IMPACT OF IMPELLER BLADE TRIMMING ON THE PERFORMANCE OF
CENTRIFUGAL COMPRESSORS
By
Daniel Swain

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering - Doctor of Philosophy
2014

ABSTRACT
IMPACT OF IMPELLER BLADE TRIMMING ON THE PERFORMANCE
OF CENTRIFUGAL COMPRESSORS
By
Daniel Swain
Centrifugal impeller blade trimming can be used to modify an existing impeller design
to meet a new flow or pressure ratio design point. Using flow trimming, the passage area is
reduced from inlet to outlet along the entire meridional length of the impeller to reduce the
flow coefficient of the impeller while maintaining the pressure ratio of the baseline impeller.
Axial trimming is a method of trimming that reduces the blade height at the impeller exit
while maintaining the shroud profile of the original impeller in order to reduce the head
coefficient of the impeller while maintaining the original flow range.
In this work, computational fluid dynamics was employed to numerically model four impellers of varying geometries, speeds, and performance characteristics in order to understand
the performance effects and limits of modifying the impeller geometry by either flow or axial
trimming. Flow trimming was found to be capable of reducing the flow coefficient by between 20 and 50% while maintaining the pressure ratio and efficiency of the baseline impeller.
The suitability of an impeller for flow trimming was found to correlate strongly with flow
incidence angle. Axial trimming could be employed to reduce the pressure ratio to between
9 and 13% of the baseline pressure ratio before choking in the passage reduced the effective
flow range. Impeller performance was found to respond differently to axial trimming based
on the diffusion ratio of the baseline impeller.

To my family. To Laura, my partner and best supporter. To Jacob and Colin, my favorite
boys. I love you all.

iii

ACKNOWLEDGMENTS

First, thanks goes to my advisor, Professor Abraham Engeda, for his support, encouragement, and guidance; and to the members of my PhD committee. I would also like to thank
Michael Cave and Solar Turbines for providing the raw geometry and data that helped set
the foundation for this work. In addition, I owe a debt of gratitude to my fellow students in
the Michigan State University Turbomachinery Lab and to my coworkers at Rolls-Royce all of whom have been generous with their time and knowledge when I have distracted them
from their own pursuits to help me hash out a problem with my research.
I would especially like to express my deepest gratitude to those teachers and professors who
went beyond stuffing my head with knowledge and inspired the curiosity and love of science
and learning that has brought me to the brink of my proudest achievement. Chief among
those is Rob Sload, a gifted teacher and scientist, who sparked a fire twenty years ago that
burns to this day. If I contribute anything to science or inspire anyone to follow my path, it
will be because Sload once taught me that physics does not suck.

iv

TABLE OF CONTENTS

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

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction and Background
1.1 History and Application . . . . . . . .
1.2 Construction and Design . . . . . . . .
1.2.1 Construction . . . . . . . . . .
1.2.2 Design Methods . . . . . . . . .
1.2.3 Design Considerations . . . . .
1.2.3.1 Sources of Loss . . . .
1.2.3.2 Limitations . . . . . .
1.2.4 Analysis . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

1
1
4
4
6
9
9
10
11

Chapter 2 Current State of Study . . . .
2.1 General Design Considerations . . . . .
2.2 Compressor Performance Modification
2.3 Importance of Impeller Trimming . . .
2.4 Types of Trimming . . . . . . . . . . .
2.4.1 Radial Trimming . . . . . . . .
2.4.2 Axial Trimming . . . . . . . . .
2.4.3 Flow Trimming . . . . . . . . .
2.5 Theoretical Effects of Trimming . . . .

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

15
15
19
20
21
22
23
25
26

Chapter 3 Modeling Methodology . . . . . .
3.1 Computational Fluid Dynamic Theory . .
3.1.1 Motivation for CFD . . . . . . . .
3.1.2 Governing Equations . . . . . . . .
3.1.2.1 Turbulence Modeling . . .
3.1.2.1.1 Baldwin-Lomax .
3.1.2.1.2 Spalart-Allmaras
3.1.2.1.3 k-epsilon . . . . .
3.1.2.1.4 k-omega . . . . .
3.1.2.1.5 Wall Funcions . .
3.1.3 Boundary Conditions . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

29
29
29
30
31
32
32
32
33
33
34

v

3.2

3.1.4 Meshing . . . . . . .
Specific Methodology . . . .
3.2.1 Geometry Modeling .
3.2.2 Meshing . . . . . . .
3.2.3 Boundary Conditions
3.2.4 Solver . . . . . . . .

. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
and Fluid Domain Properties
. . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

35
36
36
36
38
39

Chapter 4 Flow Trimming . . . . . . . . . . . . .
4.1 Impeller Baseline Characteristics . . . . . . .
4.2 Comparison of CFD Results with Experiments
4.3 Trimming Methodology . . . . . . . . . . . . .
4.4 Flow Trim Results . . . . . . . . . . . . . . .
4.5 Analysis . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

42
42
43
50
51
64

Chapter 5 Axial Trimming . . . .
5.1 Axial Trimming Methodology
5.2 Axial Trim Results . . . . . .
5.3 Analysis . . . . . . . . . . . .

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

67
67
68
76

Chapter 6 Conclusions . .
6.1 Summary . . . . . . .
6.1.1 Flow Trimming
6.1.2 Axial Trimming
6.2 Applicability to Design
6.3 Next Steps . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

79
79
80
80
81
82

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

BIBLIOGRAPHY

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

vi

LIST OF TABLES

Table 4.1:

Basic Geometry of Baseline Impellers . . . . . . . . . . . . . . . . .

43

Table 4.2:

Overall Performance for Baseline Impellers . . . . . . . . . . . . . .

43

Table 4.3:

Percent Difference Between Experimental and Computational Results
for Impeller D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

vii

LIST OF FIGURES

Figure 1.1:

Typical Operating Range of Compressor Types . . . . . . . . . . . .

2

Figure 1.2:

Cordier Line Diagram - specific diameter vs. specific speed . . . . .

3

Figure 1.3:

Features of a Centrifugal Compressor . . . . . . . . . . . . . . . . .

5

Figure 1.4:

Velocity Triangle at Impeller Inlet . . . . . . . . . . . . . . . . . . .

11

Figure 1.5:

Velocity Triangle at Impeller Exit . . . . . . . . . . . . . . . . . . .

12

Figure 1.6:

Graphical Representation of the Jet-Wake Fluid Flow Model . . . .

13

Figure 2.1:

Effect of Blade Sweep . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Figure 2.2:

Ratio of Axial Length to Exit Diameter vs. Flow Coefficient . . . .

19

Figure 2.3:

Radial Impeller Trimming

. . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.4:

Axial Impeller Trimming Profiles . . . . . . . . . . . . . . . . . . . .

23

Figure 2.5:

Axial Impeller Trimming Profiles Preserving Shroud Profile . . . . .

24

Figure 2.6:

Impeller Flow Trimming Profiles . . . . . . . . . . . . . . . . . . . .

25

Figure 3.1:

3D Representation of a Typical Impeller Geometry and Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Figure 3.2:

Impeller Meridional Profile . . . . . . . . . . . . . . . . . . . . . . .

38

Figure 3.3:

Typical Computational Mesh . . . . . . . . . . . . . . . . . . . . . .

39

Figure 3.4:

View of Mesh at Leading Edge . . . . . . . . . . . . . . . . . . . . .

40

Figure 3.5:

View of Mesh at Tip Region . . . . . . . . . . . . . . . . . . . . . .

41

Figure 4.1:

Effect of Mesh Resolution on Pressure Ratio for Impeller D . . . . .

45

viii

Figure 4.2:

Effect of Mesh Resolution on Isentropic Efficiency for Impeller D . .

46

Figure 4.3:

Experimental vs. Computational Results for Impeller D . . . . . . .

47

Figure 4.4:

Head Coefficient vs. Flow Coefficient for CFD and Experimental
Results for Impeller A . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Efficiency vs. Flow Coefficient for CFD and Experimental Results for
Impeller A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 4.6:

Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller A

.

52

Figure 4.7:

Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller B . .

52

Figure 4.8:

Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller C . .

53

Figure 4.9:

Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller D .

53

Figure 4.5:

Figure 4.10: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller A 54
Figure 4.11: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller B 55
Figure 4.12: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller C 55
Figure 4.13: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller D 56
Figure 4.14: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 4.15: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 4.16: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 4.17: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 4.18: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed
Impeller A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 4.19: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed
Impeller B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

ix

Figure 4.20: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed
Impeller C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Figure 4.21: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed
Impeller D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Figure 4.22: Circumferentially Averaged Mach Number for Flow Trimmed Impeller D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Figure 4.23: Relative Mach Number at Impeller Exit Near Choke for Flow Trimmed
Impeller B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Figure 4.24: Percent Difference Between Flow Ratio and Area Ratio for Flow
Trimmed Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Figure 5.1:

Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller A 68

Figure 5.2:

Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller B 69

Figure 5.3:

Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller C 69

Figure 5.4:

Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller D 70

Figure 5.5:

Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially
Trimmed Impeller A . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially
Trimmed Impeller B . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially
Trimmed Impeller C . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially
Trimmed Impeller D . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Exit Flow angle vs. Flow Coefficient for Axially-Trimmed Impeller D

74

Figure 5.10: Relative Mach Number at Impeller Exit Near Choke for AxiallyTrimmed Impeller D at 50% Span . . . . . . . . . . . . . . . . . . .

76

Figure 5.6:

Figure 5.7:

Figure 5.8:

Figure 5.9:

Figure 5.11: Ratio of Head Coefficient to Design Head Coefficient vs. Trim Percent 78

x

LIST OF SYMBOLS

a - speed of sound
b - blade height
c - absolute velocity
e - specific energy
h - enthalpy
k - thermal conductivity
m - mass
ns - specific speed
q - heat energy
r - radius
u - blade speed
A - area
D - diameter
M - Mach number
N - rotational speed
P - pressure
Q - volume flow rate
Rc - compressor pressure ratio
T - torque, temperature
V - velocity
W - relative velocity
xi

Z - compressor axial length dimension
α - absolute flow angle
β - blade angle, relative flow angle
γ - ratio of specific heats
η - efficiency
ν - kinematic viscosity
ρ - density
τ - shear stress
ω - angular velocity
Φ - flow coefficient
Ψ - head coefficient
Ω - rotational speed
Superscripts and Subscripts
h - hub
o - stagnation property
r - radial direction
s - shroud
u - tangential direction
1 - inlet
2 - exit

xii

Chapter 1
Introduction and Background

1.1

History and Application

In his book, Dixon defines a turbomachine as, ”a device in which energy is trasferred either
to, or from, a continuously flowing fluid.” [1] The concept of turbomachinery dates back to at
least Heron of Alexandria’s steam engine from the first century A.D. Centrifugal compressors
are used for gas compression and were first developed around the latter part of the 19th
century and were used for industrial applications such as mine ventilation or in pneumatic
conveyors. Later, centrifugal compressors were used in the first turbojet engines. [2]
The range of application for centrifugal compressors exists between those of positive displacement compressors and axial compressors in terms of pressure ratio and flow rate as
shown in Figure 1.1 [3]. The most familiar application for centrifugal compressors is paired
with a turbine in an automotive engine turbocharger, but there are many other applications
for which a radial compressor is ideally suited. While the centrifugal compressor has been
largely replaced by the axial compressor in most modern turbojet applications because axial
compressors are generally more efficient than centrifugals, the relative compactness of a centrifugal compressors means it still holds a significant advantage in turboprop and rotary-wing
applications where size and weight are of primary concern. The increase in core compression ratio and resultant reduction in corrected flow in response to demand for increased fuel
economy means that centrifugal compressors are beginning to find new life in aerospace ap1

Figure 1.1: Typical Operating Range of Compressor Types

plications as the rear stage of larger turbofan compressors. Centrifugal compressors are also
ideally suited for natural gas compression applications because they can produce a much
larger pressure ratio over a single stage than an axial compressor and have advantages over
positive displacement compressors in that they are able to handle higher flow rates, are more
efficient, and have a much simpler mechanical design.
Efficient turbocompressor designs have been found to cluster along a curve first noted by
Cordier [4] when specific diameter is plotted against specific speed as seen in Figure 1.2. According to Cordier’s work, centrifugal compressors have an advantage over axial compressors
below a specific speed of 2.
Of course, reducing the selection of a compressor type to a single parameter is too simplistic, and it is important to consider the advantages and disadvantages of a compressor
type relative to other types. For a single stage, a positive displacement compressor can
produce a higher pressure ratio than any turbo-compressor, but size increases dramatically
2

Figure 1.2: Cordier Line Diagram - specific diameter vs. specific speed

with flow rate. Compared with a reciprocating or other positive displacement compressor, a
radial compressor has a relative advantage in terms of ability to handle higher flow rates and
in mechanical simplicity, as previously mentioned. The disadvantages relative to positive
displacement compressors is in the fact that a centrifugal compressor must employ multiple
stages to produce comparable pressure ratios and in the fact that a positive displacement
compressor produces static pressure while a centrifugal compressor produces both static and
velocity pressure which usually must then be converted into static pressure - and so incur
losses - in order to be useful. When considering the relative merits of axial and centrifugal compressors, the primary considerations are flow rate and pressure rise. Either axial or
centrifugal compressors may be staged to achieve desired pressure ratios and axials hold an

3

advantage in the fact that the flow may be delivered directly from one stage to the next
without the use of elaborate, and loss-inducing, return channels such as those required for
staged centrifugal compressors. In addition, an axial compressor can comfortably handle
much higher flow rates for a given size than a centrifugal while a radial compressor can
generate a greater pressure rise over a single stage than an axial machine. A compromise is
sometimes found in a ”mixed flow” compressor which exists to fill the gap between centrifugal
and axial compressors.

1.2
1.2.1

Construction and Design
Construction

The distinguishing characteristics of a centrifugal compressor are an axial inlet, a 90-degree
turn in the flow passage, and a radial outlet. The impeller discharges into a diffuser and
then enters a volute or, in the case of a multi-stage compressor, a return channel which turns
the fluid into the inlet of the next stage. Figure 1.3 shows the primary features of a typical
centrifugal compressor.
Pressure rise in the working fluid is achieved in two ways: the fluid is diffused in the
relative reference frame within the passage, raising the static pressure, and additional pressure rise comes through accelerating the fluid in the absolute reference frame and thereby
raising the total pressure. This, and the large tangential velocity component of the fluid at
the impeller exit, creates the necessity for the diffuser.
A centrifugal compressor diffuser may be vaned or vaneless. Either way, as the fluid
travles in the radial direction, the passage area increases and slows the fluid in the relative
reference frame converting dynamic pressure to static pressure. Vanes within the diffuser
4

Figure 1.3: Features of a Centrifugal Compressor

may be used to turn the fluid direction from the circumferential to the radial direction. In
doing so, the vanes reduce the distance the fluid travels around the diffuser and thereby
reduce losses due to diffuser wall friction.
Centrifugal compressors may be constructed using either a shrouded or unshrouded design. An unshrouded impeller has blade passages open on one side that must be matched
to a stationary shroud. An unshrouded compressor offers simplicity of construction since
the impeller may be machined from a single piece of material, however the unshrouded design means there is a gap between the blade tip and the shroud across which the fluid may
leak from the pressure side to the suction side of the blade. A shrouded impeller has the
advantage of not suffering from blade tip leakage and is more structurally robust than an
5

unshrouded impeller, but is more difficult to manufacture since it must typically be made
from two pieces of material which must be carefully fit and fastened together. A shrouded
design is also more difficult to modify than an unshrouded impeller since the shroud must
be removed and replaced in order to trim the blades in the axial direction.

1.2.2

Design Methods

The flow in the channel of a centrifugal compressor impeller is extremely turbulent and
complex. High operating speeds - commonly exceeding sonic velocities - and difficulties of
visual accessibility make centrifugal compressor flows difficult to analyze. Furthermore, the
computing power required to perform numerical CFD analysis has only recently been made
available to common users. Historically, centrifugal compressor designs were done empirically
and, when a particular design proved successful, was expanded to different applications or
operating conditions using laws of similitude. Similitude allows the designer to create a
geometrically similar design to a successful design using non-dimensional parameters such
as flow coefficient, head coefficient, power coefficient, specific speed, and efficiency. While
designs are usually verified or improved using higher-fidelity analysis, similitude laws are
often applied to a previous successful design in order to give the designer a reasonable
starting point for the new design and so the application of similitude is still common practice
in industry. Similitude also gives a set of dimensionless parameters that allows the engineer
to quickly evaluate and compare the characteristics of different compressors. Some of the
most commonly used parameters are: flow coefficient (φ), head coefficient (ψ), and specific
speed (ns ). Where:
φ=

4Q
πD2 U2

6

ψ=

∆Po
ρN 2 D22

ns =

φ0.5
ψ 0.75

One-dimensional analysis is a useful tool for overall stage design. It helps the designer
select major parameters for the impeller design such as inlet and outlet diameters, speed, and
inlet and outlet blade angles. However, 1-D design relies on empirically-defined relationships,
such as those used for slip or efficiency, to predict compressor performance and does not give
the designer any guidance on how to create the blade angle distribution, hub and shroud
curvature between inlet and outlet, or how to eliminate losses within the blade passage.
Typically, the results of one dimensional analysis are improved by incorporating empirically
derived factors to account for slip and sources of loss that do not appear in 1-D equations.
There are many other important parameters related to centrifugal compressor design
and performance that affect performance, and describing them all would require turning this
work into a textbook, but one that needs highlighting is diffusion ratio. Diffusion ratio is
a ratio of the relative fluid velocities at the inlet and exit and is a measure of the amount
of diffusion that occurs within the passage. For impellers with a value less than one, there
is acceleration rather than diffusion of the fluid within the passage. [5] The equation for
Diffusion Ratio is as follows:

Dif f usionRatio =

W1
W2

Two-dimensional, or meanline, analysis is the application of computational fluid dynamics
to the meanline stream surface (measured between hub and tip) of the blade passage of a
compressor. 2-D analysis offers more insight into the flow field characteristics than 1-D

7

analysis with lower computing cost than 3-D CFD, but is obviously limited in its ability to
predict performance of the whole flow field since it has no way to directly predict hub or
shroud wall friction or tip leakage and cannot account for blade angles that vary between
hub and tip. This shortcoming can be partially addressed without excessive penalties in
computing cost by using quasi-3D analysis. This is a method by which the flow field is
broken into a number of 2-D ”slices” along the blade span which allows the flow field to be
calculated directly at a number of points along the span and the overall performance to be
calculated by interpolating between planes.
Three-dimensional computational fluid dynamic analysis used to be the exclusive domain
of high-powered computing, but modern computer technology is now allowing CFD analysis
to be performed on desktop computers. There is still significant computational cost to CFD
analysis - a typical steady-state solution can take hours or days on a PC and longer for
transient analysis. Therefore, 3-D CFD is still too computationally expensive to be used
for ”white-sheet” design, but multi-core computing brings CFD within the reach of the
everyday user. The great advantage of CFD is that a carefully constructed simulation can
give a complete flow-field analysis of an impeller and sources of loss such as flow separation
can be easily identified, located, and eliminated.
Modern design practice uses a mixture of the tools mentioned. 1-D or 2-D design tools
are still useful for initial design and/or parametric analysis. CFD is used to improve and
verify the robustness of impeller designs created using similitude or lower-fidelity analysis.

8

1.2.3

Design Considerations

1.2.3.1

Sources of Loss

As with any compressor, the goal of a successful design is to produce the desired pressure
rise at the required flow rate with the least possible work. In order to minimize work input,
the losses in the compressor must be minimized, so it is useful to know the sources of loss in
a centrifugal compressor.
One of the primary considerations is friction. Friction losses occur on the working fluid
within the fluid passage, but also on the back of the disk and in the mechanical components
of the compressor. In a shrouded compressor, there is additional loss due to fluid friction on
the non-working fluid side of the shroud.
Flow separation can occur within the blade passage on the hub wall, shroud wall, or the
blade surface. Fluid separation reduces the effective cross-sectional area of the compressor
blade passage in the direction of flow and thereby reduces the diffusion pressure rise.
Blade tip leakage is another source of loss in an unshrouded compressor. In order to keep
the impeller turning freely and allow space for material distortion due to centrifugal forces
and thermal growth, there must be a clearance gap between the impeller blades and the
compressor shroud. This gap allows fluid to pass from the pressure side of the blade to the
suction side of the blade. Compressor efficiency has been shown to decline with increasing
tip clearance by Schumann et al. [6] and by Senoo and Ishida. [7]
Centrifugal compressors typically operate at very high rotational speeds to accelerate the
fluid to create the velocity pressure, and because of this high blade speed the working fluid
experiences a sudden acceleration in the tangential direction as it enters the blade passage.
If the rotational speed and inlet diameter are large enough, and/or the velocity of the fluid

9

entering the impeller is high enough, there can be a shockwave created at the inlet of the
compressor at the leading edge of each of the blades which creates a pressure loss in the fluid
as it crosses the shockwave. An inlet may be transonic where only a portion of the inlet
(beginning at the outer edge) is affected by shock losses or fully supersonic. A sonic inlet
condition may be mitigated by the use of inlet guide vanes to turn the fluid in the direction
of rotation, but doing so reduces the tangential acceleration of the fluid within the impeller
and therefore reduces the pressure rise in the compressor. Ultimately, a designer must weigh
the relative benefits of the pressure rise that can be achieved by high impeller speeds, the
size of the inlet required by the flow rate, and the losses incurred by shock at the inlet.

1.2.3.2

Limitations

In addition to minimizing losses, a centrifugal compressor designer must consider certain limitations of compressor operation. Choke is an operating condition which limits the throughflow in the blade passage. The relative velocity of flow in the passage is limited by the
speed of sound, so the choke limitation is determined by the smallest cross-sectional area
of the flow path in the direction of flow. When a compressor is operating in a fully-choked
condition, the flow rate cannot be increased by reducing the outlet pressure (also known as
back pressure) or by increasing the rotational speed of the impeller.
On the other side of the operating line there exists a pair of limiting conditions known
as stall and surge. Both conditions occur when the flow is reduced below a certain level.
Surge occurs when the compressor flow rate is reduced by increasing the pressure it has to
work against. The result is an unsteady and rapidly varying flow rate that generates noise
and creates large cyclical stresses on the compressor components. Stall is a phenomenon
wherein the flow becomes separated from the compressor wall or blade surface. Rotating
10

Figure 1.4: Velocity Triangle at Impeller Inlet

stall is a condition in which only a portion of the circumferential area has separated flow.
The separated flow regime propagates from blade to blade at a rate on the same order as the
rotational speed of the impeller. The repeated and very rapid separation and reattachment
of the flow to the blade surface imparts high-frequency cyclical stresses on the blades that
can be damaging, especially if the blade has a sympathetic frequency mode.

1.2.4

Analysis

The Euler Turbomachinery Equation as presented in [8] demonstrates how work input creates
pressure rise through increasing the tangential velocity of the fluid.

T = (m
˙ + ∆m)(c
˙ u2 r2 − cu1 r1 ) + TF
Where ∆m
˙ is the fluid leakage and TF is torque input absorbed by disk friction. The
remainder of the torque produces a useful effect by increasing the enthalpy of the fluid.
A basic understanding of the inlet and exit flow can be had by examining the compressor
velocity triangles. While simplistic, this analysis provides a great deal of useful insight into
the overall performance of the compressor impeller. Figures 1.4 and 1.5 show inlet and exit
velocity triangles for a centrifugal compressor.

11

Figure 1.5: Velocity Triangle at Impeller Exit

Another form of the Euler turbomachinery equation relates the change in the fluid enthalpy directly to the blade and fluid velocities plus losses.

U22 − U12 W12 − W22 c22 − c21
+
+
+ hL + hF
∆h =
2
2
2
The first two terms of the equation represent the static enthalpy rise and the third term
is the dynamic enthalpy rise. hF and hL are the friction and leakage losses, respectively.
Because the compressor blades are not able to perfectly guide the flow through the
passage, the concept of slip is introduced. Slip is not a source of loss, but has the effect of
reducing the change in velocity experienced by the working fluid and thereby reducing the
pressure rise that would be expected if the fluid followed the blades exactly. A number of
empirical models have been developed to predict slip. One popular model was proposed by
Stodola [9].

cus =

πU2 cosβ2
Nblades

12

Figure 1.6: Graphical Representation of the Jet-Wake Fluid Flow Model

In this model cus represents the difference between the predicted and actual tangential
velocity and β2 is the blade exit angle.
The velocity triangle compressor model is useful for understanding the basic inlet and
exit flow conditions, but the flow in the impeller is highly non-uniform with large areas of
separated flow. A more accurate one-dimensional model is the jet-wake model proposed by
Dean [10]. In this model, visualized in Figure 1.6, the fluid exits the impeller with a highvelocity jet near the pressure side of the blade and a large area of low-velocity fluid behind
the pressure side of the blade.
The jet-wake model is a useful way to think about the passage flow; however, the flow
within the impeller passage is actually far more complex. The low-momentum fluid which
comprises the wake is influenced by Coriolis effects, tip leakage flows, and backflow from
the diffuser even when the flow field is well-behaved and does not contain large areas of
separation. These complex flows can only be modeled using computational fluid dynamics;
and, even then, imperfectly.
13

The efficiency of a centrifugal impeller can reach 95% or better, so it is possible to design
an impeller with very low levels of loss, but according to Cumpsty, ”The mechanisms of
loss generation inside the impeller are not really understood. It is not known, for example,
whether the presence of a large wake is really detrimental, nor whether its position affects
loss,” [11] and so compressor aerodynamic design still relies on the application and modification of previous successful designs. The laws of similitude allow the designer to scale an
existing design, but changing the performance of an existing impeller by way of blade trimming is less well understood. The purpose of this study is to explore the effects and limits
of impeller performance modification by way of blade trimming. In this way, designers will
be afforded an additional tool for centrifugal impeller design or performance modification.

14

Chapter 2
Current State of Study

2.1

General Design Considerations

In an early study which focused on impellers with straight radial blades, Rodgers found that,
”The impeller characteristic is primarily governed by the inducer geometry, in that maximum
flow and low efficiency will occur when the inducer chokes, and also as flow is decreased from
choke, efficiency will initially rise to a maximum value adjacent to inducer stall and then
decrease as a major consequence of inducer stalling losses.” [12] Rodgers reports that this
is due to the fact that, since the cross-sectional area of the flow passage increases in the
streamwise direction, the narrowest portion of the blade passage is found at the inlet, choke
will occur at that location. In order to reduce blade blockage at the inlet, some blades are
cut back from the inducer portion of the impeller. These partial-length blades, known as
”splitter blades” allow the designer to achieve a wider choke margin while maintaining the
number of blades necessary to guide the flow at the outlet and limit slip. Also, in low-flow
situations, surge phenomena is initiated at the inlet because that is where flow first meets
the blades and where separation is initiated.
Because the relative velocity of the leading edge varies in the inducer between the impeller
hub and shroud, modern inlet designs employ inlet blade angles that vary in the spanwise
direction to maintain an optimum incidence angle along the leading edge. In a 1988 study,
Harada found that when a constant inlet blade angle is used, ”a large incidence loss at the
15

inlet occurs as a result of mismatching between the inlet flow and blade angle. In order
to achieve a smooth inlet flow into the impeller, and to improve impeller performance, it
is necessary to incorporate leading edge twist and to vary blade angles from hub to shroud
throughout the impeller.” [13] In that study, it was found that a fully three-dimensional
blade angle distribution could improve efficiency as well as increasing both head coefficient
and choke flow rate.
As computing tools improve and computing power increases, new studies are able to
be designed and executed to improve impeller design. Because there are so many interrelated factors which factor in to turbomachinery designs, they are, ”usually multiobjective
problems, where several performance parameters must be considered in the optimization
process,” [14] so modern computational methods can be used to automate the design of centrifugal impellers. Bonaiuti and Zangeneh [14] employed a multiobjective genetic algorithm
to improve a baseline design to smooth the flow field and thereby improve efficiency across
the whole operating range of a compressor as well as extend the choke limit to a higher flow
rate.
Many modern impellers are designed with significant backswept blades. This has many
beneficial effects such as reduction in blade loading and lengthening of the passage so that
more fluid deceleration can occur within the passage rather than in the diffuser - which
operates less efficiently. Backswept blades also produce a more radial flow at the impeller exit,
and this will tend to decrease the overall pressure rise of the impeller as seen in Figure 2.1.
However, because backsweep reduces blade stress, an impeller with backswept blades may
be operated at higher speeds, therefore a backswept impeller can achieve a larger pressure
rise by the centrifugal effect which is loss-free.
As demand increases for higher pressure ratios and flow capacities, inlet sizes and speeds
16

Figure 2.1: Effect of Blade Sweep

increase; and so even well-designed inlets may have shockwaves formed at the leading edge
of the blade particularly as the flow rate increases towards choke. There is a pressure loss
incurred as the flow crosses a shockwave and so imposes a penalty to the overall pressure
ratio of the compressor. Furthermore, in a pair of studies [15, 16] Ibaraki, et al studied
the flow field of a transonic impeller and found that an unshrouded compressor suffers an
additional penalty. The result of the study found that, in an unshrouded transonic centrifugal
compressor, ”inteaction between the leakage vortex and the shock wave enhances the total
pressure loss of the vortex core pitchwise, at downstream of the shock wave.”
The clearance distance between the blade tip and shroud of an unshrouded impeller is
another key parameter impacting compressor performance. It has been shown clearly by
Schumann et al. [6] that increasing tip clearance reduces the pressure ratio of a compressor.
That finding was confirmed by Schleer et al. who also found a narrowing of the operating
range of a compressor for increased tip clearance because, ”for increased tip clearance ratio, stall occurs at higher flow coefficients while the stage pressure ratio was substantially
reduced.” [17] Intuitively, then, it would seem that decreasing the tip clearance to zero, as

17

in a shrouded impeller, would improve stall characteristics and increase the operating range,
but Harada found that, ”A shrouded impeller has inferior overall performance in the lower
flow region compared with an identical unshrouded one and also has a narrower stable range,
because rotating stall takes place at a higher flow rate.” [18]
Another design feature that has attracted some attention from researchers is the axial
length of the compressor which can also be described in terms of curvature. As discussed in
Section 1.2.1, some portion of the pressure rise in the impeller comes from velocity pressure
and some comes from diffusion within the channel. According to Al-Zubaidy, ”The so-called
50% reaction design strategy attempts to achieve approximately half the diffusion in the
impeller and half in the diffuser. In modern literature the term ”diffusion ratio” (which is
an indication of static pressure rise) is more widely used.” [19] In his mathematical study,
Al-Zubaidy proposes that there is a ratio of axial length to outlet diameter (z/D2 ) that will
produce optimum efficiency. This is due to the fact that a large value for z/D2 creates a
short, wide channel that will cause over-diffusion and flow separation within the channel,
but frictional effects become the dominant source of loss as channel length increases. An
optimum efficiency band is found to be in the range of about 0.25 < z/D2 < 0.32 and
the geometry of around 30 impeller designs is found to cluster in that range. In his book,
Aungier proposes that the best value for z/D2 is dependent on flow coefficient and proposes
a direct, linear relationship between flow coefficient and z/D2 as seen in Figure 2.2. [20]
These proposals were tested by Sorokes et al. in a computational fluid dynamic study of the
effect of shroud curvature (as a proxy for axial length) on impeller performance. A baseline
design was modeled through z/D2 values ranging from 0.21 to 0.30 and the best performance
was seen at a value of 0.30 for a flow coefficient of 0.12 [21] giving support to the findings of
Al-Zubaidy and Aungier.
18

Figure 2.2: Ratio of Axial Length to Exit Diameter vs. Flow Coefficient

2.2

Compressor Performance Modification

It sometimes occurs that the installed operating conditions differ from the design operating conditions, and is becomes necessary to change the performance characteristics of the
compressor. However, once installed, the options left to the designer are limited. One of
the simplest and most commonly available options is to change the operating speed of the
compressor. In this case, the designer needs only to be able to predict the effect of changing
the speed of the impeller. A reasonable approximation of the performance change can be
made by calculating the difference based on velocity triangles, dimensionless parameters such
as ψ, φ, and Ns or by using rules of thumb such as those described by Godse: ”Head of a
compressor varies as the square of tip speed,” and ”Flow handled by the compressor varies
with tip speed and rotor diameter.” [22] These methods are useful to a point, but only within
a narrow range. If the actual performance goal varies greatly from the original design, then
physical modifications must be performed. Furthermore, rotational speed affects both head
19

and flow, so one cannot be changed by varying rotational speed without affecting the other.
Inlet guide vanes can be used to impart pre-swirl to the working fluid before it reaches the
leading edge of the impeller. If the direction of rotation of the fluid is the same as the rotation
of the impeller, then the impeller does not accelerate the fluid as much in the tangential
direction and so the pressure rise across the impeller is reduced. The opposite is true if
guide vanes are used to turn the flow opposite of the impeller rotation. A mathematical
model for predicting the effect of changing inlet guide vane angles and comparison with
experimental results is presented by Liu et al. [23] The results of the study show that inlet
guide vanes are a useful tool for adjusting pressure ratio but that deviating from design
conditions comes with a penalty to efficiency.
These methods of performance modification serve a useful purpose in that they are inexpensive and can be applied with relative ease. However, the methods discussed in this section
suffer from penalties such as reduced flow range, inability to change head independent of
flow, or reduced efficiency. Improved methods of performance modification are available to
the designer by physically modifying the impeller.

2.3

Importance of Impeller Trimming

Compressor impeller trimming is a common industrial practice and serves a number of different purposes. In applications where low cost is more important than high efficiency, such
as in the production of automotive turbochargers which are manufactured in large quantities
and operate using waste energy from the exhaust stream, cast impellers may be machined
later in the process to produce impellers with different operating characteristics from a single
die. Similarly, a single high-flow design can be expanded to become a family of compressors

20

by successive trimming in order to cover a wide flow range.
Another important application is the reducing the outlet height of an impeller as described by L¨
udtke [8] in order to alter the exit Mach number of an impeller for different
molecular weight gases.
In addition to these, it can be very useful to modify an existing impeller design for a
new performance requirement rather than perform a ”white sheet” design on a new impeller.
As discussed previously, modification by the laws of similitude has long been used to adapt
successful designs to new operating conditions, but impeller trimming adds additional degrees
of flexibility for creating new compressor designs.
Despite these important uses, a lack of research published in open literature has been
noted by Rodgers [24] and David [25], and those studies that exist focus on trimming of a
single impeller. By examining the effects and exploring the limits of impeller trimming on
four different centrifugal compressor impeller designs, this study is an attempt to begin to
fill that hole and establish some general guidelines for design engineers who wish to employ
impeller trimming as a design tool.

2.4

Types of Trimming

In general terms, there are three primary methods by which a centrifugal impeller may be
modified by trimming. The outer diameter may be trimmed, the exit height may be reduced
without affecting the inlet, or the blade height may be reduced along the entire passage.
These will be discussed individually.

21

2.4.1

Radial Trimming

In radial trimming, the outlet of the impeller is trimmed in the radial direction reducing D2
as seen in Figure 2.3. Because the inlet is unaffected, the flow coefficient of the compressor is
left unchanged. The purpose of radial trimming is to reduce the total pressure rise across the
impeller. This method of trimming can be very useful because it is simple to apply and does
not require any modification to the compressor shroud, however reducing the outer diameter
does change the exit fluid angle (α2 ), and so a vaned diffuser would require redesign. In
terms of performance effects, according to Sapiro, radial trimming, ”increases the specific
speed of the given impeller; therefore, when applied to low specific speed wheels, it displaces
their specific speed value closer to the optimum one, while on high specific speed types, it
moves them further away into regions of even higher specific speed and lower efficiency.”

Figure 2.3: Radial Impeller Trimming

This method of trimming does not fundamentally change the flow field in the impeller passage except to shorten it, and so the effects of radial trimming are relatively well-understood
and are not addressed in this current work.
22

2.4.2

Axial Trimming

Axial trimming is a method of trimming by which the impeller blade height is reduced at the
outlet of the impeller while leaving the inducer unchanged. This is similar to the method of
trimming described by Engeda [27] and L¨
udtke [8]. Two representative examples are shown
in Figure 2.4.

Figure 2.4: Axial Impeller Trimming Profiles

However, in the current study, the outlet blade height is employed to reduce the diffusion
in the radial portion of the passage to modify the performance of the impeller rather than
to apply an impeller design to a different molecular weight gas. In order to minimize the
amount of additional hardware modification, the impeller is trimmed in such a way as to
preserve the shroud profile. Figure 2.5 shows a baseline impeller with a series of progressive
axial blade trims which preserve the shroud profile shape.
Schumann et al. studied the effects of outlet trimming on a centrifugal impeller. [6, 28]
The general trend shown by Schumann is a reduction in the pressure ratio as the outlet
of the impeller is progressively trimmed. One significant difference between Schumann’s
study and the current work is that Schumann maintained the ratio of tip gap to outlet
23

Figure 2.5: Axial Impeller Trimming Profiles Preserving Shroud Profile
blade height so that the tip gap becomes smaller as the blade height is reduced. In doing
so, the Schumann study did not isolate the effect of tip gap reduction from the effect of
blade trimming. Instead, tip gap losses were reduced as diffusion pressure rise was reduced
and channel friction increased. This is a likely explanation for the finding of a local pressure
ratio maximum and global efficiency maximum for a progressively-trimmed impeller. A later
study by Nili-Ahmadabadi et al. [29] claims to confirm the findings of Schumann’s study,
but does not reduce the tip gap in proportion to the outlet blade height, and so the studies
cannot be directly related.
In practice, because losses increase with tip gap, the clearance between impeller and
shroud is typically reduced to the greatest extent possible and is limited by the impeller exit
radius. For a given operating speed, tip gap is largely determined by material distortion
which is greatest at the outer diameter, where centrifugal forces have the strongest effect.
Therefore, tip gap can usually be considered to be a function of the impeller outlet diameter
for a fixed speed and cannot be reduced as the impeller is trimmed. For this reason, and so
the effects of blade trimming can be isolated from the effects of tip gap, it is assumed that
the blade height of the baseline design has been reduced as far as is practicable for the given
r2 and the clearance is kept constant as the impeller is trimmed.
24

Axial trimming is proposed as an alternative to radial trimming for high specific speed
impellers. Although the modification is more complicated for axial trimming, the limited
applicability of radial trimming to high specific speed machines means that axial trimming
may have certain relative advantages since reducing the outlet height will lower the specific
speed of the impeller and push a high specific speed machine toward a more favorable value
of specific speed.

2.4.3

Flow Trimming

The purpose of flow trimming is to modify the flow coefficient of a compressor while leaving
the pressure ratio unchanged. Practically, that means reducing the blade height along the
entire meridional length of the impeller. A representative meridional view of an impeller
blade and a series of flow trims is shown in Figure 2.6.

Figure 2.6: Impeller Flow Trimming Profiles

This method of trimming was studied by Rodgers [24] who found a small decline in
efficiency for a slightly-trimmed impeller that increases significantly as the impeller is progressively trimmed. In addition, Rodgers’ data shows a narrowing of the flow range as the
impeller is trimmed. Rodgers also suggests limiting flow trimming to a range of specific
speeds between 0.5 and 1.2.
25

A pair of studies by David et al. [30] and Zhang et al. [25] study the effect of trimming on
low specific speed radial compressors. Both studies confirm the findings of Rodgers that flow
trimming has a degrading effect on efficiency and flow range. It is also noted that additional
losses are due to frictional effects in the flow chanel, but Zhang’s study suggests that flow
trimming may be useful to specific speeds as low as 0.038 and recommends a limitation
of 50% blade height be used as the limit of flow trimming while David’s study does not
recommend flow trimming beyond a specific speed of 0.06.

2.5

Theoretical Effects of Trimming

The inlet of a centrifugal impeller is given much more design consideration than the impeller
exit. It has already been noted that the impeller inlet is the primary driver of compressor
performance. Furthermore, the inlet flow is usually considered to be uniform or, at least, wellbehaved; whereas the flow at the exit has thick boundary layers, tip clearance interactions,
secondary flows, and a non-uniform exit velocity described in Section 1.2.4. Axial trimming
has no effect on the inlet geometry, but flow trimming reduces the inlet area by reducing
r1s .
Assuming the inlet velocity is uniform, the mass flow can be found using the equation [1]:

mΩ
˙
3 sin2 β cosβ
= Ws1
s1
s1
ρ1 kπ
r

where k = 1 − rh1
s1

2

.

Mass flow is directly proportional to k and k decreases as rs1 is reduced. For a 2D blade,
β1s is equal to β1h and so does not change as the inlet shroud diameter is reduced. For a
3D blade, where the blade angle changes between the hub and tip β1s will also factor in to
26

the equation.
Another inlet consideration has to do with losses at the inlet. If the inlet velocity exceeds
the speed of sound at the tip, a shockwave will form ahead of the blade and the fluid that
crosses it will experience loss. The relative inlet tip Mach number can be found by

M1s,rel =

W1s
a1

and
W1s =

Ωr1s − c1 sinα1
sinβ1s

According to the above equations, the inlet relative mach number is directly proportional
to r1s , so the effect of flow trimming will be to reduce the inlet shock loss of a trans-sonic
compressor.
Flow trimming will also have an effect on frictional losses. Because the passage narrows,
the ratio of cross sectional area to hydraulic diameter for the passage will decrease and
so frictional effects would be expected to have a larger effect on the fluid. Furthermore,
since the back plate and rotating mechanical components other than the impeller remain
essentially unchanged with flow trimming, the amount of work that goes into raising fluid
pressure will decrease while mechanical and disc friction remain unchanged. Therefore, a
greater proportion of work input will go to frictional losses, and the compressor efficiency
will tend to decrease.
The effects of axial trimming are somewhat more difficult to predict, because outlet
height (b2 ) is not included in 1-D compressor design. The difficulty arises from the fact that
the fluid diffuses within the impeller passage, because the flow within the passage is highly
non-uniform. There is, of course flow separation on the suction side of the blade, and the
27

flow is certainly attached on the pressure side, but it is difficult to predict flow separation on
the hub and shroud surfaces or the amount of diffusion that occurs within the passage. If the
ratio of static to dynamic pressure rise is known for a given impeller, then some prediction
may be able to be made, but there is no generic model for predicting the effect on pressure
rise.
Since, as a rule, impellers choke and stall at the inlet, axial trimming should have no
effect on the flow range of a compressor in the initial stages. However, trimming narrows the
radial passage area, and at some point the choke point will move from the inlet to the knee
or radial passage and the choke margin will decrease. In terms of effect of axial trimming
on efficiency, any narrowing of the passage would tend to increase frictional effects, and so
efficiency would be expected to decrease with axial trimming.

28

Chapter 3
Modeling Methodology
The study that is the basis of this work was performed primarily using computational fluid
dynamics. A series of centrifugal impellers were modeled in CFX [31], a Reynolds-Averaged
Navier-Stokes (RANS) flow solver. This chapter will provide some general information on
CFD theory and the specific methodology used to model the impellers of interest.

3.1
3.1.1

Computational Fluid Dynamic Theory
Motivation for CFD

Computational fluid dynamics is an important and valuable resource for design and analysis
in the turbomachinery industry. The flow field inside a turbomachine is highly turbulent and
complex; there is therefore no analytical solution for turbomachinery problems. Engineering
correlations are used, and useful, but provide limited information. Experiments usually provide the most reliable data, but one-off hardware is extremely expensive and time-consuming
to produce, and therefore not appropriate for iterative design. Furthermore, turbomachines
operate at high speeds and are usually optically inaccessible; this places a significant restriction on their usefulness in understanding the properties of the flow field.
Forty years ago, computational fluid dynamics of a sufficient fidelity to resolve the flow
field of a turbocompressor was beyond the ability of any computer to perform. Twenty

29

years ago, CFD could only be performed on mainframe computers; but today a RANS solver
can be run on a centrifugal compressor on a desktop computer. This rapid advancement in
computing power means that more analysis can be done in less time and the design cycle
has been greatly accelerated. For these reasons, most design and a significant amount of
analysis of turbocompressors is performed using computational fluid dynamics.

3.1.2

Governing Equations

In CFD, instead of getting an analytical solution for the flow field, the complex fluid domain
is discretized into small elements with simple geometry, and the fluid dynamic equations
are applied to each one individually. As in any closed system, the laws of conservation of
mass, momentum, and energy must apply to every element. The continuity equation for any
element is:
∂ρ ∂(ρu) ∂(ρv) ∂(ρw)
+
+
+
=0
∂t
∂x
∂y
∂z
or
→
−
∂ρ
∇ · (ρ V ) +
=0
∂t
In turbomachinery applications, gravitational effects are ignored, so the momentum equations become:

ρ

→
−
Du
∂(ρu)
∂p ∂τxx ∂τyx ∂τzx
=
+ ∇ · (ρu V ) = −
+
+
+
Dt
∂t
∂x
∂x
∂y
∂z

ρ

ρ

→
−
Dv
∂(ρv)
∂p ∂τxy ∂τyy ∂τzy
=
+ ∇ · (ρv V ) = −
+
+
+
Dt
∂t
∂y
∂x
∂y
∂z

→
−
Dw
∂(ρw)
∂p ∂τxz ∂τyz ∂τzz
=
+ ∇ · (ρw V ) = − +
+
+
Dt
∂t
∂z
∂x
∂y
∂z

30

And, finally, ignoring internal heat generation and gravity, the conservation of energy
equation is:

ρ

D
Dt

e+

V2
2

−

+

=

∂
V2
ρ e+
∂t
2

→
−
∂
V =
∂x

k

∂T
∂x

+

∂
∂y

k

∂T
∂y

+

∂
∂z

k

∂T
∂z

∂(up) ∂(vp) ∂(wp) ∂(uτxx ) ∂(uτyx ) ∂(uτzx )
−
−
+
+
+
∂x
∂y
∂z
∂x
∂y
∂z

∂(vτxy ) ∂(vτyy ) ∂(vτzy ) ∂(wτxz ) ∂(wτyz ) ∂(wτzz )
+
+
+
+
+
∂x
∂y
∂z
∂x
∂y
∂z

This system of equations comprise the Navier-Stokes equations for three-dimensional, viscous
flow. The purpose of the CFD flow solver is to discretize the Navier-Stokes equations. There
are many different discretization techniques, and the particular methods are a field of study
unto themselves, but the purpose of each is to replace the derivatives with a difference
approximation and so convert the partial differential equations into a system of algebraic
equations which can be easily solved by a computer.

3.1.2.1

Turbulence Modeling

Turbulence is a major complicating factor in computational fluid dynamics because turbulent
flows are irregular, time-dependent, three-dimensional, and essentially chaotic. [32] Turbulence also operates on a wide range of time and length scales. Direct numerical simulation
(DNS) of turbulent flows is possible with CFD, but the mesh resolution and time scale must
be no larger than a viscously determined scale known as the Kolmogoroff scale. [33] Turbulence also operates on very small time scales so that the computational cost to precisely
resolve a turbulent flow field is prohibitive for any but very low Reynolds number flows.

31

Therefore, when a turbulent flow is modeled using CFD, the overall effects of turbulence are
typically approximated by a numerical model. Selecting a turbulence model is an important
consideration for CFD and can have a significant impact on the solution.
Large eddy simulation (LES) represents a compromise between accuracy and computational cost by resolving only large scale turbulent flow features and modeling small ones with
a numerical approximation; however, for turbomachinery applications, a numerical model is
typically used for turbulence at all length scales. A few of the most common models are as
follows:

3.1.2.1.1

Baldwin-Lomax The Baldwin-Lomax model is a 0-equation turbulence model

that predicts eddy viscosity as a function of the boundary layer velocity profile. Its advantage
is low computational cost, but its simplicity also limits its accuracy.

3.1.2.1.2

Spalart-Allmaras The Spalart-Almaras model [34] is a 1-equation turbulence

model that has been used in the gas turbine industry. The Spalart-Allmaras model solves a
transport equation for a viscosity-like variable. Like the Baldwin-Lomax model, it has low
computational cost, but is less accurate than more sophisticated models.

3.1.2.1.3

k-epsilon The k − model is the most commonly used turbulence model for

CFD. In it, the effect of turbulence is described as the rate at which kinetic energy of
turbulence is converted into heat. The k − model is a 2-equation model where k is turbulent
kinetic energy and

is the rate of viscous dissipation where:

q = k 1/2

32

and
≈

3.1.2.1.4

k 3/2
l

k-omega The k − ω turbulence model is another popular 2-equation model

similar to the k − turbulence model in that turbulent kinetic energy k is one of the variables
transported, but differs in that dissipation is determined by a variable, ω, which attempts
to model the length scale of the turbulence.

3.1.2.1.5

Wall Funcions In flows where the Reynolds number is very high (such as in

a turbocompressor), the viscous boundary layer is so thin that it is impractical to refine the
grid enough to accurately represent the profile. Therefore, a simplifying assumption about
the boundary layer is made by imposing so-called wall functions. Using this assumption, the
velocity profile near the wall is modeled by the equation:

u+ =

1
v¯t
= ln y + + B
uτ
κ

where
uτ =

τw
ρ

where τw is the wall shear stress and v¯t is the mean velocity parallel to the wall, κ is the
von Karman constant (κ ≈ 0.41) and B is an empirical constant which is around 5.5 for a
smooth flat plate boundary layer. See Section 3.1.4 for a definition of y + .

33

3.1.3

Boundary Conditions

Boundary conditions have a strong influence on the behavior of a CFD solution. Both the
value and the type of boundary selected can affect how the model behaves. The primary
types of boundary that must be considered are the wall boundary, and the flow inlet and
exit boundaries.
In a viscous flow problem, a no-slip boundary is imposed at the wall so that u = v = w =
0. Also, since pressure does not transfer across a wall, then ∂P/∂y = 0 for y normal to the
wall. However, that is an oversimplification because it ignores viscosity. A more accurate
wall boundary condition is:
∂ 2V
1 ∂P
=ν 2
ρ ∂y
∂y
Walls in turbomachinery are usually taken to be adiabatic, so where no heat transfers across
the wall ∂T /∂y = 0.
The inlet and exit flow boundary types require some a priori knowledge of the flow. For
the outlet, a supersonic flow boundary behaves differently than a subsonic flow boundary
because the flow across a subsonic boundary can be affected by the boundary condition
itself, however information about downstream temperatures, pressures, and velocities do not
propagate upstream in a supersonic flow. For an inflow boundary, the opposite is true if the inlet is super-sonic, then all fluid properties are fixed at the boundary because the
upstream flow cannot be affected by the flow field in the computational domain, but only
limited information may be transferred into the domain from a sub-sonic boundary. Some
assumptions about the flow field such as equilibrium in the direction parallel to the exit
plane may be imposed to reduce the distorting effects of a boundary plane, but inevitably
the boundary includes some simplifications which will distort the flow field in the region near

34

the boundary. Therefore, best practice dictates that all boundaries be placed sufficiently far
away from the domain of interest that the distorting effects are not experienced by the fluid
in the domain being studied.

3.1.4

Meshing

In CFD, the computational grid size and type must be appropriate for the system being
modeled. Uniform grids are uncommon in CFD simulations because, in order to capture
near-wall flow structures in a meaningful way, they must be refined to the point that the
grid imposes a huge computational cost on the model. Unstructured grids are useful when
the direction of flow is unknown, but in a turbomachinery application the major structures
and directions of the flow are known at least approximately, so structured grids aligned to
the direction of flow are typically used.
Because gradients in the flow are usually highest near walls, the grid is typically refined
in the near-wall region and spaced more widely away from walls to reduce computational
cost. A common metric used to measure grid quality is y + value. This is a non-dimensional
measure of near-wall grid quality, but can only be calculated after the solution is complete.
It is defined as:
y+ =

uτ y
ν

where y is the distance to the nearest wall.
A smaller value of y + corresponds to a finer near-wall grid and is a very useful measure in
turbomachinery since velocities in the flow passage are very high and consequently velocity
gradients at walls are very high.

35

3.2

Specific Methodology

All of the compressors in this study were modeled using a four step process: the geometry
was modeled, then meshed, the flow solution was generated, and finally the solution is post
processed. A detailed description of the process follows.

3.2.1

Geometry Modeling

In each case, a detailed description of the impeller geometry was obtained either from the
manufacturer, in the case of Solar Turbines, or from detailed geometry in open literature
for the remaining impellers. That geometry was entered into BladeGen software [35]. The
detailed geometry included: x,r coordinates for hub and shroud profiles, leading and trailing
edge coordinates for both the main blade and the splitter blade (where present), as well as
thickness and either beta or theta values at multiple spanwise locations along the meridional
length of the impeller blade. The result was an accurate 3D model of the passage geometry. A
representative view of the 3D geometry and a meridional profile are presented in Figures 3.1
and 3.2. As noted in Section 3.1.3, the inlet and exit boundaries are located far away from
the impeller passage in an effort to minimize the effects of the boundary on the flow field in
the region of interest.
The specific geometry of each impeller included in this study is included in Chapter 4.

3.2.2

Meshing

Once the detailed geometry was created in BladeGen, the computational mesh was generated
using TurboGrid. [36] A mesh sensitivity study was performed and the results are presented
in Chapter 5, but typical computational mesh details are presented here. The mesh is divided

36

Figure 3.1: 3D Representation of a Typical Impeller Geometry and Computational Domain
into three blocks - a stationary block for each of the inlet and exit domain and a rotating
block for the impeller. A typical mesh view is shown in Figure 3.3.
Because steady state and axisymmetric behavior was assumed for all cases studied, a
sector mesh was used to model the flow field. This is standard practice in the turbomachinery
industry and has the advantage of modeling the performance of the compressor with a greatly
reduced computational cost. If transient or non-axisymmetric behavior were to arise due to
the onset of rotating stall or surge, then the steady-state solution would fail to converge.
The geometry of the inlet and outlet blocks are simple, so a structured hexagonal mesh
was used in these regions. In the passage block, an orthogonal mesh was used and refined
near the blade surface. A view of the region near the leading edge can be seen in Figure 3.4.
Because of flow dependency, it’s not possible to set the y + value for the mesh precisely, but
generally the meshes were kept to a value of around 15 near the blade surface.
The tip clearance region is a zone of very high gradients and so there are 10 elements
between the blade tip and the shroud.

37

Figure 3.2: Impeller Meridional Profile

3.2.3

Boundary Conditions and Fluid Domain Properties

Setup of the CFD model properties was performed in CFX-Pre which is part of the CFX
suite of software from ANSYS [31]. The following applies to each of the impeller models.
The working fluid was air modeled as a non-bouyant ideal gas. The inlet and outlet
blocks were stationary with a rotating passage block with a frozen rotor interface between
the stationary and rotating blocks.
Walls were modeled as smooth and adiabatic with a non-slip boundary condition. Where
an open impeller was modeled, the shroud surface was stationary with a rotating impeller.
The fluid at the inlet boundary was subsonic, standard day air (288.2 K, 101.4 kPa) entering
with a uniform total pressure and velocity normal to the boundary with medium turbulence
intensity. The exit boundary was a subsonic static pressure boundary with the static pressure
area-averaged over the whole outlet. The k − with wall function model was used to simulate
38

Figure 3.3: Typical Computational Mesh
the effects of turbulence.

3.2.4

Solver

The solver used to compute the flow field was ANSYS CFX [31], a Reynolds-Average NavierStokes solver. The high performance compute servers at Michigan State University were used
to speed solution convergence. Each computational model was run for a minimum of 1,000
iterations and convergence was considered to be reached when the RMS residuals reached a
maximum of 1 × 10−5 and inlet and outlet mass flow differed by less than 0.1%.
Turbocompressors pose a particularly challenging computational problem because, unlike
most devices, the flow moves against the pressure field from a region of lower pressure toward
a region of higher pressure. Because of this, convergence can be difficult to achieve even for
39

Figure 3.4: View of Mesh at Leading Edge
a steady-state solution. An inappropriate value for a boundary condition will cause nonconvergence or reverse flow. A sophisticated CFD code will initialize the flow field in an
attempt to improve stability and reduce the time required to reach a converged solution,
but even with a well-initialized flow field it can be difficult to achieve solution convergence.
A common technique used by designers is to begin with a very low exit pressure so the
compressor begins operating in a choked flow condition so the solution has a chance to
converge with a lower pressure difference and establish a more accurate flow field. Once a
converged solution is realized, the pressure at the exit boundary is raised and the converged
solution is used to initialize the flow solution for the new operating point. Too great a change
in exit pressure can still cause non-convergence, so this may process may need to be repeated
several times, and in doing so, the compressor is ’walked up’ the speed line from a choked
condition, which is the most stable operating point, to near surge, which is the least stable
40

Figure 3.5: View of Mesh at Tip Region
operating point with the highest pressure difference between inlet and outlet. Because surge
is an unsteady phenomenon, it can not be modeled using a steady-state CFD model.

41

Chapter 4
Flow Trimming

4.1

Impeller Baseline Characteristics

For this numerical study, four impeller designs were studied. They were selected because they
operated at widely varying conditions. These impellers represent a wide range of nominal
performance parameters (pressure ratio and mass flow rate), non-dimensional performance
parameters (flow coefficient and pressure coefficient), as well as speed (nominal and specific),
inlet Mach number, and so forth. For the sake of simplicity, the four impellers included in
this study are labeled Impellers A, B, C, and D and the source of each impeller geometry is
as follows:
Impeller A - Solar Turbines proprietary geometry
Impeller B - Coordinates for a High Performance 4:1 Pressure Ratio Centrifugal Compresor [37]
Impeller C - Aerodynamic and Mechanical Design of an 8:1 Pressure Ratio Centrifugal
Comperssor [38]
Impeller D - The original design geometry is contained in: Splitter-Bladed Centrifugal
Compressor Impeller Designed For Automotive Gas Turbine Application [39] with modifications to the geometry as described in Effect of Area Ratio on the Performance of a 5.5:1
Pressure Ratio Centrifugal Impeller [6, 28]

42

Baseline geometry characteristics for each compressor are presented in Table 4.1 and
baseline performance characteristics are presented in Table 4.2.
Impeller
A
B
C
D
Impeller
A
B
C
D

ω[rpm]
r1,h [cm] r1,s [cm] r2 [cm]
b2 [cm]
14000
5.0
11.7
17.8
3.048
21789
4.14
10.49
21.56
1.704
75000
2.19
4.37
7.98
0.324
45337
3.20
5.63
10.15
0.757
Tip Gap [cm] β1,h [deg] β1,s [deg] β2 [deg] No. Blades
0.000
40
69
47
9+9
0.030
35.8
54.8
49.6
15+15
0.025
36.7
56.3
30
19
0.174
53.7
62.0
30.2
18+18
Table 4.1: Basic Geometry of Baseline Impellers

Impeller
A
B
C
D
Impeller
A
B
C
D

m[
˙ kg
s ]
3.29
4.07
0.86
0.93
W1 /W2
1.22
1.24
1.49
1.92

Po,2 /Po,1
1.54
4.66
10.6
5.68
M1,avg
0.46
0.68
1.03
0.76

ηisen
Φ
Ψ
Ns
0.944
0.104
1.11
0.92
0.928
0.046
1.30
1.96
0.914
0.056
1.41
0.61
0.939
0.046
1.56
0.50
M1,s β1,b,rms [deg] β1,f [deg] Incidence [deg]
0.55
57.3
56.3
-1.0
0.82
47.3
53.4
6.1
1.20
48.1
52.5
4.4
0.88
58.4
58.1
-0.3

Table 4.2: Overall Performance for Baseline Impellers

In Table 4.2, average values are defined as mass flow averaged, and incidence is defined
as the difference between the mass flow averaged relative flow angle (W1,avg ) and β1,rms .

4.2

Comparison of CFD Results with Experiments

In order to verify the validity of the CFD results, the numerical models were compared with
published experiemental results. A full set of experimental data was not available for all the
impellers, but the methodology was proven out using experimental data from two impellers
43

(Impeller A and Impeller D) and applied to the others. The experimental data for Impeller
A is proprietary, belonging to Solar Turbines, and is not available in the open literature, but
was provided for use in this study. The data for Impeller D is taken from Schumann’s study
of an axially-trimmed impeller [6, 28].
First, a mesh study was performed at design speed for Impeller D. The mesh was varied
from a coarse mesh ( 50,000 nodes in the impeller passage) to a medium mesh ( 100,000
nodes) to a fine mesh ( 250,000 nodes) and the results were compared with the experimental
results presented in [28]. For the sake of comparison with experimental results, the CFD
model was run to a prescribed mass flow rate at the exit. The results for total pressure
ratio vs. mass flow rate are presented in Figure 4.1 and the impeller isentropic efficiency
vs. mass flow rate are presented in Figure 4.2. According to these results, the coarse
mesh under-predicts the pressure ratio and, more importantly, does not follow experimental
trends as the pressure ratio declines at low mass flow rate where the experiment shows a
continued increase. Both the medium and fine meshes follow experimental trends well, and
both somewhat over-predict pressure ratio, though the medium mesh does not over-predict
to the same degree as the model with the fine mesh. In Figure 4.2, it can be seen that all
three meshes over-predict efficiency. This is to be expected, because not all the sources of
loss experienced by a real compressor are included in the numerical model. These include
disk friction on the back side of the impeller as well as mechanical friction due to bearing
losses, etc. In industrial practice, this over-prediction in efficiency is commonly accounted
for by applying a performance debit to all CFD models. The ”medium” mesh was selected
for use in the study because using it produced results most consistent with the experimental
trends and values.
A full compressor map was run using the ”medium” mesh and the design geometry. The
44

Figure 4.1: Effect of Mesh Resolution on Pressure Ratio for Impeller D
results for pressure ratio and efficiency are presented in Figure 4.3 and in Table 4.3.
Qualitatively, there appears to be a good agreement between all of the experimental data
and simulated results. Quantitatively, except for points on the 50% speed line, all predicted
pressure ratios are within 2.7% of the experimental values and all efficiencies are within 4.8%
of experimentally-measured values.
A full speed line at design speed was also available for Impeller A, as well as a series
of flow-trims of that impeller. The results of the numerical model were compared with
the experimental data presented in [24]. Figure 4.4 demonstrates good agreement with the
experimental data for the baseline and 19.3% trimmed impeller. The more-trimmed impellers
follow expected trends, but differ from the experimental data somewhat as head coefficient

45

Figure 4.2: Effect of Mesh Resolution on Isentropic Efficiency for Impeller D
rises. This difference can be attributed to the fact that the experiments were conducted
using an unshrouded impeller with tight clearances to represent a shrouded compressor.
Some tip leakage must necessarily occur when the the shroud is not attached and therefore
the difference in head coefficient for the more trimmed impellers is considered to be within
the range of expectation. The results for efficiency vs flow coefficient also follow expected
trends. Mechanical losses and windage losses on the back side of the impeller are fixed
for a given speed; they account for a greater portion of the total power consumed as the
impeller is trimmed. Therefore, differences between the numerical and experimental results
are expected to be small for the baseline impeller and increase with trimming as demonstrated
in Figure 4.5.

46

Figure 4.3: Experimental vs. Computational Results for Impeller D

N=45337
m [kg/s] Rc
0.864
2.6%
0.900
1.8%
0.949
0.3%
1.058
2.4%
N=36269
m [kg/s] Rc
0.604
1.2%
0.715
0.6%
0.795
2.0%
0.855
3.0%
0.950
1.6%

N=40803
m [kg/s] Rc
0.757
2.1%
0.846
1.7%
0.951
2.7%
1.000
0.9%
N=27202
m [kg/s] Rc
0.214
3.7%
0.311
0.0%
0.440
1.0%
0.546
1.9%
0.699
2.1%

η
2.9%
3.5%
3.4%
3.2%
η
1.7%
1.4%
1.9%
2.9%
2.1%

η
3.1%
2.7%
1.6%
4.8%
η
4.1%
3.8%
1.8%
1.6%
5.8%

Table 4.3: Percent Difference Between Experimental and Computational Results for Impeller
D

47

Figure 4.4: Head Coefficient vs. Flow Coefficient for CFD and Experimental Results for
Impeller A

48

Figure 4.5: Efficiency vs. Flow Coefficient for CFD and Experimental Results for Impeller
A

49

4.3

Trimming Methodology

Because Impeller A was related to established experimental results for a flow trimmed impeller, the trims for that impeller were modeled consistently with the experimental compressor. For the other impellers, the passage was trimmed from inlet to exit according to the
following methodology: Both the inlet and exit areas were reduced proportionally so that
the ratio of trimmed inlet area to design inlet area is equal to the ratio of trimmed exit area
to design exit area.
A1,trimmed
A2,trimmed
=
A1,baseline
A2,baseline
The blade is trimmed from the shroud side only, so that the blade height is reduced at
inlet and outlet in accordance with the following equations:

2
2
r1s,trimmed
− r1h
A1,trimmed
Area Ratio =
=π 2
2
A1,baseline
r1s,baseline − r1h

and

Area Ratio =

b2,trimmed
A2,trimmed
=
A2,baseline
b1,trimmed

For a particular area ratio, the trim was translated into a percent blade height reduction
at inlet and outlet. The spanwise blade height was also calculated at each meridional distance
and reduced in proportion to its proximity to the inlet and outlet. In mathematical terms:

b2,trimmed
T rimmed Blade Height
(r − r1h )trimmed
= M D 1s
+ (1 − M D)
Baseline Blade Height
(r1s − r1h )design
b2,design
where M D is the meridional distance fraction (0 at impeller inlet 1 at outlet).

50

For each baseline impeller design, a series of trims were applied as described until the
inlet area of each impeller was trimmed by at least 60%. Each impeller trim is abbreviated
by a letter-number designator where ”F” identifies it as a flow trim and the number which
follows is the percent area reduction, so an ”F30” trim represents a flow trim where the inlet
and exit areas have been reduced by 30%. In all cases, the tip gap was kept consistent with
the baseline tip gap. In so doing, the effects of trimming are evaluated independent of the
effects of varying blade tip clearance.

4.4

Flow Trim Results

Once the geometry for each impeller had been produced, trimmed, and each trim meshed; a
speedline was produced for each of the impeller trims. Each model was run from a choked
flow condition, and the back-pressure raised to the point that the model failed to converge.
The failure of a model to converge at high back-pressure can be assumed to be related to flow
instability due to stall or surge, however steady-state CFD has been shown to have limited
ability to predict stall in turbomachinery applications [40] [41].
Pressure ratio vs. mass flow for each of the four flow-trimmed impellers is presented in
Figures 4.6 through 4.9.

51

Figure 4.6: Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller A

Figure 4.7: Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller B

52

Figure 4.8: Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller C

Figure 4.9: Pressure Ratio vs. Mass Flow Rate for Flow Trimmed Impeller D

53

In each case, it can be seen the choked flow rate is reduced with trimming and the
maximum pressure ratio is relatively consistent for all of the impellers except for Impeller
C which demonstrates an obvious decline in max pressure ratio as the impeller is trimmed.
Most of the impellers (except Impeller C) also show a narrowing of the flow range with
progressive trimming. This is in agreement with Rodgers’ previous study of flow trimming
of Impeller A [24].
Plots of isentropic efficiency vs mass flow rate are shown in Figures 4.10 through 4.13.

Figure 4.10: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller A

54

Figure 4.11: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller B

Figure 4.12: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller C

55

Figure 4.13: Isentropic Efficiency vs. Mass Flow Rate for Flow Trimmed Impeller D

56

In order to compare the performance of each flow trim, it is useful to nondimensionalize
the performance parameters. If the flow coefficient of each trim is divided by the choked flow
coefficient then the speed line for a baseline impeller and all trims, then it is much easier to
evaluate how the performance of each trimmed impeller compares with the baseline. This
ratio will be defined as flow coefficient ratio so that:

F lowCoef f icientRatio =

Φ
Φmax

Head coefficient is plotted against flow coefficient ratio in Figures 4.14 through 4.17.

Figure 4.14: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller A

57

Figure 4.15: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller B

Figure 4.16: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller C

58

Figure 4.17: Head Coefficient vs. Flow Coefficient Ratio for Flow Trimmed Impeller D

59

These plots show the head coefficient of Impeller A and Impeller D are insensitive to flow
trimming as the flow-normalized speed lines for both impellers are nearly identical to the
baseline impeller. In contrast, the head coefficient for Impellers B and C degrades as the
impeller is trimmed. In the case of Impeller B, there is a gentle decline in head coefficient
as the impeller is trimmed, but Impeller C shows a sudden and significant decline when it is
trimmed beyond 30%.
Isentropic efficiency is also plotted against flow coefficient ratio in Figures 4.18 through
4.21.

Figure 4.18: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed Impeller A

60

Figure 4.19: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed Impeller B

Figure 4.20: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed Impeller C

61

Figure 4.21: Isentropic Efficiency vs. Flow Coefficient Ratio for Flow Trimmed Impeller D

62

It can be seen that the impeller efficiency reacts very similarly to trimming as the head
coefficient. The efficiency of Impellers A and D are relatively insensitive to trimming, but
Impellers B and C show significant penalties to efficiency when they are flow trimmed.
An examination of the flow field confirms predictions regarding the effects of flow trimming. Figure 4.22 shows the circumferentially-averaged meridional Mach number profile.
The meridional profile does not change much with trimming except in that the inlet tip
Mach number reduces with trimming and increases slightly in the passage. This effect can
also be seen in the flow field at the exit. Figure 4.23 shows the Mach number profile in
the radial portion of the passage. The flow field doesn’t change much as the impeller as
trimmed except that the Mach number increases slightly and uniformly. This is due to the
fact that the ratio of passage area to wetted perimeter reduces as the impeller is trimmed
and therefore the boundary layers occupy a larger portion of the passage area. This creates
a smaller effective passage area for the fluid to occupy and so the velocity increases. These
effects shown in Figures 4.22 and 4.23 are counteractive. Reducing the Mach number at
the inlet will tend to decrease loss at the inlet as the impeller is trimmed while increasing
velocity in the passage will tend to increase friction losses in the passage.

Figure 4.22: Circumferentially Averaged Mach Number for Flow Trimmed Impeller D

63

Figure 4.23: Relative Mach Number at Impeller Exit Near Choke for Flow Trimmed Impeller
B

4.5

Analysis

The results presented in Section 4.4 demonstrate that an impeller may be trimmed so that
the flow coefficient is reduced while the head coefficent is maintained. Furthermore, it is
shown that some designs are more tolerant of trimming than others in that the efficiency
may be maintained as the impeller is trimmed or there may be a penalty to efficiency with
trimming depending on the design of the impeller.
These results confirm Rodgers’ [24] finding that trimming narrows the flow range of a
compressor, but when the flow is non-dimensionalized by dividing it by the choked flow rate,
no reduction in relative flow range is seen. To put it another way, the surge and choke
margins do not change for the same Φ/Φmax where surge margin and choke margin are
defined as:
SurgeM argin =

Φ − Φsurge
Φ

ChokeM argin =

Φ − Φchoke
Φ

Rodgers also suggests limiting flow trimming to a range of specific speeds between 0.5
64

and 1.2, however David et al. [30] and Zhang et al. [25] studied flow trimming impellers of
much lower specific speed and found that flow trimming could be applied to compressors with
specific speeds much lower than the lower bound recommended by Rodgers. This study does
not attempt to trim an impeller with such a low specific speed, but confirms those findings
in that this study found that a flow trim could usefully be applied to Impeller B, which
has a baseline specific speed around 2.0; so no adverse effects were noted when Rodgers’
recommended upper limit for application of flow trimming was violated. There may be an
upper limit to specific speed for which flow trimming may be applied, but it was not found
in this study. Those latter studies also suggest limiting flow trimming to 50% of the original
blade height; however, Impeller D could be trimmed as much as 60% of the original blade
height while maintaining head coefficient and efficiency similar to the baseline design.
Impellers A and D demonstrated good tolerance for flow trimming in that the efficiency
did not decline very much as the impeller was trimmed. Referring back to Table 4.2, the
clearest correlation between baseline operating conditions and suitability for trimming exists
in the baseline flow incidence where incidence is defined as the difference between the mass
averaged inlet flow angle and the blade angle at the RMS inlet radius. Both Impellers B and
C have positive incidence, while Impellers A and D have neutral or slightly negative flow
incidence.
Because a centrifugal impeller typically chokes at the inducer, the choked flow rate of a
flow-trimmed impeller would be expected to scale with the impeller inlet area. In this study,
the choked flow rate for a trimmed impeller was found to scale slightly differently than the
inlet area. If ”Area Ratio” is defined as the ratio of the inlet area of a trimmed impeller
to its baseline impeller and ”Flow Ratio” is the ratio of the choked flow rate of a trimmed
impeller to its baseline impeller, then the percent difference between the two describes how
65

the flow-trimmed impeller flow rate scales with inlet area. The percent difference between
area ratio and flow ratio is plotted against area ratio in Figure 4.24. This graph shows
an approximately linear increase in the difference between area ratio and flow ratio as the
impeller is trimmed for each of the impellers studied. Impellers B, C, and D all demonstrate
similar behavior in that there is an increasing difference between the inlet area ratio and the
flow ratio as the impeller is trimmed. Significantly, Impeller A demonstrates much different
behavior than the other impellers. This may be due to the fact that Impeller A is the only
shrouded impeller studied, but there is insufficient data for shrouded impellers to draw such
a conclusion.

Figure 4.24: Percent Difference Between Flow Ratio and Area Ratio for Flow Trimmed
Impellers

66

Chapter 5
Axial Trimming

5.1

Axial Trimming Methodology

For the axially-trimmed impellers, the trimming methodology is simpler than for the flowtrimmed impellers. Because it is desirable to apply trimming to the impeller in such a way
that will require the least amount of redesign or permit as much of the original equipment to
be reused with a modified impeller, the shroud profile is maintained per the original design
and shifted axially into the impeller so that the blade height at the outlet is reduced. When
the impeller is trimmed, the clearance gap is maintained at the nominal value. This is in
contrast to Schumann’s study [6,28] wherein the tip gap is reduced in proportion to the exit
blade height. In doing so, the effects of trimming should be separated from the effects of
reducing the tip gap. The exit area of the impeller is reduced by the equation:

Area Ratio =

A2,trimmed
b2,trimmed
=
A2,baseline
b1,trimmed

For each baseline impeller design, a series of trims were applied as described until the
inlet area of each impeller was trimmed by at least 60%. Each impeller trim is abbreviated
by a letter-number designator where ”A” identifies it as an axial trim and the number which
follows is the percent outlet area reduction, so an ”A30” trim represents an axial trim where
the exit blade height has been reduced by 30%.

67

5.2

Axial Trim Results

As with the flow trimmed impellers, the baseline geometry for each impeller was trimmed
and each trim meshed. Then, a speedline was produced for each of the impeller trims from
choked flow and the back pressure raised progressively to the point that the steady-state
CFD model failed to converge.
Head coefficient is plotted against flow coefficient in Figures 5.1 through 5.4.

Figure 5.1: Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller A

68

Figure 5.2: Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller B

Figure 5.3: Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller C

69

Figure 5.4: Head Coefficient vs. Flow Coefficient for Axially Trimmed Impeller D

70

These results show certain behaviors related to trimming for each of the impellers. Initially, as the impeller is trimmed, the head coefficient reduces, but the flow coefficient remains
unchanged; in addition, the flow range of the impeller tends to increase. As was noted in
Section 4.4, CFD has limited capability to predict the surge line of a compressor, but the
consistency of the widening flow range is notable and may be predictive of real phenomena.
As the impeller is trimmed further, the choked flow coefficient of each impeller is reduced.
The point at which the choked flow coefficient reduces from the baseline value will be considered to be the limit of usefulness for axial trimming because the purpose is to reduce the
pressure rise without reducing the flow range.
In terms of how trimming affects flow and pressure ratio, the head coefficient for Impeller
A reduces with trimming, but the choked flow of the impeller begins to show reduction
when it is trimmed 30%. Impeller B loses flow range when trimmed 40% and shows a more
significant reduction in head coefficient than Impeller A. Impeller C can be trimmed up to
30% of the baseline exit blade height, and Impeller D tolerates trimming up to almost 40%
before it loses flow range.
Another criterium for the beneficial use of axial trimming is efficiency. If trimming
imposes a significant penalty to efficiency, then the usefulness is compromised and it may
prove profitable to invest in a new impeller design. Plots of isentropic efficiency vs. flow
coefficient for each of the four axially trimmed impellers are presented in Figures 5.5 through
5.8.

71

Figure 5.5: Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially Trimmed
Impeller A

Figure 5.6: Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially Trimmed
Impeller B

72

Figure 5.7: Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially Trimmed
Impeller C

Figure 5.8: Total-to-Total Isentropic Efficiency vs. Flow Coefficient for Axially Trimmed
Impeller D

73

Figure 5.8 shows efficiency performance near the baseline rate across the entire flow range
of Impeller D up through the outlet blade height trimmed by 40% and only loses efficiency
when the impeller is trimmed to the point that it can not achieve the baseline flow. Impeller
C, while it maintains its flow range through a 30% outlet trim, suffers a significant efficiency
penalty beginning at a 30% axial trim. Impeller B can only tolerate a small amount of
trimming, but demonstrates impeller efficiency equal to or better than the baseline impeller
when 10-20% of the outlet blade height is trimmed; and, although Impeller A exhibits a
similar reduction in pressure ratio with trimming to Impeller B, according to Figure 5.5, it
suffers a penalty to performance when axially trimmed as little as 20%.

Figure 5.9: Exit Flow angle vs. Flow Coefficient for Axially-Trimmed Impeller D

The mass-averaged exit flow angle (α2 ) is plotted against flow coefficient for Impeller D
74

in Figure 5.9. This plot shows a reduction in exit flow angle for the same flow coefficient
as the impeller is trimmed. This is an important parameter for outlet trimming because
the mechanism by which axial trimming reduces the head coefficient of an impeller is by
reducing the area in the radial portion of the passage causing a higher throughflow velocity
and reducing the amount of diffusion in the passage. The result is a more radial flow angle
at the exit. This relationship between exit flow angle and pressure rise can be seen in
one-dimensional analysis where; given the same gas properties, inlet and exit blade angles,
rotational speed, and efficiency; pressure ratio becomes a function of exit flow angle. [1]
γ

Po2
= 1+
Po1

(γ − 1)ηU22 tanα2 γ−1
a2o1 (tanα2 + tanβ2 )

As the throughflow velocity increases with axial trimming, there comes a point at which
the flow in the passage reaches sonic velocity and the choke point of the impeller moves from
the inlet to the radial portion of the passage. This is what causes the flow range to reduce
as the impeller is trimmed. This behavior is demonstrated in Figure 5.10 which shows a
plot of the relative mach number in the impeller passage at the mid-span near the impeller
exit. The operating point for each of these is near-choke, so the baseline case shows Mach
numbers well below sonic velocity because the choke point is at the inlet. The A30 (30%
axially-trimmed) case shows increasing velocity due to trimming at the outlet, and the A60
case demonstrates the choke point moving to a point near the exit of the passage.

75

Figure 5.10: Relative Mach Number at Impeller Exit Near Choke for Axially-Trimmed Impeller D at 50% Span

5.3

Analysis

According to the CFD study, the pressure ratio of a centrifugal impeller may be reduced
without reducing the flow range by reducing the outlet height. This ability is limited in each
case by choking in the impeller passage when the impeller is trimmed beyond a certain point,
but that point varies based on the original design of the impeller. The impeller efficiency
may or may not be maintained as the impeller is trimmed.
In his study, Schumann [6,28] performed an axial trim on a centrifugal impeller, and found
a peak efficiency as the impeller was trimmed. The current study did not find trimming to
have any benefit to efficiency; to the contrary, this study has found a potential penalty
to efficiency with axial trimming. However, as discussed previously, Schumann varied the
clearance gap as a function of exit blade height to maintain a constant ratio of tip clearance
to b2 while the current study uses a constant tip gap. Because of this, the two studies cannot
be directly related; but the findings can be combined to conclude that efficiency gains found
by Schumann can be attributed to the reduction in clearance alone and not to reduction in
exit blade height.

76

According to Figures 5.1 through 5.4; in the un-choked region, to the left of the ”knee”
in the speedline curve, the lines of Ψ vs Φ are roughly parallel for the baseline and axiallytrimmed impellers. The practical use of axial trimming is limited to those impellers which
can operate in the same flow range as the baseline impeller; therefore, considering only those
which do not choke in the passage, the reduction in head coefficient is plotted against the
reduction of outlet height in Figure 5.11. In that figure, the Ψ/Ψbaseline is taken at the
approximate center of the unchoked portion of the speedline. In each case, employing axial
trimming allowed the head coefficient to be reduced by between 7 and 11 percent before
choking in the passage limited the flow range. There was wide variance observed in the
effect of outlet trimming on head coefficient; the outlet of Impeller D was trimmed up to
40% and produced a 9% reduction in pressure ratio, while Impellers A and B realized an
11% reduction in pressure ratio with an outlet height reduction of only 20%. According to
Figure 5.11, the four impellers divide themselves clearly into two groups - Impellers A and
B exhibit a much sharper decline in pressure ratio for a smaller reduction in outlet height
while Impellers C and D may be trimmed to a greater degree while maintaining their flow
range, but to lesser effect.
Referring back to the baseline performance characteristics presented in Table 4.2, it can
be seen that impellers C and D have both lower specific speeds and higher diffusion ratios
than impellers A and B. Impellers A and B have diffusion ratios not much greater than one, so
most of the pressure rise comes from the centrifugal effect and produces only a small amount
of static pressure rise in the impeller passage. Impellers C and D achieve a much higher
ratio of static-to-total pressure rise in the impeller and this difference is reflected in how the
impellers react to axial trimming. As an alternative to radial trimming, the results for axial
trimming are promising, but inconclusive. There was not a strong correlation between high
77

Figure 5.11: Ratio of Head Coefficient to Design Head Coefficient vs. Trim Percent
specific speed and tolerance for axial trimming. The pressure ratio of the higher specific speed
impellers (A and B) were able to be reduced by axial trimming more than the low specific
speed wheels; however, Impeller A suffered a significant efficiency penalty with trimming
while Impeller be was particularly tolerant. This difference may be due to the shroud or the
simple blading of Impeller A vs the more modern design of Impeller B.

78

Chapter 6
Conclusions

6.1

Summary

In this work, two types of centrifugal impeller blade trimming were studied using computational fluid dynamics. In the first type, flow trimming, the passage area is trimmed from
inlet to exit. The purpose of flow trimming is to reduce the flow of the compressor while
maintaining the pressure ratio. The second type of trimming, axial trimming, shifts the
shroud line axially so that the exit blade height of the impeller without affecting the inlet
geometry with the intent of reducing the pressure ratio of the compressor without changing
the flow range.
Four impellers of widely differing geometric and performance parameters were selected
for study. The geometry of each impeller was modeled using ANSYS BladeGen and a series
of flow and axial trims were applied to each of the four impellers and each series of impellers
was then meshed using ANSYS TurboGrid. Finally, the performance was modeled using
ANSYS CFX and a speed line was created at the design speed for each series of trims by
varying the back pressure and running a series of steady state performance points between
choke and surge.

79

6.1.1

Flow Trimming

For the flow trimmed impellers, the results show that any of the four impellers may be
trimmed to some extent while maintaining both pressure ratio and efficiency. The flow
range decreased with trimming, but when the flow coefficient of each trim was normalized
by the choked flow coefficient, it was seen that the relative flow range (that is, the surge
margin) actually increased with flow trimming.
The choked flow coefficient of a trimmed impeller would be expected to scale in proportion
with the inlet area of the impeller, but the results showed the choked flow coefficient to scale
somewhat differently from the inlet area. The difference between the area ratio and the flow
coefficient ratio was found to increase linearly with trimming in each case, and could be a
positive or negative difference based on the baseline impeller design.
Different impeller baseline designs demonstrated different tolerances for flow trimming.
Two of the impellers could be trimmed significantly (up to about 50% of the original area)
without adversely affecting impeller efficiency or pressure ratio. The others suffered significant penalties to efficiency as the impeller was trimmed. An impeller’s tolerance for flow
trimming correlated with the flow incidence on the impeller wherein a positive incidence
correlated with poor tolerance for flow trimming and a neutral or slightly negative incidence
correlated with good tolerance for flow trimming.

6.1.2

Axial Trimming

In the case of the axially trimmed impellers, each of the impellers modeled were able to be
trimmed so that the pressure ratio was reduced without adversely affecting the flow range
or impeller efficiency. The mechanism by which the pressure ratio is reduced is reduced

80

diffusion in the passage leading to higher throughflow velocity and a more radial exit flow
angle (α2 ). In every case, the axial trimming was limited at some point by reduction of
the choked flow rate. This reduction in choked flow was due to the fact that reducing the
passage area in the radial portion of the passage moved the choke point from the inlet to
the passage.
The performance impact of axial trimming varied significantly depending on the baseline
design. Two of the impellers were able to have the head coefficient reduced by about 10%
with about a 20% reduction in outlet height. The other impellers could be trimmed by
at least 30% of the baseline outlet height, but the reduction in head coefficient was much
less. The different behaviors correlated with diffusion ratio. The two impellers with baseline
diffusion ratios around 1.2 were able to be trimmed little but to great effect. The other two
impellers had baseline diffusion ratios of about 1.5 and 1.9 respectively and were able to be
trimmed to a progressively greater degree but with progressively lesser effect.

6.2

Applicability to Design

Centrifugal impellers are simple machines with complex geometry. As such, they require
significant design effort to produce a new aerodynamic design. If a successful existing design
can be modified to produce a new impeller, then a large portion of the design effort may be
bypassed. Traditionally, this has been accomplished by applying the laws of similitude to
existing designs. The application of impeller blade trimming adds more arrows to the quiver
of design strategies, but they must be applied with careful consideration.
If an impeller is flow trimmed, then peripheral components such as the shroud and diffuser
must also be redesigned. The application of an axial trim has the advantage of maintaining

81

the same shroud profile, but the diffuser still requires redesign due to the reduced exit blade
height and change in exit flow angle for an axially trimmed impeller.
While some of the impellers investigated demonstrated high tolerance for flow and/or
axial trimming, others did not and so careful consideration should be given to whether the
baseline design is appropriate for trimming. Based on the results of this and other trimming
studies, a safe course of action would be to limit the application of trimming to small changes
of flow or pressure ratio, because each of the impellers studied were able to be both flow
trimmed or axial trimmed small amounts without significant loss of efficiency. Larger changes
may be possible, but the designer must consider whether the baseline impeller is appropriate
to be trimmed.

6.3

Next Steps

This study represents a significant step forward in the understanding of the effects of centrifugal impeller blade trimming because, whereas previous studies examined the effects of
trimming on a single design, this work compared trimming effects for four different impeller
designs. However, there are still questions that remain unanswered and the results raise
additional questions. While a sample set of four impellers is a significant improvement from
previous works, it still represents a small sample. Expanding the work to include additional
designs would help to confirm or refine the conclusions of this study. Of the four impeller
designs, only one was shrouded. It was this shrouded impeller that exhibited a different
trend of area ratio to flow ratio from the other three, and it’s not clear if that has to do
with the fact that it was a shrouded design or if it was due to other aspects of the design.
The question as to whether high specific speed designs are particularly well-suited to axial

82

trimming remains open, and there is a new question about whether shrouded compressor
designs are particularly intolerant of axial trimming. This work provides a broader overview
of the application of centrifugal impeller blade trimming than has previously been available,
but the topic still holds fruitful areas for new research.

83

BIBLIOGRAPHY

84

BIBLIOGRAPHY
[1] Dixon, S. L., 2005. Fluid Mechanics and Thermodynamics of Turbomachinery. Elsevier
Butterworth-Heinemann.
[2] Krain, H., 2003. “Refiew of centrifugal compresor’s application and development”. In
Proceedings of ASME Turbo Expo.
[3] Boyce, M. P., 2012. Gas Turbine Engineering Handbook (Fourth Edition). ButterworthHeinemann.
[4] Cordier, O., 1955. “Similarity considerations in turbomachines”. VDI Report, 3, p. 85.
[5] Japikse, D., 1996. Centrifugal Compressor Design and Performance. Concepts ETI,
Inc., Wilder, VT.
[6] Schumann, L. F., Clark, D. A., and Wood, J., 1987. “Effect of area ratio on the
performance of a 5.5:1 pressure ratio centrifugal impeller”. Journal of Turbomachinery,
109, pp. 10–19.
[7] Senoo, Y., and Ishida, M., 1987. “Deterioration of compressor performance due to tip
clearance of centrifugal compresors”. Journal of Turbomachinery, 109, pp. 55–61.
[8] L¨
udtke, K. H., 2004. Process Centrifugal Compressors. Springer-Verlag.
[9] Stodola, A., 1945. Steam and Gas Turbines. Vols. I and II. McGraw-Hill.
[10] Dean, R. C., and Senoo, Y., 1960. “Rotating wakes in vaneless diffusers”. Trans ASME
Journal of Basic Engineering, 82, pp. 563–574.
[11] Cumpsty, N., 1989. Compressor Aerodynamics. Krieger Publishing Company.
[12] Rodgers, C., 1964. “Typical performance characteristics of gas turbine radial compressors”. Journal of Engineering for Power, 86, pp. 161–175.
[13] Harada, H., 1988. “Performance characteristics of two- and three-dimensional impellers
in centrifugal compressors”. Journal of Turbomachinery, 110, pp. 110–114.
85

[14] Bonaiuti, D., and Zangeneh, M., 2009. “On the coupling of inverse design and optimization techniques for the multiobjective, multipoint design of turbomachinery blades”.
Journal of Turbomachinery, 131, pp. 021014–1 – 021014–16.
[15] Ibaraki, S., Matsuo, T., Kuma, H., Sumida, K., and Suita, T., 2003. “Aerodynamics of
a transonic centrifugal compresor impeller”. Journal of Turbomachinery, 125, pp. 346–
351.
[16] Ibaraki, S., Sumida, K., and Suita, T., 2009. “Design and off-design flow fields of a
transonic centrifugal compressor impeller”. In Proceedings of ASME Turbo Expo 2009:
Power for Land, Sea and Air, no. GT2009-59986.
[17] Schleer, M., Song, S. J., and Abhari, R. S., 2008. “Clearance effects on the onset of
instability in a centrifugal compressor”. Journal of Turbomachinery, 130, pp. 031002–1
– 031002–11.
[18] Harada, H., 1985. “Performance characteristics of shrouded and unshrouded impellers
of a centrifugal compressor”. Journal of Engineering for Gas Turbines and Power, 107,
pp. 528–533.
[19] Al-Zubaidy, S. J. J., 1992. “Axial length influence on the performance of centrifugal
impellers”. Journal of Propolsion and Power, 8, pp. 1245–1251.
[20] Aungier, R. H., 2000. Centrifugal Compressors: A Strategy for Aerodynamic Design
and Analysis. American Society of Mechanical Engineers.
[21] Sorokes, J. M., Kopko, J. A., Geise, P. R., and Hinklein, A. L., 2009. “The influence
of shroud curvature and other related factors on impeller performance characteristics”.
In Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and Air, no. GT200960109.
[22] Godse, A. G., 1989. Predict compressor performance at new conditions. Hydrocarbon
Processing, June.
[23] Liu, C., Tan, J., Qi, D., Wang, X., and Qin, G., 2011. “A practical approximate
method to predict adjusting performance of a centrifugal compressor stage with inlet
guide vane”. In Proceedings of ASME Turbo Expo 2011, no. GT2011-45195.
[24] Rodgers, C., 2001. “Centrifugal compressor blade trimming for a range of flows”. In
Proceedings of ASME Turbo Expo.

86

[25] Zhang, D., DiLiberti, J.-L., and Cave, M., 2010. “A cfd study on a base and a flowtrimmed low specific speed centrifugal compresor”. In Proceedings of the ASME 2010
International Mechanical Engineering Congress & Exposition, no. IMECE2010-40561.
[26] Sapiro, L., 1983. “Effect of impeller-extended shrouds on centrifugal compressor performance as a function of specific speed”. Journal of Engineering for Power, 105,
pp. 457–465.
[27] Engeda, A., 2007. “Effect of impeller exit width trimming on compressor performance”.
In 8th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, no. ISAIF8–00135, Ecole Centrale de Lyon.
[28] Schumann, L. F., Clark, D. A., and Wood, J. R., 1986. “Effect of area ratio on the
performance of a 5.5:1 pressure ratio impeller”. In Proceedings of the International Gas
Turbine Conference and Exhibit, no. 86-GT-303.
[29] Nili-Ahmadabadi, M., Hajilouy-Benisi, A., Durali, M., and Motavalli, S. M., 2008. “Experimental and numerical invegation of a centrifugal compressor and numerical study
of the area ratio and tip clearance effects on the performance characteristic”. In Proceedings of ASME Turbo Expo 2008: Power for Land, Sea and Air, no. GT2008-50969.
[30] David, T., Zhang, D., and Cave, M., 2006. “Frictional effects on a base and a flowtrimmed impeller of a low specific speed industrial compressor”. In Proceedings of
ASME Turbo Expo.
[31] Ansys cfx [computer software]. version 13.0. canonsburg, pa: Ansys, inc.; 2010.
[32] Zikanov, O., 2010. Essential Computational Fluid Dynamics. John Wiley & Sons.
[33] Ferziger, J., and Peri´c, M., 1996. Computational Methods for Fluid Dynamics. SpringerVerlag.
[34] Spalart, P. R., and Allmaras, S. R., 1994. “A one equation turbulence model for aerodynamic flows”. Recherche Aerospatiale, 1, pp. 5–21.
[35] Bladegen [computer software]. version 13.0. canonsburg, pa: Ansys, inc.; 2008.
[36] Turbogrid [computer software]. version 13.0. canonsburg, pa: Ansys, inc.; 2010.
[37] McKain, T. F., and Holbrook, G. J., 1997. Coordinates for a high performance 4:1
pressure ratio centrifugal compresor. Tech. Rep. NASA CR-204134, NASA.
87

[38] Osborne, C., Runstadler-Jr., P., and Stacy, W. D., 1975. Aerodynamic and mechanical
design of an 8:1 pressure ratio centrifugal comperssor. Tech. Rep. NASA CR-134782,
NASA.
[39] Pampreen, R. C., 1977. Splitter-bladed centrifugal compressor impeller designed for
automotive gas turbine application. Tech. Rep. NASA-CB-135237, NASA.
[40] Vagani, M., 2013. “Prediction of impeller rotating stall onset using numerical simulations of a centrifugal compressor. part 1: Detection of rotating stall using fixed-flow
transient simulations”. Proc. IMechE Part A: J. Power and Energy, 227, pp. 403–414.
[41] Cornelius, C., Biesinger, T., Galpin, P., and Braune, A., 2014. “Experimental and
computational analysis of a multistage axial compressor including stall prediction by
steady and transient cfd methods”. Journal of Turbomachinery, 136, pp. 061013–1 –
061013–12.

88