 Absolute Mach number contour at 3% from impeller exit (a=1.0, a=1.3, a=1.5)

127

Fig. 6- 12 Absolute Mach number contour at 6% from impeller exit (a=1.0, a=1.3, a=1.5)

128

Fig. 6- 13 Absolute Mach number contour at 9% from impeller exit (a=1.0, a=1.3, a=1.5)

129

The static pressure contours are shown in Fig. 6-12 to 6-17. It can be shown in pressure
contour that at the case of a=1.5, difference between suction side and pressure side is smaller
than a=1.0 and a=1.3.

130

Fig. 6- 14 Static pressure contour at 0% from the impeller exit (a=1.0, a=1.3, a=1.5)

131

Fig. 6- 15 Static pressure contour at 3% from the impeller exit (a=1.0, a=1.3, a=1.5)

132

Fig. 6- 16 Static pressure contour at 6% from the impeller exit (a=1.0, a=1.3, a=1.5)

133

Fig. 6- 17 Static pressure contour at 9% from the impeller exit (a=1.0, a=1.3, a=1.5)

134

Fig. 6-18 presents mass averaged flow angle, Mach number, and static pressure
distribution along the radius after impeller exit. First, in the flow angle chart, the absolute flow
angle is about 10 degree right after impeller exit in the case of a=1.5; and as flow goes
downstream, differences in flow angle between the cases goes down. Second, in the chart for
Mach number, the cases of 1.0 and 1.3 are almost the same and have a maximum Mach number
of 0.9 at the impeller exit. The case of a=1.5 shows that Mach number is a little smaller than
other cases. Finally, the pressure distribution shows that static pressure can be observed the case
of a=1.5 has maximum value throughout the impeller exit region.

Fig. 6- 18 Mass flow averaged absolute flow angle, Mach number and static pressure after
impeller exit

135

Fig.6-18 (cont’d)

136

Fig.6-18 (cont’d)

Fig6-19 presents the mass averaged flow angle distribution at the impeller exit (0%)
along the hub to shroud. The chart shows that the flow angle of the case of a=1.5 is more
uniform in the span-wise direction as increase axial length.

137

Fig. 6- 19 Hub to shroud mass averaged absolute flow angle distribution at the impeller exit

138

7. CONCLUSION
This study has presented flow characteristics, Mach number, flow angle and pressure, at
the impeller exit varying axial length based on length difference between impeller exit and
impeller inlet shroud radius. This study is focused on the effect of impeller axial length when the
flow is unstable at the impeller exit. For such a condition, the cases, which have high Mach
number and high swirl at the impeller exit, are designed. The effect of axial length on flow
conditions at impeller exit is difficult to predict. Also, it is important to check flow uniformity at
the impeller exit in order to make proper experiment conditions.
Although the flow characteristic within the impeller, especially the relative Mach number
contour, shows that the flow is unstable near the shroud, proper flow angle and Mach number at
the exit, the goal of the study, are obtained.
As a result of the cases varying axial length, the case of a=1.5 is the most stable at the
impeller exit because absolute flow angle and Mach number are uniform, although absolute flow
angle is more tangential.

7.1. 1D PERFORMANCE ANALYSIS ON TESTED IMPELLER
This study made 1D analysis code based on Ron Aungier’s model, which is empirical
based, and a Two-Zone model with TEIS model, which is physics based, and applied it to the
tested impeller. Two different 1D analysis methods are tested and compared with test data. While
Ron Aungier’s method shows better performance prediction in whole passage, Two-Zone model
shows an accurate prediction in the narrow flow range, especially around design points.
This study shows the relative pressure loss models in Ron Aungier’s model. They predict
that relative total pressure drop affects head rise through the impeller. It captures head coefficient

139

around the choking and surge region better than the Two-Zone model. Most of the total pressure
losses are affected by diffusion loss, which is a function of the relative velocity ratio across the
impeller inlet and impeller throat.

7.2. TWO ZONE MODEL WITH CORRECTED DIFFUSION MODEL
This study shows that a Two-Zone model can capture the head coefficient well around
the design point. However, because of its characteristics of predicting the diffusion ratio, this
study shows a Two-Zone model can predict a very narrow flow range. As rotational speed
increases, the differences in head coefficient around the choking point are increased. Japikse
suggested the TEIS model, which is a diffusion model, in a Two-Zone model. The TEIS model
predicts the diffusion ratio accurately. However, in every single flow rate, TEIS model uses
invariant pressure recovery effectiveness during the whole flow range so that it needs to be
changed as a function of the loss model. This study focused on pressure recovery effectiveness in
the inlet portion of the impeller. This is because the differences in head coefficient are mainly
affected by the inlet diffusion loss model in Ron Aungier’s model.
This study applies inlet diffusion loss model to the diffusion model in the Two-Zone
model. After applying the corrected diffusion model in the Two-Zone model, the 1D
performance analysis result shows a better prediction in the choking and surge region.

7.3. RE-DESIGN IMPELLER
This study showed a re-design impeller procedure and performed a 1D impeller design
using a 1D analysis code. The impeller design procedure usually starts with a 1D preliminary
design. However, if the designer could reduce variables such as the machine Mach number,

140

impeller diameter and flow rate, the 1D analysis code can replace the 1D preliminary design
code. Especially in re-design impeller work, 1D analysis code plays an important role in the
design procedure because the designer could expect impeller performance. This study made four
cases that have the same diffusion ratio, which is predicted by the 1D analysis code and
conducted CFD analysis. CFD results are compared with the 1D analysis results.

7.4. CONSIDERATION OF AXIAL LENGTH FOR IMPELLER DESIGN
With the 1D design procedure, an axial length of the impeller is determined. Impeller
axial length affects impeller curvature and blade length and theta angle. When the impeller exit
flow is unstable (high Mach number and high swirl at the impeller exit), the effect of axial length
is studied.
CFD analysis shows that the flow condition at the impeller exit is stable as the axial
length increases, although the impeller exit flow is the most tangential flow.

7.5. CONTRIBUTION
This study made two different 1D analysis codes using Ron Aungier’s method and TwoZone modeling. Two 1D analysis codes conducted a 1D analysis on the tested impeller and
showed its results, which are compared to each other. Relative loss models in Ron Aungier
model showed how much the relative pressure loss changes according to the flow rate and the
impeller rotational speeds.
A method that determines the diffusion ratio in the Two-Zone model is corrected based
on the inlet loss model in Ron Aungier’s model. By applying corrected inlet diffusion model, the
accuracy of Two Zone model has been improved.

141

This study also showed an impeller re-design procedure. The 1D analysis code finds a
trade-off point, keeping the diffusion ratio between impeller tip widths and impeller blade exit
angles. CFD results and 1D analysis results of re-designed impeller are compared and showed
similar tendency.

7.6. RECOMMENDATION
The corrected diffusion ratio used in the Two-Zone model in this study does not consider
the element “b”, which determines the diffusion ratio across the impeller throat to the exit. The
study assumed other losses do not vary with impeller rotational speed. And it is difficult to
consider all the relative total pressure losses because they are calculated in the iteration except
inlet diffusion loss and incidence loss. However, the corrected diffusion model could be more
accurate if other loss models were properly applied to element “b”.
Impeller re-design using 1D analysis code shows that 1D analysis code is useful for
finding a trade-off point between an impeller tip width and a blade exit angle. In the meridional
curve and blade angle distribution design process, it is necessary to consider changing area
distribution at mid-passage, which mainly affects velocity diffusion.
Assuming impeller slip factor is very important when starting an impeller design.
Although there are many empirical correlations for slip factor, slip factor used in the 1D analysis
is changed for the 1D analysis method. The 1D analysis method based on empirical correlations,
such as Ron Aungier’s method used in this study calculates aerodynamic blockage. However, the
Two Zone model, which is based on physics, does not calculate aerodynamic blockage. Because
of blockage problem, meridional velocities at the impeller exit are different according to 1D

142

analysis method. Therefore, in order to use both of the 1D analysis methods, a slip factor model
correlation, which is a function of blockage, is recommended between them.
This study does not include testing the re-designed impeller. Through the testing of the
re-designed impeller, the difference in performance between 1D analysis and CFD results is
understood and parasitic work that is not included in CFD analysis can be verified. For better
performance prediction, impeller slip factor can be found through the test.

143

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144

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