_ .flk/ u.
r 41.11111}.

. $3.3m; is

I ‘
.11.} u

leH
—

7

W..1.1
11.1.11u
A. ..

 

 

 

71-15819

1111111111111111111111111111111

31293 01766 3166

This is to certify that the

dissertation entitled

DESIGN AND DEVELOPMENT OF LOW-FLOW COEFFICIENT
CENTRIFUGAL COMPRESSORS FOR INDUSTRIAL APPLICATION

presented by

Jean-Luc Di Liberti

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degree in Mechanical Engineering

Al

M r professor

Date May 13, 1998

MSUi: an Afﬁrmatiw Action/Equal Opportunity Institun'on 0- 12771

 

 

LIBRARY

Mlchlgan State
Unlverslty

 

 

 

 

PLACE IN RENRN BOX to remove this checkout from your record.
' TO AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE I DATE DUE DATE DUE

11122130?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ma W14

 

DESIGN AND DEVELOPMENT OF LOW-FLOW COEFFICIENT CENT RIFUGAL
COMPRESSORS FOR INDUSTRIAL APPLICATION

By

Jean-Luc Di Liberti

A DISSERTATION
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1998

ABSTRACT

DESIGN AND DEVELOPMENT OF LOW-FLOW COEFFICIENT

CENTRIFUGAL COMPRESSORS FOR INDUSTRIAL APPLICATION

By
Jean-Luc Di Liberti

In centrifugal compressors, as in any type of turbomachinery, there exists an
optimum efﬁciency over a certain range of ﬂow. The ﬂow coefﬁcient of a centrifugal
stage can be correlated to the efﬁciency, and the optimum range is between 0.06 and
0.09. In a multistage machine, the volume-ﬂow rate decreases as the pressure of the flow
is increased through the stages. As the last stages are reached, the flow coefﬁcient (I) is,
therefore, smaller than at the inlet of the machine. The impeller and diffuser channel are
narrower to handle the smaller volume flow rate, leading to higher friction losses. Hence,
low-ﬂow coefﬁcient compressors are inherently inclined to have lower efﬁciency than
the previous stage.

A one-dimensional and a quasi-three-dimensional code were successfully
developed to design a centrifugal impeller. Ten low-ﬂow coefﬁcient compressors were
designed and analyzed numerically to analyze the effect of the design flow coefﬁcient, of
the outlet width-outlet blade angle combination, and of the blade leading edge position on

the stage performance.

The numerical analysis provided vital information regarding the internal

aerodynamic of each impeller. All the impellers exhibited a jet/wake flow pattern. The

high backsweep angle of the blade was responsible for the rather small wake area at the

design point.

The change in blade angle-blade outlet width showed that the wake was the
largest for the impeller with the largest backswept. Hence, the development of the wake
appeared more controlled by the curvature in the meridional plane and the diffusion
achieved, than by the exit blade angle. Even if the wake was larger in the high-backswept
impeller, higher performances were obtained because of the larger impeller exit area; the
lower velocity levels and the wider diffuser led to reduce the total pressure loss. No
advantage was found by using an impeller with a blade whose leading edge is located in

the radial part of the impeller.

The performances of each impeller were estimated by CFD on a stage basis for
hydraulically smooth and for rough walls. The necessary aero—thermodynamic quantities
required to calculate the performance were mass-averaged at both the impeller exit and
the diffuser exit. The numerical results were used to update the one-dimensional model
and generate the predicted performance map of each impeller. It was found that the one-
dimensional prediction with no external losses was in very good agreement with the CFD
analysis with rough walls. As expected, the performances of the stages were found to
decrease as the flow coefﬁcient was reduced. The impellers with 60 and 65 degrees
backsweep were found to be a good compromise between efﬁciency level, pressure ratio,

and stability.

To Lisa

iv

ACKNOWLEDGMENTS

The author is very grateful to his advisor, Professor Abraham Engeda, for his
guidance, support, and encouragement throughout the course of his research work.
Sincere thanks go to Professors Craig Somerton, Chang Wang and Indrek Wichman for
their advice, suggestions and interest in this work. A particular thanks goes to Mr. Mike
Flathers and Mr. Mike Cave from Solar Turbines Inc. for initiating this work, and for
their help and support.

The author is also grateful to Professor Rene Van den Braembussche without
whom he would not have been studying centrifugal compressors.

A special thanks goes to Mr. Frank Shih, Mr. Tim David, and Mr. Ed Fowler at
Solar Turbines Inc.; Mr. Georges Bache and Mr. John Crapuchettes at AEA Technology
for their helpful discussions; and to Mr. Craig Gunn at MSU for his help in preparing this
thesis. The author also appreciates the numerous criticism, assistance, and great
friendship of Naresh Amineni but also of Won Kim and all the German students that
came to study at the turbomachinery lab at Michigan State University: Joachim Fischer,
Bemd Wilmsen, and Alexander Kuppe.

The support of Solar Turbines Inc. and of the Division of Engineering Research is
greatly appreciated.

Last, but certainly not least, my family and Lisa Nicholas, for their love and

support.

TABLE OF CONTENTS

 

 

 

 

 

 

 

ACKNOWLEDGMENTS V
LIST OF TABLES IX
LIST OF FIGURES -- - X
NOMENCLATURE - - - XIII
GREEK ........................................................................................................................... xm
SUBSCRIPTS .................................................................................................................. XIV

1. INTRODUCTION 1
1.1. DEMANDS ON AND CHALLENGES FOR CENTRIFUGAL COMPRESSORS ....................... 1
1.2. THE NEED FOR Low-FLOW COEFFICIENT COMPRESSORS ......................................... 3
1.3. STRUCTURE AND OBJECTIVES OF THE WORK ........................................................... S

2. GENERAL DISCUSSION OF TURBOMACHINES 7
2.1. ELEMENTS OF A CENTRIFUGAL COMPRESSOR STAGE ............................................... 7
2.2. VELOCITY TRIANGLES .............................................................................................. 9
2.3. THE EULER EQUATION ........................................................................................... 10
2.4. HEAD RISE ............................................................................................................. 13
2.5. EFFICIENCY ............................................................................................................ 14
2.6. WORK REDUCTION FACTOR AND SLIP FACTOR ...................................................... 16
2.7. OPERATING MAP AND COMPRESSOR RANGE .......................................................... 18
2.8. DIMENSIONLESS PARAMETERS ............................................................................... 20
2.8.1. Flow and head coefﬁcient ............................................................................. 20
2.8.2. Speciﬁc diameter and speciﬁc speed ............................................................. 21

3. FLOW IN THE IMPELLER AND DESIGN PARAMETERS 24
3.1. FLOW IN THE IMPELLER .......................................................................................... 24
3.1.1. Qualitative Description of the Flow in the Impeller ..................................... 24
3.1.2. Inviscid Flow Analysis .................................................................................. 25
3.1.3. Secondary Flows ........................................................................................... 29
3.1.4. Effect of Rotation and Streamline Curvature on Boundary Layer Stability .31
3.1.5. Experimental Observation ............................................................................. 33

3.2. LOSSES IN A CENTRIFUGAL COMPRESSOR .............................................................. 40
3.2. 1. Internal Losses .............................................................................................. 40
3.2.2. External Losses ............................................................................................. 44

3.3. ONE-DIMENSIONAL DESIGN PARAMETERS ............................................................. 46
3.3.]. Overall View of the I -D Design Parameter .................................................. 46
3.3.2. Inducer Hub and Tip Radius ......................................................................... 47
3.3.3. Exit Width ...................................................................................................... 49
3.3.4. Velocity Ratio ................................................................................................ 50
3.3.5. Slip Factor ..................................................................................................... 50

4. ONE-D COMPRESSOR MODELING AND DESIGN PARAMETERS .............. 52

4.1. DESCRIPTION OF THE l-D METHOD ........................................................................ 52

 

 

4.1.1. Two—zone Model ............................................................................................ 53
4.1.2. Impeller Diffusion Level ................................................................................ 54
4.1.3. Oﬁf-Design Performance ............................................................................... 56
4.2. ONE-DIMENSIONAL ANALYSIS ............................................................................... 58
4.2.1. Design Constraints ........................................................................................ 58
4.2.2. Modeling Assumptions .................................................................................. 58
4.2.3. Inﬂuence of the Modeling Assumptions on the Choice of the Geometry ....... 59
4.2.4. I -D Impeller Design ...................................................................................... 65
4.3. STREAMLINE CURVATURE THROUGHFLOW METHOD ............................................. 72
4.3.1. Description of the Quasi-3D Method ............................................................ 73
4.3.2. Deﬁnition of the Blade Geometry .................................................................. 73
4.3.3. Impeller Blade Shape for the Five Impellers ................................................. 75
4.3.4. Velocity Distribution and Blade Loading for the Five Impellers .................. 77
5.DESIGN AND ANALYSIS OF FOUR LOW-FLOW COEFFICIENT
COMPRESSORS 78
5.1. DESIGN REQUIREMENTS ......................................................................................... 78
5.1.1. Inlet Design Conditions ................................................................................. 78
5.1.2. Impeller Sizing ............................................................................................... 79
5.2. STREAMLINE CURVATURE THROUGHFLow ANALYSIS ........................................... 82
5.2.1. Geometry ....................................................................................................... 82
5.2.2. Streamline Curvature Analysis ...................................................................... 84
5.3. CFD ANALYSIS ...................................................................................................... 85
5.3.1. Presentation of the Commercial Software TASCﬂow ................................... 85
5.3.2. Equations Solved ........................................................................................... 86
5.3.3. Boundary Conditions ..................................................................................... 88
5.3.4. Grid Speciﬁcations ........................................................................................ 89
5.3.5. Convergence Criteria .................................................................................... 91
5.4. QUALITATIVE ANALYSIS OF THE FLOW FIELD ........................................................ 91
5.4.1. Flow at Design Point in the AA Impeller ...................................................... 91
5.4.2. Flow at Oﬂ-design Conditions (AA Impeller) ............................................. 101
5.4.3. Comparison Between AA and AD ............................................................... 106
6. COMPARISON OF THE VARIOUS DESIGNS 109
6.1. POST-PROCESSING TECHNIQUE ............................................................................ 109
6.2. QUANTITATIVE ANALYSIS OF THE AA, AB, AC, AND AD IMPELLERS ................ 1 10
6.2.1. Performance at Design Point ...................................................................... 110
6.2.2. Consequence for the I -D analysis ............................................................... 1 I 1
6.2.3. Determination of the TEIS Models .............................................................. 115
6.2.4. Oﬁ—design Performance (AA impeller) ....................................................... 116
6.2.5. Off design performance of the AA, AB, AC and AD stages ......................... 118
6.3. VALIDATION OF THE LOADING ESTIMATION ........................................................ 120
6.4. ANALYSIS OF THE EFFECT OF EXIT BLADE ANGLE AND WIDTH ........................... 121
6.4.1. Qualitative analysis ..................................................................................... 121
6.4.2. Quantitative analysis ................................................................................... 127
6.5. INFLUENCE OF THE BLADE LEADING EDGE .......................................................... 131

Vii

7. CONCLUSIONS AND RECOMMENDATIONS - 133

 

 

 

BIBLIOGRAPHY 136
APPENDIX A 141
APPENDIX B 143
APPENDIX C 152

 

APPENDIX D 166

viii

LIST OF TABLES

TABLE 3.1 SUMMARY OF THE LIMITATION ON SOME MAIN 1D PARAMETERS ...................... 47
TABLE 4.1 GEOMETRY OF THE FIVE SELECTED IMPELLERS ................................................. 72
TABLE 5.1 IMPELLER MAIN DIMENSIONS ............................................................................ 81
TABLE 5.2 DESIGN VELOCITY TRIANGLES .......................................................................... 81
TABLE 6.1 AA, AB, AC, AD STAGE PERFORMANCE PREDICTION .................................... 1 10
TABLE 6.2 IMPELLERS EXIT CONDITIONS .......................................................................... 1 12
TABLE 6.3 TEIS MODELS FOR THE AA, AB, AC, AND AD IMPELLERS ............................ 1 16
TABLE 6.4 IMPELLER EXIT CONDIITONS ............................................................................ 127
TABLE 6.5 TEIS MODELS ................................................................................................. 129
TABLE 6.6 TASCFLow ANALYSIS (SMOOTH WALLS) — EFFECT OF THE LEADING EDGE
POSITION ................................................................................................................... 131

ix

LIST OF FIGURES

FIGURE 2.1 CENTRIFUGAL COMPRESSOR STAGE ................................................................... 8
FIGURE 2.2 SHROUDED IMPELLER ........................................................................................ 8
FIGURE 2.3 INLET AND EXIT VELOCITY TRIANGLES ............................................................ 10
FIGURE 2.4 EFFECT OF EXIT BLADE ANGLE ON THE ENTHALPY RISE ................................... 12
FIGURE 2.5 REPRESENTATION OF A COMPRESSION PROCESS IN A T-S DIAGRAM ................. 15
FIGURE 2.6 IMPELLER EXIT VELOCITY TRIANGLE WITH AND WITHOUT SLIP ........................ 17
FIGURE 2.7 INFLUENCE OF THE BLADE NUMBER AND THE EXIT BLADE ANGLE ON THE SLIP
FACTOR ...................................................................................................................... 18
FIGURE 2.8 TYPICAL CENTRIFUGAL COMPRESSOR HEAD-FLOW CURVE .............................. 19
FIGURE 2.9 MAXIMUM COMPRESSOR AND PUMP EFFICIENCIES AS FUNCTION OF SPECIFIC
SPEED (BALJE 1981) ................................................................................................... 23
FIGURE 2.10 TEST IMPELLER EFFICIENCIES VERSUS AVERAGE SPECIFIC SPEED (ROGERS
1980) .......................................................................................................................... 23
FIGURE 3.1 SCHEMATIC OF COMPRESSOR STAGE FLOW REGION ......................................... 25
FIGURE 3.2 FORCES ACTING ON A FLUID ELEMENT INSIDE AN IMPELLER PASSAGE ............. 26
FIGURE 3.3 VELOCITY DISTRIBUTION IN IMPELLER PASSAGE — INVISCID ANALYSIS ........... 27
FIGURE 3.4 FLOW IN THE MERIDIONAL PLANE NEAR IMPELLER EXIT — INVISCID ANALYSIS
(AFTER KRAIN 1984) .................................................................................................. 28
FIGURE 3.5 MOVEMENT OF LOW P* FLUID IN THE BOUNDARY LAYER IN AN IMPELLER
PASSAGE (AFTER JOHNSON 1978) ............................................................................... 31

FIGURE 3.6 SIMPLIFIED REPRESENTATION OF THE FLOW BETWEEN Two RADIAL BLADES... 32
FIGURE 3.7 DEFINITION OF STABLE AND UNSTABLE CASES (AFTER JOHNSTON AND EIDE

1976) .......................................................................................................................... 33
FIGURE 3.8 OPTICAL MEASUREMENT PLANES (ECKARDT 197 6) ......................................... 34
FIGURE 3.9 NON-DIMENSIONAL VELOCITY DISTRIBUTION AT FIVE STATIONS ALONG THE

IMPELLER (ECKARDT 1976) ........................................................................................ 36
FIGURE 3.10 NORMALIZED MERIDIONAL VELOCITY IN BACKSWEPT IMPELLER (ECKARDT

1980) .......................................................................................................................... 37
FIGURE 3.1 1 ROTARY STAGNATION PRESSURE AT IMPELLER EXIT, BELOW DESIGN POINT

(JOHNSON AND MOORE 1983) .................................................................................... 38
FIGURE 3.12 ROTARY STAGNATION PRESSURE AT IMPELLER EXIT, AT DESIGN POINT

(JOHNSON AND MOORE 1983) .................................................................................... 38
FIGURE 3.13 ROTARY STAGNATION PRESSURE AT IMPELLER EXIT, ABOVE THE DESIGN POINT

(JOHNSON AND MOORE 1983) .................................................................................... 38
FIGURE 3.14 EFFECT OF INLET MACH NUMBER ON Loss COEFFICIENT (DOUBLE CIRCULAR

ARC BLADE CASCADE) LIEBLEIN 1965 ........................................................................ 41
FIGURE 3.15 IMPELLER MAIN GEOMETRICAL PARAMETERS ................................................ 46

FIGURE 3.16 IMPELLER MAIN DIMENSION (AFTER PAROUBEK ET AL. 1994) ....................... 48

FIGURE 3.17 IMPELLERS EFFICIENCY (PAROUBEK ET AL. 1994) ......................................... 49

FIGURE 3.18 SLIP FACTORS (PAROUBEK ET AL. 1994) ........................................................ 51
FIGURE 4.1 COMPARISON OF ACHIEVABLE PRIMARY FLOW DIFFUSION IN INDUSTRIAL 2D
IMPELLERS AND IN STRAIGHT INLET, 3D IMPELLERS (BENVENUTTI 197 8) .................. 55
FIGURE 4.2 TEIS MODEL — DIFFUSING PASSAGES (AFTER JAPIKSE 1984) .......................... 56
FIGURE 4.3 EFFECT OF x AND MR2 ON THE WAKE WIDTH .................................................. 60
FIGURE 4.4 EFFECT OF x AND MR2 ON v=W25/W2p ........................................................... 61
FIGURE 4.5 EFFECT OF x AND MR2 ON C=W2MIWZP AND ON 52.. ........................................ 63
FIGURE 4.6 EFFECT OF x AND MR2 ON THE IMPELLER EFFICIENCY .................................... 63
FIGURE 4.7 EFFECT OF x AND MR2 ON THE IMPELLER PRESSURE RATIO ............................ 64
FIGURE 4.8 INFLUENCE OF THE HUB-TO-SHROUD INLET RADIUS RATIO ON THE SHROUD
INLET RELATIVE MACH NUMBER ................................................................................ 66
FIGURE 4.9 STAGE TOTAL-TO-TOTAL PRESSURE RATIO AS A FUNCTION OF IMPELLER EXIT
WIDTH AND IMPELLER EXIT BLADE ANGLE .................................................................. 67
FIGURE 4.10 IMPELLER EXIT TANGENTIAL VELOCITY VERSUS IMPELLER WIDTH ................ 67
FIGURE 4.1 1 RELATIVE VELOCITY RATIO As A FUNCTION OF IMPELLER EXIT WIDTH AND
IMPELLER EXIT BLADE ANGLE ..................................................................................... 68
FIGURE 4.12 IMPELLER EXIT ABSOLUTE FLOW ANGLE VS. IMPELLER EXIT WIDTH AND
IMPELLER EXIT BLADE ANGLE ..................................................................................... 69
FIGURE 4.13 DIFFUSER PRESSURE RECOVERY AS A FUNCTION OF IMPELLER EXIT WIDTH AND
EXIT BLADE ANGLE ..................................................................................................... 70
FIGURE 4.14 DIFFUSER TOTAL PRESSURE LOSS COEFFICIENT AS A FUNCTION OF IMPELLER
EXIT WIDTH AND EXIT BLADE ANGLE .......................................................................... 71
FIGURE 4.15 DEFINITION OF THE LEAN ANGLE ................................................................... 74
FIGURE 4.16 VIEW OF THE IMPELLER TIP AND DEFINITION OF THE RAKE ANGLE ................. 75
FIGURE 4.17 IMPELLERS MERIDIONAL CONTOUR ................................................................ 76
FIGURE 4.18 BLADE ANGLE DISTRIBUTION FOR THE AA50, AA55, AA65, AND AA7O ..... 76
FIGURE 4.19 BLADE-TO-BLADE LOADING FOR THE IMPELLERS ........................................... 77
FIGURE 5.1 MERIDIONAL CONTOURS OF IMPELLERS ........................................................... 83
FIGURE 5.2 BLADE ANGLE DISTRIBUTIONS ......................................................................... 83
FIGURE 5.3 RELATIVE VELOCITY DISTRIBUTION AT HUB (LEFT) AND SHROUD (RIGHT) FOR
THE AA IMPELLER AT DESIGN POINT .......................................................................... 84
FIGURE 5.4. BLADE-TO-BLADE LOADING ........................................................................... 85
FIGURE 5.5 MERIDIONAL VIEW OF CALCULATION GRID (IMPELLER ONLY) ......................... 90
FIGURE 5.6 BLADE-TO-BLADE VIEW OF THE CALCULATION GRID (IMPELLER ONLY) ........... 90
FIGURE 5.7 VELOCITY VECTOR AT MID PITCH (AA IMPELLER, DESIGN POINT) ................... 92
FIGURE 5.8 VELOCITY VECTOR NEAR THE SUCTION SIDE (AA IMPELLER, DESIGN POINT) .. 93
FIGURE 5.9 CALCULATING STATIONS ALONG THE IMPELLER CHANNEL .............................. 94
FIGURE 5.10 NORMALIZED MERIDIONAL VELOCITY AT STATION 1 (AA IMPELLER, DESIGN
POINT) ........................................................................................................................ 96
FIGURE 5.1 1 NORMALIZED MERIDIONAL VELOCITY AT STATION 11 (AA IMPELLER, DESIGN
POINT) ........................................................................................................................ 96
FIGURE 5.12 NORMALIZED MERIDIONAL VELOCITY AT STATION HI (AA IMPELLER, DESIGN
POINT) ........................................................................................................................ 97
FIGURE 5.13 NORMALIZED MERIDIONAL VELOCITY AT STATION IV (AA IMPELLER, DESIGN
POINT) ........................................................................................................................ 97

xi

xii

FIGURE 5.14 NORMALIZED MERIDIONAL VELOCITY AT STATION V (AA IMPELLER, DESIGN
POINT) ........................................................................................................................ 98

FIGURE 5.15 FLOW ANGLE DISTRIBUTION AT STATION V (AA IMPELLER, DESIGN POINT). 98

FIGURE 5.16 REDUCED RELATIVE TOTAL PRESSURE (NORMALIZED BY P0 AT INLET) (AA

IMPELLER, DESIGN POINT) ........................................................................................ 100
FIGURE 5.17 TKE DISTRIBUTION AT STATION V (AA IMPELLER, DESIGN POINT) ............. 101
FIGURE 5.18 NORMALIZED MERIDIONAL VELOCITY AT STATION V (AA IMPELLER, 70%

FLOW) ....................................................................................................................... 103
FIGURE 5.19 VELOCITY VECTORS AT MID-PITCH (AA IMPELLER, 60% MASS-FLow) ....... 103

FIGURE 5.20 REDUCED RELATIVE TOTAL PRESSURE (AA IMPELLER, 60% MASS FLow) .. 104
FIGURE 5.21 N ORMALIZED VELOCITY DISTRIBUTION (STATION V, A IMPELLER, 60% MASS

FLow) ....................................................................................................................... 104
FIGURE 5.22 REDUCED RELATIVE TOTAL PRESSURE DISTRIBUTION (AA, 140% MASS FLow)

................................................................................................................................. 105
FIGURE 5.23 N ORMALIZED VELOCITY DISTRIBUTION AT STATION V (AA, 140%

MASSFLow) .............................................................................................................. 105
FIGURE 5.24 TKE AT STATION V (AB IMPELLER, DESIGN POINT) .................................... 107
FIGURE 5.25 TKE AT STATION V (AC IMPELLER, DESIGN POINT) .................................... 108
FIGURE 5.26 TKE AT STATION V (AD IMPELLER, DESIGN POINT) .................................... 108
FIGURE 6.1 PERFORMANCE PREDICTION FOR THE AA, AB, AC, AND AD STAGES ........... l 13
FIGURE 6.2 DIFFUSER LOSS COEFFICIENT AND PRESSURE RECOVERY ............................... 1 15
FIGURE 6.3 CALCULATED PERFORMANCE MAP FOR THE AA IMPELLER ............................ 1 17
FIGURE 6.4 PERFORMANCE MAP PREDICTION OF THE AA, AB, AC AND AD STAGES ...... 119
FIGURE 6.5 ISENTROPIC BLADE-TO-BLADE LOADING (AB IMPELLER, DESIGN POINT) ...... 121
FIGURE 6.6 NORMALIZED MERIDIONAL VELOCITY (AASO, DESIGN POINT) ...................... 123
FIGURE 6.7 NORMALIZED MERIDIONAL VELOCITY (AA55, DESIGN POINT) ...................... 123
FIGURE 6.8 NORMALIZED MERIDIONAL VELOCITY (AA65, DESIGN POINT) ...................... 124
FIGURE 6.9 NORMALIZED MERIDIONAL VELOCITY (AA70, DESIGN POINT) ...................... 124
FIGURE 6.10 TKE CONTOUR AT STATION V (AASO, DESIGN POINT) ................................ 125
FIGURE 6.1 1 TKE CONTOUR AT STATION V (AA70, DESIGN POINT) ................................ 125
FIGURE 6.12 RELATIVE FLOW ANGLE AT STATION V (AA55, DESIGN POINT) ................... 126
FIGURE 6.13 RELATIVE FLOW ANGLE AT STATION V (AA65, DESIGN POINT) ................... 126
FIGURE 6.14 PERFORMANCE AT DESIGN POINT ................................................................. 128
FIGURE 6.15 DIFFUSER PERFORMANCE AT DESIGN POINT ................................................. 129
FIGURE 6.16 PERFORMANCE PREDICTION COMPARISON ................................................... 130
FIGURE C. 1 COORDINATE SYSTEM FOR THE GENERAL EQUATIONS ................................... 153
FIGURE C.2 DEFINmON OF THE GEOMETRY AND NOTATIONS .......................................... 154
FIGURE C.3 EXAMPLE OF BEZIER CURVE .......................................................................... 159
FIGURE C.4 EXAMPLE OF CALCULATION GRID .................................................................. 160
FIGURE C.5 LOCAL COORDINATE SYSTEM ........................................................................ 161
FIGURE C.6 TYPICAL BLADE ANGLE VARIATION ............................................................... 163

FIGURE D. 1 MERIDIONAL VELOCITY PROFILE AT IMPELLER INLET ................................... 166

Greek

ZZEBZZZW“””’EW
‘5

Nazcawaewmowm

Q

NOMENCLATURE

area

width or distance along a quasi-orthogonal
absolute velocity

dissipation coefﬁcient

friction coefﬁcient

speciﬁc heat at constant pressure
pressure recovery

diameter

hydraulic diameter

diffusion ratio

distributed blade force

dissipative blade force

gravitational constant

head

enthalpy

rothalpy or node number in the streamwise direction
incidence (blade angle minus ﬂow angle)
node number in the pitchwise direction
node number in the spanwise direction
Mach number

Mach number ratio

Relative mach number

distance along the meridional direction
mass-ﬂow rate

rotational Speed

speciﬁc speed (US deﬁnition)

pressure

volume flow rate

heat exchanged

radius or gas constant

radius

radius of curvature

entropy

temperature

peripheral Speed

relative velocity

work exchanged
blade number

blade or ﬂow angle (relative to meridional) or slope angle of calculating
station

xiii

blade or ﬂow angle (relative to meridional)

isentropic exponent or lean angle

deviation or thickness

area ratio between secondary zone and outlet area or slope angle of
meridional direction

(”004W

7] effectiveness or efﬁciency

9 circumferential coordinate

1.1 work reduction factor

p density

6 slip factor

1: torque or exit rake angle

(1) flow coefﬁcient

x ratio of the mass in the secondary zone to mass flow

‘I’ head coefﬁcient

(.0 rotational speed

(2, streamwise vorticity
Subscripts

blade

front stage

ideal or inlet

isentr0pic

leading edge

middle stage

meridional

outlet

on previous calculating station
pressure side

rear stage

on current calculating station or radial
suction side

tip

throat

tangential

axial direction

stage entrance or total conditions
impeller exit

diffuser pinch

diffuser exit

exit bend

exit return vane

exit stage

OOQOUIANONI:9"$H;UEUOBEE5'T"TIE

xiv

 

1. INTRODUCTION

1.1. Demands on and Challenges for Centrifugal Compressors

A centrifugal compressor is a high Speed rotating device that adds energy to a
ﬂuid passing through it. The flow enters the impeller axially through the impeller “eye”
and leaves the impeller radially at higher pressure and absolute velocity. The range of
applications for centrifugal compressors is large. They can be found in various industries:
from the process industry to petrochemical industry, from gas turbines to turbochargers
for internal combustion engine, and from aeronautical applications to air conditioning
units.

For the early gas turbine application, a single-stage centrifugal compressor was
sufﬁcient. After World War H, as the need for larger power grew, axial ﬂow compressors
appeared more suitable for large engines. At the time, multistaging of centrifugal
compressors, especially for aircraft application, seemed inappropriate because of the
necessity to turn the ﬂow from the radial back to the axial direction to enter the next
stage. The additional ducting required contributed to the increase in size and weight of
the machine, which does not occur in axial machines. Consequently, most development
funding went towards the axial machine, leaving the centrifugal compressor on the side.

By the late ﬁfties, it became clear that smaller gas turbines would have to use

centrifugal compressors. Centrifugal compressors appeared to be more suitable for small

volume flows. On a Single stage application (and for the same duty), they have a shorter
axial length than their axial counter part and a better resistance to foreign objects. They
are also able to operate over a wider flow range for a constant rotational speed.

For turbochargers, the centn'fugal compressor is the ideal candidate with its wide
margin between surge and choke, which allows it to match the continuous range of
automotive engines, from idle to full power.

In the process industry, because Size and weight are of secondary importance, the
centrifugal compressor can be used in multistage applications. The return systems
between each impeller have seen their aerodynamics improved, and higher efﬁciency can
now be achieved.

The design of the impeller depends mainly on the type of application. For
centrifugal compressors used in gas turbines or turbocharger applications, the impeller is
unshrouded: the tip of the moving blade passes near the casing where a small clearance is
maintained. Generally, the impeller is predominantly axial at the inlet. This axial portion
is called the inducer. The aerodynamic performance will be improved by using a well-
designed inducer.

For industrial applications, multistage centrifugal compressors are used. To limit
the overall length of the machine, the impellers have a smaller axial inlet. They are
generally Shrouded, that is, a cover is ﬁxed on the blades and rotates with them. This
removes the need for clearance, but the additional weight generates stresses that limit the
maximum rotational speed that can be used.

In this thesis, centrifugal impellers for an industrial multistage machine will be at

the center of attention.

1.2. The Need for Low-Flow Coefﬁcient Compressors

In centrifugal compressors, as in any type of turbomachinery, there exist an
optimum efﬁciency over a certain range of flow. The flow coefﬁcient (I) of a centrifugal

stage can be correlated to the efﬁciency, and is deﬁned by

.=_g_
a) U

tip tip

(1.1)

where Q is the inlet volume ﬂow rate, de is the impeller tip diameter, and Utip is the
peripheral speed at impeller tip. The optimum range is for a ﬂow coefﬁcient between
0.06 and 0.09. In a multistage machine, the volume-ﬂow rate decreases as the pressure of
the flow is increased through the Stages. As the last stages are reached, the ﬂow
coefﬁcient (1) is, therefore, smaller than at the inlet of the machine. The impeller and
diffuser channel are narrower to handle the smaller volume ﬂow rate. The friction losses
are thereby increased, and a smaller efﬁciency is to be expected. Also, the disk friction
losses and the leakage losses (see Chapter 4) have been shown to be inversely
proportional to the volume flow rate. Hence, low-ﬂow coefﬁcient compressors are
inherently inclined to have lower efﬁciency than the earlier Stage.

If efﬁciency is directly related to flow coefﬁcient, one would think that the only
way to improve the performance of such compressors is to increase their ﬂow coefﬁcient.
Looking back at equation (1.1), this would imply decreasing Dﬁp or reducing Uup. In both
cases, this would lead to a decrease in the work input of the machine (see Section 2.3),
which is not desirable.

The literature covering the subject of low-ﬂow coefﬁcient compressor is limited.

Four main sources have been found. The ﬁrst one is the solution proposed by Rusak

(1974) in his patent. He proposed to reduce the friction losses by enlarging the impeller
exit area while continuously increasing the thickness of the blade from inlet to exit in
order to maintain the same diffusion. This leads to a larger hydraulic diameter. The
invention was claimed for very low-ﬂow coefﬁcient impellers only. Very little
performance data were provided. Efﬁciencies around 55% at a flow coefﬁcient of 0.008
were quoted.

The second reference is by Paroubek et al. (1994). They investigated the effect of
the hub-to-outlet diameter ratio and the exit width to exit diameter ratio. They found that
higher performances were Obtained with the smallest hub diameter and with the wider
impellers. Unfortunately, the hub diameter in a multi-stage machine has already been
optimized, and is more or less set by rotordynamics considerations. The limited
geometrical input and the non-dimensionalized experimental results reduced greatly the
interest in this paper.

The third reference is by Koizumi (1983). He mainly showed the large impact of
the Reynolds number on the performance of the impeller and the need to maintain the
surface roughness of all the passages as low as possible.

The last reference is by Casey (1990). He showed experimentally that impellers
with higher backward-swept blades and a larger impeller exit width could lead to better
performance even with their smaller work input. The optimum conﬁguration was difﬁcult
to evaluate with the conventional prediction methods and required an experimental
evaluation to determine the best design. The large ﬂow angle (from radial) at the diffuser
entrance led to instabilities in the vaneless diffuser at low flow rates, and the use of a

vaned diffuser was recommended by the author.

1.3. Structure and Objectives of the Work

In this thesis, centrifugal compressor stages with ﬂow coefﬁcient below 0.022
will be considered. No attempt has been made to minimize the seal losses or the disc
friction losses. Only a better seal design could reduce the seal friction losses, which is
beyond the scope of this thesis. For the disc friction losses, the only recommendation will
be to manufacture the impeller with the best surface ﬁnish on the back disk.

The purpose of the work will be to optimize the geometry of the impeller in order
to keep the friction losses as low as possible. For this, different combinations of blade
outlet angles and exit impeller widths have been studied to fulﬁll the design requirement
of an industrial application.

The return channels between each stage have been designed and analyzed, but the
results will not be presented here.

This dissertation starts with a general chapter on centrifugal compressors to deﬁne
the notations and the speciﬁc vocabulary used through this thesis.

Chapter 3 presents the general ﬂow characteristics in a centrifugal impeller. An
inviscid analysis explains the forces applied to a fluid element travelling through a blade
passage. The results of secondary ﬂow theory are described, and experimental results are
Shown to help Visualize the flow ﬁeld. The different types of losses and the one-
dimensional parameters are then reviewed.

The ﬁrst step in the work was to develop two codes to design a centrifugal
compressor: a one-dimensional code and a streamline curvature through-ﬂow code.

These codes are presented in Chapter 4. They have been applied to design ﬁve impellers

satisfying the same design requirements but having a different outlet blade angle and
outlet width. The detailed CFD analysis of these impellers is presented in Chapter 6.

Chapter 5 presents the design of four low-ﬂow coefﬁcient compressors as an
extension of an existing family of impeller. The one-dimensional code and streamline
curvature code have been used to select the geometry. A commercial CFD code,
TASCﬁow, has then been used to qualitatively describe the flow ﬁeld inside the
impellers. The effect of the mass flow rate on the flow ﬁeld of the ﬁrst impeller is Shown.
The flow ﬁeld between the four impellers is then compared.

Chapter 6 attempts to obtain quantitative results from the CFD in order to
compare all the designs developed in Chapter 4 and 5. The analysis of two more
impellers (one with a long inducer and one with no inducer) is presented.

The objective of this work is to use CFD as a way of artiﬁcially testing various
designs, as CFD can provide information on many details unreachable through
experiments. The quantitative results should provide the necessary information to select

the best suitable design.

2. GENERAL DISCUSSION OF TURBOMACHINES

The purpose of this chapter is to deﬁne some general parameters and basic
concepts that are commonly used in the area of turbomachinery and to introduce the

notation that will be used throughout this thesis.

2.1. Elements of a Centrifugal Compressor Stage

A centrifugal compressor Stage consists of an impeller (the rotating part) and a
diffuser (non-rotating). Inlet guide vanes are sometimes used in front of the rotor to direct
the ﬂow to the impeller eye (or inducer, i.e. the axial part of the impeller). The impeller is
used to impart energy to the ﬂow by increasing the velocity and pressure of the fluid. The
diffuser is used to convert the kinetic energy available at the impeller exit into static
pressure by decelerating the ﬂuid. The diffuser is followed by a collector, a scroll
(volute), or a return channel depending on the type of application.

For a multistage machine, as the flow leaves the impeller radially, a return
channel is used at the diffuser exit to turn the ﬂow back in the axial direction. Return
vanes are used to remove any residual swirl before entering the next stage. In the last
stage, the diffuser discharges into a volute that leads the ﬂuid to the exit piping system.

Figure 2.1 Shows a typical centrifugal compressor stage.

Diffuse

 

 

Impell :

 

Figure 2.1 Centrifugal compressor stage

The impeller is the essential element of the centrifugal compressor stage. There
are two types of impellers: unshrouded, which does not have a front cover, and shrouded,

which has a cover. Figure 2.2 shows a meridional View of a shrouded impeller.

    
 

Shroud

Hub

 

Figure 2.2 Shrouded impeller

By using a shrouded wheel, the secondary flow arising from tip clearance can be

avoided. A trade-off must be made between the tip clearance losses of the unshrouded

 

wheel, and the cover friction losses of the shrouded wheel. The high stress generated by
the front cover limits the use of this type of impeller to relatively low rotational speed

machines.

2.2. Velocity Triangles

Velocity triangles are used to represent the velocity components (absolute or
relative) at any station of a turbomachine. The velocity measured in a ﬁxed frame is
called absolute velocity and is denoted by C. The velocity measured with respect to a
rotating frame is denoted by W. The blade speed relates both velocities, and the following
vector relation holds:

C=W+U
where U is the blade speed, also called the peripheral speed.

Figure 2.3 shows a typical velocity triangle with the notation that will be used
throughout the thesis. The velocity vectors Shown are those for the mean ﬂow at a
Speciﬁed radius at inlet and at exit. The flow direction within the rotor passage or on the

blade is not Shown in the diagram. 0t represents the ﬂow angle between the absolute
velocity and the meridional plane, and B is the flow angle between the relative velocity

and the meridional plane. The component of velocity in the tangential direction is the

projection of C (or W) in the tangential plane and is denoted Cu (or Wu).

lO

 

 

 

 

 

uouoaup [enuaﬁuel

 

 

   
 

Q

 

 

 

Figure 2.3 Inlet and exit velocity triangles

2.3. The Euler Equation

The Euler equation relates the head change in a turbomachine to the velocities at
the impeller inlet and exit. The Euler equation is derived from the conservation of linear
momentum: the rate of change in linear momentum of a volume moving with the ﬂuid is
equal to the surface forces and body forces acting on the ﬂuid.

By the conservation of momentum principle, the change of angular momentum
(obtained by the change in tangential velocities) is equal to the summation of all forces
acting on the rotor, i.e. the net torque of the rotor.

Mathematically, between inlet (1) and exit (2), this can be written as
T = m(rlCul — rZCuZ)
where C“; and Cuz are the tangential velocity at inlet and exit.

The rate of change of energy transfer is the product of the torque and the angular

velocity, and therefore the total energy transferred is

11

E = Ta) : m(rleul _ r‘ZwCRZ): n.1(UlCul — UZCuZ)
where U; and U2 are the peripheral velocity at I] and r2.

The energy transferred per unit of mass ﬂow is equal to the change in total

enthalpy ho and hence
Aho : UZCuZ —U1Cul = hoz Thor = E12(Toz "T011
where Ep is the average Specific heat between inlet and outlet.

It is useful to relate the enthalpy change to the exit blade angle. Using the velocity

triangle at the impeller exit and assuming no prewhirl, the enthalpy rise is given by
Cu2 2 U2 + Cm2 tan B2
with the convention that B2<0 for a backward swept impeller (the word backswept is used
to describe impellers with vanes inclined backward at the outlet).
Introducing the theoretical head rise \II :95”;- and the ﬂow parameter (p: £1172 to

make a non dimensional representation of the enthalpy rise versus mass ﬂow rate, the

following relation is obtained:

W=l+¢tanl32

which is plotted in Figure 2.4.

12

 

1.81
1.6 .
1.4 ~
1.2 ~

Bz=+45°

2:00

 

0.8 ~
0.6 1
0.4 e
0.2 .

132='45°

 

 

 

Figure 2.4 Effect of exit blade angle on the enthalpy rise

The theoretical head rise is seen to decrease with increasing mass ﬂow rate for a
backward-swept impeller, independent of the mass ﬂow for a radially—ending impeller,
and to increase with mass ﬂow for a forward-leaned blade impeller. The increasing head
with decreasing mass ﬂow for backward-swept impellers gives a more stable Operating
stage characteristic and consequently a greater range.

It is worth noting that the enthalpy change in the machine can also be expressed in

the following way:

Aho =%{U§ -Uﬁ]+ [111,2 —w,2]+[c,2 -ch} (2.1)
after having introduced the relations inside the inlet and exit velocity triangles
w2 = C2 +112 —2UC,
Hence, it is seen that the enthalpy rise in a centrifugal machine is due to three

different contributions. The ﬁrst term in Equation (2.1) represents the change of energy

of a particle as it moves from one radius to another. This contribution by the centrifugal

13

forces represents about one-half of the work input, and is accomplished without loss
(Casey and Marty 1986). The second term represents the rise in enthalpy due to the
diffusion of the relative ﬂow. This term represents approximately one-ﬁfth of the total
work input (depending on the impeller shape). Assuming that the velocity at the exit
stage is equal to the velocity at the inlet of the stage (C1=C4), the last term represents the
diffusion of the absolute velocity in the diffuser and accounts for about one-third of the

work input.

2.4. Head Rise

The adiabatic head Had represents the energy transferred through a turbomachine
per unit mass of ﬂuid. If a compressor produces 1 m of head, it means that l kg-force will
be required to elevate 1 kg Of gas mass to a height of 1 meter. For an incompressible
ﬂow, the head is expressed by

AH, =£
P

where AP is the pressure rise, and p the density.

For compressible ﬂow, the average density between inlet and exit must be used in
the equations. An exact relation for a compressible ﬂow, considering an ideal gas and an

isentropic process, is given by

where, for a compression process

14

r—l

ATad : 71 fl —1
P1

The head, the enthalpy rise, or the pressure ratio are quantities used to express the

energy exchange into the impeller. They can all be used interchangeably.

2.5. Efficiency

The ﬁrst principle of thermodynamics in its complete form relates the work input

W and heat exchanged Q to the change in total enthalpy and altitude 2 as follows:

gg=lh+£21+gZI zlcpTo+ng

Neglecting the change in altitude, the heat loss, and considering a perfect gas, the
change in energy is directly related to the change in total temperature. It is then natural to
represent the compression process in a T-S diagram.

It is easy to show (e.g. Wilson 1984) that the Slope of each line increases with

increasing temperature. A typical compression process is Shown in Figure 2.5.

 

 

1 5
T 11
P02
T02
T023
P01
T01

 

 

Figure 2.5 Representation of a compression process in a T-S diagram

The isentropic efﬁciency compares the actual enthalpy change during the
compression process to the one that would have taken place for an isentropic process. It is

deﬁned by

’12-’11
hzrh.

 

"is :

State 1 generally refers to the total conditions at the inlet. State 2 corresponds to the exit
conditions. If total conditions are used at state 2, the total-tO-total isentropic efﬁciency is
obtained. If static conditions are used, it is the total-tO-static isentropic efﬁciency. The
difference between these two efﬁciencies will be large for stages having a large exit
velocity. Depending on the type of application, one efﬁciency or the other will be used.
Generally, if the exit velocity is used for a purpose (as in a jet-engine for example) the
total-to-total efﬁciency is used; if the exit velocity is not used, the total-to-static
efﬁciency is used. For a perfect gas with constant speciﬁc heat, the expression for the

efﬁciency becomes

16

7-1

a. i,
___ Cp (T021: ”7111): Pm
Cp (T02 “T011 £92-]
T01

 

77.

Note that the isentropic efﬁciency depends on the inlet conditions and the pressure ratio.

A representation of the isentropic efﬁciency is shown in a T-S diagram in Figure 2.5.

2.6. Work Reduction Factor and Slip Factor

The ﬂow at the exit of the impeller does not follow the blade. A parameter is
introduced to compare the actual absolute tangential velocity to the one obtained in the
case of an inﬁnite number of blades where the ﬂow would follow the blade. This

parameter is called the work reduction factor It

”-2...
Ci}

This quantity directly inﬂuences the estimation of the enthalpy rise in the compressor.

The Euler equation is now rewritten as follows:
AH = ”U 2C1;

assuming no inlet prewhirl. The corresponding velocity triangles are shown in Figure 2.6.

l7

 

 

 

 

 

Figure 2.6 Impeller exit velocity triangle with and without slip

The other parameter commonly used in the literature is the slip factor 0' deﬁned

 

where C C; — Cu2

slip =

The work coefﬁcient and the slip factor are related through the following relation:

C
0'=l—1——-—L— l—
[ Uztanﬂz} I”)

Many correlations are available in the literature to predict the value of the slip factor. One

of the most commonly used is the one by Wiesner (1967)

J COS .32.

O. = 1 — 20.7
where [321, is the exit blade angle and Z the blade number.

This equation is a data ﬁt of the theoretical curves obtained by Buseman (1928)

for inviscid ﬂow in an impeller with logarithmic-spiral shaped blades. The validity of this

18

expression was checked by Wiesner with over 60 compressors and pumps with various
blade angles and blade numbers, and compared to other correlations. The agreement was
found to be fairly good, but this relation may be off by a few percents for some machines.

The effect of the blade angle and blade number on the Slip factor is Shown in
Figure 2.7. The slip factor is seen to be smaller for a radial impeller that for a backward-

swept impeller. It increases with increasing blade number.

 

1.0

0.95

 

.0
o
o

 

Slip factor 0'
O
on
0‘)

p
on
.0

 

.O
\I
Q‘

 

 

 

0.70

 

Exit blade angle [32b

Figure 2.7 Inﬂuence of the blade number and the exit blade angle on the slip factor

2.7. Operating Map and Compressor Range

The compressor performance consists of a plot showing the efﬁciency and
pressure ratio versus. the mass-ﬂow rate for different rotational Speeds, or the head versus
the volume-ﬂow rate, or in a dimensionless form, the head coefﬁcient versus the ﬂow

coefﬁcient. A typical curve for a centrifugal compressor stage is Shown in Figure 2.8.

19

1.4

 

1.2 4,

08 4»

06..

0.4 o

0.2 0

 

 

 

.6... ., ...
Figure 2.8 Typical centrifugal compressor head-flow curve

Three different regions may be distinguished. On the right is the choke limit
obtained when sonic conditions are reached in the minimum section of the machine
(impeller throat or diffuser throat). The choke limit appears as a vertical line in the
compressor map because the efﬁciency and the pressure ratio drop drastically.

On the left is the stall region and surge line, which corresponds to a region of
instability. Rotating stall is an unstable phenomenon often preceding surge. It can take
place in the impeller or in the diffuser. Rotating stall is a local instability in which zones
of separated ﬂow, called cells, rotate in one component of the machine. When operating
at a high pressure level, the pressure ﬂuctuations corresponding to rotating stall generate
mechanical vibrations that can severely damage the machine. Operation in this area
should therefore be avoided. In the case of vaneless diffusers, the onset of this
phenomenon has been correlated to the diffuser inlet ﬂow angle (Senoo 1978).

During surge operation, oscillations of mass ﬂow rate between the compressor
exit and inlet occur that may damage the machine especially for high-density gas. Van

den Braembussche (1984) Showed that for a negative value of d‘P/dtl) (i.e. increasing

2O

pressure for decreasing mass ﬂow) any perturbation will be damped out, resulting in a
stable operating point. As soon as d‘I’/d¢ is positive, the ﬂow may become unstable

depending on the characteristic of the global system (i.e. the compressor, but also the
inlet and exit piping). The compressor should not be operated in this region.
At medium ﬂow coefﬁcient is a region of high efﬁciency, generally close to the

design point, where the compressor should be operated.

2.8. Dimensionless Parameters

As in any physical process, dimensional analysis and similarity considerations can
be applied to turbomachinery. They state that for machines that are geometrically similar,
similar velocity triangles will be obtained. Among the dimensionless groups that appear,
the head coefﬁcient, the ﬂow coefﬁcient, the efﬁciency, the speciﬁc speed, and the
speciﬁc diameter are used.

Dimensional analysis can be used to compare data from different machines to
predict the performance of an actual machine when the testing has been performed on a
scale model, and to predict the performance at different speed line or operating ﬂow

point.

2.8.1. Flow and head coefficient

The ﬂow coefﬁcient is deﬁned by

 

and the head coefﬁcient by

 

 

lili-

pa
me
the

for 2

21

,, = _H_

N21)2

where Q is the volume ﬂow rate, H is the head, N is the rotational speed, and D is the
impeller diameter.

These two parameters are generally used to predict the off-design characteristic of

a machine. The Similarity does not hold close to the choke line and close to the surge line

due to compressibility and viscous effect.

There are various deﬁnitions of (I) and ‘1’. The following deﬁnitions are used

. : —Qin
m it. DtIpUZ
and
H.
[P : IS

 

U22

23
where Qin is the inlet volume ﬂow rate, His is the isentropic head produced by the

machine, U2 is the peripheral speed at rotor tip, and g is the gravitational constant.

2.8.2. Specific diameter and specific speed

The speciﬁc diameter and the Speciﬁc speed are another group of dimensionless
parameters. They are a combination of the head and ﬂow coefﬁcient. They are used
mainly for stage selection applications. Numerous correlations are available and allow for
the selection of the type of machine (positive displacement, centrifugal, axial) required
for an application.

The speciﬁc diameter is defined by

 

N — W05
3 .75
H 3.1
D 0113,25
5 “0.5

in which V1 is the inlet volume ﬂow rate.

The expression Of the speciﬁc speed used in the text is

where N is in rpm, Q,“ in ft3/sec, His in ft-lbf/lbm.
These deﬁnitions are not truly dimensionless. Letting (0 be in round per second,

and replacing Had by (gHad), we get the following dimensionless deﬁnitions.

. -_aw

S _ (gHad )075

and

d your”

’ JV.

n, and d, have been correlated for best efﬁciency between different type of machines.

Figure 2.9 taken from Balje (1981) shows the relation between peak efﬁciency and

speciﬁc speed, and the application domain Of different types of turbomachines.

 

23

 

 

 

 

 

 

 

 

 

 

 

 

1.0
IFII 1111 1111 III! V
Mixodflow
as /‘>\
Ip/p-Z) \
Multilobe Radial / CemrIput \
as -
/ / I \ \
11.... ._...._ Axial
Partial emission /\ cm‘m
a4
Drag /+ I INK R0’ -2 X 10'
" (La'-0.3l s/h-aozAxw
Q2 :lb, - 1102 1mm __
s/D ' 0.001 Drag, Multilobo
La'M-OJ
01111 L111 1411 llllll
2 ea 2 4 e a 2 4 e 3 2 4 a a 2 4
10” 10" 10" 1 10

(a)

Figure 2.9 Maximum compressor and pump efficiencies as function of specific speed

(Balje 1981)
Figure 2.10 shows the empirical relation between efﬁciency and speciﬁc speed

for centrifugal impellers of the same family with similar geometrical layout (Rodgers,

1980). It clearly shows that there exists an Optimum Speciﬁc speed range.

 

van

1mm rumor-1c enmity, as
8

 

03,0”
0132mm
(332551309

 

 

 

0.1

0.2 0.3 0.4

0.5 0.0 0.7 0.. 0.0

is AVERAGE DENSITY DIMENSIONLESS SPECIFIC SPEED (LOG SCALE

1.0 1.2 1.4

Figure 2.10 Test impeller efﬁciencies versus average specific speed (Rogers 1980)

 

 

 

 

3.]

3.]

fun

diff

The

24

3. FLOW IN THE IMPELLER AND DESIGN PARAMETERS

The ﬁrst part of this chapter gives initial insight into the detailed ﬂow ﬁeld in a
centrifugal impeller. The Chapter starts with a review of the forces applied on a ﬂuid
element, followed by the typical secondary ﬂows encountered. Experimental
Visualization of the ﬂow ﬁeld taken from the literature is then presented. It is also shown
that Computational Fluid Dynamics (CFD) has become a valuable tool capable of
qualitatively predicting the ﬂow. In the second part, the different types of losses
encountered in centrifugal compressors are summarized, and the main one-dimensional

design parameters are reviewed.

3.1. Flow in the Impeller

3.1.1. Qualitative Description of the Flow in the Impeller

The ﬂow in a centrifugal impeller is highly complex. It is three-dimensional,
turbulent, viscous, and unsteady. The ﬂow at the impeller exit is highly nonuniform and

differs drastically from the one-dimensional picture presented in the previous chapter.

The ﬂow ﬁeld in a typical centrifugal compressor is shown in Figure 3.1.

 

25

 

Figure 3.1 Schematic of compressor stage flow region

The ﬂow enters the impeller through the inducer in which the relative ﬂow is
turned from the inlet direction to almost axial. The ﬂow then enters the axial-to—radial
bend. In this region, boundary layers and secondary ﬂows have developed. AS will be
seen later, Coriolis forces and streamline curvature contribute to the destabilization of the
ﬂow. Two regions develop in the ﬂow ﬁeld: one with a high relative Mach number called
a jet and one with a low relative Mach number called a wake. After exiting the impeller,

the two regions mix rapidly due to the difference in angular momentum and enter the

diffuser.

3.1.2. Inviscid Flow Analysis

Although the viscous effect can not be neglected when analyzing the ﬂow in a
centrifugal impeller, an inviscid analysis provides good insight into the forces applied to
a ﬂuid element moving into a blade channel. Simple but important physical knowledge

can be gained.

 

110

26

The equation of motion for the relative ﬂow, assuming an inviscid ﬂow with no

body force, is given by

DvTr ‘ g g _. Vp

—Dt_ + 2coxw + (DXu = -—

p

1 l. l 1.
Relative . Coriolis . Centrifugal Pressure
acceleration acceleration acceleratlon gradient

In this equation, Wrepresents the relative velocity, ii is the peripheral Speed and
(To is the angular velocity.

In a streamline coordinate system (s, n, b), where s is along the streamline, n is
normal to the streamline in the blade-to-blade surface, and b is the binormal

perpendicular to s and n, the projection of the equation of motion in the n-direction is

2
—lv—+20)w-(DuCOSB =la—p
R pan

'1

The different forces applied to a ﬂuid particle are shown in Figure 3.2.

 

 

Figure 3.2 Forces acting on a ﬂuid element inside an impeller passage

 

27

By combining the energy equation and the equation of motion, Krain (1984)
derived the following approximate relation, which relates the velocity difference between
the pressure Side and the suction side of the blade to the mean velocity, the rotational

speed and the radius of curvature (Appendix A).

w” —wp, =[2m-r’ﬁﬂkn (3.1)

This approximate relation Shows that
0 the velocity varies almost linearly from pressure to suction side
0 the velocity gradient results from the contribution of the blade curvature and of the
Coriolis forces
0 the velocity difference between pressure Side and suction side is smaller for a
backswept impeller (Rn>0), than for a radial impeller (Rn=oo) and for a forward

leaning impeller (Rn<0). This is illustrated in Figure 3.3.

 

Rn>o Rn: 00

    

I
rx‘” rx‘” A”
k’
l
Fomard leaning impeller

 

 

 

Baekswept impeller Radially ending impeller

Figure 3.3 Velocity distribution in impeller passage - Inviscid analysis

(adapted from Krain 1984)

Equation 3.1 has another important consequence. If we consider a blade passage
with a sufﬁciently high radius of curvature, the last term on the right—hand side can be

neglected, leading to

wss - wps : (2mm

 

28

This Shows that there exists inside the passage a forced vortex of strength 2m rotating in a
direction opposite to the rotor. The force created by this vortex explains why the ﬂow
does not follow the blade at the impeller exit.

The equation of motion in the meridional plane is given by

2 2
lap. zi-‘CLCOSE (3.2)
p dz Rz r

where 8 is the local angle of the streamline considered with the axis of rotation. Close to

the impeller exit, the streamline is almost radial (i.e. 8 E 90°); the last term in (Eq.3.2)
can be neglected with respect to the others. The equation of motion then expresses the
balance between the streamline curvature force and the pressure gradient. The radius of
curvature of the meridional contour is positive. The pressure gradient with respect to z is

then positive, and the expected meridional velocity distribution is shown in Figure 3.4.

 

 

Figure 3.4 Flow in the meridional plane near impeller exit - Inviscid analysis (after
Krain 1984)

29

This distribution of meridional velocity is not what is necessarily Observed, as
will be seen in the next sections. A more advanced treatment is required to predict the

exit ﬂow ﬁeld.

3.1.3. Secondary Flows

Secondary ﬂows are deﬁned as the difference between the full three-dimensional
inviscid solution and the real ﬂow occurring in a component of the compressor (Van den
Braembussche 1985). The ﬂuid particles inside the boundary layer have a lower velocity,
but they are submitted to the same pressure gradient as the main ﬂow. This imbalance
between the normal pressure gradient and the centripetal acceleration will therefore move
the ﬂow inside the boundary layer in a different direction than the main ﬂow. This is
called a secondary ﬂow by contrast with main or principal ﬂow.

Secondary ﬂows induce cross-ﬂows, and therefore secondary vorticity.
Hawthorne (1974) developed an expression for the development of the streamwise

vorticity in an incompressible and frictionless ﬂow.

 

 

X W -pw2 Rn db w dz
In Equation (3.3), 52, represents the streamwise vorticity, (Dis the rotational speed,
2 is the axial direction, and p' = p +.'3pW2 —%pU 2 the rotary stagnation pressure. The

rotary stagnation pressure is constant along a streamline for a Steady, inviscid,
incompressible ﬂow similarly to the total pressure along a streamline for a steady
incompressible frictionless ﬂow. The rotary stagnation pressure reduces only because of

losses. As shown by Johnson (1978), the secondary ﬂows tend to move the low

3O

momentum ﬂuid towards regions of stable location which correspond to minimum
reduced static pressure P, = P — izpmzr2 (or relative Mach number).

From Equation (3.2), it can be seen that streamwise vorticity will be generated
when a gradient of rotary stagnation pressure in the binormal direction (together with
curvature) or in the axial direction (together with rotation) is present. Hence, even if the
secondary ﬂows are essentially due to the presence of the boundary layer, they can be
accounted for in an inviscid analysis if the gradients of rotary stagnation pressure present
at the inlet are introduced into the model.

Johnson (1978) investigated the ﬂow in rotating bends using secondary ﬂow
theory. The movement of the low p* ﬂuid in the boundary layer of the impeller passage is
shown in Figure 3.5. In the inducer, the boundary layers are thin, and only small
secondary ﬂows may occur. If the contribution of the blade curvature is large enough,
then some vorticity will be created along the streamwise direction on the pressure side
and suction Side that will take the low p* ﬂuid from the boundary layer to the shroud
(Figure 3.5). It is only when the ﬂow starts to deviate from the axial direction to the
radial direction that the secondary ﬂows become consequent.

In the axial-tO-radial bend, both Coriolis and streamline curvature are acting on
the ﬂow with equivalent strength. The boundary layers have now developed. The

meridional curvature on the blade surface is in the direction of vorticity. Secondary ﬂow

transports the low-momentum ﬂuid from the hub to the shroud.

 

 

 

p11

C01

gm.

31

shroud
shroud

 

 

 

 

p
s p 5 r
U s r s u s e I
c . e , c 1 S d
' s ' t S
t d
d s d i U 8
1 e e e
o u o r
n r n 6
e hub
hub MID PASSAGE
EARLY PASSAGE
shroud
s p
U 5 wake ; s
c l s i
3 d s d
I
o e U e
n T r
hub e

LATE PASSAGE

Figure 3.5 Movement of low p* ﬂuid in the boundary layer in an impeller passage
(after Johnson 1978)

In the radial part of the impeller, Coriolis forces dominate, which results in
secondary ﬂow into the blade-tO-blade plane. An accumulation of low-momentum ﬂuid
may take place in the shroud-suction region.

The ﬂow characteristics revealed by secondary ﬂows have been experimentally

conﬁrmed, as will be seen in section 3.1.5.

3.1.4. Effect of Rotation and Streamline Curvature on Boundary Layer Stability

Rotation and curvature are responsible for the stability of the boundary layer. The
boundary layer on the suction Side is stabilized by Coriolis forces while the one on the
pressure side is destabilized. To understand qualitatively this phenomenon, let us
consider a particle in the blade-tO-blade plane that is in equilibrium between the pressure

gradient and the Coriolis forces in a radial impeller with straight blades.

32

Boundary layer

 

Figure 3.6 Simplified representation of the ﬂow between two radial blades
Figure 3.6 represents a ﬂuid particle in equilibrium in the center of the blade

passage. Assume that a particle near the pressure side is perturbed and moved to an
adjacent layer on the left while retaining its original velocity. Its velocity will then be
lower than the velocity of a particle in equilibrium, and the Coriolis force applied to this
particle will not be able to balance the pressure gradient. The particle will therefore move
towards the suction side. For a particle moved out of the boundary layer on the suction
side, the pressure gradient will send it back to the suction side. We see that, qualitatively,
all low momentum particles will accumulate on the suction side.

The boundary layer on a convex plane is stabilized (i.e. laminarized) by
centrifugal forces, whereas the one on a concave plane is destabilized (i.e. more
turbulent). The laminarized boundary layer is more likely to separate under the action of

the adverse pressure gradient than the one at the hub.

33

The same reasoning would explain the stabilization/destabilization in the
meridional plane due to centrifugal forces, and therefore, why low momentum ﬂuid may

accumulate on the shroud side. Figure 3.7 summarizes the different possible cases.

 

 

 

 

 

 

 

 

STABLE UNSTABLE
0
Y y T
R<0
U U
WALL
CURVATURE / \
CONVEX nso CONCAVE
y 41 y I
SYSTEM U U
ROTATION
mo (50

 

 

 

 

Figure 3.7 Definition of stable and unstable cases (after Johnston and Eide 1976)

Two non-dimensional numbers can be used to express the Sign and magnitude of
each effect. They are the gradient Richardson numbers given by

Ric = 2(%)/[%Vl and R1,, = 294%]

/ /

Ri>0 corresponds to a destabilization (i.e. a reduction in turbulent kinetic energy
compared to the normal state of ﬂow with no rotation and curvature, Ri=0). Ri<0 has the

opposite effect.

3.1.5. Experimental Observation

Quantitative information at the impeller exit is very difﬁcult to obtain.
Measurements with classical probes like Pitot tubes are only approximate due to their

intrusive character (especially in narrow passages) and to the unsteadiness of the ﬂow. It

34

is only since the mid-seventies that laser velocimetry measurements have been performed
in a high-Speed impeller beginning with Eckardt (1976). Figure 3.8 shows the optical

measurement planes (denoted I—V) where the velocities have been measured.

 

 

THROTTLE
111140 6 _ ,

 

000*
I

 

 

9
Id
i— VMLESS DIFFUSH G]

+3
(I0
C
I
"’o
la

 

 

a? "-7

A A

IMPELLER

 

 

I-OPIMEAS

1- $1.955 __.__l_._...

 

 

 

Figure 3.8 Optical measurement planes (Eckardt 1976)
Figure 3.9 shows for the radial impeller at its design point the evolution of the

meridional velocity distribution (referred to the exit peripheral speed) at ﬁve different
stations from impeller inlet to exit. For the sake of clarity, each measurement section
along the blade has been represented as a trapezoid in which the blade spacing and width
are equally spaced. From station I to H, the ﬂow follows the blade and develops as
predicted by the inviscid theory. Starting at station III, a ﬁrst distortion can be noticed in
the shroud stream-tube. The deﬁcit in velocity develops rapidly in section IV and V. The
area in the shroud suction Side comer, often referred to as “wake”, is characterized by
- a low mass-ﬂow component of the order of 15% of the total mass ﬂow

- a high-ﬂuctuation intensity (up to four times higher than in the main stream)

35

- a steep, relatively stable velocity gradient to the surrounding ﬂow characteristic

of the separated ﬂow.

These measurements conﬁrmed an earlier impeller model proposed by Dean and
Senoo (1960) stating that the ﬂow does not remain attached to the blade and is in fact
composed of two regions: one at high speed (jet) and one of low-momentum ﬂuid
(wake). The suction-side separation is an accumulation of low energy ﬂuid and does not

show any return ﬂow.

36

 

 

 

 

   

 

' 01
I: /
I. , as
‘ o

 

 

 

 

 

 

'5‘

g :3 oz- '
3. 32 1 / / n: 1%
E
u u / 01

0 0 510.000

o m In as or on

PS I" —. 55

 

 

 

 

 

 

 

 

 

® , I I
I
l l
‘1“ | |
. I l
1' .
Q“ 'I
.. . l
3 1
s. ~/
2‘ ” S
.0 m
o o C1 9’ ‘5 ‘7 O 1.0
PS v" --~* 55

Figure 3.9 Non-dimensional velocity distribution at ﬁve stations along the impeller
(Eckardt 1976)

 

37

Further studies showed that in a backswept impeller, the curvature of the blades
minimizes the jet/wake phenomenon (Eckardt 1980 or Farge and Johnson 1990). Eckardt
(1980) measured the ﬂow in an impeller with 30° backsweep. Figure 3.10 shows the
meridional velocity at measurement planes IV and V. Compared to Figure 3.9, it can be
seen that the jet/wake pattern was also present but appeared to be weaker. Krain (1988)
conﬁrmed this trend. Better conditions were provided at the entrance of the diffuser,

therefore improving the performance of the diffuser.

1V

   
    

 

 

 

 

 

 

 

 

 

    

   

         

r
051

l ' 1 : I A

5' 0‘ '1 | 1 ' A 1.0
l or. 2 l. I as

N. I . /

i .1 a:
5' “ or "
a. I / 0.5
p'
'i ‘1’

a 01 q, q; g, Q! ‘9 snow 0 01 OJ as Q? as 1.0 ‘ motto
PS 1" —-0- 55

Figure 3.10 Normalized meridional velocity in backswept impeller (Eckardt 1980)

Johnson and Moore (1983) studied the inﬂuence of the mass-ﬂow rate on the
jet/wake structure. They measured pressure and velocity at ﬁve different stations of an
impeller similar to the one used by Eckardt. The probes were rotating with the impeller.
They found that the wake was located on the suction surface at low ﬂow rate, at the
suction surface/Shroud comer at design point, and on the shroud at higher mass-ﬂow rate.

The relative strength of the secondary ﬂow was related to the Rosby number R0 = ﬁ ,

which characterized the ratio of Coriolis to inertial forces in the relative ﬂow.
Figure 3.11 to Figure 3.13 show the rotary stagnation pressure below, at, and

above the design ﬂow rate. The regions of low p* can be seen in Figure 3.11 to Figure

38

3.13. They indicate the location of the wake. Thus, it is seen that the relative magnitude

of secondary ﬂows due to curvature and rotation govern the position of the wake.

 

-.
C}

 

 

'suciiou '

 

 

 

Figure 3.11 Rotary stagnation pressure at impeller exit, below design point (Johnson
and Moore 1983)

 

 

 

Figure 3.12 Rotary stagnation pressure at impeller exit, at design point (Johnson
and Moore 1983)

 

 

 

 

Figure 3.13 Rotary stagnation pressure at impeller exit, above the design point
(Johnson and Moore 1983)

39

Casey et al. (1992) investigated numerically the ﬂow ﬁeld on an impeller typical
of those found in multistage centrifugal machine and for which detailed experimental
data were available. They used two 3-D viscous codes, the Denton LOSSBD code and the
Dawes BTOB3D code. Both codes overestimated the static pressure rise, especially at
low mass-ﬂow rate; but they were overall in very good agreement with the
measurements. Concerning the impeller ﬂow ﬁeld, the magnitude of the exit wake at the
impeller outlet was well predicted although its position was closer to the suction Side than
its measured position. The calculated secondary ﬂows showed the tendency of the ﬂow
on the blade surfaces to move towards the shroud. The main stream moved slightly
toward the hub in order to compensate for the shift in mass ﬂow in the surface boundary
layers. This pattern was very Similar to the one developing in rotating axial-to-radial bend
(Johnson 1978) where secondary ﬂows are dominated by centrifugal forces (Figure 3.5).
CFD was therefore able to provide detailed information in regions where accurate
measurements are very difﬁcult to make.

Hathaway et al. (1993) measured the ﬂow in a low-speed centrifugal impeller
experimental facility. They used laser anemometry to measure the three-dimensional
velocity ﬁeld. Their optical measurement data were asserted by means of 5-hole probes,
hot-wire anemometry, and surface ﬂow visualization. The Dawes code was used to
analyze the impeller. The movement of the low momentum ﬂuids near the blade surface
towards the tip of the blade predicted by the CFD was conﬁrmed by the laser
measurements. This showed the capacity of the CFD to capture the ﬂow physics. These

results obtained on a low-speed impeller were shown to be in qualitative agreement with

40

the measurement performed by Krain (1988) on a subsonic high-speed 4:] pressure ratio

impeller.

3.2. Losses in a Centrifugal Compressor

The losses in a centrifugal compressor can be classiﬁed into two categories:
internal and external. The internal losses are those due to incidence, diffusion, friction,

blade loading, wake mixing. The external losses are disk friction losses and seal losses.

3.2.1. Internal Losses

3.2.1.1. Incidence

There is only one operating point at which the ﬂow will move smoothly into the
blade. Away from this point, there exists a difference between the ﬂow angle and the
blade angle. There is generally a range of incidence for which the losses are minimum.

Figure 3.14 Shows the losses with respect to incidence for a two-dimensional cascade as

reported by Lieblein (1965).

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i=3

.- . 4

.9 2 (c)

3:

0

O

0

m .I6

3 ‘18
a)

h .7

3 08 '6 R \
(I)

g -4 5\

a

E o

,2 -15 '10 -5 o 5 10 IS

Incidence angle, i, deg.

Figure 3.14 Effect of inlet Mach number on loss coefficient mouble circular arc
blade cascade) Lieblein 1965

Note that the optimum incidence is a function of the Mach number and has a

tendency to increase as the Mach number increases.

3.2.1.2. Friction Losses
The viscous boundary layers are responsible for the friction losses. In a ﬁrst
approximation, the expression developed for pipe ﬂow can be used to estimate the loss in

total pressure due to friction. The classical expression is

APO =4cfi-w2
p 20

 

where Apo is the loss in total pressure, p iS the density, L is the passage length, D is the
hydraulic diameter, W is the velocity, and Cf is the friction coefﬁcient

The main problem is to determine the value of Cf. For pipe ﬂow, the Moody chart
is used to ﬁnd the friction coefﬁcient as a function of the Reynolds number and the
relative roughness. Various expressions have been proposed in the literature and are not
necessarily more accurate.

The interest of this relation is to Show the importance of

42

0 decelerating the relative ﬂow as soon as possible
0 maintaining the ﬂow passage as short as possible
0 keeping the hydraulic diameter of the passage as large as possible
Maintaining the passage as smooth as possible has also been shown to be beneﬁcial,

especially in narrow passages.

3.2.1.3. Diffusion and Blade Loading Losses

The boundary layers in compressors are subject to an adverse pressure gradient.
They are therefore more likely to separate as the ﬂow develops along the blade. In order
to prevent or to limit the separation of the ﬂow, the diffusion of the ﬂow must be limited.

A criterion commonly used to measure the diffusion level is

D ___ Wmax: W2
W

A blade loading coefﬁcient is used to quantify the amount of secondary ﬂow. It is

deﬁned by

W—W

BL = 55 ps

w” +wpt

2

 

The optimization of the velocity distribution and hence of the blade loading on a blade is
a difﬁcult and controversial issue. Dallenbach (1961) proposed a set of guidelines to
apply when designing a conventional radial impeller based on some aerodynamic
considerations on an airplane wing.

From boundary layer calculations on an airplane wing with various constant
velocity gradients, it has been shown that “when decelerating at a slow rate, the boundary

layer displacement thickness is considerably larger than when the same ﬁnal velocity is

43

reached at a fast rate of deceleration. Since a thick boundary layer is unstable, particularly
for entry into the diffuser, an initial rapid rate of deceleration seems to be desirable”
(Dallenbach 1961). Thus, the preferred condition is a rapid deceleration of the mean
velocity at the shroud followed by a slower rate near the outlet. This will lead to a
decrease in the velocity at the hub that could generate reversed ﬂow at the hub depending
on the rate of deceleration used and the meridional shape of the impeller (curvature).

Also, it is worth mentioning that, as the friction losses are proportional to the
square of the relative velocity, one Should reduce the velocity as soon as possible.

AS the boundary layer gets rapidly thinner on the pressure side due to secondary
ﬂows, a rapid deceleration should not be a problem (Dallenbach 1961). The ﬂow inside
the impeller will generally separate on the suction Side when it gets closer to the impeller
exit, limiting the pressure rise. This jet-wake ﬂow pattern (in which no further pressure
rise is achived by diffusion) will be established quickly if the blade to blade pressure
difference inside the channel is large. Consequently, a light loading should be used close
to the impeller exit. To achieve the desired pressure ratio, the higher loading should then
be applied before the expected separation point. As the boundary layer gets bigger and
more sensitive to separation as the ﬂuid moves towards the impeller outlet, the higher
loading should be applied shortly after the inlet to avoid boundary layer separation and
together with a reduction of the mean velocity to maintain the peak velocity on the
suction side at the minimum possible value. The maximum loading should not exceed a

value around 0.65.

3.2.1.4. Separation Losses

As mentioned in the previous section, high friction losses will take place as the
boundary layers develop along the blade passages. The boundary layer may even
separate. No further pressure rise will be achieved in the separated area, and higher losses
are expected.

Nevertheless, some designers (e.g. Dean 1974) consider that a high-performance
impeller will have a separated zone, which must, therefore, be modeled into the design

process. The two-zone model presented in Chapter 4 uses this approach.

3.2.1.5. Secondary Flows

Secondary ﬂows in centrifugal compressors have been discussed in section 3.1.3.
They contribute to the deterioration of the performance by different means
(Lakshminarayana 1996). They generate cross-ﬂow velocities enhancing the three
dimensiOnality of the ﬂow ﬁeld. They contribute to the destabilization of the ﬂow, hence
eventually promoting the separation of the ﬂow near the shroud-suction Side region. They

also affect the turning of the ﬂow and in turn the associated pressure rise in the machine.

3.2.2. External Losses

3.2.2.1. Disk Friction
The torque required to rotate the impeller increases due to the presence of ﬂow
between the impeller disk and the non-rotating casing. Daily and Nece (1960) have

investigated the torque required to rotate a smooth plane disk into an enclosed chamber.

45

Their results give a correlation between the torque coefﬁcient, the relative roughness, and

the Reynolds number calculated at impeller exit

0.1
37%] Re“ for Re < 3*105
r2

 

r2

O.l
0.102(3) Re“?- for Re > 3 *105

where Re=U2r3/V
The head loss due to disk friction is then expressed as

_ C C
AqDF = kpUzrz2 '7:— z kU2r22 A

where k is a coefﬁcient between 0.13 and 0.25, depending on the author.

In a multi-stage machine, the volume ﬂow rate decreases as the ﬂow passes
through the machine. The disk friction losses are therefore higher in the last stages, i.e. in
the low-ﬂow coefﬁcient stages. At a design level, it can only be recommended to

maintain the disk surface as smooth as possible.

3.2.2.2. Leakage Losses

Leakage losses represent the head loss due to the ﬂow passing through the
running clearance between the rotating element and the stationary casing part. It takes
place between the casing and the impeller at the impeller eye, between two consecutive

stages in a multistage machine.

46

3.3. One-Dimensional Design Parameters

3.3.1. Overall View of the 1-D Design Parameter

Figure 3.15 Shows the main geometrical parameters deﬁning the impeller:

b2

 

 

 

 

Blade leading edge

 

 

 

 

 

Figure 3.15 Impeller main geometrical parameters

0 hub radius R”,
1- shroud (or tip) radius R],
0 blade angle distribution from inlet to exit and from hub to Shroud
0 impeller exit radius R;
0 impeller exit width b2
0 blade number Z
0 impeller axial length Lax
The main aerodynamic parameters are:

0 velocity ratio W2/W1

47

0 exit ﬂow angle
0 impeller exit Mach number
0 choice of a slip factor for performance prediction
Table 1 shows some of the conventional limits on the inducer and impeller

parameters given by Van den Braembussche (1987).

Table 3.1 Summary of the limitation on some main 1D parameters

 

 

 

 

 

 

 

 

 

 

Parameters Critical range
0.5<R1./R2<0.8 Low values leads to long channel (friction)
/ High values gives unsatisfactory meridional contour
bz/Rz Smaller values imply high friction and clearance
losses
0.3<R1h/R1,<O.5 Balance between blockage and high inlet velocities
Bn<70° To avoid blockage and high turning in the inducer
65°<Ot2<80° To avoid too tangential velocities at impeller exit
W2/W1>0.6 Smaller values will lead to boundary layer separation

 

Some of the design parameters will now be presented with respect to the

development of low-ﬂow coefﬁcient compressors.

3.3.2. Inducer Hub and Tip Radius

The hub and tip radius of the inducer are determined as a function of the mass
ﬂow rate. Once the hub ratio is ﬁxed, the tip radius is detennined in order to minimize
the Mach number at the tip. It will be seen in Chapter 4 that there always exists an
optimum value of R1,.

The hub radius is determined most of the time by rotordynamics considerations.
The effect of hub-to-outlet diameter ratio on the performance of a low-ﬂow coefﬁcient
Stage has been investigated by Paroubek et al. (1994). Three stage-conﬁgurations with a

ﬂow coefﬁcient of 0.007 (A, B, and C) and three with a ﬂow coefﬁcient of 0.021 (D, E,

48

and F) have been tested. The A, B, and C impellers have the same exit width but a
different inlet hub radius. Their meridional shape, blade geometry, and blade number
have been chosen to maintain similar aerodynamic performance between them. The D, E,
and F impellers have been derived from the A, B, and C impellers by increasing the blade

width. All impellers have 60° backward-swept angle.

A, D (Z=13) B, E (2:11) C, F (2:9)

DZ

“— / /
011 ll
01 /
D111

 

 

 

 

 

 

 

 

Figure 3.16 Impeller main dimension (after Paroubek et al. 1994)
Figure 3.17 shows the impeller efﬁciency for the tested stages. The hub-to-exit

diameter ratio was found to have a stronger effect on the narrow impeller than on the
wide stages. The impellers with the smallest hub diameter had the best performance: for
the same relative outlet ﬂow capacity, the deceleration is higher (i.e W2/Wu is lower) for
impellers with greater hub diameters while it must be achieved on a shorter meridional

COUIOUI‘.

L
kn.

49

 

 

 

 

 
   

1.5' .
T .. -_. stage
71"“ r-"T. 91-! + A.D
.‘FE I' ' J ,3 at 8E
T ; . o
1.ai_ . . . . . :3; 1; . C» ..
W: '

 

IIIIIII

 

MUZ 1’ 0.9

I'l‘l'l‘l‘

 

 

 

 

 

L I A L4 1 A l L Lgl L l n

 

1.8

in

 

a Dynamic stability
limit "

Figure 3.17 Impellers efﬁciency (Paroubek et al. 1994)

3.3.3. Exit Width

The exit width is determined based on the volume-ﬂow rate passing at impeller
exit. Based on the estimated density and the required head, the corresponding width can
be determined with the continuity equation.

The effect of impeller width on the performance of low-ﬂow coefﬁcient stages
has been studied by Paroubek et. al. (1994). The performances of the wider impellers
(Figure 3.17) were, as expected, higher than the performances of the narrower ones due
to the reduced friction losses. At high mass ﬂow, the efﬁciency decreases faster for the
wider stages. Their operating range is much smaller than the operating range of the

narrow stages.

50

Casey (1990) suggested increasing the impeller width for a given impeller (the
exit tip radius is kept identical) while adjusting the exit blade angle in order to keep the
head rise capacity. The velocity levels are lower due to the larger passage area. The
reduction of the friction losses due to the lower velocity is counterbalanced by the
increase in blade length. The improvement of the performance is greater for impellers

with lower ﬂow coefﬁcient.

3.3.4. Velocity Ratio

The maximum deceleration is an important parameter that can be correlated to
boundary layer separation. The limiting value for W2/Wlmm varies between the authors

but generally ranges between 0.5 and 0.7.

3.3.5. Slip Factor

The slip factor is estimated from empirical correlation or theoretical calculation.
The main correlations have been developed as a function of blade number and exit blade
angle. The slip factor is a function of many more parameters and should be considered as
a one-dimensional approach.

The slip factor depends on the mass ﬂow rate. Eckardt (1980) showed for his
radial discharge impeller that the slip factor increased for decreasing mass ﬂow rate. For
his backward impeller, an opposite trend was found. Figure 3.18 Shows the measured Slip

factor by Paroubek et al. The slip factor is almost constant around 0.81-0.82 for the

51

narrow stages. For the wider Stages (D, E, F) the slip factor increases for decreasing mass

ﬂow rate Similarly to the radial discharge impeller of Eckardt.

 

 

 

 

 

 

 

 

 

 

1.00
[I
0.90
g 0.00
2
.2
F 0.70 71::— \
a,“ \\\
u:qu A.D
0.00 .9
..F \ 8,5
e.
0 Cf

 

0.50 ~ - s - E -
0.50 1.00 1.50 ‘fnr 2.00

flow coeliicienl

* AD 0 B.E ° C.F

Figure 3.18 Slip factors (Paroubek et al. 1994)

It does not seem possible to establish any design rule for the choice of the slip
factor. Nevertheless, the low slip factor values (around 0.8) obtained by Paroubek on his

highly backward-swept impeller should be kept in mind.

52

4. ONE-D COMPRESSOR MODELING AND DESIGN

PARAMETERS

A one-dimensional method and a streamline curvature throughﬂow method have
been used to design the impellers. Both methods have been adapted from the open
literature and are described in detail in Appendices B and C. This chapter presents a
parametric study to Obtain the geometry of ﬁve impellers satisfying the design
constraints. Each impeller must perform the same duty but has a different outlet angle-
outlet width combination. The one-dimensional analysis and the determination of the

geometry are presented hereafter. The CFD results will be discussed in Chapter 6.

4.1. Description of the 1-D Method

Various methods exist to design a centrifugal impeller. A dimensionless approach
based on the established relations between speciﬁc speed, speciﬁc diameter and
efﬁciency like the approach of Balje (1980) can be used for the preliminary sizing of the
impeller. Other types of approach are the ones described by Galvas (1973) or Aungier
(1995). These methods rely on the equations of ﬂuid dynamics onto which loss models

are added to describe each type of loss (incidence, friction, etc.). A third approach

53

proposed by Dean (1974) tries to reduce the number of correlations by introducing a
diffusion level parameter to characterize the performance of the impeller.

All of these methods require the introduction of the some empirical correlations.
They may therefore be more adapted to one type of impeller or another, depending on the
origin of the experimental data. The two-zone model was chosen hereafter because some
analysis had already been done with this model on the existing stages of the impeller

family.

4.1.1. Two-zone Model

The two-zone model (Japikse 1985) is an evolution of the jet-wake model
proposed by Dean and Senoo (1960) and Dean (1974), and schematically shown in
Figure 3.1. Japikse, after reviewing the basic assumptions and the derivation of the
equations governing the jet and the wake, used the term two-zone model to personalize
his approach. The author assumes that:

(1) an isentropic core zone and a non isentropic secondary zone coexist,

(2) the diffusion of the core ﬂow and development of the second zone is Similar to other
diffuser problems

(3) the tip static pressure may be assumed the same for each zone or corrected by a small
factor,

(4) the deviation of the second zone has a small effect on modeling; deviation of the
primary zone is critical in setting overall slip,

(5) a mixed-out State can be computed at the impeller exit for thermodynamic State

evaluation and for determining average conditions.

54

Assumption (1) is fundamental. It is based on earlier experimental observations
showing that these two regions co-exist. The jet being isentropic, all the losses can only
originate in the wake. Experimental works (e.g. Rothe and Jonston 1976) in rotating
diffusers have shown similarities with the ﬂow in centrifugal impellers and have led to
assumption (2). Assumption (3) is based on the condition that the pressure side and the
suction side velocities must be equal at impeller exit (the Kutta condition). The
experimental observations at the impeller exit showed the larger extent of the primary
zone and justiﬁed assumption (4). Assumption (5) allows one to compute a mixed-out
state at the impeller tip, even if in reality the mixing between the two zones will be
accomplished further downstream.

The equations governing the two zones and the mixed-out conditions are
presented in Appendix B. The two-zone model has been used to represent the ﬂow in the
low-ﬂow coefﬁcient compressors presented in this study. Its validity will be shown in

Chapter 6.

4.1.2. Impeller Diffusion Level

The diffusion of the relative ﬂow that can be achieved without separation is
limited. Among all the parameters that can be used to describe this limit, the impeller

Mach number ratio MR2 introduced by Dean (1974) is used. It is deﬁned by

MWlt

 

MR, =
M

W2primary zone
i.e. as the ratio of the relative Mach number at inducer tip over the relative Mach number

of the primary zone (jet) at impeller exit.

55

An ideal Mach number ratio, which intends to account for the geometry of the

impeller channel is deﬁned by

MWlt

 

MR2i =
M

W2primary zonc.idcal

i.e. the ratio of the relative Mach number at the inducer tip over the relative Mach number
of the primary zone at the impeller exit in an ideal state assuming no blockage, no
friction, and no deviation.

AS stated by Dean, “MR2 indicates the actual Mach number reduction the impeller
achieves in its cover (or shroud) stream tube (which is the critical one)” MR2 and MR2,
have been correlated to obtain a state-of-the-art curve. A ﬁrst general correlation between
MR2 and MR2i was proposed by Benvenutti (1978) for three-dimensional impellers and
for industrial two-dimensional impellers (Figure 4.1).

2

 

 

 

SDlmpel rs

 

MR2

 

1.4 /

ﬂame My (20)

 

 

 

 

 

 

 

Figure 4.1 Comparison of achievable primary ﬂow diffusion in industrial 2D
impellers and in straight inlet, 3D impellers (Benvenutti 1978)

56

At a preliminary stage, MR2 can be set to the value corresponding to the state-of-
the-art. A more recent correlation proposed by Japikse (1996) was used hereafter. It will

be referred to it as the “state-of-the-art” or SOA.

4.1.3. Off-Design Performance

The TEIS model (Two-Elements-In-Series) was introduced by Japikse (1984) to
describe the performance of an impeller and predict the overall map of a machine. In this
model, the impeller is viewed as two diffusers in series: the ﬁrst one extends from the
inlet to the throat and acts as a diffuser or a nozzle depending on the ﬂow conditions; the

second one, from the throat to exit, is the impeller passage (Figure 4.2).

Diffuser "b"

Diffuser "a"

~
“
“
‘

 

 
 

 
 

' Ain (low
mass-flow

rate)

 

Actual element “b”

Actual element "a"

Figure 4.2 TEIS model - Diffusing passages (after Japikse 1984)

57

An effectiveness is used to characterize the diffusing ability of each passage. It is
the ratio Of actual pressure recovery to the ideal pressure recovery obtained for an
inviscid ﬂow.

The parameter T], represents the effectiveness of the impeller inlet portion, which

can be considered as a diffuser or a nozzle depending on the incidence. In the following
equation, Cp is the pressure recovery between the inlet and the throat, and AR is the ratio
of the throat area to the inlet area. Cpa is the actual pressure recovery taking place
between inlet and exit. Cpaj the ideal pressure recovery that would take place based only

on the geometrical area ratio.

 

2
C
Ila z—Ka—WhereCpa'. :1.— 12 =1— COSBI
Cpaj . AR“ COS B11)

At high mass ﬂow, the area ratio ARa =A..JA,,, is larger than 1, and Cpay is negative. At
low mass ﬂow rate, ARa is lower than 1, and Cpaj is positive. Therefore an increase of 1],,
makes Cpa more negative at high mass ﬂow rate and reduces the diffusion, whereas it has
an opposite effect at low mass ﬂow rate where Cpa is made more positive.

110 represents the effectiveness of the passage portion. Cpb represents the pressure

recovery between the throat and the exit.

 

 

2
A
11b 2 Cp” whereCpb, =1— 12 =1— ——’"—
Cpbj ‘ ARb
Cpb'j is only function of the geometry and not of the ﬂow condition. Therefore, 11., will
affect the impeller diffusion over the whole operating range.

From '112, and 1’10. the diffusion ratio DR=Wn/W2p can be calculated (Appendix B),

and all the subsequent aero-thermodynamics variables.

58

A maximum diffusio ratio DRmax (or DRsml) is used to limit by a reasonable
physical value, the maximum diffusion achieved at low mass-ﬂow rates in the impeller.
It sets the diffusion to a constant value for mass ﬂow rate below the limiting point.

Once the values of these three parameters have been determined, the performance

of the impeller at various rotational speeds and various mass ﬂow rates can be predicted.

4.2. One-Dimensional Analysis

A one-dimensional Fortran code has been developed to solve the equations
describing the two-zone model of Japikse (1985) for the case of a perfect gas. The ﬂow in
the vaneless diffuser is calculated following the approach of Johnston and Dean (1966).

In this chapter, the code has been applied to design ﬁve impellers with different
outlet blade angles and outlet widths fulﬁlling the design requirement. All the designed
impellers have the same inlet ﬂow coefﬁcient. The objective was to investigate the effect

of the exit blade angle on the impeller ﬂow ﬁeld and on the impeller performance.

4.2.]. Design Constraints

The exit conditions of an existing stage provide the inlet conditions (total
temperature and total pressure) of the stage to be designed. The impellers must pass the
required mass ﬂow at the design rotational speed and provide an increase of head
between 7,000 and 8,000 ft-lbf/lbm, which corresponds to a pressure ratio around 1.18 at
speciﬁed inlet conditions. Geometrically, the hub diameter, exit diameter, and impeller

axial length are ﬁxed.

59

4.2.2. Modeling Assumptions

When solving the system of equations for the two-zone model and mixed-out
conditions, a few parameters have to be assumed in order to equate the number of
unknowns with number of equations:

(1) The slip factor is set by the Wiesner correlation. As it was observed that these values
were generally high, a value 0.03 lower than the correlation was used.

(2) The impeller Mach number ratio was set by the state—of-the-art curve.

(3) The ratio x Of the mass contained in the secondary zone to the total mass ﬂow was set
to 0.15, which is in the range of the recommended values.

(4) The inlet blockage and inlet distortion were ﬁrst guessed, and then reevaluated with
the help of CFD (Appendix E). A blockage of 5% and a hub-to-shroud distortion of 1.3
were used.

This ensures the closure of the system of equations and allows the calculation of
all the thermodynamic conditions at inlet and outlet. The validity of assumption (2) and

(3) is studied in the next section.

4.2.3. Influence of the Modeling Assumptions on the Choice of the Geometry

The inﬂuence of the secondary ﬂow mass fraction x=9fhﬁ and of the Mach
number ratio MR2 on the performance prediction has been investigated.
The two-zone model recommends using a value of x between 0.02 and 0.30.

Considering the AA impeller conﬁguration, whose geometry will be given in Chapter 5,

60

the value of x has been varied between these two extremes for various values of MR2 but
at a constant slip factor. As the geometry is ﬁxed, MR2 sets the diffusion into the primary
zone.

Figure 4.3 shows the effect of x and MR2 on the area ratio 8 between the
secondary zone and the outlet area. At low values of MR2, the wake area is nearly
independent of the secondary zone mass fraction. For a ﬁxed value of MR2, greater than
1.0, diffusion occurs into the jet; and as expected, the wake area increases as the assumed
mass ﬂow into the wake increases (increasing values of x). For a given x, the velocity in
the wake increases as MR2 increases; and because the mass ﬂow in the wake is constant,
the area of the wake must reduced. It will be seen in Figure 4.5 that the range of values
covered by x and MR2 is limited. In particular, the lower left end of the curves MR2=1.2

and MR2=SOA corresponds to 0 aerodynamic blockage and Should not be considered.

 

 

 

 

0.6
E
05 4 b‘.\\‘\\‘
0.4 ~ +MR2=1.0 l
\/ ‘ +MR2=1,1
0.3 « .
+MR2=1.2
0.2 . —+—MR2=SOA '
0.1 «
O T I I I I I
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 4.3 Effect of x and MR2 on the wake width

61

 

 

 

 

 

 

 

1.25

MR2

Figure 4.4 Effect of x and MR2 on v=W7.,/W2p

Figure 4.4 shows how MR2 and x influence the velocity ratio between the jet and
the wake. For small values of x, the velocity of the wake remains small and does not
exceed 20% of the jet velocity. x has a large inﬂuence on v for MR2 larger than 1.0. As
more diffusion occurs in the jet, the velocities between the two zones get closer,
especially for large values of x. Values of MR2 greater than 1.2 should not be considered
as is explained below.

Figure 4.5 shows the effect of MR2 and x on C=W2m/W2p and on the deviation
into the primary zone. C is an indication of the amount of aerodynamic blockage at
impeller exit. C varies linearly with MR2 because the velocity in the primary zone is
directly proportional to MR2 and independent of x. The mixed-out velocity W2,n is only
slightly inﬂuenced by x. As the diffusion of the primary zone increases, the relative

velocity in the wake increases, affecting in turn the deviation in the primary zone. It is

62

assumed that the secondary zone follows the blade. The deviation of the mixed-out
conditions is ﬁxed by the slip factor. Therefore, the deviation of the primary zone is
adjusted to satisfy the continuity equation. In reality, the primary zone will contain most
of the mass flow and will deviate from the blade by a few degrees, reducing the work
input. Hence, all values of 82,, above —5° should be excluded. The lower horizontal line in
the ﬁgure indicates the values of x and MR2 corresponding to a deviation of the primary
zone by 5°: this corresponds to values of MR2 greater than 1 for x=0.3, and greater than
1.15 for 1:0.02. Also, the velocity ratio between the secondary and the primary velocity
should remain lower than 1.0 to satisfy the initial assumption of a high speed zone (jet)
and a low-momentum zone (wake). Otherwise, most of the mass ﬂow would be passing
into the wake, and there would be hardly any jet left anymore. Furthermore, at the exit of
the impeller there should be a minimum of 10% of aerodynamic blockage (Pampreen
1981). Only values of C smaller than 0.9 should therefore be considered. The upper
horizontal line in the ﬁgure indicates the values of x and MR2 corresponding to a
blockage of 10%. An upper limit for MR2 can then be set around 1.13 (with the value of
the assumed slip factor). This maximum value is the same for all the range of values

covered by x because C is independent of MR2.

63

 

 

 

 

 

 

 

 

1.10 30
. . 52»
MR2 limit to have . 25
1 00 j at least 10% blockage
0.90 3:33 » 15.___,_-z-_ _.
' -x=0.02
I * 101 —x=0-05 !
98°: ,5 41.0.1 1
1 l +x=0.15
070: \ t . 0 1 +x=0-2 1
k x 59"5 5? +14)” 1
050 :\\\:§:§\ 52:55 - ‘I:X=O3s
1 -10
0.50 I T v v r v -15
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

M82

Figure 4.5 Effect of x and MR2 on C=W;.../W2p and on 82,,

1.00 - AA
0.98 -
0.96 ;
|
0.94 l
2 0.92 - 1,. -— -
.§ 0.90 ’ +x=0-05
2 . +x=0.l .
+x=0.2
0.86 +X=0-25
0.84 4 +Xﬂ'§_
0.82 4 i
0.50 AA- -- . ., -A-A ~ ~ A - AA —--A A
1 1.05 1.1 MIR 1.2 1.25 1 3

Figure 4.6 Effect of x and MR2 on the impeller efficiency

'01

64

Figure 4.6 shows the effect of MR2 and x on the impeller efﬁciency. As MR2
increases, the relative area of the wake decreases. Because all the losses are generated
into the wake and into the mixing process between the jet and the wake, the impeller
efﬁciency approaches 1.0. Consequently, the pressure ratio in the impeller increases with
MR2 as shown in Figure 4.7. The effect of x is small in the useful range of MR2 (for a
constant slip factor). The cases corresponding to MR2 values greater than 1.2 are

unrealistic for the given geometry as they lead to very small aerodynamic blockage.

 

1.21 ~

 

6 1 j /” 5 l +x=0.52~'
E ' 3 +x=0.05
1.19 3 . ' +x=°J°

‘ +x=0.15

+x=0.20

‘ -°- 1:0.25
_ “r 159-.39-

L_-n1A

 

 

 

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35
MR2

Figure 4.7 Effect of x and MR2 on the impeller pressure ratio

These systematic calculations have shown that MR2 should be chosen between 1
and 1.15 for the chosen slip factor. The x ratio should be kept between 15 and 25%.
When calculating with a constant slip factor, the deviation of the primary zone should lie

in the expected range of -5 to —10°. The implication of these results on the choice of the

TEIS model will be discussed in Chapter 6.

65

4.2.4. l-D Impeller Design

4.2.4.1. Inducer
The objective of the inducer is to guide the ﬂow from axial to radial, increasing its
angular momentum. A designer will draw particular attention towards:
0 minimizing the inlet relative Mach number (keep it subsonic)
0 choosing the inlet blade angle in order to get a slightly positive incidence at design
point (Rodgers, 1964)
0 checking the throat area to ensure the passage of the required mass flow rate at design
Figure 4.8 shows the calculated inlet tip relative Mach number Mwlt for various
inducer tip-to—hub diameter ratios. For a small value of the inducer tip radius, the axial
component of velocity is large and the tip peripheral speed is low. As the tip radius is
increased, the area increases, hence reducing the axial velocity but increasing the inducer
tip peripheral speed. Therefore an optimum radius ratio always exists, where both effects
compensate and for which Mwlt is minimum.
Once the velocity triangles are obtained at hub, mean, and shroud, the blade angle
can be set to obtained the desired incidence which was chosen to be slightly positive at

hub, mean, and shroud.

0.46

66

 

Hm!

0.44 ..

0.42 1

0.4 <

0.36 --

0.34 ~

0.32 1

 

0.3

 

 

 

1.15

1.2

1.25

R1VFI1h

1.3

1.35

Cm1

U11
W1t

Small R1t/R1 h

Cm1

U11

W11

 

1
Large R1t/R1h

Figure 4.8 Inﬂuence of the hub-to-shroud inlet radius ratio on the shroud inlet

4.2.4.2.

The determination of the optimum impeller exit width and exit blade angle is
much more difﬁcult as it involves many more parameters. The following calculations
have been made for a constant slip factor assumed to be three points lower than the slip
factor of Wiesner’s correlation. The exit tip radius is ﬁxed. The mass ﬂow in the
secondary zone is kept constant at 15% of the total mass-ﬂow. The Mach number ratio is
set to the value of the state-of-the-art. By doing so, there is a minimum value of b2 that
can be used, below which the area of the secondary zone becomes smaller than 10% of
the exit area and falls out of the assumptions of the model. The width of the diffuser is set
at 75% of the impeller exit width to ensure a larger inlet ﬂow angle at the diffuser

entrance. Figure 4.9 shows the predicted stage total-to-total pressure ratio at the design

Impeller Exit Width and Exit Blade Angle

point for the 54 cases run.

relative Mach number

67

1.26

 

1.25 -
1.24 4
1.23 ~
1.22 4
1.21 -

Pan
1.2 a

B2=‘65

 

 

 

 

0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48 0.52 0.56
b2ID2

Figure 4.9 Stage total-to-total pressure ratio as a function of impeller exit width and
impeller exit blade angle

Figure 4.10 shows the exit tangential velocity, which expresses directly the work

done by the impeller.

0.75 -H, iAAA—A , 7A 2 __. ________
CuZ/UZ [
0.7 1
1
0.65 ~
132=-30° 3
0.6 - {sf-35 \
‘ l
l ‘
,1 Bz=40° B.=-45° 1
0.55 . B2=‘55°
52-14500 1
0.5 e 1
132F600 ‘
‘ i
0.45 j BF-650 / ’
‘ 1327-40
0.4 1' “A, AA AA AA - -A A - A A 7A» » A— ..
0.010 0.015 0.020 0.025 0.030 0.035 0.040
b2/D2

Figure 4.10 Impeller exit tangential velocity versus impeller width

68

A line of constant Cu2/U2 corresponds to a line of constant work input. The
choice of the geometry was based on the stage pressure ratio rather than the impeller
work input in order to include the diffuser in the design process.

The relative velocity ratio W2/Wmp is shown in Figure 4.11. It characterizes the
relative diffusion on the shroud streamtube. It remains higher than 0.6 for most cases.
When the velocity ratio becomes smaller than 0.6, boundary layer separation can be

expected and higher losses will occur.

 

 

    
  

 

 

 

 

WW1
Bz='65° BF'7OO
0.9 « _ o
0.=-so° '32-‘50 \
BF 13 _45
0.8 " 322-35" 2- B2='55°
52:.
0.7 «
(W2/W11m
0.5 4
0.4 I T I I I
0.010 0.015 0.020 0.025 0.030 0.035 0.040

Figure 4.11 Relative velocity ratio as a function of impeller exit width and impeller
exit blade angle

Figure 4.12 shows the absolute ﬂow angle at impeller exit (12.... The exit impeller
ﬂow angle should not be too tangential to avoid high losses and to maintain an acceptable
range in the vaneless diffuser. The curve representing acme.“ vs. b2/D2 is adapted from

Senoo (1978). If the ﬂow angle at the diffuser entrance is higher than the critical ﬂow

69

angle, rotating stall inside the diffuser should be expected. At the design point, the ﬂow at
the impeller exit should be at least 10° smaller than the critical ﬂow angle to ensure an

acceptable operating range.

 

 

 

 

 

85

a2m
01cm (Vaneless diffuser stall

80 . - Senoo's correlation)

75 -

70 <

65 -

60 1 1 1 1 1

0.010 0.015 0.020 0.025 0.030 0.035 0.040

Figure 4.12 Impeller exit absolute ﬂow angle vs. impeller exit width and impeller
exit blade angle

The performance prediction for the vaneless diffuser must be taken into account
when sizing the impeller. Indeed, the best compressor performance can only be achieved
if the impeller and the diffuser reach their maximum performance at the same mass-ﬂow
rate. Figure 4.13 and Figure 4.14 show the pressure recovery and the total pressure loss
coefﬁcient between diffuser inlet and diffuser exit. The method used to calculate the ﬂow
in the vaneless diffuser is presented in Appendix B. The friction coefﬁcient is determined
as a function of the diffuser inlet Reynolds number. It increases as the diffuser becomes
narrower and as the inlet velocity decreases. Hence, as the diffuser width is reduced, the
pressure recovery decreases and the loss coefﬁcient increases. One would therefore

believe that the widest diffuser should be used to maximize pressure recovery and

70

minimize the total pressure loss. It should be kept in mind that the ﬂow in a wider

diffuser will be more tangential, and therefore more likely to become unstable.

Furthermore, lower inlet ﬂow angles lead to longer ﬂow path and higher friction losses.
The pressure recovery predicted by this method is generally optimistic. A slightly

lower value should be expected in the actual machine.

0.52
Cp4-5

0.6 <

 

0.58 -

0.56 -

0.54 <

0.52 < 52:400
132=-35°
Bz='30°

 

 

 

 

0.5 1 1 1 1
0.01 0.015 0.02 0.025 0.03 0.035

Figure 4.13 Diffuser pressure recovery as a function of impeller exit width and exit
blade angle

71

 

LC 4-5
0.34 ~
0.32 1

529300

132=-35°

0.3 '* [SF-40°

0.28 « [12:450
BF-
0.26 ~

 

 

 

 

 

 

024 ~
Bf’soN - - =
0.22 « [32:55? ' #-
Bf'7oo r,“
0.2 W Y Y I
0.01 0.015 0.02 0.025 0.03 0.035
52/02

Figure 4.14 Diffuser total pressure loss coefficient as a function of impeller exit
width and exit blade angle

Based on this analysis, the geometry of five candidate impellers able to provide a
total-to-total pressure ratio around 1.19 was selected. The exit blade angles were chosen
between —50 and —700 because of the relatively low pressure ratio that needs to be
achieved. An impeller with a more radial outlet blade angle will require a too narrow
outlet width. The corresponding geometry (obtained from Figure 4.9) and some of the
performance parameters are given in Table 4.1.The head coefﬁcient, the efﬁciency, and
the pressure ratio (PR) are given for the stage (impeller and diffuser) and are based on
total-to-total conditions. The denomination used is AA, followed by the value of the
outlet blade angle. The AA70 has a smaller work input than the four other impellers, but
the expected total-to-total pressure ratio is close to 1.19 due to the reduced losses in the

diffuser.

72

Table 4.1 Geometry of the ﬁve selected impellers

 

 

 

 

 

 

 

 

 

 

 

 

 

AA50 AA55 AA60 AA65 AA70

b2 0.30 0.34 0.40 0.49 0.58

132 -50 -55 -60 ~65 —70

0' 0.84 0.845 0.85 0.86 0.87
cam/U2 0.549 0.537 0.533 0.539 0.522
Wyn/W1; 0.82 0.82 0.80 0.77 0.79
(12 66.0 68.2 71.0 74.5 76.4
82 -6l.6 -65.0 -68.6 -72.0 -75.2
CP45 0.55 0.56 0.58 0.59 0.61
LC45 0.27 0.25 0.24 0.22 0.21
(1115).. 0.825 0.835 0.838 0.83 0.841
11.. 1.198 1.195 1.194 1.195 1.189

 

 

 

 

 

 

 

 

This performance prediction at design point will be revised in Chapter 6 based on

the CFD results.

4.3. Streamline Curvature Throughﬂow Method

As seen in Chapter 3, an inviscid analysis is generally not able to predict
accurately the ﬂow in a centrifugal impeller especially near the impeller exit.
Nevertheless, it is useful to establish the initial geometry based on such a method. Many
aerodynamic criteria have been developed for inviscid ﬂow that allows the designer to
adjust the meridional contour and the blade angle distribution. A streamline curvature
code has been developed to analyze the impeller. The method requires the introduction of

relaxation factors to ensure numerical stability.

73

4.3.1. Description of the Quasi-3D Method

A streamline curvature method is an iterative procedure for obtaining a solution of
the ﬂow ﬁeld starting from an initial guess of the streamsurface shape. It assumes an
axisymmetric ﬂow. As mentioned by Denton (1978), this may be regarded as being
obtained by circumferentially averaging all ﬂow properties or by solving for the ﬂow on
a mean blade-to-blade streamsurface whose thickness and inclinations are determined by

the geometry of the blade rows. The method is described in details in Appendix C.

4.3.2. Deﬁnition of the Blade Geometry

The impeller geometry is deﬁned by the shape of the meridional contour at the
hub and at the shroud. Bezier polynomium have been used to describe the contours.
Details on this procedure can be found in Appendix C.

When designing any impeller, the blade angle distribution at the hub and the
shroud is prescribed as a function of the relative distance along the meridional contour.

The local circumferential position 0 at hub and shroud is then calculated by
9 ___ J tan ﬂ m
r d
where r is the radius along the meridional contour.
The 0 distribution (also called wrap angle distribution) can be adjust at the hub

and at the shroud by adding any constant at inlet, leading to

tan ﬁ h h 11 tan ﬂ shroud i
Bhub = (ghub entry +1 r u m and ashram! = (ashram! entry +I m

r

74

The 0 distribution then determines the lean angle distribution 7, which
characterizes the inclination of the blade (i.e. how tilted the blade is with respect to the

radial direction). Mathematically, y is expressed by

tanyzr—

8b
where b is the distance along a quasi-orthogonal. The convention adopted herein for the
sign of the lean angle assumes that the blade leans forward in the direction of rotation for
positive lean.
Letting (Gsmud)cnuy=0, and (ehub)cm_ry=(6ref), the lean angle at inlet is shown in

Figure 4.15.

Blade

V

  

 

 

Figure 4.15 Deﬁnition of the lean angle
The lean angle at the impeller exit is called the rake angle 1' (Figure 4.16). High

values of the rake angle are undesirable in narrow impellers because they will induce

more aerodynamic blockage. The entry value of the wrap angle and the blade angle must

 

75

therefore be adjusted to obtain a low lean angle. The validity of the lean angle

distribution will be determined in the end by stress analysis.

2 A
(@hub

 

Blade tip

 

 

 

“ashram
Figure 4.16 View of the impeller tip and definition of the rake angle
The blade thickness distribution normal to the blade is prescribed as a function of
the relative distance along the meridional contour. Using the local blade angle, the wrap

angle on the pressure side and on the suction side of the blade can be evaluated.

4-3.3. Impeller Blade Shape for the Five Impellers

To adjust the blade contour of the impeller and the blade angle distribution, a
commercial software available at Solar Turbines Inc. was used. The resulting geometry is
a compromise between the blade loading distribution, the lean angle distribution, the and

smoothness of the curvature distribution.

76

AA50 AA55

Figure 4.17 Impellers meridional contour

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A55
-40 -40
B B
.45 45 1
Hub Hub

-50 . I -50 1
-55 - 5m“ -55 1 Shroud 1
~60 1 .50 .

I I
55 4 -55 1
'70 Y r 7 ﬁ' ‘70 Y 1 Y r

0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00
1. meddonal dleunce 95 meddoml deuce
M65 M10

-40 -40
ﬂ 6
.45 . -45 J
-50 d “u” .50 1 Hub
.55 1 .55 4
‘60 1 .60 4

l Shroud 1 Shroud
-65 J j -65 J
'70 v r V Y '70 r 1 I I

0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00
11. Milan-l distance 96 Melon-l deﬁne.

Figure 4.18 Blade angle distribution for the AA50, AA55, AA65, and AA70

77

4.3.4. Velocity Distribution and Blade Loading for the Five Impellers

The blade angle distributions, the meridional contours, and the hub wrap angle at

entry were adjusted to satisfy acceptable velocity distributions, blade loading, and lean

angle distribution. The momentum at the exit of the blade was prescribed, based on the

assumed slip factor. The following blade-to-blade loadings were obtained. The maximum

loading was located between 50 and 70% of the meridional length, and its maximum

value was kept below 0.7. Thirteen blades were used.

0.80

M50

 

0.70 1
0.60 1
0.50 1
i 0.40 1
0.30 1
0.20 1

0.10 1

0.00

 

Hm)

 

 

0.00

0.80

0.40 0.60
96 Moral We.

 

0.70 1
0.60 1
0.50 1
0.40 1
0.30 4
0.20 1

0.10 1

Hub

Shroud

coding

 

0.00

0.00

 

0.40 0.60
95 merldloml distance

 

 

0.00
0.70
0-60 ‘ Mean
0.50 ‘
0.40 1
0.30 1
0.20 1

0.10 1

 

 

o-m T I 1 7
0.00 0.20 0.40 0.00 0.80 1 .00
96 maiden-l distance

 

0.80

 

0.70 1

0.60 1 Hub

0.40 1
0.30 ‘
0.20 1 ’

0.10 1

 

 

Cam I l V T
0.00 0.20 0.40 0.60 0.80 1 .00
96 Wood dist-lice

 

Figure 4.19 Blade-to-blade loading for the impellers
These impellers have been analyzed with CFD and will be reviewed in Chapter 6.

78

5. DESIGN AND ANALYSIS OF FOUR LOW-FLOW

COEFFICIENT COMPRESSORS

This Chapter presents the one-dimensional, the streamline curvature, and the CFD
analysis of four stages that have been designed as an extension of an existing family of
impellers. The ﬂow is analyzed qualitatively at design and at off-design conditions. The
ﬂow between the four impellers is compared to determine the effect of the blade passage

width.

5.1. Design Requirements

5.1.1. Inlet Design Conditions

In a multi-stage compressor, the exit conditions of one stage provide the inlet
conditions of the next stage. All the impellers have the same rotational speed and the
mass-ﬂow rate passing through the machine is the same for all stages.

The existing family of impellers consists of ten shrouded. The four new stages
that will be added will be denoted AA, AB, AC and AD. The AA impeller has the largest
ﬂow coefﬁcient, followed by AB, AC, and AD.

The existing family of impellers has been designed for a made-up gas

corresponding to the one used in the ﬁeld of application of the machine. The speciﬁc

 

79

gravity, the pseudo-critical pressure and temperature of the gas are given. The speciﬁc
heat, the viscosity and the compressibility can be determined as a function of pressure
and temperature.

The inlet conditions of the AA impeller are set to the reference conditions, which
are an inlet total pressure of 750 Psia (52 bars) and 580°R (322 K). An equivalent mass-
ﬂow rate and equivalent rotational speed have been calculated, which conserve the

volume ﬂow rate corresponding to the exit conditions of an A1 stage.

5.1.2. Impeller Sizing

5.1.2.1. Choice of the Outlet Blade Angle
The one-dimensional method used to design the impeller has been described in

Chapter 4. It has been shown that various combinations of exit blade angle and exit blade

width will satisfy the design requirements. For the four impellers designed as part of the

existing family, the exit blade angle was set to —60° from radial. Many reasons led to this
decision:

0 The low-ﬂow stage will be added to an existing family. It is very important to ensure
an adequate matching between all stages to ensure maximum range, head, and
efﬁciency. Keeping a similar exit blade angle will ensure a similar head-ﬂow curve.

0 The wake appears to be weaker in a high-backswept impeller as noticed by Farge and
Johnson (1990) or Chen et al. (1996). The wake-mixing losses should, therefore, be
minimized.

0 The stability of the compressor is improved by increasing the exit blade angle (see

Chapter 2)

 

 

 

80

0 With a high backward-swept angle, the absolute velocity at the diffuser entrance is
lower than the one provided by a radial impeller at an equal power input. A lower loss
takes place in the diffuser, improving the stage efﬁciency.

All this considerations lead to the design of impellers with an exit blade angle of —

60°.

5.1.2.2. Modeling Assumptions

The performances of the impeller are also affected by the distortion of the
incoming ﬂow from hub to shroud. This is an important concern for multistage machines
because each impeller (except the ﬁrst one and last one) may play the role of a front stage
or of a rear stage. For a front stage, the ﬂow is coming out of a radial inlet and is rather
uniform from hub to shroud.

In case of a middle or a rear stage, the ﬂow entering the impeller exits from a
return channel. The last turn from radial to axial and the residual swirl at the exit of the
return vane lead to a distorted ﬂow. The curvature of the hub and shroud side highly
inﬂuences the velocity proﬁle. The incidence at the shroud increases, whereas the one at
the hub decreases. Figure D.1 in Appendix D shows the velocity proﬁle at the impeller
inlet obtained from a CFD simulation. A parameter AK=Cm_.,hmud/Cmmm is used to
characterize this distortion. AK increases as the curvature of the bend increases, i.e. as we
go towards lower ﬂow coefﬁcient compressors.

The objective for the sizing of these impellers was to maintain similar velocity
triangles between the four impellers at inlet and exit. This procedure does not necessarily

minimize the inducer tip Mach number, but the Mach number value is not really an issue

 

81

for these low ﬂow stages (Mwltip==0.4). Also it was shown in Chapter 4 that the
minimum Mach number as a function of tip-to-hub ratio was relatively ﬂat.

Table 5 .1 shows the inlet conditions and dimensions used for each impeller.

Table 5.1 Impeller main dimensions

 

 

 

 

 

 

 

 

 

AA AB AC AD
P00 (psi) 750 888.9 1049 1263
T00 (R) 580 607 635 664
R1t(“) 4.18 4.13 4.06 4.00
13111 (°) -64.49 -65.7 -65.1 -65.0
01: 1°) -62.66 -63.4 -63.5 -63.0
52 (“) 0.4 0.36 0.32 0.28
82 (°) -60 -60 -6O -60

 

 

 

 

 

 

A constant deviation of 4° for the primary zone and an MR2 value corresponding
to the state of the art lead to the velocity triangles presented in Table 5.2. These assumed
values are slightly different from those used in Chapter 4. The main concern here was to

ensure continuity between these four designs and the existing family of impellers.

Table 5.2 Design velocity triangles

 

 

 

 

 

 

 

 

 

 

 

AA AB AC AD
Cm1/112 0.218 0.210 0.208 0.209
Cm2/U2 0.183 0.181 0.179 0.183
Cuz/Uz 0.573 0.578 0.585 0.581
012", 72.3 72.6 73.0 72.5
W2/U2 0.465 0.459 0.452 0.457
Wz/me 0.74 0.75 0.75 0.76
Can/Cy 0.84 0.86 0.86 0.88
o (slip factor) 0.890 0.893 0.894 0.899
711.1 rotor (w/o leak) (%) 93.4 93.4 93.3 93.2

 

 

 

 

 

The revised performance prediction will be presented in Chapter 6.

 

82

5.2. Streamline Curvature Throughﬂow Analysis

5.2. 1 . Geometry

The meridional contour and blade angle distribution were set to satisfy the
following criteria:
0 Smoothness of the curvature and slope distribution of the meridional contours
o Acceptable velocity distribution
a Acceptable blade loading
0 Reasonable lean angle distribution

The blade thickness distribution was kept identical to the A1 impeller. The
impeller exit rake angle was set at 15°. The choice of the velocity distribution and the
blade loading have been discussed in 3.2.1.3.

The meridional contour and blade angle distribution for the AA, AB, AC, and AD
impellers are shown in Figure 5.5 and in Figure 5.2. They have been obtained after a few

iterations between the streamline curvature throughﬂow analysis and the CFD analysis.

83

 

 

AA AB AC AD

Figure 5.1 Meridional contours of impellers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AA impeller AB Impeller

~40 -40
B a
’45 " _‘5
-5o « “U” -50 . Hub
'55 ‘ ~55J
'60 ‘ 51mm ' -60 « Shroud 1

I

I
~65 1 -65 4
°‘°° 02° “0 06° 03° ‘-°° 0.00 0.20 0.40 0.60 0.80 1.00
A0 Impeller AD Impeller
-40 .\ -40
3 11
*5 ‘ -45 1
Hub

'50 ‘ Hub '50 ‘
‘55 “ -55 4
-60 < Shroud I —60 1 Shroud '

I l
'65 1 -65 4
-70 <L\ r T v r '70 ' I Y I

0'00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00
95 merldonel (lemme 98 merldonel Get-nee

Figure 5.2 Blade angle distributions

84

5.2.2. Streamline Curvature Analysis

The velocity distributions along the impeller at the hub, mean, and shroud are

shown in Figure 5.3 for the AA impeller.

 

 

 

 

 

 

 

 

M impollor AA impeller
500 700
450
600 -
A 400 '1 Suction 35 A
i ' 5
350 1W ; 500 ‘ sum" “do
3 W
300 - E 400 ‘
250 « . 3 ‘ '
20° .. Pressure Side 2 300 4W
i150_ 5200‘ Pressureslde
100 ..
100 ~
50 «
o v r r I o r r 1 v
0.00 0.20 0.40 0.60 0.80 1 .00 0.00 0.20 0.40 0.60 0.80 1.00
96 meddlonel «ﬁnance % meridional dietence

Figure 5.3 Relative velocity distribution at hub (left) and shroud (right) for the AA
impeller at design point

The blade-to-blade loadings obtained with the streamline curvature program for
the four low-ﬂow stages are shown in Figure 5.4. The method assumes a linear variation
thween pressure and suction side. They were calculated based on the assumption of a
Slip factor of 0.85, from which the exit momentum was calculated. The values of the
maximum loading are located around 50% of the meridional length and are all below
0-55. The apparent discontinuity in the blade loading results from the imposed outlet

momentum, which forces the tangential velocity to match the prescribed value.

i

0. 70

AA Impeller

 

o.eo«
o.sO<
0-40<
0-301
0-20 «

0.104

 

0.00

Hub

Shroud

 

 

0.00

0.70

0.20

0.40
36 meridionel deuce

Ac Impeller

0.60

0.80

 

0-60 *
0-50 *
0-40 ‘
0.30 1

0-20 ~

 

Shroud

 

 

5.3.

0.20

0.40

f

0.60

56 meridional distance

0.80

85

 

 

 

 

 

 

 

 

AB Impeller
0.70
0.60 <
Hub

0.50 .
'5
3 0.30 1

0.20 <

0.10 ~

Gem 7 Y Y I

0.00 0.20 0.40 0.60 0.80 1.00
95 merldlonel diet-nee
AD lmpeler
0.60
0.50
l Hub
0'40 ‘ Shroud :
l

f 0.30 l

0.20 <

l
0.10 4
0.00 . . T .
0.00 0.20 0.40 0.60 0.80 1.00
as merldlonel diet-nee

Figure 5.4. Blade-to-blade loading

CFD Analysis

5-3.1. Presentation of the Commercial Software TASCflow

The 3-dimensional viscous code used for the flow ﬁeld analysis is the
C0Intriercially available code TASCflow (version 2.7) from AEA Technology. TASCﬂow

is a general purpose CFD software package for a wide range of industrial applications. It

can solve incompressible as well as compressible flows. The code has been used by many

researchers. Its capability to describe the flow characteristics in centrifugal machines has

86

been shown by various authors (Cooper et al. 1994, Howard and Ashraﬁzaadeh 1994, or
Flathers and Bache 1996). Dalbert and Wiss (1995) compared the numerical simulations
of the NASA compressor Rotor 37 performed with the Dawes code (BTOB3D) and with
TASCflow to the detailed experimental measurement performed at NASA. Both codes
provided good predictions of overall values. The efﬁciencies were predicted to within +/-
1.5%. The total—to-total pressure ratio was slightly over-predicted especially at off-
design conditions. The turbulence model of TASCflow was considered to be superior to
the classical Baldwin-Lomax model used in the Dawes code. Based on these encouraging

reports, TASCﬂow was chosen to model the impellers.

5.3.2. Equations Solved

The equations solved are the Reynolds—stress averaged N avier-Stokes equations in
primitive variable form, and the time-averaged mass and energy equations in the rotating
frame of reference. The equations are expressed in a ﬁnite-volume formulation, which is
fully conservative. In the mean form, the equations solved are as follows.

The conservation of mass:

a). a _
at+dxj(puj)—O

The conservation of momentum:
a 8 3p 8 Bu. all].
_ . _ . . : —— — —' — S .
at (put )+ axj (pa/u!) axi + axj {#€ﬂ[axj + axi ]} + at

Where liefFllﬂll. Sui is the momentum source term. For flows in a rotating frame, the

effect of Coriolis and centripetal forces are modeled in the code by

87

S.“ =—2§2xl7 —Qx(fl><'r’)

The energy equation, in which the work of the viscous forces is neglected, is

solved

a 91» a 9 31 ,u ah
— H—— — .H =— — —'— S
at”) at+axj u, ) axl axj+pr,axj]+ E

+_a_ u _ai+§_u_’. __2. _a_ul.6
Bx]. "ﬂ' ax, ax, 3”'ax, '7

where the total enthalpy is deﬁned by H = h + guilt, + k.

2 2
is advected in the

 

In the rotating frame of reference, the rothalpy I = H —

energy equation in place of the total enthalpy. (l) is the rotational speed, and R the local
radius.

The computational technique used to solve these equations is classiﬁed as
pressure based, as opposed to density based time marching. The discretization scheme is
second-order accurate. It uses a directionally sensitive, upwind discretization scheme
known as Linear Proﬁle Skew upwinding (LPS) (Raithby 1976) combined with a
Physically based correction term known as Physical Advection Correction (PAC) (Van
Doormaal et a1. 1987). These features retain false diffusion and false total pressure losses
that would occur due to flow directionality and streamwise gradients but at a very low
level,

The turbulence model used is a standard k-e model combined with a wall function

aPPI'Oach. The wall function approach eliminates the necessity of discretely resolving the

large gradients in the thin near-wall region. The wall function equations have been

88

derived under the assumption that the nearest grid point to the surface has a y+ value

pywu‘r

between 30 and 500, where )2+ = . Here, yw is the distance from the wall and ur is

the friction velocity. This condition was very important to produce grid independent

results.

5.3.3. Boundary Conditions

The boundary conditions used were as follows:

0 The inlet total pressure and temperature were speciﬁed at the inlet upstream of the
impeller (see the meridional view in Figure 1.1). The flow was assumed to be swirl
free.

0 The mass-flow rate is given as an exit boundary conditions at the exit of the diffuser.

0 A periodic boundary condition is used for the side walls because only one impeller
passage is modeled.

‘ The calculations are made in the rotating frame of reference. The whole grid is
rotating. Because the inlet walls and the diffuser walls are stationary in reality, they
are counter-rotating in the CFD calculations. They therefore appear stationary.

° A symmetry condition is used at the inlet and at the exit to represent, in a ﬁrst
approximation, the ﬂow that may recirculate through the clearance of seals. No shear
forces are applied to the fluid in this small region.

' The walls were modeled as hydraulically smooth.

89

The labyrinth seals have not been modeled in this study. It is believed that the
flow will not be affected by their presence at the design point. It is understood that this

approximation becomes less valid at off-design conditions.

5.3.4. Grid Speciﬁcations

The computational domain consists of part of the inlet duct, a whole blade
passage with the blade in the center, and a sector of the vaneless diffuser. The length of
the axial duct is about twice the axial length of the impeller, and the diffuser outlet radius
is twice the impeller exit radius. The impeller grid uses an H-grid topology.

In all the models used, the computational grid consisted of 127 nodes in the
streamwise direction with 70 nodes inside the impeller, 25 nodes in the pitchwise
direction (J) and 25 in the spanwise direction (K). Different node numbers and grid
densities were tested. It appeared that the main condition needed to obtain a grid
independent solution was to respect as much as possible the criteria on y+ (Di Liberti,
1997). Once an adequate node distribution close to the node surface was established, it
was found on a qualitative basis and on a one-dimensional basis that the results become
independent of the grid node number (75,000 and 100,000 nodes).

The meridional view in Figure 5.5 is a projection in the meridional plane as
explained in Chapter 4. The blade-to-blade view in Figure 5.6 corresponds to a projection

in the m-B plane.

9O

 

 

.3.—2....Cz:

 

  

,. I”

  

.4

 

a
66 I

I
”e 9 ’66,”?

     

Blade trailing edge

 

o
It IIIIIO”
9 ’ I a
’66,

        

Blade Leading edge

-leee-'

   

 

 

ller only)

'd (impe

ion gri

Figure 5.5 Meridional view of calculat

 

 

 

 

 

d (impeller only)

1011 gri

Figure 5.6 Blade-to-blade view of the calculat

91

5.3.5. Convergence Criteria

The convergence was attained when the maximum residual for the pressure and
for the three components of velocity was smaller than 1E-04. This represents a
convergence by more than three orders of magnitude. It was found that the solutions were
identical both qualitatively and on a one-dimensional basis to the solutions obtained with
a maximum residual of lE-OS (Di Liberti 1997).

Each calculation takes approximately 9 to 12 hours of CPU time on a Sun Ultra

Sparc workstation.

5.4. Qualitative Analysis of the Flow Field

All four impellers have been modeled with TASCflow. The ﬂow will now be
qualitatively described in the meridional and blade-to-blade plane in terms of meridional
velocity to detect zones of low-momentum, relative flow angle (to compare the deviation
of each zone), and of turbulent kinetic energy. High values of the turbulent kinetic energy

can be found in regions of mixing and in regions of unsteady flow (Pinarbasi and Johnson

1994).

5.4.1. Flow at Design Point in the AA Impeller

The velocity vectors in the meridional plane show a ﬂow following the meridional
contour in all the surfaces between the blades (Figure 5.7). On the pressure side and on

the suction side, the secondary flows show a tendency of the flow to move towards the

 

92

shroud (Figure 5.8). The curvature of the meridional contour is responsible for these
secondary flows. For this impeller with no inducer, the curvature of the axial-to-radial
bend dominates the effect of the blade curvature.

In the blade-to-blade plane (not shown), the flow remains attached to the blade

because the incidence levels are still low. The flow is well guided by the blades from hub

to shroud.

 

SPEED
2.H20£'I2
2.2908002

 

17°!9I2

“ll ll"

857l°I2

l

 

Illllllllllllll
llillllllllllll

"
unuunuuu

QSOZ’OZ

 

 

 

 

 

 

 

 

 

 

.IISEVOI
[Outelz
S75E°IZ
«$220.2
.331E902
ZIOE'IZ
IGOEOIZ
OOZEOI;
M7LE9IL
ZOLIOCL
OSLK'OL
IHLEOOL

GSOE’IL

 

“2.2.31

ZlCEOIL

 

I 0' N U C a V C O I- I- 0‘ II. P D- M

.IOOCVOC

 

 

 

 

Figure 5.7 Velocity vector at mid pitch (AA impeller, design point)

 

 

SPEED
2.HZO!'.2

2 29989.2

  

.IISE'II
.IGHE’IZ
.5732002
.NSIE..2
.3313‘02
.2102'02
,IOOI902
.OGZI‘II

.ZGLEOUL
.CSLC’UL

OHLE¢OL
.03.!‘01
.M2IEOIL

.ZIOEOCL

 

 

1
l
1
1
1
1
1
O
O.N7LI°.1
7
6
u
3
2
1
U

 

 

.UOOK'II

 

Figure 5.8 Velocity vector near the suction side (AA impeller, design point)

The ﬂow has been analyzed at ﬁve stations along the impeller channel located at
10, 30, 50, 70, and 95% of the meridional contour. The position of the ﬁve stations
(denoted respectively I, II, III, IV, and V) is shown in Figure 5.9. The objectives were to:
o analyze the development of the ﬂow as it goes through the impeller channel from
inlet to exit
0 identify the possible presence of a secondary zone on the shroud suction comer

0 determine the influence of mass flow rate on the location and on the extent of it

 

94

 

 

 

 

 

Figure 5.9 Calculating stations along the impeller channel

In Figure 5.10 to Figure 5.14, the shroud is at the top of the picture, the
suction side is on the right, the hub is at the bottom, and the pressure side is on the left.
As the calculations have been made with the blade in the center of the passage, one of the
surfaces has been rotated to obtain a picture of the ﬂow as it will appear between two
blades. It should be mentioned that each station corresponds to a line of constant I
(streamwise direction), and the resulting plane is not normal to the ﬂow.

For the impeller at the design ﬂow, the meridional velocity distribution at inlet is
nonuniform from hub to shroud. The curvature of the inlet section generates a higher
velocity at the shroud than at the hub. This can be seen in Figure 5.10, which represents
the normalized meridional velocity at station I. Also, as expected, the velocities are

higher on the suction side than on the pressure side.

 

p...»—

95

This pattern can still be seen at station 11 (Figure 5.11) with a tendency for the
higher velocities to be located on the right side of the passage. On the shroud suction
corner, a small region can be seen with velocity level of the same order of magnitude
than near the pressure side.

At station III (Figure 5.12), the velocities near the hub suction side get higher, as
expected from the streamline curvature analysis. The small region on the shroud suction
side corner can be seen to extend slightly more. This trend continues to develop at station
IV. The flow is dominated by the curvature in the meridional plane. As the exit of the
impeller is reached, the curvature is less and less important.

The flow pattern changes quickly between station IV (Figure 5.13) and station V
(Figure 5.14). The particles of largest velocities can now be found near the pressure side,
and to a lesser extent on the hub suction side. A region of low momentum has now
clearly developed on the shroud suction side. Its extent will be deﬁned more precisely
later as the reduced relative total pressure loss contour will be shown (Figure 5.16).

Figure 5.15 shows the flow angle distribution at station V. It is seen that only a
thin region at very high velocities near the pressure side is following the blade. Near the
suction side, the flow is under-tumed close to the surface at the hub, and following the
blade near the shroud. In most of the passage, the average flow angle is between —65° and

—70°, conﬁrming the expected large deviation at the impeller exit.

 

96

 

 

 

 

 

 

 

tn
2: LJIIK‘I).
2| 3.3251-31
u 3.15-I'D].
u 2.975z-u
L7 2.0Ill'l1
LI I.IZSI'IL
u LMSIE-IL
u 2.275141
u 2.1IIl-u
.Lz i.ezu-u
u 1.75OI'01
il 1.57sl-u
9 i. «all-I;
a 1.125l-IL
7 i.esn—u
6 aJsu-ez
S 7.000E-IZ
-l 5. 2591-”
I 3.5-91-02
2 1.75.142
.l. It’ll

 

 

Figure 5.10 Normalized meridional velocity at station 1 (AA impeller, design point)

 

 

 

 

 

 

 

C11
21 3.5-It'll.
2| 3. SZSI-IL
1° 9.15.1'I1
18 Z. D75E-I1
l7 Z.IUIE-Il
.ll 1.3232'I1
15 2.MSIE-I1
1‘4 2. Z7SI'I1
15 Z. llllrll
12 1.925ﬁ-I1
11 I‘VSCI'IA
i. 1.57SE-Il
9 1.10.1‘01
B 1. IISI-Il
7 l ISIl—Il
6 I. ’SIE‘IZ
S 7 IDII'II
‘1 5. ZSIE-IZ
3 3. EDIE-.2
2 ) 75IE—IZ
1 ..IBIE'II

 

Figure 5.11 Normalized meridional velocity at station [1 (AA impeller, design point)

97

 

 

 

 

CH
2] 3.5IIE‘I1
2| 5.315l'll
1.9 3.15Il-Il
ll 10753-01
1’ 2.00.1'01
1' 2.IZSI‘I1
1! Inﬁll-ll
1'4 1 27ll-01
1, Z liIE‘IL
.12 J DISC-IL
11 I 75II—I1
LI 1 57IE-Il
9 I «III-I1
a I ZZSE-IL
7 I ”OI-.1
C ! 75.!‘IZ
S 7 CIII-IZ
N 5 ZSIE-IZ
Y 5 I SIDE—I2
e} 2 1 75'l-I2
1 I "Il'l.

 

 

 

 

Figure 5.12 Normalized meridional velocity at station HI (AA impeller, design
point)

 

 

 

 

CH

2: 3.5-02"].
2. I. SZSI-Il
19 5. XSIE'IL
1,3 2 . 975E-I1
1’ 2. I'll-I1
15 I UZSE‘IL
15 2 “SHE-I1
1‘! Z 2752-..1
15 Z XIII-IA
12 I 925E-I1
ll 1 TSII‘II
ﬂ 1 5752'”.
D l «ODE-.1
O I 22§£‘01
l ISIl-Il

O C ’SUK'IZ
5 7 IBII‘II
'1 5. ZSIE-IZ
3 3. SUIE'IZ
! Z l ’Sll-IZ
.l U. 'IIE‘II

 

 

 

 

Figure 5.13 Normalized meridional velocity at station IV (AA impeller, design point)

98

 

 

   

 

 

 

 

CH

21 3.503E'U1
2- 3.!25E-I1
15 5. 15.1-01
13 2 9755-11
17 1. CODE-I1
15 2.525E'I1
1S 2.I45IE-I1
1H 2. 27SE'I1
13 2.1..2‘01
12 1.9255'O1
11 1.75ll‘l1
1D 1.57SE-I1
9 L‘lOBE'I1
a 1. ZZSE‘I1
7 1.8532‘01
O H.75'K'I2
S 7 I'll-I2
-l 5. ZSIE'IZ
Y ! 3.5855-I2
F l l.’5.£-IZ
1 U.IIIE'IG

 

Figure 5.14 Normalized meridional velocity at station V (AA impeller, design point)

 

 

 

 

8519

21 '5.5l.!'.1
2. '5.III£001
1° '5.7OHE¢I1
1B '5. 83'5‘91
17 '5 BUUE‘OL
1U ‘B.ICCE'U1
15 '8 1005.01
1'1 '8. ZODE‘OJ.
19 V'D. EIBE‘I1
12 'B.HIOE‘I1
11 ‘5.5IOE'I1
1i 'B.§UIE‘I1
9 '5. 7OIE‘BL
I -I.BIUE*IL
7 '5.BUUE‘IL
0 ‘7.IGIE'I1
5 '7 JIIE¢I1
I1 '7 ZGIE'IA
Y 3 '7. SIDE-l1
r 2 -7 klﬂl°ll
1 ".SDIE'I1

 

 

 

 

Figure 5.15 F10w angle distribution at station V (AA impeller, design point)

99

Hence, the two flow regions predicted by the two-zone model do not appear at the
exit as distinctly as expected in terms of meridional velocity difference. The secondary
zone is located in the shroud suction comer at the design point and seems to follow the
blade. Close to the pressure side, the ﬂow follows the blade; whereas the relative eddy
seems to be responsible for the deviation of the flow over the entire passage. ‘

The entropy production along the impeller passage is related to the loss in reduced

relative total pressure (Eckardt 1975):

\ t“.

 

rel S i
P R .

to!

1P"! 5 = 8
tot inlet

(1331),,“ = (P0 )mm for a flow with no inlet prewhirl.

The reduced relative total pressure at any grid point is calculated by
re! 31—1
P4? = infer)
T
/
The development of the wake region can clearly be seen in Figure 5.16. A value
of 1.0 corresponds to an isentropic ﬂow. A region of small losses can be seen along the

suction side at the inlet due to a slightly positive incidence. Then the wake region

develops in the shroud suction side comer as explained earlier.

 

L...

 

 

 

 

 

PTOTRLJiEL
1.!1II‘II

1 . .1“...

1 .UIIE‘IC

D . 0.05-I1

9 . DICE-.1

9.7IIE-ﬂl

9 . QUIZ-.1

I . SIR-II.

 

 

Figure 5.16 Reduced relative total pressure (normalized by P0 at inlet) (AA

impeller, design point)

The turbulent kinetic energy at station V is shown in Figure 5.17. The TKE is the
largest near the hub and the shroud contour and the lowest near the pressure side and in
the middle passage. Intermediate values are found in the secondary zone. The regions of

high TKE are found in the boundary layer and in the secondary zone. It is seen that the

boundary layer near the pressure side is very thin due to the high velocities.

In the meridional plane, on the suction surface, the TKE levels can clearly be seen

to decrease near the shroud at approximately 80% of the meridional length as the ﬂow is

laminarized.

101

 

 

 

 

 

 

 

TKE

21 s.eee:ve«
2. n. 75.50.41
19 ~.seeeoe~
to «.2seeoeu
17 «.eeet-eu
15 3.7seeveu
15 s.see¢oeu
in 3.25ezoeq
13 ’ s.eee:oe~
12 2.7seeoeu
11 2.3eeeveu
re 2.2seeveu
e 2.eeeEoe«
a 1.7seeoeu
1.:eezeeu

e 1.23ec'eu
s 1.IOIE¢IH
« 7.see:oes
a s.eeecoes
*1 z 2. scene:
1 e.eee:vee

 

Figure 5.17 TKE distribution at station V (AA impeller, design point)

5.4.2. Flow at Off-design Conditions (AA Impeller)

A simulation of the flow in the AA impeller from 60% to 140% of the design
mass-ﬂow rate has been performed. The evolution of the ﬂow will be discussed
qualitatively. Only the most relevant ﬁgures will be shown in order to keep this section to
a reasonable length.

At 80% mass-ﬂow, the ﬂow in the meridional plane at mid pitch is still clean. The
ﬂow near the pressure side and the suction side of the blade now tends to move towards
the shroud. Near the inlet, 10% away from the suction surface, a very small recirculation
bubble can be seen on the shroud contour, which does not extend very much

circumferentially. In the blade-to-blade view, near the shroud, this small disturbance of

 

102

the flow can be noted. Because only the velocity vectors lying on the surface are plotted,
the fact that there is a recirculation along the ﬂow path can not be seen from this plot.

At station V, the secondary zone has moved towards the hub. A region of constant
velocity extends to most of the passage, the highest velocity still being located near the
pressure side. The flow angle shows the highest deviation near the hub-suction side
comer (around -13°). The flow near the pressure surface and near the suction surface
follows the blade.

At 70% mass-flow rate, the recirculation on the shroud contour is very large on
the suction surface and can be felt right up to the mid-passage. The velocity distributions
at section I, H, and III are strongly affected by this recirculation. Low velocities can now
be seen on the shroud at section I, mainly close to the suction surface. The incidence has
become clearly positive at the hub. The high turbulent kinetic energy (T KE) occupies half
of the passage towards the suction side. Low values of TKE are associated with the
primary zone. A large area with a high deviation angle occupies the hub of the blade
passage.

At 60% of the design mass-ﬂow, the ﬂow is separated on the shroud side. The
separation extends from the shroud surface, past the mid-channel (Figure 5.19). The

secondary zone occupies now most of the area near the shroud side from hub to shroud.

 

103

 

 

   

 

 

 

 

CNN

21 5.SBI£'I1
Z. I IZSE‘IL
1. 3.15.!“I1
1D Z.I75!'l1
1’ Z.IUI!'I1
.15 2. BZSE'IA
15 Z.N5I!'.1
1H 2.2751'I1
13 1.10ll-l1
12 1.3251—01
11 1.7SUE‘I1
1| 1.575E'IA
9 1.4.lE'I1
a l 225E'I1
7 l.l5ll'I1
6 6.7SIE'II
5 7 I'll-I2
u 5 ZSIE'IZ
V 3 3 SllE-IZ
11 2 1.75'I'I2
1 IAIICIVUI

 

Figure 5.18 Normalized meridional velocity at station V (AA impeller, 70% flow)

 

     

SPEED
1.59ll'll

 
  
 

5122.32

   

.NIDI'IZ
sauce-z
22920.2
1:»!902
e~I£~ez
.QSIEVIL
SIIIOIL
SSME‘DL
eleever
S’BE’CL
.YIDZ'IL
182E¢I1
cancel:
IDLE’IL

“56!...

 

I a v w a t u o v n n u n v v w v

.IUDE'U.

 

 

 

 

Figure 5.19 Velocity vectors at mid-pitch (AA impeller, 60% mass-ﬂow)

 

 

 

 

 

 

 

P TOIRL_”E

e lell'll

a i.e.ucvel

1 i.eeezvee

e LBIII'O1

s menu-u

« I.7ll!-l1

: 9.60lt'l1

iv 2 LSIIZ'II
1 IJOODZ'I1

 

Figure 5.20 Reduced relative total pressure (AA impeller, 60% mass flow)

 

 

 

 

 

 

 

CHN
Z1 3.5..l‘i1
2| 3.325l'l1
1° 3.15-I'l1
1O 2.075E'I1
17 2.0.0! '1
1! 2.325E‘I1
15 2.“!Il'l1
I." 2. 17SI'I1
13 1. 1III'I1
12 1.02§!-I1
11 1.73IC‘O1
1| 1.57SK'I1
I 1.!OIE—I1
B 1.225E‘I1
7 LISIE‘IJ.
G H.75IK'IZ
S 7.IIII‘I2
q 5. ZSDE-IZ
3 3.5lll-IZ
‘5'} I l 7501‘52
1 I.I.I!'Iﬂ

 

Figure 5.21 Normalized velocity distribution (Station V, A impeller, 60% mass flow)

105

 

 

 

 

 

 

 

 

 

 

   

pYOl’FlLJiE
11 1.IZIE'..
1. 1.01IEOII
9 LDIIE‘I.
D D.D.IE-l1
7 3.3IIE'01
I H.7OOE'I1
5 9 SDOE'I1
H D.SIUE'I1
5 B.HOIE'I1
: I D. Sill-I1
1 9.2lll‘l1
Figure 5.22 Reduced relative total pressure distribution (AA, 140% mass ﬂow)
CNN
21 5.DOIE‘IL
2. Id. 75Il-I1
1D ~.SOIE'I1
1. ‘4. ZSUE'IL
.17 “Jolt-IL
15 3.739E'I1
15 S. SIDE-I1
.A._.... """‘-'-.---“ H 1N 3.15lE'91
' ' 1! 3 ODIE'IL
“1.....M » L2 2 ’su—u
“'3‘", 11 Loner-u
1| 2 2505-01
9 13032-01
I 1 ’SIE'I1
1 LSIII'IL
6 1.250E'I1
5 1 I'll—I1
I1 7 SOIE‘II
3 5.990E'32
F) 2 2 SIDE-'2
1 8.0.3E'IB

 

 

 

 

 

Figure 5.23 Normalized velocity distribution at station V (AA, 140% massﬂow)

106

Above the design point, the secondary zone becomes smaller. Figure 5.22 shows
the reduced relative total pressure (normalized by the inlet total pressure) along the
impeller for the AA impeller at 140% mass flow rate. The secondary zone is located very
close to the shroud and near the suction side. At the inlet, positive incidences on the
pressure side can be noticed.

The normalized meridional velocity is shown in Figure 5.23. The velocity
distribution is very similar to the one at the design point.

This detailed qualitative analysis of the flow ﬁeld showed the presence of a
primary zone and of a secondary zone at the impeller exit. The secondary zone is clearly
seen as a zone of low momentum with no backflow. Its position is seen to be close to the
shroud surface above and at the design point and to extend towards the hub as the mass
ﬂow rate is reduced. The development of the wake is similar to the one reported by
Johnson and Moore (1983) except near the exit, where the high backsweep angle reduces

the wake development.

5.4.3. Comparison Between AA and AD

The AA, AB, AC, and AD impellers have been designed to be aerodynamically
similar in terms of velocity triangle and blade loading. The inlet and exit impeller widths
have been adjusted in order to maintain the meridional velocity level at the inlet and
outlet. Hence, the flow is qualitatively similar in the impellers, and the normalized
meridional velocities can be directly compared. They appear almost identical for the four
impellers. The analysis revealed that the incidence at the hub was slighlty too positive for

the AC and AD impellers. The curvature of the inlet duct used for the calculation

107

increases as the inducer tip was reduced leading to a more distorted velocity proﬁle for
the lowest ﬂow coefﬁcient impellers. The same effect would be observed if the impellers
were run as middle stages, and it was therefore decided to reduce the incidence rule at the
hub.

In all the impellers, the development of the ﬂow from inlet to exit is similar to the
AA impeller because the curvature distribution in the meridional plane and the blade
angles distribution are similar (Figure 5.1 and Figure 5.2).

In the AB impeller, the boundary layers at hub and shroud occupy a larger relative
area than for the AA impeller but are not merged (Figure 5.24). In the AC and AD
impellers, the zone of high TKE extends from hub to shroud near the suction side (Figure
5.25 and Figure 5.26). This will generate higher mixing losses, and hence, lower

performances should be expected.

 

 

 

 

 

 

 

TKE
21 s.eee:veu
2e n.7sezoeu
1e ~.seezoe~
10 «.2seloeu
i: u.eee:oeu
15 3.7SOE'I«
is :.see¢oeu
in 3.2sezoeu
is e.eeeeoe«
12 2-7seeoeu
11 2.3eeeveu
1e 2-2seeveu
e 2.eeetoe~
e i.7sezoe«
r 1.see£oe~
o 1.25ecveu
s 1....E9IH
« 7.5eeeoe:
s s.eee¢oes
YY 2 2.5005.”
1 e.eeecvee

 

Figure 5.24 TKE at station V (AB impeller, design point)

 

108

 

 

 

 

 

IKE

2i 5 east-u
29 M "SBE‘BM
15 u sees-ea
iii u zseE-eu
i7 n uses-3.4
16 3.75954”
is a sans-en
1L 3 zsaz-eu
L! a ”up“
I: ? 7SBE'IH
ii awake-4
1i 2 ZSEE'IH
9 2 aura-4
5 i rseE-eu
7 i SBBE'IH
s i neg-nu
5 1 IBBE‘B“
1. 1.5aez-u
v s 5 ”new:
Y 2 : ueE-u
i tau-u

 

 

Figure 5.25 TKE at station V (AC impeller, design point)

 

 

 

 

 

 

 

TKE

Z) 5.DEGE'UH
2n n. rsllvnu
19 ~.500!‘II
18 M 25UE'IH
U u GOIE‘U“
16 3.’SGC’IH
15 3.50.10.54
1" 3 ZSIE‘IH
1! 3.50.2‘IH
12 2.75050.“
11 1.303C'I‘4
19 2 Z5IE’OM
9 2.IOI£~O~
ﬂ 1.7SIE'I‘4
7 1 5851'.“
6 1.25IK’I‘4
S 1 loatvlu
N 7 SBIE‘DS
y 3 5 CUIE‘IS
Y Z I SBBI‘IB
1 ILBGBE‘IB

 

 

Figure 5.26 TKE at station V (AD impeller, design point)

 

109

6. COMPARISON OF THE VARIOUS DESIGNS

This chapter presents the quantitative analysis of the AA, AB, AC, and AD
impellers and the method used to reﬁne the off-design performance prediction. The CFD
analysis of the impellers designed in Chapter 4 are presented qualitatively and
quantitatively. The inﬂuence of the exit blade angle and exit width on performance is
discussed. Two additional impellers (one with a straight leading edge and one with a

leading edge located in the radial part) are presented.

6.1. Post-Processing Technique

The CFD calculations are done in a rotating cartesian coordinate system. The

velocity in the absolute frame is obtained using the relation discussed in section 2.2:
C = W +17 . The components of velocity (Cx, Cy, C2) are then converted to cylindrical
coordinates (Cr, Cu, CZ). Finally, the meridional velocity is calculated from the radial and
the axial component through C m = W . The static pressure and temperature are

directly obtained from the CFD. Hence, all relevant aero-thermodynamic quantities can

be calculated.

 

‘Tr- In..- _

110

All quantities can then be mass-averaged at any desired location along the
meridional contour and in particular at the stage inlet, at the impeller exit (one node after

the trailing edge), and at the diffuser exit.

6.2. Quantitative Analysis of the AA, AB, AC, and AD Impellers

6.2.1. Performance at Design Point

The performances of the AA, AB, AC, and AD stage have been calculated in
terms of total-to-total pressure ratio and isentropic total-to-total efﬁciency based on stage
inlet conditions and diffuser exit conditions. The diffuser has been included into the
calculation because the performance requirements were deﬁned on a stage basis. Also, it
would be difﬁcult to compare an impeller efﬁciency based on stage inlet conditions and
on impeller exit conditions with its one-dimensional equivalent. The one-dimensional
method assumes that an instantaneous mixing between the jet and the wake occurs at
impeller exit. In the CFD calculation, the mixing of the jet and the wake takes place after
the trailing edge and expands into the diffuser.

The TASCﬂow model does not include the seal leakage, the disc friction losses

and assumes a smooth blade surface. High efﬁciencies are therefore predicted as shown

 

 

 

 

in Table 6.1.
Table 6.1 AA, AB, AC, AD stage performance prediction
AA AB AC AD
(nis)tt 0.885 0.883 0.878 0.874
IItt 1.184 1.175 1.170 1.158

 

 

 

 

 

 

 

The friction losses are very important in narrow impellers. Koizumi (1983)

showed that the wall roughness had a very large inﬂuence on the impeller efﬁciency. In

5'3: .."

11]

order to reﬁne the performance prediction, the TASCflow analyses have been rerun for
rough walls. An equivalent sand grand roughness of a 120 uinches was assumed for all

walls. It corresponds to the value of the rms roughness in the impeller. The wall functions
used in the CFD calculations are modiﬁed by replacing the constant in the log-law region
of the boundary layer by the classic Prandtl-Schlichting sand-grain roughness relation
(e.g., White 1991). The same care for the y+ values of the ﬁrst grid node shall be taken as
for the case of smooth walls.

The use of rough walls can be justiﬁed using an analogy with a turbulent flow in a
pipe. For a pipe of diameter D and roughness k, the roughness is considered as
unimportant for (k/D)*ReD<10 whereas the ﬂow is considered fully rough for
(k/D)*ReD>1000 (White 1991). In case of the AA impeller, the Reynolds number based
on the diffuser entrance is of the order of 4.106, which leads to a (k/D)*Rep of
approximately 600. This value is in between the two limiting cases mentioned by White
and justiﬁes the use of rough walls to improve the CFD predictions.

The qualitative results were similar to the one obtained for hydraulically smooth
wall. The regions of high TKE were found to be thicker as the boundary layers were
more deve10ped. The predicted efﬁciencies dropped by 4 points for the AA impeller and
up to 6 points for the AD impellers compared to those obtained with hydraulically

smooth walls. The pressure ratio dropped to 1.177, 1.168, 1.162, and 1.149, respectively.

6.2.2. Consequence for the 1-D analysis

The quantitative analysis also provided some information on the conditions of the
flow at impeller exit. Table 6.2 shows the mass-average velocities, pressure and

temperature at the impeller exit.

 

112

Table 6.2 Impellers exit conditions

 

 

 

 

 

 

 

 

 

AA AB AC AD
Cu; 374.8 377.9 386.9 382.2
Cm; 140.8 139.8 137.7 141.0
0t; 69.4 69.7 70.4 69.7
52 -7.85 -7.78 -7.53 -7.34
P02 900.3 1059.4 1268.1 1482.9
P2 850.1 1002.3 1200.5 1408.6
T02 604.2 631.3 658.9 687.2

 

 

 

 

 

 

This quantitative information has been used together with the analysis of section
4.2.3 and the qualitative analysis of section 5.4 to reﬁne the performance prediction of
each impeller. The qualitative analysis of the CFD results has shown that the exit relative
ﬂow angle was rather uniform over the exit section. It was therefore decided to use the
two-zone model with a deviation of the primary zone equal to the one calculated by
TASCflow. The Mach number ratio MR2 was then adjusted to match the primary zone
static pressure with the average exit static pressure calculated by TASCﬂow. This is
justiﬁed by the fact that the two-zone model assumes that the static pressures of the two
zones are equal. Following the discussion of section 4.2.3, the ratio x of the mass in the
secondary zone to the total mass flow was increased to 0.20 in order to obtain an
aerodynamic blockage a of at least 10%.

Using these, new model parameters, the total-to-total pressure ratio and isentropic
efﬁciency were calculated. Also, in order to be able to compare directly these predictions
with the CFD analysis, the one-dimensional model was run with no external losses (i.e.
the disc friction is set to zero and the seals are not taken into account). Figure 6.1 shows

the performance prediction at design point obtained with the one-dimensional model

 

1., e'

113

(with and without external losses), with TASCﬂow (with hydraulically smooth walls and

with rough walls).

 

 

 

 

 

1.3 1
1.28 4 (Tlieln
ABAA
1.26 ~ 0:3,...” ' 0-9
1.24 - :gf': o a
122 4 :0-1-D prediction (including?
1 2 1 extemal losses) 1,
Fl . +TASCflow - Hydraulically-E
“1.18 J smooth walls -
.1 0.6 1+TASCflow - Rough walls
1.16 (0.00012)
1 14 1 +1-D prediction (no
' 0.5 . ,._extema','98§3§z ____ __
1.12 1
1.1 r T r 0.4
0 0.01 0.02 0.03 0.04

Figure 6.1 Performance prediction for the AA, AB, AC, and AD stages

As already mentioned, the efﬁciencies predicted by TASCﬂow for hydraulically
smooth walls are too optimistic. The efﬁciencies predicted by the one-dimensional
method were the lowest as they include the external losses. Nevertheless, very good
agreement can be found between the TASCﬂow analysis with rough walls and the 1-D
analysis with no external losses, the difference in efﬁciency being lower than one
percentage point. This gives conﬁdence in the method adopted to determine the two-zone
model parameters.

In terms of total-to-total pressure ratio, there is no difference between the two
one-dimensional methods because the external losses inﬂuence only the total temperature

at the rotor exit. The one-dimensional prediction lies in between the two TASCﬂow

 

114

analyses, i.e. it over-predicts the total pressure at diffuser exit. The slip factors obtained
when setting the deviation of the primary zone to the average value calculated by
TASCﬂow are approximately 0.85. Paroubek et al. (1994) had a slip factor value of
approximately 0.80 for impeller with an identical exit blade angle suggesting that the
calculated slip factors would be too high.

In terms of diffuser performance, Figure 6.2 shows the diffuser pressure recovery
and loss coefﬁcient calculated with TASCflow and the one-dimensional method. The
TASCﬂow calculation with wall roughness predicts loss coefﬁcient 50% greater than for
hydraulically smooth walls. It is encouraging to notice that the TASCflow prediction with
rough walls agrees well with the one-dimensional prediction method.

The pressure recovery of the 1-D analysis is higher than the one predicted by
TASCflow with rough walls. This tendency of the one-dimensional method of Johnston
and Dean (1966) to over-predict the pressure recovery is known. A calculation of the
diffuser flow with an integral boundary layer method would have helped to obtain a
better prediction of the static and total pressure at the same time.

These results show that the TASCflow results obtained with the rough walls

mode] are most accurate and validate the performance prediction method.

 

115

 

 

 

 

0.700
0 600 AD AC AB AA
Pressure M
recovery CP45 /
0.500 .
‘ +TASCflow - Hydraulicallym
smooth walls '
040° ‘ + TASCflow - Rough wall
+ (0.00012)
0300 q Loss +1-D analysis
coefficient L045 ——— — ~ —
0.200 . \\
0.100 1
0.000 . T r
0 0.01 0.02 0.03 0.04

 

Figure 6.2 Diffuser loss coefﬁcient and pressure recovery

6.2.3. Determination of the TEIS Models

The use of the TEIS model described in Chapter 4 for the prediction of the
operating map requires a number of inputs, which can not be known precisely at the early

design stage:

the effectiveness of the inlet portion 1],.

0 the effectiveness of the passage portion m,

o the diffusion limit DRsmn

o a distribution of recirculation losses from surge to choke

Based on the experience acquired from the existing family, a constant T]a of 0.78 for

all the stages was chosen. No recirculation was assumed at choke and design. 3%

116

recirculation was assumed at the surge point. The impeller passage effectiveness was
determined to match the Mach number ratio at the design point. The maximum diffusion
ratio was set to the value of the Mach number ratio incremented by 0.02. The TEIS

models presented in Table 6.3 were then obtained.

Table 6.3 TEIS models for the AA, AB, AC, and AD impellers

Tia “b 0'2 DRstall }
AA 0.78 0.44 0.846 1.151 -
AB 0.78 0.42 0.848 1.141
AC 0.78 0.44 0.853 1.141
AD 0.78 0.37 0.853 1.111

 

 

 

 

 

 

 

 

 

 

 

 

 

6.2.4. Off-design Performance (AA impeller)

The flow in the four impellers has been seen in Chapter 5 to be qualitatively
similar. Therefore, CFD calculations at off-design were done only for the AA impeller.
The impeller has been run (with hydraulically smooth walls) at off-design conditions
between 60 and 140% of the design conditions. The performances in terms of pressure
ratio and efﬁciency have been obtained for the stage and are shown in Figure 6.3. This
ﬁgure also includes the performance prediction obtained with the TEIS model with and

without external losses.

117

    
 

 

 

 

1.35
1 ; ("10)11
+ 0.95
l
1.3 . A + 0.85
+ 0.75
1.25 1
(”5 +055
1.2 ~ + 0.55
* ’“_“ -r 0.45
+1-D Prediction (including '
“5 , external losses) I, ,
+ TASCflow - Hydraulically
smooth walls ‘ 'T 0-35
+1-D prediction - No
external loss
1.1 7‘ ""1"" 7 “r r 0.25
0 0.01 0.02 0.03
(I)

Figure 6.3 Calculated performance map for the AA impeller

The minimum flow rate for the one-dimensional prediction was set by the onset of

rotating stall in the vaneless diffuser based on Senoo’s criterion.

The shapes of the efﬁciency curve obtained by the three methods are similar.

Based on the conclusion of section 6.2.2, the efﬁciency curve predicted by the one-

dimensional method with external losses is probably fairly good. The pressure ratio is

 

118

likely to be slightly over-estimated because a constant slip factor was used for the one-
dimensional prediction. This slip factor was calculated at the design point in order to
match the deviation of the primary zone with the value calculated by TASCﬂow, and
then kept constant throughout the operating map.

It also appears that the criterion used to set the impeller range was not adapted.
The qualitative analysis has shown that the impeller will exhibit the ﬁrst sign of a
recirculation at 70% mass flow, and that a large recirculation will take place at 60% mass
ﬂow rate. The minimum flow coefﬁcient at which the impeller can be operated will
therefore be around 0.015, leading to a useful range of approximately 2. Because the
diffuser would stall at a lower flow coefﬁcient, a smaller pinch ratio could have been

used, leading to a wider diffuser.

6.2.5. Off design performance of the AA, AB, AC and AD stages

Using the TEIS model determined in section 6.2.3, the performance maps for the
four impellers were determined. They are shown in Figure 6.4 in terms of isentropic head
coefﬁcient and total-to—total isentropic efﬁciency. The one-dimensional method includes
the external losses. As already mentioned in the previous section, the last two operating
points (on the left of the map) should correspond to unstable conditions and therefore be

avoided.

 

119

 

 

 

 

 

2 1
his».
1 8 — ~~ 0.9
1.6 , 0.8
1.4 J . 0.7
1.2 ~ 0.6
(Wis),t
1 — ~ 0.5
0.8 ~ ~~ 0.4
0.6 - - 0.3
o 4 4 ~ 0 2
0.2 — -. 0.1
O l l l O
o 0.01 0.02 0.03
(b

Figure 6.4 Performance map prediction of the AA, AB, AC and AD stages

The isentropic head coefﬁcient and the efﬁciency decreases from the AA to the
AD impeller. This decrease in performance was expected because the impeller exit width
was reduced. This is also in agreement with Figure 2.10 showing that impeller efﬁciency

decreases as the speciﬁc speed decreases.

 

120

Following the discussion of the previous section, the efﬁciency prediction should

be accurate up to one point. The isentropic head coefﬁcient is slightly over-estimated.

6.3. Validation of the Loading Estimation

The isentropic relative velocity has been calculated on the pressure and suction
surface of the blade to calculate an inviscid loading. The main purpose was to compare
the loading obtained by TASCﬂow with the one predicted by the streamline curvature
throughﬂow method. The separation is expected to occur for blade loading higher than
065-070.

To calculate the isentropic relative velocity, a streamline from impeller inlet to
impeller exit is considered. For a steady isentropic flow, the rothalpy is conserved along
this streamline. The isentropic relative velocity is one that will take place on the blade if
an isentropic transformation will bring the particle from the inlet total conditions to the

local static pressure (calculated by TASCﬂow).

 

121

0.7

 

0.6 J

    

0.5 .
Shroud
0.4 . ‘
0.3 «

0.2 «

0.1 4

 

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1|

 

-O.2 ,

 

Figure 6.5 Isentropic blade-to-blade loading (AB impeller, design point)

The maximum loadings compare well with the predicted maximum loading of

section 5.2.2.

6.4. Analysis of the Effect of Exit Blade Angle and Width

6.4.1. Qualitative analysis

The four impellers designed in Chapter 4 (AA50, AA55, AA65, and AA70) were
analyzed with TASCflow at design point with hydraulically smooth walls and with rough
walls. They cover a range of outlet blade angles between —70 and —50°.

The four impellers exhibited similar secondary flows in the meridional plane to
the one discussed for the AA impeller. At the design point, the ﬂow follows the contour,

and no zone of recirculation can be seen.

 

122

Figure 6.7 to Figure 6.9 show the meridional velocity contour at station V for the
AA50, AA55, AA65, and AA70. The zone of high meridional velocity near the hub gets
smaller from the AA50 to AA55 and disappears in the AA65 and AA70. In these last two
impellers, the zone of low meridional velocity extends from hub to shroud. The TKE
contour shows that the extent of the zone of higher TKE becomes larger as the
backsweep angle is increased (Figure 6.10 and Figure 6.11). The development of the
wake, therefore, seems more controlled by the curvature in the meridional plane than by
the curvature of the blade in the blade-to-blade plane. The wake mixing losses may
therefore be higher in the AA65 and AA70 than in the AA50 and AA65. On the other
hand, the velocity levels in the narrow impellers are much smaller, which will counter
balance the effect.

In terms of deviation, the relative flow angle plots show that the maximum
deviation is located near the center of the blade passage for the AA50 and AA55 and

moves towards the shroud for the AA65 and AA70.

 

 

 

123

 

 

”N

21 3.5.0E‘01

2| 3.325E-IL

1. 3.15.!‘01

2.07SE‘CL
2

.OOIE‘IL

p.-
a
N

.OISC'II

.—
II
N

.QSIE-IL

.27SI‘IL

N

.llIE-Il

 

H

.OZSE-OL

h

.7SOE‘OL

 

 

hi

.S7SE‘OL

0
.-

.ﬁOOE-CL

C
.-

.ZZSE-Il

‘0
p

.OSCE-Ol

.7505‘02

ti 0
V

.IOOE-IZ

8

.25.!‘02

.5008-02

s
a s

F‘ 2 1. 7502-02
2 o

.UIOC'OI

 

 

 

Figure 6.6 Normalized meridional velocity (AA50, design point)

 

 

 

 

 

 

 

 

CNN
21 3.3002-01
2| 3.3252-01
LI 3.1502-01
1. 2.97SE-IL
27 2.0002-01
is 2.02:2-01
L5 2.usaz-02
LN 2.2752-01
13 2.1-02-02
12 1.9252-01
11 1.7SOE-IL
LO 1.5752-01
e 1.QOIE-IL
a 1.2252-01
r 1.0502-01
e 3.75-2-02
s 7.00IE-IZ
~ 5.2502-02
a 3.500c—02
T 2 1. 7502-02
1 0.00-evo-

 

 

Figure 6.7 Normalized meridional velocity (AA55, design point)

 

124

 

 

CNN
21 I. Sill-l].
2| 3. SISE-IL
1' {1502-01
18 2. .7SI‘I1
1’ 2. IIIl-Il
16 2. ﬂlﬂﬂ’ll
1.! I. “Sit-I].
1“ 2. 275l-IL
13 2. XIII-I].

 

«a

 

 

rum:

0. IIOC'I'

 

 

 

Figure 6.8 Normalized meridional velocity (AA65, design point)

 

 

 

 

 

CNN
21 2.3.02'01
2. 2.27II-I1
1. 2.2!II‘IL
1. 2.1251-IL
.l’ Z.I.IC-IL
18 1.615E'I1
15 1.7SIE-Il
1'1 1.I2§E-Il
A! 1.5ll! IA
12 1.87SI-Il
ll 1.25.2‘l1
1. 1.1251'IL
9 1.00IE-I1
I 5.7SIE‘I2
7 7.5..E-CZ
6 3.25IK'IZ
S 5....E-I2
N ! 75.1‘02
V 3 2.530E‘I1
E“ Z I. ISIS—'2
l I.UIIE'II

 

 

 

 

Figure 6.9 Normalized meridional velocity (AA70, design point)

125

 

 

 

21
2.
LI
1.
17
18
15
an
1!
12

 

11

LI

gnu

 

 

TKE
3.0.0t'IM
n.7SIEqu
~.80I!°lu
«.2SOIOIQ
u.OOOI‘IH
3.750¢'I«
3.30.!00u
3.250l'0H
8.000(00u
2.760loou
2.5.IEVIM
2.ZSII+IH
2.0..l60n
1.7BIEOIM
1.30.10.“
1.250C'IH
1.00.290u
7.50.2003
$.0IDEOII
2.800(003
I.II.£?II

 

Figure 6.10 TKE contour at station V (AA50, design point)

 

 

 

21
2|
1.
1.
p
18
15
u
L!
12
L1
1.

 

ﬂ 0

l

 

 

TKE
5.0.IC’UQ
u.7SIE¢IH
H.300890H
9.280(00M
“.000100“
3.7SOE'IH
3.5.019.“
3.25IIOIH
3....l’ﬂﬂ
2.7'IIOIH
2.3..K'IM
2.23IEOIM
2.00.1.0“
1.75.10.“
1.5.020.“
1.23.!‘Ik
1.00.29.“
7.5..(‘03
5.0ICI003
2.500(903

0.0.08'00

 

Figure 6.11 TKE contour at station V (AA70, design point)

 

 

 

126

 

 

 

 

 

 

BE TR
21 '5.III!’I1
2. '5. 10.19.11
1’ 'S.2III°I1
13 ‘§.3IDZ'I1
1' 'S.NOIE‘I1
16 “5.5.02'l1
15 ‘5.IBIEOI1
1‘! ‘5, 7IIE‘I1
1.3 . '5.BIIE‘IA
13 ‘5.DIIE‘I1
11 ‘5.'.UE'I1
1. ‘B. IOIEVII
9 ‘5. ZIIE'II
I ‘5. SUIﬂ'll
7 *B.M.IE’I1
0 'B.5.IC'I1
I ‘E BOIEVII
It -6.7.IE0I1
5 -I.I'II‘I1
Fl 1 'I.IIII‘C1
1 ‘7.IIIEYI1

 

 

 

Figure 6.12 Relative flow angle at station V (AA55, design point)

 

 

 

 

 

 

BETH

21 'l. IIII’I1
II -I. I75E’I1
1D “I. 1SII'I1
1! -I. 225l‘l1
.17 'I. SUIK‘Il
1B '0. 3730'01
15 -I. “5.10.1
1'4 ‘3. 525l’01
13 '6. OIIE‘II
12 >5 . C7SE‘C1
11 '6. 7SIE'I1
1. -E. GZS!*I1
B -S. DIIE‘I1
C ‘3. 97SEOI1
‘ 7 . ISIE'UI

6 ‘ 7 . IZSK'C‘I.
5 ‘7 . 2III'I1
q *7 . Z75l*.1
5 -7 . SSIE‘CJ.
P‘ 2 -7 _ «25!...1
1 ‘7 . Slﬂt'ﬂl

 

 

Figure 6.13 Relative flow angle at station V (AA65, design point)

127

6.4.2. Quantitative analysis

A similar analysis to the one performed for the AA, AB, AC and AD impellers
was done for the AA50, AA55, AA, AA65, and AA70 impellers. The quantitative results

obtained from the CFD analysis are given in Table 6.4.

Table 6.4 Impeller exit conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AA50 AA55 AA AA65 AA70
Cu; 396.6 375.1 374.8 372.7 362.4
sz 185.3 165.3 140.8 117.1 101.6
a; 65 66.2 69.4 72.6 74.3
52 - 10.2 -9.4 -7.85 -6.4 42
P02 909.6 899.5 900.3 899.5 894.7
P2 849.8 847.1 850.1 851.6 850.2
T02 605.6 604.2 604.2 604.1 603.4

 

 

 

Figure 6.14 shows the performance of the ﬁve impellers obtained from
TASCﬂow with hydraulically smooth walls and with rough walls. The one-dimensional
performance predictions at the design point with and without external losses were
obtained after updating the two-zone model. The MR2 levels were adjusted to match the
static pressure calculated by TASCﬂow.

The efﬁciency prediction obtained with TASCﬂow rough walls is very close to
the 1-D prediction with no external losses. The difference is less than one percentage
point for all stages. The total-to-total pressure ratio predicted by the 1-D method is higher
than the one predicted by TASCflow for the case of rough walls. The apparent jump in
pressure ratio between the AA50 and the subsequent impellers is due to the blade angle-
exit width combination chosen at design: the absolute tangential velocity at impeller exit

is higher than the other impellers (see Table 4.1).

128

1.25 1
(T113111

0.9

 

1.23 ~

0.8

 

+1- D prediction (including
external losses)

1.21 1

+0.71

._w——o——o
11.. I
1.19 ~ smooth walls 1
0'6 +TASCrow- Rough wall 1
W i (0. 00012) .

 

                     

 

0 5 +1-D prediction (no
' external losses)

 

 

 

 

1.15 w r r 0.4
0 0.01 0.02 0.03 0.04 0.05

132102
Figure 6.14 Performance at design point
Figure 6.15 shows that the loss coefﬁcient predicted by TASCﬂow for the case of
rough walls is in good agreement with the one-dimensional theory. The pressure
recovery obtained from TASCﬂow is 10% lower than the one predicted by the one-
dimensional method. Since the 1-D method is known to over-predict the pressure
recovery, these results leads us to believe that the prediction obtained with TASCﬂow
rough walls is the most accurate once external losses are added to it. It also nearly

corresponds to the one-dimensional analysis with external losses.

 

129

 

 

 

 

 

 

0.700
0.600 ~
Pressure
recovery CP45
0.500 -
l +TASCrow-
draulical smooth 1
0.400 « 2y. s N i
+ Tgécﬂow -Rough walls ;
(0.00012) 1
0.300 ~ 1 , 1
Loss , +1-D analysns l
coefficient LC45
0.200 -
0.100 «
0.000 1 r . r
0 0.01 0.02 0.03 0.04 0.05

Figure 6.15 Diffuser performance at design point
The TEIS model for each impeller was determined following the method

described in section 6.2.3. The models are described in Table 6.5.

Table 6.5 TEIS models

”3 le 02 I)Rstall
AA50 0.78 0.46 0.853 1.15
AA55 0.78 0.38 0.841 1.13
AA 0.78 0.44 0.846 1.151
AA65 0.78 0.475 0.845 1. 17
AA70 0.78 0.52 0.853 1.149

 

 

 

 

 

 

 

 

 

 

 

 

 

The corresponding performance map for the ﬁve impellers is shown in Figure
6.16. The highest efﬁciency is achieved for the widest impeller (AA70). At the design
point, the highest total-to-total pressure ratio is obtained for the impeller with the smallest
backsweep angle (AA50). The pressure ratio curves for the AA55, AA60, and AA65 are
very similar. Considering the exit flow angle at the rotor exit, the AA60 or the AA65

should give the best compromise between stability, efﬁciency, and pressure ratio.

 

130

 

1.5 0.85
(iii-“1)::
1.4 ~
~ 0.75
1.3 —
1.2 —
0.65
1.1 -

 
 

 

 

(‘Pish
1 — -- 0.55

0.9 -
, 0.45

0.8 -

0.7 —
2 —~ 0.35

0.6 -
0.5 . . 1 . 0.25

 

0.005 0.01 0.015 0&2 0.025 0.03 0.035

Figure 6.16 Performance prediction comparison

131

6.5. Inﬂuence of the Blade Leading Edge

To investigate the effect of blade leading edge position, the AB impeller was
selected. Two additional impellers were then designed. A “straight” impeller (ABstraight)
where the leading edge at hub and shroud is at the same axial coordinate and a “radia ”
impeller (ABradial) where the leading edge starts in the radial part of the impeller. The
meridional contours and blade angle distributions were adjusted to obtain acceptable and,
if possible, similar relative velocity distributions.

The main problem associated with each conﬁguration is of structural concern. For
the “straight” impeller, the GM has to be large to ensure an acceptable rake at exit. The
lean angle at inlet then becomes very large because of the small tip-to-hub radius ratio.
This may increase the aerodynamic blockage and may not be suitable for stress
consideration, the blade having to support the impeller cover. For the radial impeller, the
blade length being much smaller, the maximum stress will have to be checked to ensure

that the blade can hold the cover.

Table 6.6 TASCﬂow analysis (smooth walls) - Effect of the leading edge position

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AB ABstraight ABradial
Cuz 377.9 369.3 380.7
sz 139.8 140.4 147.9
012 69.7 69.2 68.8
52 -7.8 -8.2 -6.7
P02 1059.4 1055.2 1054.5
P2 1002.3 1000.4 994.7
T02 631.3 630.8 631.6
Husmge 1.175 1.171 1.171
(Tlis)n mg; 0.883 0.882 0.85

 

The AB impeller exhibits the best stage performance. The ABstraight and the AB

are very similar. The flow in ABstraight is better guided because the blade leading edge

 

 

132

is positioned before the one of the AB, but the additional blade length generates friction
losses. In the AB radial, the blade length is much shorter, but the flow is less well guided.

Separation took place on the blade, which explains the drop in efﬁciency.

133

7. CONCLUSIONS AND RECOMMENDATIONS

A one—dimensional and a quasi-three-dimensional program were successfully
developed to design a centrifugal impeller. Ten low-ﬂow coefﬁcient compressors were
designed and analyzed numerically. Four of them were designed as an extension of an
existing family (AA, AB, AC, and AD), and this allowed a study of the inﬂuence of ﬂow
coefﬁcient on the stage performance. Four other impellers (AA50, AA55, AA65, and
AA70) were designed as a variation of the AA impeller: they have a different outlet
width-outlet blade angle combinations but are aimed at providing the same stage total-to-
total pressure ratio. Finally, two impellers were derived from the AB impeller to study the
effect of the blade leading edge position: the AB straight with an axial leading edge and

the AB radial with a leading edge located in the radial part of the impeller.

The numerical analysis provided vital information regarding the internal
aerodynamics of each impeller. All impellers exhibited a jet/wake ﬂow pattern. The high
backsweep angle of the blade was responsible for the rather small extent of the wake area
at the design point. The wake was found to extend towards the hub suction side region as
the mass ﬂow was reduced for the AA impeller. At the design point, as the ﬂow
coefﬁcient was reduced (i.e. narrower impeller), the meridional velocity distribution and
the wake pattern in the exit plane were very similar. The wake occupied a larger relative

area because of the smaller impeller exit width.

134

The change in blade angle-blade outlet width showed that the wake was the
largest for the impeller with the largest backsweep. Hence, the development of the wake
appeared to be controlled more by the curvature in the meridional plane and the diffusion
achieved, than by the exit blade angle. Even though the wake was larger in the high-
backswept impeller, high performances were obtained because of the larger impeller exit
area: the lower velocity levels and the wider diffuser led to a reduction in the total

pressure loss.

No advantage was found by using an impeller with a blade whose leading edge is
located in the radial part of the impeller. It is possible that a more careful adjustment of
the inlet blade angle could have led to similar performance than the AB impeller or the

AB impeller with straight leading edge.

The performances of each impeller were estimated by CFD on a stage basis for
hydraulically smooth and rough walls. The necessary aero-thermodynamic quantities
required to calculate the performance were mass-averaged at the impeller exit and
diffuser exit. The numerical results were used to update the one-dimensional model and
to generate the predicted performance map of each impeller. It was found that the one-
dimensional predictions with no external losses were in very good agreement with the
TASCﬂow analysis with rough walls. The one-dimensional model with internal losses is
therefore believed to be capable of predicting the efﬁciency but it had a tendency to
slightly over-predict the pressure ratio. The qualitative trend, nevertheless, was very

accurate.

 

 

 

COG

of

113.

61

1'6

[11

C11

135

As expected, the performances of the stages were found to decrease as the ﬂow
coefﬁcient was reduced. The impellers with 60 and 65 degrees backsweep were found to
be a good compromise between efﬁciency level, pressure ratio, and stability. An increase
of the exit blade angle with a wider outlet width would probably be helpful for the

narrowest stages (AB, AC, AD).

The performance predictions quoted herein can only be quantiﬁed by an
experimental testing program of research. Considering the levels of efﬁciency obtained
with the existing family, the predicted levels of efﬁciency for the stages appear to be

reasonable.

As an extension of the work, the back disk and the seals could be included into
the CFD model. This would allow a more accurate prediction of the ﬂow ﬁeld at off-
design performance on a qualitative and on a quantitative basis as recirculation of the
ﬂow through the seals will be possible. The one-dimensional model would also beneﬁt

from such an analysis.

A quantitative CFD analysis at off-design could be performed to validate the
assumptions made in the one-dimensional off-design prediction method (constant slip

factor, constant secondary ﬂow mass fraction).

Even though more numerical work and experimental work remains to be done,
this study has shown that numerical analysis is able to provide qualitative and
quantitative information to help improve the design of low-ﬂow coefﬁcient compressors

with high efﬁciency and a large operating range.

 

BIBLIOGRAPHY

 

136

BIBLIOGRAPHY

Aungier, R., 1995, “Mean Streamline Aerodynamic Performance Analysis of
Centrifugal Compressors”, J. of Turbomachinery, Vol.117, 1995, pp.360-366

Balje, O.E., 1981, “Turbomachines a Guide to Design, Selection and Theory”,
John Wiley & Sons, New York

Benvenutti, E., 1978, “Aerodynamic Development of Stages for Industrial
Centrifugal Compressors. Part 2: Test Data Analysis, Correlation and Use”, ASME Paper
78-GT—5

Bosman, C. and Marsh, H., 1974, “An Improved Method for Calculating the Flow
in Turbomachines Including a Consistent Loss Model”, J. Mech. Eng. Sci, Vol.16, No.1,
pp.25-31

Casey, M., 1983, “A Computational Geometry for the Blades and Internal Flow
Channels of Centrifugal Compressors”, J. of Eng. for Power, Vol.105, pp.288—295

Casey, M. and Roth, P., 1984, “A Streamline Curvature Throughﬂow Method for
Radial Turbocompressors”, I.Mech.E. Publications 1984-3, Computational Methods in
Turbomachinery, London

Casey, M. and Roth, P., 1984, “A Streamline Curvature Throughﬂow Method for
Radial Turbocompressors”, I.Mech.E. Publications 1984-3, Computational Methods in
Turbomachinery, London

Casey, M. and Matty, F., 1986, “Centrifugal Compressors - Performance at
Design and Off-Design”, Proc. of the Institute of Refrigeration, London, pp.7l-80

Casey, M.V., Dalbert, P. and Schurter, E., 1990, “Radial Compressor Stages for
Low Flow Coefﬁcients”, IMEch. E., Den Hague

Casey, M., Dalbert, P. and Roth, P., “The Use of 3D Viscous Flow Calculations in
the Design and Analysis of Industrial Centrifugal Compressors”, ASME Paper 90-GT-2

137

Cooper, P., Graf, E., Luce, T., 1994, “Computational Fluid Dynamical Analysis
of Complex Internal Flows in Centrifugal Pumps”, Proceedings of the Eleventh
International Pump Users Symposium, pp.83-94

Cumpsty, NA, 1989, “Compressor Aerodynamics”, University of Cambridge,
Longman Scientiﬁc Technical, UK

Daily, J.W. and Nece, R.E., 1960, “Chamber Dimension Effects on Induced Flow
and Frictional Resistance of Enclosed Rotating Disks”, J. of Basic Engineering, pp.217-
232

Dalbert, P., Wiss, DH, 1995, “Numerical Transonic Flow Field Predictions for
NASA Compressor Rotor 37”, ASME Paper 95-GT-326

Dallenbach, F., 1961, “The Aerodynamic Design and Performance of Centrifugal
Compressors and Mixed Flow Compressors”, Society of Automotive Engineers,
Technical Progress Series, Vol.3, pp.2-30

 

Dean, RC. and Senoo, Y., 1960, “Rotating Wakes in Vaneless Diffusers”, J.of
Basic Engineering, Sept. 1960, pp.563-574

Dean R.C., Jr., 1974, “The Fluid Dynamics Design of Advanced Centrifugal
Compressors”, Lecture notes, presented March 25-29, 1974 at the Von Karman Institute,
Brussels, Belgium

Denton, J .D., 1978, “Throughﬂow Calculations for Transonic Axial Flow
Turbines”, J. of Eng. for Power, April 1978, Vol. 100, pp.212-218

Di Liberti, J .-L. and Engeda, A., 1996, “Low-Flow Coefﬁcient Compressors,
Phase I: Literature Survey”, Contract report, February 1996

Di Liberti, J .-L., Engeda, A., Flathers, M., 1996, “Low-Flow Coefﬁcient
Compressors: Phase H: Preliminary Design of Four Low Flow Coefﬁcient Compressors”,
Contract report, December 1996

Di Liberti, J.-L., 1997, “Validation of the new pre-lpost- processor of TASCﬂow
for impeller analysis”, Gas Compressor Engineering Report GCER , Solar Turbines

Eckardt, D., 1976, “Detailed Flow Investigations Within a High-Speed
Centrifugal Compressor Impeller”, J. of Fluids Eng, Sept. 1976, pp.390-402

Eckardt, D, 1980, “Flow Field Analysis of Radial Backswept Centrifugal
Compressor Impellers - Part 1: Flow Measurement Using Laser Velocimeter”, In
Performance Prediction of Centrifugal Pump and Compressors, ASME Fluid
Conference.

138

Farge T. and Johnson, M., 1990, “The Effect of Backswept Blading on the Flow
in a Centrifugal Compressor Impeller”, ASME Paper 90-GT—231

Flathers, M., and Bache, G., 1996, “Aerodynamically Induced Radial Forces in a
Centrifugal Gas Compressor - Part 2: Computational Investigation”, ASME Paper 96-
GT-352

Galvas, M. R., 1973, “Fortran Program for Predicting Off-Design Performance of
Centrifugal Compressors”, NASA TN D-7487

Hathaway, M.D., Chriss, R.M., Wood, J .R. and AJ. Strazsiar, A.J., 1992,
“Experimental and Computational Investigation of the NASA Low-Speed Centrifugal
Compressor Flow Field”, ASME Paper 92-GT-213

Howard, J .H., Ashraﬁzaadeh, M., 1994, “A Numerical Investigation of Blade
Lean Angle Effects on Flow in a Centrifugal Impeller”, ASME Paper 94-GT-l49

Jansen, W. and Kirschner, A., 1967, “Impeller Blade Design Method for
Centrifugal Compressors”, Proceedings of symposium on Fluid Mechanics, Acoustics
and Design of Turbomachinery at Penn State University, NASA SP-304, Vol.2, pp.537-
563

Japikse, D., 1984, “A Critical Evaluation of Stall Concepts for Centrifugal
Compressors and Pumps - Studies in Component Performance, Part 7”, Presented at
ASME Winter Meeting, New Orleans

Japikse, D., 1985, “Assessment of Single- and Two-Zone Modeling of Centrifugal
Compressors - Studies in Component Performance: Part 3”, ASME Paper 85-GT—73

Johnson, M.W., 1978, “Secondary Flow in Rotating Bends”, J. of Eng. for Power,
Oct. 1978, Vol. 100. PP.553-560

Johnson, M.W. and Moore, J ., 1983, “The Inﬂuence of Flow Rate on the Wake in
a Centrifugal Impeller”, J. of Eng. For Power, Vol. 105, pp.33-39

Johnston, J .P. and Dean, R., 1966, “Losses in Vaneless Diffuser of Centrifugal
Compressor and Pumps: Analysis, Experiment, and Design”, J. of Eng. for Power,
January 1966. Pp.49-62

Johnston, JP. and Bide, S.A., 1976, “Turbulent Boundary Layer on Centrifugal
Compressor Blades: Prediction of the Effects of Surface Curvature and Rotation”, J. of
Fluids Eng.. PP.374-381

Katsanis, T., 1964, “Use of Arbitrary Quasi-Orthogonals for Calculating Flow
Distribution in the Meridional Plane of a Turbomachine”, NASA TN-2546

 

139

Koizumi, T., 1983, “Experimental Studies on Performance of Centrifugal
Compressors with Very Small Flow Coefﬁcient”, Mitsui Sozen Technical Review 117,
pp.28-39

Krain, H., 1984, “Experimental Observation of the Flow in Impellers and
Diffusers”, VKI LS 1984-07

Krain, H., 1988, “Swirling Impeller Flow”, J. of Turbomachinery, Vol.110,
Jan.l988, pp.122-128

Lakshminarayana, B., 1996, “Fluid Dynamics and Heat Transfer of
Turbomachinery”, John Wiley & Sons, Inc.

Lieblein, S., 1965, “Experimental Flow in Two-Dimensional Cascades”, in
“Aerodynamic Design of Axial-Flow Compressors”, NASA SP-36, pp.183-226

Marsh, H., 1971, “The Uniqueness of Turbomachinery Flow Calculations Using
the Streamline Curvature and Matrix Through-Flow Methods”, J. of Mech. Eng. Science,
V01.13, N069 pp-376'379

Paroubek, J. et al., 1994, “Experimental Investigation and Performance
Analysis of 6 Low Flow Coefﬁcient Centrifugal Compressor Stages”, ASME Paper 94-
GT—43

Pinarbasi, A. and Johnson, M.W., 1994, “Detailed Flow Measurements in a
Centrifugal Compressor Vaneless Diffuser”, Journal of Turbomachinery, Vol.116,
pp.453-46l

Raithby, G. D., 1976, “Skew Upstream Differencing Schemes for Problems
Involving Fluid Flow”, Comp. Meth. Appl. Mech. Eng, Vol.9, pp.153-164

Rodgers, C., 1964, “Typical Performance Characteristics of Gas Turbine Radial
Compressors”, J. of Eng. for Power, April 1964, pp. 161- 175

Rodgers, C., 1980, “Efﬁciency of Centrifugal Compressor Impellers”
AGARD CP 282, 7-9 may 1980

Rusak, V., 1974, “Low Speciﬁc Speed Compressor”, United States Patent
37888765, Jan.29, 1974,Inventor: Vasil Rusak, Assigned: Delaval Turbine Inc., Trenton,
N.J.,Filed: Nov.l8, 1971

Sapiro, L., 1995, “Centrifugal compressors - Aerothermodynamic concepts for
selection and performance evaluation”, Internal note, Solar Turbines Inc., 1995

Schlichting, H., 1979, “Boundary-Layer Theory”, 7‘h ed., McGraw-Hill, New
York

 

140

Senoo, Y. and Kinoshita, Y., 1978, “Limits of Rotating Stall and Stall in Vaneless
Diffuser of Centrifugal Compressors”, ASME Paper 78-GT-l9

Stanitz, JD. and Prian, V.D., 1951, “A Rapid Approximate Method for
Determining Velocity Distribution on Impeller Blades of Centrifugal Compressor”,
NASA TN 2421

Van den Braembussche, R., 1984, “Surge and Stall in Centrifugal Compressors”
in VKI Lecture Series 1984-07, “Flow in Centrifugal Compressors”

Van den Braembussche, R., 1985, “Description of Secondary Flows in Radial
Flow Machines”, in “Thermodynamics and Fluid Mechanics of Turbomachinery”, Edited
by AS. Ucer, P. Stow and C. Hirsch, Martinus Nijhoff Publishers, pp. 665-684

Van den Braembussche, R., 1987, “Centrifugal Compressors - One Dimensional
Design”, VKI CN 134, May 1987

 

Van den Braembussche, R., 1990, “Design and Optimization of Centrifugal
Compressors”, VKI CN 141, Oct. 1990

Van Doorrnaal, JP. and Raithby, GD, 1987, “The Segregated Approach to
Predicting Viscous Compressible Fluid Flows”, J. of Turbomachinery, Vol.109, pp.268-
277

Wiesner, F., 1967, “A Review of Slip Factors for Centrifugal Compressors”, J. of
Eng. for Power, Oct. 1967, pp.558-572

Wilson, D.G., 1984, “The Design and Efﬁciency of High Efﬁciency
Turbomachinery and Gas Turbines”, MIT Press

Wennerstom, A.J., 1974, “On the treatment of Body Forces in the Radial

Equilibrium Equation of Turbomachinery”, Traupel-Festschrift, Juris-Verlag, Zurich,
pp.351-367

White, F. M., 1991, “Viscous Fluid Flow”, 2“d Edition, McGraw-Hill, New York
Wilkinson, D., 1970, “Stability, Convergence, and Accuracy of Two-Dimensional

Streamline Curvature Methods Using Quasi-Orthogonals”, Proc. Inst. Mech. E., 1969-70,
Vol.184, Pt3G, Paper 35, pp.108-l l9

 

APPENDICES

 

APPENDIX A

141

Appendix A

The energy equation in rotating passages for inviscid ﬂow is expressed by the

conservation of rothalpy as follows:

dhm, = i:- + d(%)- d(-,i)= 0 (A. 1)

Taking the derivative with respect to the n-direction of

w—-u—=——— (A. 2)

The derivative with respect to the n-direction of the peripheral speed is given by

Bu 0r
E—n- — (ea—n — (ecosB (A- 3)

Plugging Eq.(A.3) into to Eq.(A.2), gives

w— — (DucosB = __1__0£
n p an
The equation of motion on the n-direction is given by
2
—3—+ 2(0w—(0ucosB =13,-
Rn p an

Combining the Eq.A.4 and Eq.A.5 leads to

BW w
= (D

E R

 

142

This equation can be approximately integrated along the n-direction to give

w
w” — WW 2 20)-—’"“’"
Rn

_".

 

APPENDIX B

143

Appendix B

One-dimensional analysis of centrifugal compressors

B.1 Inputs

B.1.1 Geometrical

The one-dimensional geometry as shown in Figure 3.15 must be known. This
includes inducer hub and tip diameter, exit width and exit diameter, and inlet and exit

blade angles at hub and shroud.

B.1.2 Aero-thermodynamic parameters

The operating point is given (mass-ﬂow rate, rotational speed) and the inlet total
pressure and temperature are known.
If the inlet ﬂow is coming from a previous stage, the distortion of the inlet

velocity proﬁle is speciﬁed by the parameter Cmshmud/Cmmm. An inlet boundary layer

 

144

blockage is also introduced to take into account the development of the boundary layer
between the inlet and the leading edge of the impeller. Any residual swirl must be

indicated.

B.2 General equations I

In a one-dimensional procedure, the following relations are used:

0 the equation of state .

 

P = pRT
o the Mach number deﬁnition
M __. _C_
yRT

 

0 the continuity equation
m = Ap.,C..
which can be expressed in terms of Mach number and inlet total pressure and temperature

by

145

. __71'_
mm zJiM 1+‘Y—1M2 2(7-1)
Ad] p() R 2

Am is the area normal to the incoming velocity C.

o the geometrical relations in the velocity triangle (Figure 2.3 of Chapter 2).

B.2.1 Inducer

The continuity equation allows solving for the Mach number at mean diameter.
From the isentropic relations, the static pressure and temperature are calculated. All
velocities and thermodynamic quantities can then be evaluated.

The conditions at hub and at shroud are evaluated through the assumed
Cm,hmd/Cmmean ratio. The incidence at hub, mean, and shroud can then be calculated.

The blade thickness is not taken into account.

B.2.2 Impeller exit condition

The program uses the two-zone model proposed by Japikse (1985). It is then
assumed that the ﬂow near the impeller exit is made of two regions called the primary
and secondary zones and is equivalent to the jet and wake as described in Chapter 3.

The obtained system of equations contains more unknowns than equations. A
closure is obtained by assuming

0 the primary zone deviation or the slip factor

 

146

o the diffusion level MR2
o the ratio between the mass-ﬂow in the secondary zone and the total mass-ﬂow x

The relations for calculating the conditions in the primary zone, in the secondary
zone, and for mixed—out conditions will now be given for a perfect gas. They are

presented the way they have been implemented.

B.2.2.1 Relations for the primary zone

The Mach number at inducer tip Mwlt has been calculated in the inducer section,

and the MR2 value is obtained from the MR2 vs. MRZI proposed by Dean.

 

 

M
MW, = A Deﬁnition of MR2
P MR,
wl 2
IOl = cpTl, + —:2'—’— — £2"— Inlet rothalpy
I +122—
2,, 0' 2 2 Conservation of rothalpy
cp + yRszp
T 1:1
P2,, = P415) primary zone is isentropic
lr /
p = P2”
2” RT”,

sz = Mw2p1/YRTZP
sz : B21: + 82p

szp =W2p 903132,,

Cu,” =U2 +W2p srnB2p

147

 

 

2p 2p
Cu
012,) = a tan #1
mZm /
2
T02p = T2,; + 2p
2cp
m 1— . .
8 =1 ( X) (6 IS the area ratio between the secondary zone and

prWZPAexit COSBZp

must be greater than 0.1)

B.2.2.2 Relation for the secondary zone

The assumptions for the secondary zone are:
0 The secondary zone follows the blade 825:0
o The static pressure in the primary and in the secondary zone are equals
A correlation must be introduced to evaluate the power dissipated by the friction
in the front cover.

The relations used are now given.

p25 : p2p
B25 :13»: +525
I02s : 101 + Pfruntcover

The set of following equations is solved by iterations where p3,, Cmgs, T2,, and

W2, are the unknowns:

 

148

 

"'1
(1-x)p2pcm2p +Zplscm25 :7—
W25 = cmZ-T
€05st
2 2
U
1025 CpTzs + 25 -—22’

The complete velocity triangles are calculated by

Cub 2 U2 + CmZS tan B25

 

C2: = qu225 + cm225

All the remaining thermodynamics quantities in the secondary zone M25, T025,

P02s are calculated.

B.2.2.3 Mixed-out conditions

Solve by iterations on pm the following set of equations:

 

 

szm = _m__
znrzbzpzm
Cm + 1- Cm -Cm
P2," = X 2p ( X) . 23 2m +P2p
21:5me
szm + C1472,"
T2». : TOZm — 2 2 .
cp
P2":
p2m — RT

2m

Then all the remaining velocities and thermodynamics quantities are calculated at

state 2m, especially:

149

[Cum ]
a2", = atan —
szm
C
[32m = —a cos[-ﬂ3’—"-]
W2”:

52»: = B211: ‘sz
And the slip factor:
G=l——-Shl : CH2”. —Cm2m tanBZb

 

U 2 U 2
If the slip factor was input, then a new deviation is speciﬁed, and the calculation

iterated.

B.3.4 Off-design conditions

The effectiveness of the impeller inlet portion 1],, and of the passage portion 11b,

and the maximum diffusion DRsmn are given.
2
C,“ 2 1- [3%)
' cosB”,

2
l A
C .=1— =1— -—"'—

C

 

 

Cpa : nanaJ
Cpb : anpr
DR, 1 1

1" nanaJ 1" anpr

If DR2>DRsmn, then DR2=DRsmn

150

8.3.5 Vaneless diffuser analysis

Following the meanstreamline analysis of Johnston and Dean (1966), the flow
entering the diffuser is assumed to be steady, uniform, and axisymmetric. All
thermodynamic quantities ”depend only on the meridional coordinate m. An apparent wall

friction is introduced to provide a loss in total pressure. It is assumed that the wall friction
is directed along the local velocity and proportional to the local dynamic head p—Czi. The

corresponding governing equations for the flow in a vaneless diffuser are:

Conservation of mass
ti: = 21tbrpCm = const
Conservation of angular momentum

Pme M = —rCprCu
dm

Conservation of linear momentum

 

 

 

—iI:-Cfp CC” = pCm dC'" —psin0LC“2
dm b dm r
Equation of state
P = pRT
Energy equation
h+£2- =——Y——+—(C,fI +Cu2)= h0 = const

151

The flow is assumed to be uniform in both axial and circumferential direction. By

 

differentiating and combining these four relations, one can solve numerically for p, Cm,

Cu, P and T along the diffuser meanstreamline. A ﬁnite-difference method using a ﬁxed

increment Am along the meridional direction has been used in the code.

 

APPENDIX C

152

Appendix C

Streamline curvature throughﬂow method

C.l Model

C.l.l General equations

The equation of motion in the relative frame for an axisymmetric isentropic flow
with no body force was given in Chapter 3. The flow would in fact be axisymmetric only

if an inﬁnite number of blades were used. The force acting between the ﬂuid and the

blade are added to the equation of motion as a body force ﬁeld I? .

The equation in the absolute frame of reference is

2£=_ZE+IE
Dt p

Using the coordinate system shown in Figure C1, the equations of motion become
(Wennerstrom 1974):

Streamline direction

 

2
Cm ac," —C—“sin8 = -l—a£’-+ Fm
am r p 8m

Streamsurface direction

153

 

2 2
i— C“ cost»: = _la_p+ F"
’2 r p n
Circumferential direction
91 8(rCu ) = F“
r am

The notations of Figure C] are used.

 

~

Figure C.l Coordinate system for the general equations

154

Shroud
contour

    

 

Figure C.2 Definition of the geometry and notations
The derivation of the body forces in the meridional plane has been done by

Wennerstom (1974). Introducing the energy equation, the enthalpy-entropy relation, the
relation between stationary and rotating frame, and after rearrangement, Equation (C. 1) is

obtained in which Cm is the unknown. The notations of Casey (1984) are used hereafter.

Cm%=RET+SCT+BFI'+DFF (C.1)

where:
o RET stands for simple radial equilibrium:

RET = %_T§£_L2_d_(rcu)2
db db 2r db

0 SCT represents the streamline curvature and convective acceleration terms:

2

SCT = Em-sin w + Cmggm-cos w
r dm

C

o BFI‘ are the blade force terms, given by

155

BFI‘ = 2“ tanydiﬁCu) in the bladed region
r m

and by
BFT=O in the bladeless regions

0 DFI‘ represents the dissipation force terms given by
2 . d5 . .
DFT = (cos \Vcos [3+ tan ysm Bcos [ME- 1n the bladed region
m

and by

DPT = cosw les. in the bladeless region.
m

C.1.2 Continuity equation
The integrated mass ﬂow rate along a station is given by
m = jam—25mg, sin wdb

where Cmsinw represents the velocity normal to the quasi-orthogonal.

A quasi-orthogonal has been deﬁned by Katsanis (1964) as “any curve that

intersects every streamline between the flow boundaries exactly once, as does an

orthogonal to any streamline.” The quasi-orthogonal (Q.O.) are not perpendicular to the

streamline. They are ﬁxed at the beginning of the calculation and remain unchanged

regardless of any change in the streamline position.

156

C.1.3 Additional equations

C.1.3.l Angular momentum
The angular momentum is calculated by assuming that the flow follows the blade.
(rCu )q : (rCm tanﬂbl +02);

In reality, the flow does not follow the blade at inlet due to incidence level and at exit due

to slip. The following modiﬁed equation proposed by Casey et a1. (1984) is used:

(rC. 1, = 7.061.). + r. (rC. ). + (1 - 7. X1 - 7. 10¢. tan ﬂu + ”2 ),
(rCu)i represents the inlet momentum due for example to prewhirl. (rCu)o represents the
exit momentum that can be calculated once a slip factor is assumed. y, is a parabolic
function varying between 1 at the leading edge and 0 at a quarter of the meridional
length. Similarly, yo is a parabolic function varying between 0 at three-quarter of the

meridional length and 1 at the trailing edge. These functions allow the angular

momentum to vary smoothly between inlet and exit.

C.1.3.2 Enthalpy change

The change of enthalpy along a streamline between two points p and q is

calculated using the conservation of enthalpy along a streamline.
(ho)q=(ho)p

In a rotating component, it is the rothalpy I which is conserved
I=ho-(1)rCu

The following equation can be used in all cases, with (0:0 for stationary elements:

157

(ho). = (ho). +wl(rC. ), — (r6. ),1

The angular momentum is calculating as mentioned in C1.3.1.

C.1.3.3 Loss models

The program offers to deal with isentropic flow or to introduce a loss model. An
entropy change and an angular momentum variation due to friction are used along the
streamline in order to obtain a more realistic prediction of the flow ﬁeld. The two
relations used to calculate the change in entropy and the change in momentum are due to

Traupel and Eckert and Schnell

 

3
d8 =Cdl W dm
Dh TCm
and
2W C
d(rC,)= c,——erm
Dh Cm

C.1.3.4 Density calculation

The density is calculated as a function of the enthalpy and the entropy by the

relation proposed by Marsh (1971)

 

 

2 11
km - C" "gl s-so
_P_ z 2 2 e R
Po ho;
L -

 

 

For rotating parts

158

— -1_I_

 

 

 

2 2 2 ‘1-1
I a) r —E1(l+tan2B) 3-30,
L = 2 2 e_ R
p01 hOi

 

 

In these two relations, p0, and ho, represent the total density and the total enthalpy

at inlet.

C.2 Representation of the impeller meridional contour

The hub and shroud contours of the impeller in the meridional contour are deﬁned
by means of Bezier curves. A Bezier curve is a parametric representation of a curve in
space. A set of N+1 points (called polygon points) are required to construct a Bezier
curve of degree N. Only the ﬁrst and the last points effectively lie on the curve. For

example, the expression deﬁning a second -order Bezier curve is
6A7 = (1—u)23ﬁ| +2u(l-u)5F2.+uzb?;
where O is the origin of the coordinate system, M is a point on the curve, P1, P2, P3 are

the polygon points, and u is a parameter varying between 0 and 1 along the curve. For

u=0, M=Pl; and for u=l , M=P3. The corresponding curve is shown in Figure C3.

 

159

Bezier
polygon of
the curv

 
  
 
     
   
 

  

Seco :
order Bezie
curve

 

O

 

Figure C3. Example of Bezier curve

The point P2 does not lie on the curve, but the tangents at P1 and P3 are in the
direction of P2.

The general expression for a Bezier curve of degree N is

_, N+l _,
0M = 23,? (u)0P

k=l

- N _ N k-l _ N-k+I_N(N_l)-"(N_k+2) k-l _ N-k+l
With 8,, (m—[k-l} (1 u) — “—1)! u (1 u)

 

Any Bezier curve can be used in the program provided that the polygon points

deﬁning it are given.

C.2.l First guess of the streamlines

The number of streamlines and stations is chosen by the user. As a compromise

between accuracy and time calculation, 8 streamlines and 15 stations are chosen.

 

160

When the hub and shroud contours are deﬁned, the quasi-orthogonals can be
constructed by varying linearly the parameters u on the hub and shroud curves and
joining the corresponding points.

The position of the streamlines for the ﬁrst iteration is obtained by dividing the
length of each quasi-orthogonal by the number of streamlines. The corresponding points
on each Q.O. are then joined together to form a ﬁrst guess of the streamline. An example

of the corresponding calculation grid is given in Figure C4.

 

 

Figure C4. Example of calculation grid

C.2.2 Slope and radius of curvature

In order to solve the equation of motion, the slope and radius of curvature have to

be known at each point. A ﬁnite-difference method based on three points does not give a

161

good estimation of the radius of curvature, especially if the slope of the curve is higher
than 1 (Katsanis 1964).
In order to solve the problem, a local coordinate system is deﬁned at each station.

The new coordinate system is deﬁned in Figure C5.

 

 

 

 

Figure C5. Local coordinate system
To calculate the slope and the radius of curvature at Aj, a parabola passing through

AH, Aj, A,“ is constructed which has the following characteristics:

' f(YA,_.)=O
‘ f(X0, )= YA,
° f(YA,..)=O

where Y=f(X)=aX2+bX+c and (X,Y) are the coordinates in the local coordinate system.

The slope at Aj is then given by

 

162

dY
tans»:= — =b
[dX ]X=O

and the radius of curvature by

(a; )x=o Za

This method gives a good estimation of the radius of curvature. The values of the

.5
RC: (Hf—”Extw) (1+bzy'5

radius of curvature calculated analytically on the hub and shroud contour and the ones
obtained by the approximated method have been compared. The difference is less than
one thousandth. At the end points, this method is less accurate and can be in error by 10%

to 20%.

C.2.3 Blade angle and blade thickness distribution

The blade angle is deﬁned by a third-order polynomial in m/mT where m
represents the meridional distance along the hub (or the shroud) and m1 the total length of
the meridional hub (or shroud) contour. A typical blade angle distribution is shown in

Figure 4.4.

 

163

Blade angle dlstrlbutlon (Huh I. Shroud)

 

 

 

 

-30
-4o ‘ 3
-50 -
+ Hub

? + Shroud
-60
-70 ' I ' f ' r ' l '

0.0 0.2 0.4 0.6 0.8 1.0

Dimenelonleu Heddlonal Distance

Figure C6. Typical blade angle variation
Similarly, the blade thickness at hub and shroud are deﬁned by means of third-

order polynomial.

C.3 Method of solution

The momentum equation is ﬁrst solved using a second-order Runge-Kutta scheme

 

dC
as proposed by Katsanis (1964). Considering the equation d ’" = f (C m ,b) ,
m

Cm at the 0+1)st streamline can be estimated by

 

164

 

Cm;+1 =ij +[ﬂ] Am
dm j
mm}; =ij +[i'1C—m) Am
dm j+l
Cm . : Cm;+, +Cm;:,
L N 2

 

The continuity equation is then integrated along a quasi-orthogonal using a spline
ﬁt curve (piece-wise cubic interpolation). As shown by Marsh (1971), the “family of
velocity proﬁle obtained by solving [the momentum equation] cannot intersect”.
Assuming a value of the velocity at the hub, the momentum equation is solved along the
Q0. The corresponding mass ﬂow rate through the QC. can then be evaluated with the
continuity equation. If the calculated mass flow is different from the input mass ﬂow, the
assumed velocity at the hub is modiﬁed accordingly until convergence is reached. The
procedure is repeated for all the QC. Once the meridional velocity has been calculated,
the position of the streamlines is updated until convergence.

Once convergence has been reached at each station, the positions of each
streamline can be updated. To avoid over-correction, only a fraction of the new solution

is used, leading to the following relation:

ym = you + f * (yam - you)
where f is a damping factor (e.g.0.1).
The slope and radius of curvature is updated together with the enthalpy, angular
momentum and velocity gradients. The calculations can then be repeated until
convergence is reached on the streamline positions.

The ﬂow in the blade-to-blade plane can also be quickly evaluated from

 

165
dW _ Cm d(rC,)
d6 W dm
and the assumption that the relative velocity varies linearly in the blade to blade

direction (Stanitz and Prian 1951).

 

APPENDIX D

166

Appendix D

Estimation of the Inlet Blockage and of the Hub-to-Shroud Velocity

Variation

The AA impeller was analyzed with a radial inlet. The inlet boundary condition
(pressure, temperature, velocity direction, a, TKE) was derived from the return vane
analysis of the Al return system. The meridional velocity proﬁle from the hub to the

shroud in front of the leading edge is shown in Figure D. 1.

 
 
   

Cm (Wt)

seaport

Figure D.1 Meridional velocity profile at impeller inlet (radial inlet)

167

The velocity proﬁle in Region 1 and 3 can be represented by the following curve
ﬁt:
In region 1, Cm=227.6 xo'914 (correlation: R2=0.994)
In region 3, Cm=3l3.03x0'0877 (correlation: R2=O.932)
The displacement thickness 8* in region 1 is calculated as follows.
a 0.226
5' =I(1-b)dy = 1(1-%7§Y°"°” )1y
0 o
where y, 5, and 5‘ are normalized by the height of the passage, and correspond to
the distance from the surface, the boundary layer thickness, and the displacement
thickness. The velocity at the edge of the boundary layer in Region 1 is 168 ft/s.The
deﬁnition of the displacement thickness used is similar to the one on a flat plate. As

mentioned by Schlichting (p.239), the formula is still valid in an axisymmetric case as

long as the y-axis is normal to the surface.

We get 530.033 for region] and 830.0038 for region 3.

The inlet blockage is deﬁned by B = “’7

Ageum

 

In ﬁrst approximation, neglecting the blade cut-back, we have

 

Rf—RZ
B: r12 _r; (13.1)
s h

where R, = r, —6‘ b , R, = r, —5,:u,,b and b=rs-rh

shroud

 

168

Introducing the expression of Rs and R., in equation (D. l ), and neglecting

(5,?” - 551mm, )JWlth respect to (rh+rs), we get

i t
B 2:. 1 _ 61ml) — 6shr0ud

Numerically, we get B=95.3%

Considering the approximations made (y-axis normal to the surface, no cut-back),

the assumption of 5% blockage for the design is perfectly valid.

The AK value can be estimated by

= cmshmud z 251 =125

Cm 200

mean

 

AK

The calculations have been repeated upstream of the leading edge to ensure that the blade

blockage did not affect signiﬁcantly the evaluation of AK and B.