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Imperial College of Science, Technology and Med; cine

UNIVERSITY of LONDON

DESIGN AND TESTING OF A HIGHLY LOADED

MIXED FLOW TURBINE

by

Miloud ABIDAT

June 1991

This thesis forms part of the requirements for the Doctor of

Philosophy degree of the University of London and the Diploma of

Imperial College.

AKNOWLEDGEMENTS

The author wishes to thank Dr N.C. BAINES for his

supervision, help and advice during the completion of this project.

The author also wishes to thank J.DAVIS for his minutious

work in the rig instrumentation, A.K.AWAN for his contribution in

machining the volute, Holset Ltd for providing the necessary

hardware and H.CHEN for his help in the experimental work.

The author is also gratefull to his wife and Dr Z. BENHEDDI

for their support.

Abstract

A method of designing a new generation of highly

loaded mixed flow turbines for turbocharger appl icat ion is

described. A review of the published work concerning radial

turbines and closely related to mixed flow turbines is presented.

A 1-D design method was developed. It is used to

def ine the overal l turbine d imensions and to analyse its

performance at the off design conditions. The method is applicable

to both radial and mixed flow turbines. A series of designs had

been produced and then analysed by the off design performance

prediction method. The effects of several geometrical parameters

on the performance of the designs were investigated.. This had led

to the selection of an optimum rotor design for further analysis.

An analyt ica l method based on the Bezier

polynomials is used to define the three dimensional blade

geometry. The rotor geometry is optimised by means of a quasi-

three-dimensional method for the flow analysis. The effect on the

flow inside the rotor of three factors inf luencing the blade

geometry has been investigated. These consist of the rotor blade

angle variation along the leading edge, the rotor length and the

blade curvature.

Two mixed flow turbine prototypes have been

manufactured and experimentally tested. These differ mainly in the

rotor inlet, which is a constant blade angle in one case, and a

notionally constant incidence angle at design conditions in the

other case. The former turbine showed signif icant ly higher

efficiencies across the operating range, and possible reasons for

this are discussed. The experimental analysis concerns the

11

measurement of the turbine overall performance, the pressure

distribution along the rotor shroud and the flow field downstream

of the rotor exit.

I l l

Table of Contents

Aknowledgements i

Abs t rac t i i

Table of Contents 1

List of Symbols 5

List of figures 9

1. INTRODUCTION 1 8

2. BIBLIOGRAPHICAL REVIEW 2 5

2.1. Mixed Flow Turbine Survey 2 5

2.2. One-Dimensional Design 27

2.2.1. Calculation of Overall Dimensions 27

2.2.2. One-Dimensional Flow Analysis 30

2.3. Blade Geometry Design 34

2.4. Flow Field Analysis 3 6

3. ONE-DIMENSIONAL DESIGN 51

3.1. Introduct ion 51

3.2. Design Conditions Analysis 52

3.2.1. Rotor Inlet and Scroll 53

3.2.2. Rotor 57

3.3. Losses 60

3.3.1. Usefuk Work 60

3.3.2. Scroll Losses 61

3.3.3. Disk Friction Losses 63

3.3.4. Rotor Friction Losses 64

3.3.5. Rotor Blade Loading Losses 6 5

3.3.6. Leakage Losses 65

3.3.7. Exhaust Losses 66

3.3.8. Incidence Losses 66

3.4. Turbine Performance 69

3.4.1. Ef f ic iency 69

3.4.2. Net Output Power 70

3.5. Off-Design Performance Predictions 70

3.5.1. Casing Analysis 72

3.5.2. Rotor Analysis 73

3.5.3. Off-Design Performance Characteristics 75

3.6. Conclusions From the One-Dimensional Design 76

3.6.1. Effect of the Diameter Ratio D1/D2 76

3.6.2. Effect of the Volute Exit Flow Angle 77

3.6.3. Effect of the Blade and Cone Angles at Rotor Inlet 78

3.6.4. Effect of the Diameter Ratio D2/D3 78

3.6.5. Selection of the Design 79

4. BLADE GEOMETRY 101

4.1. Bezier Polynomial 102

4.2. Blade Geometry Generation 104

4.2.1. Hub and Shroud Profile Generation 105

4.2.2. Blade Curvature 107

4.2.2.1. Radial Fibre Blade 107

4.2.2.2. Camberline Generation 108

4.2.3. Examples of Bezier Polynomial Apllications 1 12

5. FLOWFIELD ANALYSIS 118

5.1. Streamline Curvature Method 120

5.1.1. Meridional Surface Calculation 121

5.1.2. Blade-to-Blade Surface Calculation 124

5.2. Fnite Volume Method 125

5.2.1. Governing Equations 125

5.2.2. Grid Generation 127

5.2.3. Finite Volume Discretisation 127

5.2.4. Corrected Viscosity Scheme 128

5.2.5. Boundary Conditions 130

5.2.6. Initial Conditions 131

5.2.7. S tab i l i t y 1 32

5.2.8. Control Volume and Surface Calculation 1 32

6. TURBINE DESIGN 1 39

6.1. Casing 1 39

6.2. Rotor Design 141

6.2.1. Influence of the Leading Edge Shape 142

6.2.2. Influence of the Rotor Length 143

6.2.3. Influence of Blade Curvature (63) 1 44

6.2.4. Selection of the Prototype 1 45

6.3. Analysis of Rotor A and B 146

6.3.1. Blade Geometry 1 46

6.3.2. Flow Analysis of Rotor A and Rotor B 146

7. EXPERIMENTAL INVESTIGATION OF TWO MIXED FLOW

TURBINES 181

7.1. Description of the Test Rig 181

7.2. Performance Measurement 1 82

7.2.1. Test Rig Conditions 1 83

7.2.2. Mass Flow Measurement 1 85

7.2.3. Turbine Performance Characteristics 1 86

7.2.4. Nozzle Pressure Measurement 1 89

7.3. Exhaust Turbine Flow Measurement 1 90

7.3.1. Calibration Factors 191

7.3.2. Flow Parameters Calculations 1 93

7.3.3. Performance Calculation 193

7.4. Experimental Results 1 94

7.4.1. Overall Performance 1 95

7.4.2. Traverse Measurements 1 97

7.4.3. Shroud Pressure 200

7.4.4. Incidence Angle at Rotor Inlet 202

7.5. Conclusion From the Experimental Investigations 203

8. CONCLUSION 240

8.1. Summary of the Design Model 240

8.2. Experimental Analysis 241

8.3. Results of Experimental Analysis 241

8.5. Suggestions and Future Work 242

Appendix 244

References 247

LIST OF SYMBOLS

A Area, coefficient

B Independent term of the Euler system of equations, coefficient

Bs Blade solidity

b Blade height

C Absolute velocity, coefficient, approach velocity factor

Cf fr ict ion coeff ic ient

Cfx Coef f ic ient

Cp Specific heat coefficient

D Diameter (rotor)

d Diameter (scroll section)

E Diameter ratio, compressibility factor

e Energy, thickness

F Convective flux term of the Euler system of equations

G • Convective flux term of the Euler system of equations

H Convective flux term of the Euler system of equations

h Enthalpy

i Quasi-orthogonal number

j Stream surface number

k bladewise surface number

Kgp Coefficient of losses

K coefficient

L Channel length

M Mach Number

m Mass flow rate

m f r Non dimensional mass flow rate

Nd Rotational speed

NQ Rotational speed in the case of a cold test

N|_| Rotational speed in the case of a hot test

n Coef f ic ient

P Pressure

p Coefficient, total pressure recovery factor

Q Loss coefficient

q Coefficient, distance along quasi-orthogonal, gas flow factor

R Gas constant

Rex Reynolds number

r Radius, temperature recovery factor

S Swirl coeff icient

T Temperature

t Time

U Peripherical velocity

u Coef f ic ient

V Absolute velocity

W relative velocity, work

X Axial coordinate, left probe pressure factor

Y Right probe pressure factor

a Absolute flow angle, discharge coefficient

P Relative flow angle, orifice plate to pipe diameter ratio

5 Cone angle

ri Ef f ic iency

X Prewirl

|i Dece lera t ion /acce lera t ion ratio, c inemat ic v iscos i ty ,

numerical viscosity

X) Dynamic viscosity

Q,o) Rotational frequency

\\i Rotor loss coefficient, azimut angle

p Density

Pd Degree of reaction

y Specific Heat ratio, cone angle

a Vector of the independent variables in the Euler equations

e Angular coordinate, probe angle

^ Scroll loss coefficient

^ Coef f ic ient

t Torque

SUBSCRIPTS

b Blade

C Cold test

c cold test

c I Clearance

d Downstream

e Exit

B< Exit, exhaust

exh Exhaust

f Friction, front pressure tapping

H Hot test

h Hot test

i In le t

in Incidence, inlet

is Isentropic

L left probe tapping

I Blade loading

m Face number of control volume, measured

p Probe

R Right tapping

r Radial

s Isentropic, static

sc Scro l l

th Work

t s Total to static

I I Total to total

u Peripheri0al, upstream i

w Relative frame of coordinates

X Inlet, axial

y Exit

e Tangential

0 Scroll inlet control surface, reference point

1 Scroll exit, vaneless inlet channel control surface

2 Rotor inlet, vaneless exit channel control surface

3 Rotor exit

4 Turbine exit

* Absolute stagnation state

+ Relative stagnation state

SUPERSCRIPTS

Updated parameters

t t i m e

LIST OF FIGURES

Chapter 1

Fig.1.1 Variation of static pressure and absolute flow angle

at the volute exit periphery (44)

Fig.1.2 Mixed flow turbine design

Chapter 2

Fig.2.1 Comparison of camberlines (34)

Fig.2.2 Mixed flow turbine components

Fig.2.3 Influence of rotor geometry on maximum efficiency

of radial turbines (48)

Fig2.4 Maximum attainable design efficiency for radial

turbines (48)

Fig.2.5 Effect of specific speed on stator blade height for

maximum static efficiency (48)

Fig.2.6 Effect of specific speed on tip-diameter ratio

corresponding to maximum static efficiency (48)

Fig.2.7 Effect of specific speed on optimum stator-exit

angle (48)

Fig.2.8 Effect of specific speed on optimum blade-jet speed

ratio (48)

Fig.2.9 Loss Distribution along curve of maximum static

efficiency (48)

Fig.2.10 Comparison of high specific speed radial and mixed flow

turbines static efficiency (35)

Fig.2.11 Chart of maximum efficiency for radial turbines (36)

Fig.2.12 Effect of rotor blade geometry on radial turbine

performance (36)

Fig.2.13 Comparison of test and computed turbine characteristics

(37)

Fig.2.14 Variation of turbine losses for different radial rotor

configurations (50)

Fig.2.15 Effect of volute geometry on radial turbine performance

(45)

Fig.2.16 Effect of volute geometry on radial turbine performance

(56)

Fig.2.17 Comparison of predicted and measured flow angles at

casing exit (7)

Fig.2.18 Measured flow angle along the exit casing periphery (7)

Fig.2.19 Nozzle loss coefficient (5)

Fig 2.20 Entropy generation. 3D viscous flow analysis (54)

Fig.2.21 Computed total pressure loss and Mach number in the

rotor channel of a radial turbine (3D viscous flow

analysis) (54)

Fig.2.22 Rotor loss coefficient (45)

Fig.2.23 Effect of axial and radial clearances on efficiency (45)

Fig.2.24 Definition of meridional channel by means of Bezier

surfaces (47)

Fig.2.25 Use of patches for meridional channel definition (60)

Chapter 3

Fig.3.1 Velocity triangle at rotor inlet

Fig.3.2 Mixed flow turbine ( overall dimensions )

Fig.3.3 Velocity Triangle at rotor exit

Fig.3.4 Expansion process in a mixed flow turbine

Fig.3.5 Velocity Triangles at rotor inlet (NASA model for

incidence loss calculation)

Fig.3.6 Incidence loss model (30)

Fig.3.7 Off-design performance prediction

10

Fig.3.8 Expansion Process

Fig.3.9 Comparison between measured and computed total to

static efficiencyof the H2D X17Q3 turbine

Fig.3.10 Effect of diameter ratio D1/D2 :

Turbine characteristics at 98000 rpm

Fig.3.11 Effect of diameter ratio D1/D2 : Turbine characteristics

at X17Q3 turbine running conditions

Fig.3.12 Effect of volute exit absolute flow angle:


Fig.3.13 Effect of volute exit absolute flow angle ; Turbine

characteristics at X17Q3 turbine running conditions

Fig.3.14 Effect of blade and cone angles at rotor inlet ;


Fig.3.15 Effect of blade and cone angles at rotor inlet : Turbine

characteristics at X17Q3 turbine running conditions

Fig.3.16 Effect of diameter ratio D2/D3 :


Fig.3.17 Effect of diameter ratio D2/D3 : Turbine characteristics

at X17Q3 turbine running conditions

Fig.3.18 Total-to-static efficiency vs velocity ratio UC

Fig.3.19 Total-to-static efficiency vs pressure ratio PR

Fig.3.20 Mass flow rate characteristics (design) and Holset

turbine X I 7 0 3 swallowing capacity

Fig.3.21 Comparison between the design and Holset turbine X I703

swallowing capacity

Fig.3.22 Absolute flow angle at rotor inlet

Fig.3.23 Incidence flow angle at rotor inlet

Chapter 4

Fig.4.1 Nth degree Bezier polynomial

Fig.4.2 3rd degree Bezier polynomial

11

Fig.4.3 Hub and shroud profiles generation by a Bezier

polynomial

Fig.4.4 Mixed flow rotor ; overall dimensions

Fig.4.5 Radial fibres blade element

Fig.4.6 Camberline generation

Fig.4.7 Examples of camberline generation by a Bezier

polynomial

Chapter 5

Fig.5.1 Rotor channel discretisation

Fig.5.2 Volume discretisation

Fig.5.3 Velocity triangle in the meridional plane

Fig.5.4 Domain of numerical dependence (CFL condition)

Fig.5.5 Control volume transformation

Fig.5.6 Surface transformation

Chapter 6

Fig.6.1 Scroll channel

Fig.6.2 Casing design

Fig.6.3 Blade camberline

Fig.6.4 Blade surface velocity distribution : Effect of the

leading edge shape (camberlines A)

Fig.6.5 Blade surface velocity distribution : Effect of rotor

length (camberlines B). (Constant blade angle along the

leading edge)

Fig.6.6 Blade surface velocity distribution ; Effect of rotor

length (camberlines D). (Constant incidence angle along

the leading edge)

Fig.6.7 Blade surface velocity distribution ; Effect of the

tangential coordinate of the trailing edge (camberline C )-

12

Constant blade angle at inlet

Fig.6.8 Turbine A (constant blade angle at rotor inlet)

Blade geometry : Camberline and meridional blade

surface projection

Fig.6.9 Turbine B (constant incidence angle at rotor inlet)

Blade geometry : Camberline and meridional blade

surface projection

Fig.6.10 Blade angle along streamlines

Fig.6.11 Streamline projection on a (r,r8) plane

Fig.6.12 Blade surface projection on a (r,8) plane

Fig.6.13 Turbine A :

Meridional surface flow calculation (S.L.C)

ND = 50000. rpm, UC = 0.61, To* = 923. °K

Fig.6.14 Turbine A ;


ND = 75000. rpm, UC = 0.61, To* = 923. °K



ND = 98000. rpm, UC = 0.61, To- = 923. °K



ND = 110000. rpm, UC = 0.61, To' = 923. °K


Blade surface velocity (S.L.C)

ND = 50000. rpm, UC = 0.61, To* = 923. °K



ND = 75000. rpm, UC = 0.61, To- = 923. °K



ND = 98000. rpm, UC = 0.61, To- = 923. °K

13



ND = 110000. rpm, UC = 0.61, To- = 923.

Fig.6.21 Turbine B;


ND = 50000. rpm, UC = 0.61, To- = 923. °K

Fig.6.22 Turbine B ;


ND = 75000. rpm, UC = 0.61, To- = 923. °K

Fig.6.23 Turbine B :


ND = 98000. rpm, UC = 0.61, To- = 923.



ND = 110000. rpm, UC = 0.61, To- = 923. °K



ND = 50000. rpm, UC = 0.61, To- = 923. °K



ND = 75000. rpm, UC = 0.61, To- = 923. °K



ND = 98000. rpm, UC = 0.61, To- = 923. °K



ND = 110000. rpm, UC = 0.61, To- = 923. °K

Chapter 7

7.1 Mixed flow turbine test rig

14

Fig.7.2 Shroud pressure tapping

Fig.7.3 Exhaust turbine duct flow measurement by a traversing

probe mechanism

Fig.7.4 Traversing probe :

Flow parameters and performance calculation

Fig.7.5 Mixed flow turbine total to static efficiency vs velocity

ratio U/C

Fig.7.6 Mixed flow turbine total to static efficiency vs velocity

ratio U/C and rotational speed

Fig.7.7 Mixed flow turbine total to static efficiency vs pressure

ratio and rotational speed

Fig.7.8 Non dimensional mass flow rate characteristics

Fig.7.9 Torque as a function of the non dimensional mass flow

rate and rotational speed

Fig.7.10 Comparison between the measured and predicted

performance of turbine A and B (50% equivalent speed)











Fig.7.16 Comparison between the total to static efficiency

obtained from the exhaust duct traversing measurement

and that obtained from the compressor work

measurement (turbine A)

15

Fig.7.17 Comparison between the mass flow obtained from

the exhaust duct traversing measurement

and that measured by means of an orifice plate

(turbine A)

Fig.7.18 Comparison between the total to static efficiency

obtained from the exhaust duct traversing measurement

and that obtained from the compressor work

measurement (turbine B)

Fig.7.19 Connparison between the mass flow obtained from

the exhaust duct traversing measurement

and that measured by means of an orifice plate

(turbine B)

Fig.7.20 Traversing measurement :

Total to static efficiency

Total to total efficiency

Fig.7.21 Turbine exhaust duct flow measurement by a traversing

probe mechanism. Turbine A (50 % speed )








probe mechanism. Turbine B (50 % speed )







16

Fig.7.29 Mixed flow turbine shroud pressure measurement

( 50 % equivalent design speed )






( 100 % equivalent design speed

Fig.7.33 Comparison between the absolute flow angle at rotor

exit and at the exhaust duct cross section (50 % speed)

Fig.7.34 Comparison between the absolute flow angle at rotor

exit and at the exhaust duct cross section (90 % speed)

Fig.7.35 Comparison between measured and computed shroud

pressure distribution ( Turbine A )

3-D inviscid calculation (74)

2-D streamline curvature method (25)

Fig.7.36 Computed incidence angle along the rotor inlet

Fig.7.37 Computed incidence angle along the rotor inlet

Fig.7.38 Mixed flow turbine optimum velocity ratio and total to

static efficiency as a function of rotor speed

Fig.7.39 Comparison between high specific speed mixed flow

turbines and radial turbines operating at maximum

efficiency (48)

17

Chapter 1

1. INTRODUCTION

Turbochargers are widely used in diesel engines as

a means of increasing the output power. They were used principally

in the marine propulsion field at their early apparition and became

in recent years commonly used for road transport applications.

With the growing importance of turbochargers,

their design and manufacture is becoming a rather specialised field

of some educational and industrial institutions,

Turbochargers with radial compressors and

turbines are the most commonly used because of their ability to

deliver/absorb more power in comparison to axial ones of similar

size. Radial turbines are mainly used for automotive engine

applications and have the advantage of retaining a high efficiency

when reduced to small sizes. They can operate at high expansion

ratio. On the other hand, axial turbines, which are used for large

turbocharger (marine and railway) engines, are made of single

stage or of several stages.

The turbine which is an important component of a

turbocharger, consists essentially of a casing and a rotor. The

casing, whose function is to convert a part of the engine exhaust

gas energy into kinetic energy and direct the flow towards the

rotor inlet at an appropriate flow angle, can be vaneless or fitted

with a stator. In the second case, the turbine has a good

aerodynamic performance at design conditions but poor efficiency

18

at off design condition compared with a vaneless stator. This is

probably due to the fact that the flow can not adjust itself with

the changing operating conditions resulting in high incidence

losses. On the other hand, a turbine with a nozzleless spiral casing

correctly designed is slightly less efficient than the above

mentioned, at design condition but remains fairly efficient over a

wide range of operating conditions. Ideally, the flow angle at

volute exit does not change with operating conditions and is

function only of the geometry. It is however less uniform around

the periphery and especially near the volute tongue ( figure 1.1).

Vaneless turbines which have reasonable

performance and low cost , are the most used in turbochargers for

automotive engines, while vaned turbines can be used in

appl icat ions requiring high eff ic iency at f ixed operating

conditions.

Radial turbines have been adopted for small engine

applications because of their simplicity, cost, reliability and

relatively high efficiency. The turbine requirements in highly-

loaded turbocharged engines are changing. Higher air/fuel ratio

required for emissions and the use of intercoolers result in

significantly lower exhaust temperatures. This together with the

fact that more power required for boost pressure has to be taken

from the exhaust has resulted in smaller turbine housings being

used, which reduces the turbine efficiency. The turbine speed is

limited by stress, so that the requirement is for a turbine stage

with maximum efficiency at a lower U/C (tip speed / spouting

velocity ) than the usual value of 0.7 to which the conventional

radial turbine is constrained ( see appendix A).

19

The most feasible way to do this is to make the

inlet blade angle positive as opposed to the usual value of zero.

This means that the rotor inlet can not be radial, but must be

mixed, so that inlet streamlines in the meridional plane have radial

and axial components. It is then possible to have non zero (i.e. non

radial) inlet blade angles while retaining radial blade fibres (the

projection of the mean blade surface on a reference cylinder is

represented by a unique curve). This type of blade geometry has the

advantage of avoiding additional stresses due to bending.

What is not clear at present is how such a turbine

will perform aerodynamically, particularly in comparison with an

equivalent radial inflow machine. Intuitively, it might be expected

that the reduced curvature of the shroud profile in the meridional

plane would be beneficial, but against this must be balanced the

additional turning in the blade-to-blade plane because of the

"bucket-shaped" blades, and the largely unknown properties of the

curved accelerating passage forming the stator. The development

of aerodynamic design tools to handle a mixed flow geometry, in

terms of both overall dimensions and performance predictions , and

flowfield calculations, will be presented in this thesis. The design

procedure of a mixed flow turbine can be summarized by the chart

in figure 1.2.

Chapter 2 contains a literature survey of the work

carried out on small turbines and closely related to mixed flow

turbine design.

Chapter 3 deals with the one dimensional design. In

a first step, the overall dimensions of the mixed flow turbine are

obtained. This is done by taking into account the fixed parameters

20

(total pressure and temperature, net output power or mass flow,

rotational speed and housing A/R ratio). In a first approach, results

from previous works concerning especially radial turbines are

used. Correlations for loss prediction and optimum geometrical

characteristic ratios are assumed to be also valid for mixed flow

turbines. The second part deals with the off-design performance

prediction of the selected design. It enables the performance

characteristics of the design such as efficiency, swallowing

capacity, etc to be predicted. These are compared with those of a

radial turbine having the same housing A/R ratio.

Chapter 4 deals especially with the three

dimensional blade shape design. An analytical method based on the

Bezier polynomials is used to define the hub, shroud and the

camberline profiles of the blade. Special care is given to the

leading edge where the flow direction varies from hub to shroud.

The influence of rotor length and blade curvature are also

investigated.

Chapter 5 presents the two-dimensional flow

analysis (streamline curvature method) both in a meridional plane

and a blade-to-blade surface. The method is widely used in

turbomachinery design. The combination of the method presented in

chapter 4 and the flow analysis by a S.L.C method permits a rapid

analysis of the design and its modification whenever it is

necessary until an optimum one is obtained. Several methods used

in turbomachine flow analysis are also reviewed. The time

dependent solution of the Euler equations is presented in more

detai ls.

Chapter 6 deals with the rotor geometry design. A

21

series of rotor geometry designs were produced and analysed by

the quasi-three dimensional streamline curvature method. The

effects (on the internal flow) of the blade angle at rotor inlet, the

rotor length and the blade curvature were investigated. Two rotor

designs were selected for the experimental testing.

Chapter 7 is concerned with the experimental

analysis of two mixed flow turbines (turbines A and B). Turbine A

is designed to have a constant blade angle along the leading edge,

while turbine B is designed for a constant incidence angle at design

conditions along the leading edge. Rotor inlet and exit flow

parameters and pressure distribution along the shroud are analysed

at different conditions. Thus a comparison between predicted and

experimental performance is obtained giving supplementary

understanding of the mixed flow turbine behaviour.

Stress analysis and blade thickness definition as

well as the manufacturing of the two prototypes were undertaken

by Holset Engineering Co. Ltd. This work is a part of a collaborative

research and development programme between Holset, Bath

University and Imperial College. The primary aim of the research

programme is to develop the technology for new, high pressure

ratio, high efficiency turbochargers for the next generation of

turbocharged automotive-type diesel engines. Holset is mainly

concerned with the programme definit ion, mechanical and

aerodynamic design support and prototype manufacturing. Bath

University on the other hand is involved in the centrifugal

compressor design and testing, while the design and testing of the

mixed flow turbine is the task of Imperial College.

22

Frcntrc Tapping* m tOr mm T«rbine Vohrte Section lor Geometrically SimiUr T i r t m

I 0

t -10

-20

a -JO

= - 4 0

% -50

MfpfewNt Neon Pfww* Acreii 1 uro*»c f \ i - ^ 1

_ o-OU iNsmI I AOLER W*1 D

ISO 2 * 0 M O 0 < 0 1 2 0

Azimuth Angle «*

180

.10

2 0

-10

TEV.

% 10t4 m#i TwrWn*

^ -*• #7 ?)MML TI*W»

— 'jO I •*».'

"9 — 0 ..-•*

.1 »$. JL_-0-72, T *Ti

* P 'I .ni iN.wui I

T«T»» CLIMOHL. _ C ON HTMRND I

ISO 240 300 0 ( 0 120

Azimuth Angle •*

ISO

Fig.1.1 Variation of static pressure and absolute flow angle at the volute exit periphery.

( Scrimshaw, 44)

23

f START )

DESIGN PARAMETERS

1-D DESIGN

1-D OFF-DESIGN ANALYSIS

Y

2-D OR 3-D FLOW FIELD ANALYSIS

•<CGOOD R1 E:SULTS2>

Y

PROTOTYPE MAJnJFACTURING

EXPERIMENTAL ANALYSIS

GOOD RESULTS ?

< D

/ • " \

3-D BLADE GEOMETRY

i * \

BLADE THICKNESS

" 0

O

O

o

< D

FIG.1.2 MIXED FLOW TURBINE DESIGN

24

Chapter 2

2. BIBLIOGRAPHICAL REVIEW

A literature survey related to the work carried out

on small turbines for turbocharger application is presented with

special reference to those on mixed flow turbine design. The

interest in this study is given to the one-dimensional design of

radial/mixed flow turbines with an emphasis on the different loss

models applied so far, to the the method used for blade geometry

generation and to the numerical methods ( two and three

dimensional ) used in turbomachine flow analysis.

2.1. Mixed Flow Turbine Survey

The mixed flow turbine concept received attention

in the 1970's [20], [38] and [2], It was demonstrated [2] that the

point of peak efficiency was indeed reduced below 0.7, and that the

steady flow eff iciencies of the mixed flow turbine were

comparable with those of a radial turbine of similar size, it was

also shown that a mixed flow turbine could deliver a slightly

higher mass flow than a comparable radial inflow turbine.

Okapuu [15] presents results from a research

programme on mixed-flow gas generator turbines. Four prototypes

have been tested. They all have their leading edge oriented at 30

degrees relative to the axis of rotation in the meridional plane, a

mean blade angle of 25 degrees and the same hub and shroud

profiles. Three of them have the same camberline with radial fibre

elements, differing only on the number of blades or rotor length

25

while the fourth has non-radial blade fibres at the leading edge.

Peak efficiencies higher than 90% (total-to-total) were obtained

with each turbine but for a lower pressure ratio than design. It was

also observed that the optimum incidence angle is around -20

degrees. The author suggests that increasing the rotor inlet blade

angle may move the point of maximum efficiency towards the

design pressure ratio.

Yamaguchi et al [34] analysed four mixed flow

turbine rotors whose blade camberlines are presented in figure 2.1.

From a quasi-three dimensional flowfield analysis, the rotor which

has camberline "C" was selected for experimental tests which

showed an improvement in the flow at the shroud. The author

showed also that a substantial improvement in engine performance

was obtained with a mixed flow turbine.

In his paper Gibbs [35] showed an improvement in

mixed flow turbine efficiency in comparison with a radial turbine

similar in size and having similar mass flow rate characteristics.

An increase in the air-fuel ratio as well as a diminution in engine

smoke and gas exhaust temperature were noticed with the mixed

flow turbine.

Rodgers [36] suggests that using mixed flow

turbines as an alternative to radial turbines to overcome the

limiting constraints - stress due to high tip speed combined with

high temperature and rotor inertia - in high specific speed

turbochargers and turbojet application.

26

2.2. One-Dimensional Design

Some work on radial and mixed flow turbines

(figure 2.2) is discussed. It deals either with the calculation of

overall dimensions or with the off-design performance prediction.

2.2.1. Calculation of overall dimensions

A considerable number of • aerodynamic and

geometric parameters influence the design of a turbine. Their

selection must lead to an efficient turbine having the desirable

performance characteristics at a wide range of operating

conditions.

The aerodynamic parameters are generally related

to the flow condit ions (pressure ratio, inlet stagnation

temperature and mass flow or power) and rotor speed (rotational

speed and velocity ratio). Geometrical parameters are selected

according to the accumulated data and experience on this type of

machine (clearance, rotor flow path length, etc.) or as a result of

the design constraints. The combination of some parameters

provides a dimensionless parameter expressing aerodynamic and

geometric similarity for a variety of turbine geometries or flow

conditions. Examples of such parameters are :

- dimensionless mass flow:

mfr = 111]

P. A V y

- velocity ratio

27

U 71 N D U/C ^ [2.2]

0 * / l - Y

7 6 0 \ / 2CpTo, ( l -P rT ' )

- Specific speed expressed either as a function of the rotational

speed ND (equation 2.3a) or as a function of the rotational

frequency co (equation 2.3b)

• NS = NDQ3^'2 / h3'4 [2.3a]

NS = coQ3 '2 / H3/4 [2.3b]

Rohlik (48) analysed the effect of specific speed on efficiency. By

substituting or co , Q3 and H as follows :

CO = 2. U2 / D2 Rotational frequency

60 U N „ = Rotational speed

K D j

Q, = 7C D , b V, Volume flow at rotor exit ^ 3 m 3 3

y 2 H = — ^ ) Ideal work based on inlet and exit total pressure

2 AH .d

It was obtained The following expression for the specific speed

N = K [3.4] U3 Dj AH'.d

Where:

K : constant ( KND = 6 0 (2 G)^'"^ / AND Kw = 2 " ^ )

K = KND ( when british units are used )

28

K = Kco ( when SI units are used )

• 2 Rotor inlet diameter

• 3 ^ : Mean rotor exit diameter

bg : Rotor exit passage height

V3 Rotor exit velocity ( axial at design conditions )

VQ* : Jet speed

AH' : Ideal work based on inlet total and exit total pressures

AH : Ideal work based on inlet total and exit static pressures

A variety of combinations of the terms in equation

[2.4] were analysed in reference (48) to optimise the design of

radial turbines. Results in figures 2.3 show the variation of total-

to-static eff iciency for a wide range of specific speeds

corresponding to different combinations of the parameters in

equation 2.4 and a constant stator exit flow angle. Similar curves

to those of figure 2.3 corresponding to different values of the

stator exit flow angle were used to plot the curves of maximum

total-to-static and total-to-total efficiencies (figure 2.4) and are

referred to as the optimum geometry curves. Figures 2.5 to 2.9

show the effect of specific speed on the different aerodynamic and

geometric characteristics of the turbine. These results were

obtained from analytical analysis and therefore are only as

accurate as the loss models on which they are based. Recent

developments of high specific speed radial and mixed flow turbines

show that optimum static efficiencies recorded are greater than

those given by Rohlik optimum geometry curves as shown in figure

2.10.

Rodgers in his paper (36) relates the maximum

efficiency attainable by radial turbines to similar parameters as

those used by Rohlik (48). The analysis was based on experimental

29

performance measurements of a series of radial turbines and the

results are presented in figures 2.11 an 2.12. Figure 2.11 shows the

effect of exit velocity ratio (Cm3/U2), velocity ratio (U2/V0*) and

non dimensional speed Ns on the maximum turbine static

ef f ic iency.

Ng a [2.5]

The expression of the non dimensional speed (equation 2.5) from

reference 36 is only a relation of proportionality and thus can not

be used to calculate the Ns as it gives a different value from that

of equation 2.3b ( for the same turbine and the same conditions).

Figure 2.12 on the other hand gives the influence of the diameter

ratio D2/D3 and blade solidity Z L/D2 ( which characterise the rotor

geometry ) on the maximum turbine static efficiency.

2.2.2. One Dimensional Flow analysis

The flow through a turbine is three-dimensional

and consequently very complex and not yet fully understood

especially in the rotor channel. This makes any one-dimensional

model questionable because of the numerous assumptions in the

flow description. Different methods have been proposed to model

the one-dimensional flow through a turbine (2), (21), (37), (38),

(43), (45) and (48). The flow properties are solved along a mean

streamline on some key stations (38) using momentum, energy and

continuity equations. These equations are used in combination with

loss coefficients in order to model the real flow. They generally

differ from each other by the loss correlation employed. Wallace

(21) states that a simple isentropic treatment of the flow provides

30

results with an accuracy of 10%. More sophisticated analysis are

reviewed by Wallace et al (38), Rohlik (48) and Rodgers (37). In

reference (38), the authors describe a method using a unique

equation which can be used to solve the flow in the different parts

of the turbine (volute, nozzle, interspace, rotor or diffuser). The

terms of this equation which is presented in section 3.5 (equation

3.41) are a combination of the flow governing equations and terms

taking into account the losses in the turbine component considered

(volute, nozzle, interspace, rotor or diffuser). The method is well

suited for programming and needs only the specification of the

inlet conditions (mass flow and stagnation temperature and

pressure), the geometry at component exit, the losses and the blade

speed when the rotor is considered in order to define the exit flow

parameters. It is applied for both stationary and rotating ducts.

Rodgers (37) describes an analytical method for the performance

predictions of a radial turbine. Experimental results showed a good

agreement between predicted and measured efficiency and mass

flow for incidences ranging from -40 to +40 degrees. Although the

method relies heavily on empirical loss coefficients, it has been

claimed to achieve an accuracy of 2% on efficiency and 3% on mass

flow . These results are presented in figure 2.13 from reference

(37). Rohlik (48) gives a detailed analysis of losses penalising the

turbine performance and which are used in the one-dimensional

flow analysis as illustrated by figure 2.9. A similar illustration of

loss magnitudes for different turbines is presented in figure 2.14

from reference (50).

In the volute, the flow is modelled with a variety

of extensions to the free vortex equation. Several simple

correlations, which take into account the losses in the scroll, have

been adopted to solve the flow in this part of the turbine. The

31

effect of boundary layer growth and secondary flow is translated

by means of blackage factor B introduced in the continuity equation

(2.6) at volute exit.

m = p2 C'2\n^2 [2-6]

Friction losses are introduced by means of a swirl coefficient S in

the free vortex equation 2.7 or by a total pressure loss coefficient

Y in equation 2.8.

r2 C62 = S rO C80 [2.7]

Y = ( PO* - P2* ) / ( PO* - PO ) [2.8]

Japikse (29) mentioned values of 0.05 ~ 0.15 for the blockage

coefficient B, 0.85 ~ 0.95 for the swirl coefficient S and 0.10 ~

0.30 for the total pressure loss coefficient Y.

The design of the volute is of great importance

because it influences the turbine performance as demonstrated in

figure 2.15 by Barnard and Benson (45) and figure 2.16 by Tennant

(56). Several works have been conducted to investigate the flow

behaviour in the volute of radial turbines (7), (11), (51), (52) and

(53). Bhinder (7) shows in figure 2.17 the relationship between the

flow angle at casing exit and the A/R ratio (ratio of the area and

the radius of the centroid at inlet volute section). For one

dimensional design analysis, the flow angle at casing exit is

assumed constant. However it has been demonstrated from

experimental measurements that it has a non uniform distribution

in the periphery as shown in figure 2.18.

32

Nozzle losses are generally Included in the scroll

losses and when taken separately, they can be sometimes of equal

magnitude as those of the scroll (37). Benson et al (5) gives a

quantitative evaluation of the nozzle losses which are reproduced

in figure 2.19 and show high values for low mass flow rates.

Losses in the interspace are generally neglected provided that it

has a short length otherwise they are calculated as those of a

straight duct (37).

In the rotor, losses are usually subdivided into

different parts (incidence losses, friction losses, clearance losses

etc.), and determined by means of loss correlations (4), (13), (37),

(38) and (48).

Several methods were devised to predict the losses

due to the flow incidence which plays an important part in the drop

of turbine efficiency at off-design conditions. The methods are

discussed in detail by Whitfield (33) with suggestions for mixed

flow turbine applications. These method are applied for loss

calculation in both vaned casings and rotor. It is agreed that

optimun efficiency occurs at negative incidence angle in the range

of -10.0 to -40.0 degrees. A three dimensional viscous flow

analysis of a radial flow turbine by Kitson et al (54) showed that

the rotor can cope with high negative incidence (0.0 to -55.0

degrees) without serious penalty in turbine efficiency. However the

turbine performance deteriorates rapidly as the flow incidence

moves positively. Illustration of the losses taking part in the rotor

are illustrated by figures 2.20 ( enthropy increase ) and 2.21 (total

pressure loss). Rotor loss coefficient(ratio of energy loss and

average relative kinetic energy in the rotor) variation with respect

to the incidence flow angle and the effect of clearance on

efficiency drop are presented respectively in figures 2.22 and 2.23

33

from reference 45.

2.3. Blade Geometry Design

The one-dimensional design analysis provides

information only on some stations through the turbine . The flow

analysis is made along a mean streamline without considerations

of the fluid property variations outside this mean line. Therefore

the flow variation on planes normal to the mean streamline is

omitted. From the one-dimensional analysis, very little

information about the rotor blade geometry is known except the

mean blade angle, hub diameter and shroud diameter at rotor inlet

and rotor exit. The design has to be completed by defining the blade

shape (hub and shroud profiles, camberline and blade thickness) in

order to progress in the design process (two and three dimensional

flow analysis, stress analysis and vibration analysis). Several

approaches have been used for this purpose. One of them is to start

from a prescribed blade surface or meridional plane flow

distribution and develop a unique blade shape by a blade to blade

solution as presented in (55). This implies that there is only one

solution for a given blade shape and vice versa. A similar method to

that used by Novak et al (55) is described by Zangeneh in reference

(62) and (64). In this design method referred to as "inverse

method", the averaged swirl velocity rVe is prescribed on the

meridional plane and the corresponding blade geometry is

calculated iteratively. Although the method seems to be attractive,

it has not been often used by turbomachine designers. Another

method consists of specifying a detailed blade and channel

geometry which is analysed (stress and flow analysis) and

34

repeatedly refined until the shape results in acceptable

aerodynamic performance and sat isf ies the mechanical

requirements such as low stress and rotor inertia. It is this

method which is mostly used in the three dimensional blade design

(2), (23), (34), (35) and (42).

Because the blade design is obtained after an

iterative process , it requires the blade geometry to be easily

modified and therefore needs to be analytically formulated. Such

task can be achieved by a series of two dimensional projections of

the blade surface on some reference plane or surface (23) and (61)

or by a complete three dimensional surface (60). A review of these

methods is presented by Whitfield and Baines (58). Whitfield (23)

and Wallace (61) describe a method using Lame' ovals to represent

the hub, shroud and camberline profiles for radial and mixed flow

compressors with radial or non-radial blade elements. The method,

although well suited for turbomachine design, gives only a limited

degree of freedom in the choice of the curves representing the

blade geometry. This is due to the difficulty arising in finding the

coefficients involved in the equations describing these curves.

Three dimensional techniques in the design and

analysis of complex surfaces are described by Merryweather (60)

and Casey (47). The surface is subdivided into a series of

parametric patches and the coordinates within the four corners of

the patch are expressed by means of parametric (two parameters)

bi-cubic polynomials (60) or Bezier polynomials (47). An example

of the use of these methods in the blade design are presented in

figure 2.24 and 2.25. A detained analysis of the use of Bezier

polynomials in the definition of surfaces is presented by Forrest in

reference (59).

35

2.4 Flowfield Analysis

The flow behaviour in turbomachines has always

been a matter of interest for designers as it affects the

performance of the machines. In the last decades, enormous

progress has been made in the development of methods which solve

the flow in turbomachinery components. This has been boosted by

the apparition of more capable computers with large memory and

increasing speed. The 1950's have seen the appearance of two-

dimensional methods for solving incompressible potential flows,

while in the 1960's finite differences began to be used in two-

dimensional calculation of inviscid and subsonic flows. Streamline

curvature and stream function methods were used to solve the flow

in a meridional or a blade-to-blade surface. Time marching

solutions of Euler equations appeared in the 1970's and were used

to solve the two-dimensional (in both meridinal and blade-to-blade

surfaces) and the three dimensional flow in turbomachinery blade

row. A detailed review of these methods is presented by Mc Nally

(40).

Katsanis devised a streamline curvature method,

solving the flow in the meridional plane (25) of blade passage,

which was latter extended to solve the flow in a blade-to-blade

surface (26). The method has been extensively used by

turbomachine designers and is referred to by Baines (2),

Cartwright (8), Wallace (20), (49) and (61) and others. The method

is also discussed by Wilkinson (27) and Hearsey (57).

Another method being largely used for the flow

analysis and blade design consists of the solution of the time

36

dependent Euler equations with a f inite volume method

discretisation of the domain of flow calculation. Several schemes

have been developed to deal either with subsonic or transonic

flows such as the one by Denton (10) or Van Hove (19) and Arts (1).

Details of streamline curvature methods and time dependent Euler

equation solution by a finite volume method are presented in

chapter 5. Recently, solutions of the Navier-Stokes equations have

been employed to solve the flow in turbomachines. Denton (65)

developed an explicit method for viscous flow calculation which is

based on the technique presented in reference (10). In this method,

the viscous effects are modelled by adding a viscous shear term

and a shear work term respectively to the Euler momentum and

energy equations rather than solving the full Navier-Stokes

equations. Dawes (66) presents a three dimensional finite volume

method which time marches the Navier-Stokes equations.

Satisfactory agreement between computed and measured results

were obtained by this method for d i f ferent types of

turbomachinery. Applications of this method to radial turbines are

also presented by Zangeneh et al (67) and Kitson et al ( 54).

37

3000

et di'

ini/m*)

LEADING EDGE

AXIAL LENCTM •

Fig.2.1 Comparison of camberlines. ( Yamaguchi et al, 34)

Fig.2.2 Mixed flow turbine components, (schematic)

38

.4

.3

Dial Tieler

itio.

Tl"*!

1 1 1 1 Limiting tip-diameter Tieler

itio.

Tl"*!

dllU. "t.2' 1. '7

1 0.60-,

f/ - U r n iting

0. (0,

ait-d

/Dt'z

ameti

0.4

•r

Lotvesl stator

hfinhi _ y

! rati

iting

0. (0,

ait-d

/Dt'z

ameti

0.4

•r

hj. 0.04 i w f 20

/ / / 20 « 60 80 100

Specific speed, N.. r p m M H ^ ^ t s e c ' ^ 1

120

J I I J L 0 .1 .2 .3 .i .5 .6 .7 .8 .9

Specific speed. rad/(rTi"^)(lcg"^((sec''^)(P'^l

Fig.2.3 Influence of rotor geometry on maximum efficiency of radial turbines. ( Rohlik, 48)

I I I I I I I I Total efficiency corresponding to

curve of m a x i m u m static efficiency-

Stator-exit Curve of m a x i m u m

J static efficiency flow angle.

100 120 140 160 180

Specific speed, Nj, rpmAfl^'^Ksec"^!

I I I I I I .2 .« .6 .8 1.0 Specific speed, Nj. radf(m^4(kg^S(sec^^l(J^^

1.2 1.4

Fig.2.4 Maximum attainable design efficiency for radial turbines. ( Rohlik, 48)

39

! •5 °

.16

.12

.08

.04

/ /

X

20 « <10 » 100 I2D 140

Specific speed. Nj. rpm/(n"^(sec"^l

I I I I I

160

I .2 .i .6 .8 1.0 1.2

Specific speed, N,. ra(l/(ni^^Kkg"^Ksec''^Kj"^)

180

L4

Fig.2.5 Effect of specific speed on stator blade height for for maximum static efficiency. ( Rohlik, 48)

.7

ir. c T . 6

e .5

•E 4

.3

.2

/ / /

/ /

/ / / /

/ 20 ilO 60 80 100 120 1 «

Specific speed, N,. rpm/(ti^'^)isec^'^)

160 ISO

1.4 0 .2 .4 .6 .8 1.0 1.2 Specific speed, N^. rad/lm^^nkg^'^Ksec^'^Kj'"')

Fig.2.6 Effect of specific speed on tip-diameter ratio corresponding to maximum static efficiency.

( Rohlik, 48)

' 8 1 0 1.2 Specific speefl. dimensionless

" ® ^ S ioo Tro 140 160 m Sofc.iic ipeed, Nj,

Fig.2. 7 Effect of specific speed on optimum sator-exit angle. ( Rohlik, 48)

40

. J 4 | —

.70

.66

I

.50 .2 .< .6 .8 1.0 1.2 1.4

Specific spted. Nj. ditnensionless

I 1 I I I I I J 0 20 40 60 80 100 120 L« 160 180

Specific speed, Nj, ((t^^Hlt)ni^î^(min((sec"îilDf"^p

Fig.2.8 Effect of specific speed on optimum blade-jet speed ratio. ( Rohlik, 48)

# "

E .9

.8

.7

.5

efficiency

C l e a r a n c e

Eiil velocity

I I I I 2 .4 .6 .8 1 Specific speed. Nj. dimensionless

I I L_L J 20 40 60 80 100 200

Specific speed, N^. (ll''^MIhni^^îininiKec"^MItil^î

Fig.2.9 Loss distribution along curve of maximum static efficiency. ( Rohlik, 48)

41

M 100 t30 UO «M ItO 200 220 240

Specilic Speed

— Rohlik (48)

t s r s . " , ; ? " " - " " " » Ymmmguchi (34)

(35;

Fig.2.10 Comparison of high specific speed radial and mixed flow turbines static efficiency.

( Chou et al, 35)

TOTAL-STATIC EFFICIENCY INCLUDING EXHAUST DIFFUSER

TURBINES 120 TO 260 MM TIP DIA INLET BLADE ANGLE 0* PRESSURE RATIO 3.0 TO 5.0 AXIAL CLEARANCEAIP WIDTH 6% Ug TIP SPEED Cm3 ROTOR AXIAL EXIT VELOCITY VQ ISENTROPIC SPOUTING VELOCrTY (2gJCp ATJsen)

I O g cc

o 2 2

i

8 6 % r | T

0.4 0.4 0.5 0.6 0.7 0.8 0.9 1 0

m3 AXIAL EXIT/TIP SPEED VELOCITY RATIO

Fig.2.11 Chart of maximum efficiency for radial turbines ( Rodgers, 36)

42

t.10

1.00

I 0.90

0.80

Da |3RMS

L

IMPELLER TIP DIAMETER ROOT MEAN SQ. EXOUCER DIAMETER IMPELLER BLADE No. LENGTH OF MEAN MERIDIONAL FLOW PATH

-^i^BLAOE SOUDITY X S N S : ^ ° 2

1

3.0 4.0 4.55.0®-°

t i l l

1.5 2.0 2.5

DIAMETER RATIO Dg/Dpws

3.0

Fig.2.12 Effect of rotor blade geometry on radial turbine performance. ( Rodgers, 3 6)

21.3 28.4 35.6

21.3 28.4

1.5 2.0 2.5 3.0

PRESSURE RATIO (P0/P3)

3.5

Fig .2.13 Comparison of test and computed turbine characteristics. ( Rodgers, 37)

4 3

1.00

.90

.83

.70 —

.<0

T Scrod-slator viscous losses

R d o r incidence loss

kinetic energy loss

^-Olher rolor losses

J L

L O O

.90

.83

.25 . 30 . 35 . « .45 . 50 . 55 . 60 . 65 Specific speed. Nj. dimensionless

I I I I 1 _ _ « 50 eo 70 80

Specific speed. Nj. ( r p m X M ^ ^ W s e c ^ '

(a) Design rdor.

- Scfoll-stator viscous losses

.70

Rotor incidence loss 7

- Exit kinelic energy loss

-Other color losses

.2^ .2S .32 .36 . « . « .48 .52 .56 Specific speed. Nj, dimensionless

1.00 -

.96

.92

.8S

•ft

35 « 45 50 55 60 65 Specific speed. Nj, (rpmXft^'l/sec^^

tbi Rotor extension.

Scroll-stator viscous losses^

70

• Rolor incidence loss

OIner rolor losses

fill kinetic energy loss

3s .0? .66 .?0 It Sp-.<'!'£ Nj. rtiniDnsionloss

I I I I J .« S5 90 95

Si' 1; , irpmiil|5' ifs-x)''? 1(1 Cut'jjc- 'OtOf

Fig.2.14 Variation of turbine losses for different radial rotor configurations.

( Kofsey, 50 )

.70

90

44

Iv

li-^ i ! ! 1

i 1 1

OMwoiOM ft* no # f' a . I w m r imtT iwmr Iwg titrm

'P 'P V w

Fig.2.15 Effect of volute geometry on radial turbine performance. ( Barnard, 45)

STAGE EFFICIENCY - *

P

15 l« 17

EXPANSION RATIO

Fig.2.16 Effect of volute geometry on radial turbine performance. ( Tennant, 56)

45

AO"

35"

30"

25"

20"

15"

.fA o Casing A

A Cosing B

-0 Cosing C

0 4 0 6 0 8 10

Fig.2.17 Comparison of predicted and measured flow angles at casing exit. ( Bhinder, 7)

60'

50'

40"

l o -

20"

10"

0"

// // / /

— 0\

/ /

\

w H I

=" f

^ Cosing A — Cosing B — - C o s i n g C

120° 240" <P

360"

Fig.2.18 Measured flow angle along the exit casing periphery. ( Bhinder, 7)

46

.S:0-10f

% O OSf

010 0-15 (Ib/sl ( d e q K )

Ibf/ir

020

Fig.2.19 Nozzle loss coefficient. ( Benson et al, 5)

Fig.2.20 Entropy generation. 3D Viscous flow analysis. ( Kitson et al, 54)

47

0.2S

0.2

P L O 0.15 S

0 . 1

o.os

- 2 0

0.8

0.6: c H

0.4 H O

0.1

30 40 60 8 0 100 UO 140 PERCENT MERID. DIST

Fig.2.21 Computed total pressure loss and Mach number in the rotor channel of a radial turbine ( 3D viscous flow analysis ). ( Kitson et al, 54)

WEA&UACD LOSS COCrnCttNT — WCRlDfONAL toss COEFFICIENT MEAN BLADE-TO-BLADE LOSS

C O E r r i c i E N t

— AVERAGE B L A D E - T O - * L A D f L O S S C O E f f l C l E N ? .

$ r

5" V El o

52 o

ol -30 "WO - S O

mOTOR INCIDENCE

Fig.2.22 Rotor loss coefficient. ( Barnard et al, 45)

48

trrcci or tHdouo 11*01*1 A N D **i&i C L C A M A N C C

ON KA* TOIAL MCAO (FFIOCNCT

I I e e

66

% 64

62

60-

00* 0 4 0 06 0 00 MCC — In

O 10

hadial now

TURDINC

No or VANE) — 12

IIP OIA. — 5 4 In

SPecO— 65000 r#v/mln

RAOIA C L C A * O 015

OCVCLOPCO View Of V A N E

.CLCAmANCC — 0 075 In

AAOXAL CLCARANC£ O 030 In

Fig.2.23 Effect of axial and radial clearances on efficiency. ( Barnard et al, 45)

1£RIDIC\AL CHANNEL DEFINITION

Fig. 5 Definition of meridional channel by m e a n s of Bezier surfaces

Fig.2.24 Definition of meridional channel by means of Bezier surfaces. ( Casey, 47)

Radius r

Focus of patch boundaries

7 / Patch

Patch

Axial Distance z

Fig.2.25 Use of patches for meridional channel definition. ( Merryweather, 60)

50

Chapter 3

3. ONE DIMENSIONAL DESIGN

3.1. INTRODUCTION

The design of a highly loaded mixed flow turbine for

turbocharger applications and satisfying the design conditions is

undertaken. The one dimensional design procedure is aimed at

defining the overall turbine dimensions and its performance at

design and off-design conditions. Engine exhaust stagnation

pressure PQ* and temperature TQ*, mass flow m , rotational speed

ND and optimum velocity ratio DC are the principal parameters

which define the design point of such turbine, while the discharge

pressure is the atmospheric pressure.

To* = 923. K

PQ* = 2.91 Bar

P 4 = 1. 00 Bar

m = 0.414 kg/s

UC = 0.61 ( for radial turbines UCoptimum = 0.70)

ND = 98000. rpm

Different combinations of flow angle a? . cone angle

62, blade angle p2b (Figure 3.1), diameter ratio D1/D2 (inlet duct

diameter to rotor mean inlet diameter ratio) as well as the

diameter ratio E ( E = D2 / Da : Inlet mean diameter to exducer mean

root diameter ratio ) are analysed.

51

E = 1.3 ~ 1.5

p2b = 0.0 ~ 20.0 deg

0.2 = 10.0 ~ 20.0 deg

D1/D2 = 1.30 ~ 1.45

§2 = 0.0 ~ 40.0 deg

They provide a series of one-dimensional designs whose off-design

performance is analysed. High efficiency for a wide range of

operat ional condi t ions and a specif ied mass f low rate

characteristic are required for the design. Except for the cone angle

at rotor inlet, the rotor length and the blade curvature which will

be analysed by a two or three dimensional flow analysis, the

remaining parameters are the product of the one dimensional design

calculation (see figures 3.2 and 3.3). They consist of:

Ao Volute area at scroll inlet

Ro . Distance from the axis to centroid of the inlet scroll

section

b2 and D2 Blade height and mean diameter at rotor inlet

DasandDsH Shroud and hub diameters at the rotor exit

(33 Relative flow angle at the mean root diameter of the

exducer

3.2. DESIGN CONDITIONS ANALYSIS

The turbine dimensions are defined for the design

point condition already mentioned. The flow is then considered to

be approaching and leaving the rotor at optimum conditions. The

flow deviation ( incidence angle ) from the blade direction at rotor

inlet must be taken within the range of optimum incidence angle at

which the turbine is considered to be running at its highest

efficiency while the flow kinetic energy at rotor exit is minimized

52

by considering the absolute velocity to have zero tangential

component. A correlation proposed by Stanitz (68) and

originally used to define the compressor slip factor

( C82/U2 = 1 - 0 . 6 3 T I / Z 2 ) is commonly used to define the

optimum relative flow angle /Sa at rotor inlet and hence the

optimum incidence angle i(S2 for radial turbines. In the

absence of data concerning mixed flow turbines, the optimum

incidence angle used in this design procedure was taken to

be equal to -20.0 degrees.

Mean rotor inlet diameter is determined by the

following equation.

D 2 = 6 0 U C Vo* / 7T ND [ 3 . 1 ]

where Vo* is the jet speed corresponding to an isentropic

expansion from the stagnation conditions, defined by Po* and

To*, to the atmospheric pressure P4 as it is shown in figure

3.4.

V o * / 2 C p To* [ 1 - (]%^/P4) ] [3.2]

3.2.1 Rotor inlet and scroll

a ) Rotor inlet

Because the scroll dimensions are unknown at the

beginning of the design process, the flow properties are

assessed first at rotor inlet. The optimum incidence angle

i/32 (ijS2 = -20.0°), the absolute flow angle 0:2 and the rotor

5 3

inlet peripheral speed U2 define the velocity triangle at

rotor inlet (figure 3.1) and hence the absolute velocity V2.

The relative flow angle ^2 is given by the

following expression :

/32 = /32b + i/32 [3.3]

While the absolute velocity components are derived from the

velocity triangle of figure 3.1 as follows :

V2U = U2 / ( 1 - tana2 tan/32 )

V2m = V2U tana2 [3.4]

V2 = V V2U^ + V2m^

The energy equation applied at the rotor inlet

gives the static temperature T2.

T 2 = To* - V 2 / 2 Cp [3.5]

As a result of the energy losses in the volute,

the actual velocity V2 is less than the isentropic velocity

V2s. These losses can be expressed in term of a loss

coefficient which is the ratio of the kinetic energy

losses to the isentropic kinetic energy.

2

Csc = 1 - ( V 2 / V 2 s ) [ 3 . 6 ]

Details of the method used to calculate (sc are given in

section 3.3. The isentropic velocity Vzs is then derived

from equation 3.6 as :

54

V 2 s = V 2 / -j 1 - (sc [ 3 . 7 ]

The isentropic static temperature Tzs and the pressure P2

are obtained as follows :

T 2 S = To* - V2S^/ 2 Cp

P2 = Po* (T2S / To*) ("3 -1) [3.8]

The remaining flow parameters at rotor inlet (density p2,

Mach number M2, stagnation temperature To* and pressure P2*

are given by the following expressions :

p2 = P2 / R T2

M2 = V2 / J y R T2

T 2 * = To* [3.9]

P2* = Po* _ \ y / ( y - i )

1 - 0 . 5 (y-l) M2 1 — sc '

The continuity equation applied to the rotor inlet gives the

blade height b2.

b2 = m / (n 02 p2 V2 sina2 ) [3.10]

b ) Scroll

The volute geometry is characterized by its inlet

cross section area Ao and the distance from the rotor axis

to the centre of the inlet section as shown in figure 3.2.

The diameter Di (equation 3.11) corresponding to the inlet

vaneless nozzle upstream of the rotor is fixed so that a

55

relationship between Ao and Ro is established.

Di = k D2 [3.11]

The coefficient k is varied during the design process until

the mass flow characteristic specified in the design is

satisfied. Let Rsc be the distance between the centre of the

volute inlet section and the inlet of the vaneless nozzle.

The radius Ro can then be expressed as follows :

Ro = 0.5 Di + Rsc [3.12a]

The relationship between Rsc and Ao is determined by the

scroll inlet cross section type. Equation 3.12b is used in

the simple case of a circular section.

Rsc = V AO/TT [3.12b]

The flow in the volute is assumed to be in a free vortex in

the case of an isentropic expansion from Po to P2.

Therefore, the inlet velocity Vo is related to the rotor

tangential absolute velocity Vzu in the following form :

Vo = R2 V2U / ( Ro T/ 1 - Csc ) [3.12c]

The static temperature To is given by the energy equation.

To = To* - Vo^/ 2 Cp [3.12d]

The density po is calculated using the isentropic equation

for a fluid expanding from the stagnation conditions

(Po*,To*) to the static conditions corresponding to To. Thus

po = po* (To/ To*) [3.12e]

56

The continuity equation at the volute inlet leads to :

Ao = la / ( po Vo ) [3.12f]

A n iterative method based on the successive approximations

of Ro has been used to solve the system of equations ( 3.12a

to 3.12e ). The method is summarized below.

Select RA and RB so that RA < Ro < Re

Let RA = Di/2 and RB = 2 Di as a first estimation. The

domain delimited by RA and RB is refined until the

difference between (RB - RA) is less than the error

allowed on Ro.

1) Estimate Ro

Ro = ( R A + RB) /2

2) Calculate Vo (eq.3.12c)

Calculate To (eq.3.12d)

Calculate po (eq.3.12e)

Calculate Ao (eq.3.12f)

Calculate Rsc (eq.3.12b)

3) Calculate the radius RON given by equation 3.12a.

4) Compare Ro with RON.

During the iteration process, RA and RB are updated as

follows :

Ro > RON # RB = Ro , RA unchanged

Ro < RON => RA = Ro , RB unchanged

The iterative procedure is repeated from step 1 until a

converged solution is obtained ( Ro = RON ).

57

Because the scroll losses are not known at the

start of the design process, an iterative calculation

(starting from equation 3.7) is repeated so that the rotor

inlet conditions, the volute geometry and the loss

coefficient (sc are determined.

The relative velocity components are deduced from

the velocity triangle (figure 3.1) at rotor inlet by

equations 3.13.

W2m = V2 sinaa

W2U = V2U - U2 [3.13]

W2 = N W2m W2U ^

3.2.2. Rotor

The relative total temperature T2+ and relative

total pressure P2+ at rotor inlet are calculated by

equations 3.14, where M2+ is the relative Mach number.

M2+ = W2 / ] y R T2

P2+ = P2 I 1 + 0.5 (y-1) M2+

T2+ = T2 [ 1 + 0.5 (y-1) M2+^ ] [3.14]

The combination of the Euler turbomachinery equation (3.15a)

and the specific work transfer equation (3.15b) in terms of

enthalpy change leads to the concept of constant rothalpy

(I) in a rotating blade row in the form of the expression

58a

3.15c.

AHu = U2 V2U - U3 V3U (U3 = U2/E) [3.15a]

AHu = (Cp T2 + V2^/ 2) - (Cp T3 + V3^/ 2) [3.15b]

I = Cp T + W^/ 2 - U^/ 2 [3.15c]

and T+ = Cp T + W^/ 2 relative stagnation temperature.

T h e isentropic relative flow p a r a m e t e r s at rotor

exit are g i v e n by the following equations in t e r m s of those

at r o t o r inlet.

2 2

T3+ = T3+S = T2+ 1 1 - — — 2 Cp T 2

P3.S = P., f 1 -^ 2 Cp T 2 +

T3S = T3+ (P3/P3+S) [3.16]

So t h e r e l a t i v e velocity corresponding to an isentropic

e x p a n s i o n in the rotor is :

W3S = 4 2 Cp T3+ ( 1 - T3S/ T3+) [ 3 . 1 7 ]

Since losses occur in the rotor, the relative velocity W3 is

smaller t h a n W3s.

W3 = J W3s -2 U2 ( Qf + Qi ) [ 3 .18 ]

58b

where Qf is the loss ratio due to friction in the rotor, (eq.3.31)

and Qj is the loss ratio due to blade loading (eq.3.32).

Because friction losses in the rotor and the velocity

W3 are dependent on each other, an iterative calculation is needed

to set their final value. At the design point, the tangential

component of the absolute velocity at the rotor exit is zero. From

the velocity triangle at exit (figure 3.4), where the absolute flow

angle a3 is also , zero, the relative flow angle P3 at the mean root

diameter is given by equation 3.19.

SIN P = - — [3.19]

The temperature at rotor exit is then given by equation [3.20]

T3 = T35 + - 4 ^ [3.20] P

and the density pg is calculated by the state equation.

The continuity equation applied to the exducer exit cross section,

enables the area A3 to be calculated. That is

A, = [3.21] COS

Several iterations are needed to adjust the parameters at rotor

exit so that the hub diameter must have an acceptable value with

the following condition enforced.

71 A3 < i [3.22]

59

3.3. LOSSES

3.3.1 Useful Work

Stagnation conditions at inlet and outlet of the

turbine in figure 3.2 are designated respectively by 0* and 3*. The

drop of total enthalpy between these two states represents the

specific energy Hu transferred by the fluid in the rotor.

) + 0.5 ( ) [3.23]

Combining this equation with equation 3.15 (for constant rothalpy

in the rotor) and velocity triangles at both rotor inlet and exit,

leads to the following expression of Hu.

Hu = U2 V2U - U3 V3U [3.24]

Since for design conditions, it is desirable to keep the exit velocity

small and thus minimize the exit kinetic energy losses, the

tangential component Vsu is zero and the useful work reduces to;

Hu = U2 V2U [3.25]

For radial turbines at design conditions, the flow direction is

assumed to be radial (this assumption is used only in loss

coefficient definition as the optimum flow angle is different from

the blade angle). This leads to the useful work being expressed as

follows :

Hu = U2^

and equation 3.26 is used to non-dimensionalise the specific energy

60

transferred and the losses. The non dimensional useful work in the

general case is then given by the following equation:

Q,h = 2 [3.27]

with V3U = 0 at design point.

3.3.2 Scroll losses

A circular equivalent cross section of this scroll is

considered. That is only to simplify the procedure, losses in the

scroll have to be calculated. Assuming that the mean section (Am)

corresponds to the azimuth angle \|/ = 180. degrees, it can be

expressed for a uniform flow distribution around the exit scroll

periphery.

A = ^— and m = m ( 1 ) V p v V 360.

^ U V

For a free vortex in the scroll, r Vy = constant. So

( A ) = 1 ^ f o r \\t = 1 8 0 . °

r m 2 n,

The diameter dm for the mean section is :

A dm = ( f

Ag 1 + / 1 + 8 71 r , / ( — )

0 / 2 K

The distance rm from the axis of rotation to the centroid is m

L = ^ ^ / ( " T ) ^ 0

The equivalent solidity ( L/dm) is :

61

dm

where L is the mean scroll channel length .

Using the Balje method (applied for curved pipes in reference 3) to

calculate the scroll loss coefficient, the following parameters are

defined.

Boundary layer momentum thickness ( 8/L ) :

8 _ A - " -L D l/(n-

^e2 1 - ( 1 /M ) ( 4 + 2 / n )

n + 1

where A and n are functions of Reynolds number Re2- Suitable

values for A and n are as follows :

( A , n ) Flow nature

( 0.0076 , 6 ) Turbulent flow (used in the

calculation)

( 0.0160 , 4 ) Turbulent, near separation flow

( 0.4600 , 1 ) Laminar flow

10, = V2 / VQ is the deceleration (or acceleration) ratio.

Rg2 = V2 D2 /1)2 is the Reynolds number and V2 is the dynamic

viscosity as defined in section 3.3.3.

The boundary layer shape factor H can be approximated as follows:

e V, H = 2.5 + where R =•

R(,U43 e

finally the loss coefficient Csc is given by the following equation:

62

;sc = K K

SF 1

[3.28]

Where m

T 2 d

(7 r -2 )d^ + Tj +

is the aspect ratio of the channel. It takes into account the straight

part of the channel.

K = 2 ( f ) ) ]

m

And Kgp is an empirical coefficient which takes into account

losses due to the secondary flow. The value of Kgp was adjusted by

testing the one dimensional performance prediction model using

available data for radial turbines. A comparison between

experimental and computed performance of an X17Q3 radial turbine

will be presented latter in this chapter.

The scroll energy losses are then given by the

following equation;

Qsc = - ^ sc

V,

U-13.29]

3.3.3 Disc friction losses 'w

A small proportion of the power developed by the

rotor is absorbed by the fluid in overcoming the friction between

the rotor back and the housing. These losses of energy result in a

reheating of the fluid. Several correlations expressing the disk

friction losses are presented in the literature (31), (37), (38) and

(62). The disk friction losses in a non-dimensional form from

63

reference 31 is given by equation 3.30.

Qw - '3.30J

where Cf is the coefficient of friction defined as

Cj. = ^ ^ and Cj. = 0.05 approximately.

K x

^2 D, R = is the Reynolds number and

D. = 17.2 10 ^ / 273.16 is the dynamic viscosity

3.3.4 Rotor friction losses : Qf

Rotor friction losses, although not always separated

from other losses occurring the rotor passage, are sometimes

expressed separately. An analogy between the rotor channel

passage and curved pipes is often used to express the rotor skin

friction. These losses are proportional to the averaged relative

velocity between inlet and exit of the rotor. The non-dimensional

rotor friction losses Qf are written in the following form:

Qf - -^AVG

^2 [3.31]

where Wavg is the average velocity in the rotor defined as:

^AVG

64

and is the rotor friction losses coefficient obtained either

from experimental data or by empirical formulae such as that given

in reference (58).

Where Cf is a coefficient of friction depending on the surface

roughness and Reynolds number and LH and DH are respectively the

hydraulic length and mean hydraulic diameter of the blade passage.

3.3.5 Rotor blade loading losses : Q|

These losses are the result of the complex flow

pattern inside the blade passage. The blade geometry must be

designed in such a way as to prevent any sudden acceleration or

deceleration which causes high increase in entropy and thus in

energy losses. A simple correlation of these losses have been

suggested by Rodgers (37) and is given below.

/ v

Where Bg = Zg L2 / D2 is the blade solidity,

Z2 and L2 are respectively the rotor blade number

and the rotor channel length.

As shown in Reference17, Bg = 6 corresponds to an optimum

efficiency for radial turbines.

3.3.6 Leakage losses : Qcl

The clearance gap between blades and the housing is a

65

source of additional loss in the work transferred to the rotor due to

the leakage of the fluid from the pressure side to the suction side

of the blade. In addition to that a proportion of the fluid flows

through this c l e a r a n c e gap and leaves the rotor without doing any

work. The effect of the clearance is more important for small

turbines as far as the drop in turbine efficiency is concerned.

Clearance losses depend on both axial clearance at inlet and radial

clearance at exit and can be either obtained from available

experimental data or by means of empirical correlations. Futral and

Nusbaum (32) present these losses, for a single shrouded rotor, as

dependent on clearance 62 and tip blade height b2-

V 2 Qa = f I I3 M]

b,

In this case , ~ = 0.05

3.3.7 Exhaust losses : Qexh

For a non-shocked rotor, exhaust losses are

V.

QEXH - T ^ i t ^

While for a shocked one, losses due to the shock have to be

added. These correspond to an expansion from pressure P3 to

pressure P4.

3.3.8 Incidence losses

Incidence losses at design conditions are considered

to be zero while at off-design conditions these losses increase

66

sharply with the Increase of the flow deviation from the optimum

one and are considered to be the main cause of the drop in

efficiency. Different incidence loss models have been developed and

used in combination with the other losses occurring in the turbine

stage to predict its performance characteristics. Constant pressure

loss model (33) has long been used for radial turbines and is based

on the assumption that the flow is redirected to follow the blade

direction at constant pressure (figure 3.6a) and resulting in an

increase of entropy. The model has not been found suitable when the

incidence angle is negative because it leads to a decrease in

entropy. The NASA model (figure 3.6b) developed by Futral and

Wasserbauer (30) is based on the assumption of the tangential

component of the relative kinetic energy being destroyed but this is

accompanied by a change in pressure. The latter so far applied for

radial turbines by the authors have been extended by Whitfield and

Wallace (33) to deal with mixed flow turbines for which the blade

angle at inlet is different from zero. This model is presented

below.

From figures 3.5 and 3.6, the tangential kinetic energy

loss is given by the following equation.

AHIN = 0.51 w r SIN^ p. + k Wf2 SIN^ (3 | [3.35]

The coefficient k is defined as follows:

k = -1 when Pi and pf are of the same signs and

k = 1 when pi and pf are of opposite signs.

From the diagram T-S of figure 3.6 representing the expansion

process at rotor inlet and equation 3.35, the following relation is

derived.

67

TR = T . f* s = 1. -

2.

7 - 1

1 + 3 ^ Mf I

SIN^ Pj + k MJ SIN^

where Mi = Wi / ai , Mf = Wf / ai and ai = (y R Ti)° ^ jg the speed of

sound. Subscripts i and f refer respectively to the station just

upstream of the rotor inlet and to the station just downstream of

the rotor inlet.

Using the equation giving TR and the continuity

equation between stations i and f leads to the following expression

of TR.

, 2

Tr = 1 - -(y - i )M.

SIN pr+k A p.

AfPr cosp. TANp^

Constant total relative enthalpy between station i and f yields.

T = ' + - 1 - 1 H - M, [W i - mJ ] f 1 , y - l | . . 2 . .2

While the density ratio is obtained from figure 3.6 as follow;

Pf _ ^y / ( 7" ' ) ~ R

Pi T.

] / ( ? - ! )

An iterative calculation is needed to solve these equations with the

parameters at station i already defined. Mf is estimated to find the

remaining parameters and is adjusted until convergence is reached

(i.e the equations for TR giving the same value).

68

The incidence loss coefficient Qlnc is then calculated

by the following expression:

Qinc = AHIN /

3.4. TURBINE PERFORMANCE

3.4.1 Efficiency

The specific energy A H S T and A H T (figure 3 . 8 ) ,

resulting from an isentropic expansion from the turbine inlet

stagnat ion condit ions ( Po- , To-) to respectively the exit

(atmospheric) static pressure P4 and the exit total pressure P4',

are defined as follows ;

^ S T = CpT, 1 -

- Cp Ty,

• - 1

Y

[3.36]

The difference between A H S T and A H T corresponds to the exhaust

losses. Total-to-total efficiency rij, is calculated by equation 3.37.

Hy -

A

[3.37]

While total-to-static efficiency rits 'S given by the following

equation;

69

^.s = "" [3.38J

Where Z A H E X corresponds to the external losses such as disk

friction losses and leakage losses. For single stage turbines like

radial or mixed flow turbines, the total-to-static efficiency is

more significant as it characterises the net output torque delivered

on the turbine shaft. For multi-stage turbines, the exhaust stage

energy is used by the next stage and therefore the total-to-total

efficiency is more characteristic for the single stage whereas the

total-to-static efficiency characterises the whole machine as it is

for one stage turbines.

3.4.2 Net Output Power

The net power delivered by the turbine is given by the

following expression:

= "H -ys rn AHg.j. [3.39]

and the net torque at the turbine shaft is calculated as follows ;

-CsH = [3.40] 0)

where co is the rotor rotational frequency.

3.5 OFF-DESIGN PERFORMANCE PREDICTIONS

Once the overall dimensions of the design have been

defined, an analysis of its behaviour at off-design conditions is

carried out in order to establish the performance characteristics

map (efficiency, mass flow rate and so on ). This analysis is also

based on the loss correlation models presented in section 3.3 with

the incidence losses being taken into account as they contribute

70

highly in the loss of efficiency when the turbine is running at

conditions far from the design point. The input parameters used for

the analysis are the stagnation pressure and temperature at turbine

inlet and the rotational speed. The other parameters such as mass

flow, efficiency, power and velocities are the product of the

calculat ion.

The procedure used to solve the one-dimensional flow

in the turbine stage is summarized by the chart in figure 3.7. The

mass flow is estimated at first and then the flow parameters are

computed at each station. Application of momentum, energy,

continuity and state equations for each component leads to the

calculation of the exit parameters provided that the inlet ones are

already known. A relation combining these equations is often used.

It was developed by Wallace et al (38) and can be used either for

rotating or stationary ducts.

V (1 +X i l 1My ) ^ ( 1 PV. SIN A,. 2 - 1

[3.41]

X» Y ^ ^ X Y

2 2 1 0.5 (Y+I)/(Y-I)

- U\

2CpT,.

where X refers to the inlet and Y to the exit. The stagnation

temperature Ty* and total pressure Py* at exit are obtained from

the following equations.

71

TY, = T. X*

1 -Ux-Uy

2 CP T X *

Py* = M y ) 1 - L XY

1 -" x - U y

^ ^X*

Y-1

[3.42]

where the subscripts X* and Y* indicate the stagnation conditions

at inlet and exit and refer to the absolute conditions when U X = U Y = 0

and to the relative ones in the rotating part.

3.5.1 Casing Analysis

At turbine inlet (station 0 ), the flow is assumed to

be expanding isentropically from stagnation state (Po',To-) to the

Inlet static pressure Po which has to be calculated. The state 0

(Po,To) can only be defined by means of equation 3.41 if the mass

flow is known. The maximum mass flow allowed at scroll inlet

corresponds to sonic conditions at station 0 and is used as a

maximum value for the estimated mass flow.

Solution of equation 3.41 gives the inlet Mach Number

Mo which is used to find the other parameters (To,Vo,Po and po) at

scroll inlet.

At station 2 just upstream of the rotor inlet, an

iterative calculation combining equations 3.41 and 3.12c enables the

flow angle a2, Mach number M2 and the loss coefficient Csc ^0 be

found. These three parameters and the free vortex law enable the

condition at the upstream rotor inlet and particularly pressure P2,

temperature T2 and absolute velocity V2 to be defined. Velocity

triangle (figure 3.1 ) at rotor inlet is used to define the relative

72

velocity W2, the relative flow angle P2 and the incidence flow ip2.

That is

^2U = ^2 COS Kg - U2

= ^2U SIN a

^2 = [3.43]

w,,, P = A T A N ( ^ )

iP^ = P^ - P^^ (incidence angle)

3.5.2 Rotor Analysis

As explained in section 3.3.8, the flow parameters

just downstream of the rotor inlet (station 2p) are generally

different from those just upstream (station 2) because the flow is

being redirected to follow the blade direction. This process is

accompanied with energy losses which are assessed in addition to

the flow parameters at station 2p by means of the method outlined

in section 3.3.8. This process is represented in figures 3.6b and 3.8.

Stagnation conditions at rotor exit (T3 + , Pa + s)

corresponding to an isentropic expansion from stagnation

conditions at station 2p of rotor inlet (T2P + , P2P+) are obtained by

use of equations 3.42 with ^XY = 0.

The static exit pressure Ps is taken equal to P4 when

the flow at the rotor exit is subsonic whereas at sonic conditions ,

it is taken equal to the critical pressure which corresponds to an

73

isentropic relative Mach Number equal to the unity. The critical

pressure PSCR is given by the following expression:

p = p 3CR ^ 3 + S

2

Y+ 1

7 - 1

[3.44]

Thus.

P3CR > PA =========» Ps = PscR and

PsCR < P4 =========>> Ps ='P4

Flow parameters (Tss, MSRS and Was) corresponding to

an isentropic expansion from the stagnation conditions (P3+s,T3+) to

P3 being defined, an assessment of disk friction losses and blade

loading losses can be made and therefore the relative velocity Ws

is calculated by means of equation 3.18. The process is repeated

until convergence in the calculation for Wa is reached.

The mass flow m' obtained from equation 3.45 is

compared with that used to calculate the flow in the turbine (m)

and a new estimation of the mass flow m is made in order to

adjust the flow parameters which leads to m = m'.

m ' = p 3 W3 A 3 COS (33 [ 3 . 4 5 ]

The velocity triangle at rotor exit is used to compute the absolute

velocity V3.

V3A = w , COS p,

^2U = ^3 SIN P + u,

74

3.5.3 Off-Design Performance Characteristics

Tota l - to - to ta l e f f ic iency r i t t and tota l - to-stat ic

ef f ic iency r \ \ ^ versus pressure ratio (or velocity ratio) are

calculated as shown in section 3.4.1 and including incidence losses

at rotor inlet. A relation between pressure ratio and velocity ratio

is given by equation 3.46.

TT N D U/C = ^ ^ [3.46]

6 0 ^ 2 C p \ , (1.- )

where PR = PQ* / P4

The dimensionless mass flow rate ; mfr is obtained by equation

[3.47]

-J ^ ^ m

mfr = / • [3.47] 0" 0

The net output power P^ at the turbine shaft is given by equation

3.39. In addition to these characteristics, absolute flow angle a 9

and incidence flow angle ip2 at rotor inlet are calculated by the

off-design performance prediction code.

The predicted (by this method) and measured total to

stat ic ef f ic iency character is t ics of a radial - inf low turbine

(X17Q3) plotted against the reduced speed N/SQRT(To.) for lines of

constant pressure ratios are presented in figure 3.9. The predicted

and measured efficiencies are in reasonable agreement for the high

pressure ratios but showed a small difference for the low ones.

75

3.6. CONCLUSIONS FROM THE ONE-DIMENSIONAL DESIGN.

The method presented in section 3.2 has been used to

design a mixed flow turbine satisfying the design requirements set

in section 3.1. A series of designs has been produced and then

analysed by the off-design performance prediction method outlined

in section 3.5. The effects of several parameters on the

performance of these designs were investigated. These parameters

consist of :

- D1/D2 ratio (inlet straight duct diameter to rotor inlet mean

diameter ratio) = 1.30, 1.37 and 1.45

- Volute exit absolute flow angle a2 = 10.0, 15.0 and 20.0 degrees

- Blade angle and cone angle at rotor inlet

((32b,52) = (0.0,0.0), (10.0,20.0) and (20.0,40.0)

- Diameter ratio D2/D3 = 1.30, 1.40 and 1.50

The optimum incidence angle was kept the same (ip2opt = -20.0

degrees ) for each case of analysis.

3.6.1. Effect of the Diameter Ratio D1/D2

The effect of the diameter ratio D1/D2 was analysed

by means of three designs having 1.30, 1.37 and 1.45 as diameter

ratios respectively. The other parameters , which were kept

constant, are as follows ;

a.2 = 15.0 degrees

(P2b,62) = (20.0,40.0)

D2/D3 = 1.40

The overall dimensions of these three designs are

presented in table 3.1, while a comparison of their performance

76

characteristics (at constant rotational speed, No = 98000 rpm),

consist ing of the total- to-stat ic ef f ic iency and the non

dimensional mass flow rate, is shown in in figure 3.10. From the

results of table 3.1, it can be shown that increasing D1/D2 leads to

an increase of the scroll inlet area Ao and the radius Ro, while the

A/R ratio and the rotor geometry remain almost unchanged. This

common features on turbine geometries resulted in identical

predicted curves of efficiency (figure 3.10a) while the difference

in scroll inlet areas resulted in different non dimensional mass

flow rate characteristics (figure 3.10b). Figure 3.11a shows the

predicted efficiency curves of these three designs and figure 3.11b

shows a comparison of the non dimensional mass flow rate

characteristics of the same designs and that of the X17Q3

reference radial turbine. The condit ions for which these

characteristics (figures 3.11) were obtained, correspond to those

of the X17Q3 "matching line" for engine running conditions (the

matching line does not correspond to a constant speed). Data for

X17Q3 were provided by Holset and were set as a target for the

design as far as the non dimensional mass flow rate is concerned

and therefore, the turbine having D1/D2 = 1.37 is closer to the

desired design. This design will be used in the following analyses

and will be referred to as "Turbine TD".

3.6.2. Effect of the Volute Exit Flow Angle a2

Table 3.2 shows the overall geometry of turbine TD

and two other designs obtained for the same parameters of turbine

TD except for the volute exit flow angle az. The increase of of the

flow angle (a2 = 10.0, 15.0 and 20.0) results in a decrease in the

rotor inlet blade height B2 while its effect on the other turbine

geometrical characteristics is less important as shown in table

77

3.2.

From figures 3.12 and 3.13, it can be seen that the

di f ference between the turbine performance characterist ics is

small and therefore the flow angle variation in this range

(10.0-20.0 degrees) has little effect on the design performance

calculated by means of the off-design performance prediction

developed in chapter 3.

3.6.3. Effect of the Blade and Cone Angles at Rotor Inlet

The three designs (one is turbine TD), used to show

the effect of the blade angle P2b and the cone angle 62 at rotor inlet

on the design performance, were obtained for three different

parameters (p2b,62) [ (0.0,0.0), (10.0,20.0) and (20.0,40.0) ]. The

other parameters were kept the same as those of turbine TD. The

overall geometry of these three designs are presented in table 3.3.

An increase in the turbine efficiency is predicted

when the blade and cone angles are increased (figure 3.14 and 3.15).

For radial turbines, the optimum diameter ratio D2/D3 is around

1.70 (figure 2.12) and the optimum velocity ratio is 0.70. Therefore

a poor efficiency, for the radial turbine in this case ( D2/D3 = 1.40

and UC = 0.61 ) , is expected.

3.6.4. Effect of the Diameter Ratio D2/D3

The overall dimensions of the three designs described

in table 3.4 were obtained for three different diameter ratios

(D2/D3 = 1.30, 1.40 and 1.50) while the other parameters were kept

identical to those of turbine TD. The three designs obtained have

the same geometry up to the rotor inlet and differ only by their

78

exducer shape as shown in table 3.4.

Predicted efficiency and non dimensional mass flow

rate characteristics of these designs for constant rotational speed

(98000) are presented in figures 3.16. Figure 3.17a, on the other

hand , shows the efficiency curves of these designs obtained at the

X17Q3 engine running conditions. Figure 3.17b represents the non

dimensional mass flow rate characteristics, of the three designs

and the X17Q3 turbine, at the same conditions of figure 3.17a.

3.6.5. Selection of the Design

From the series of the designs analysed, one turbine

design (turbine TD) was selected for detailed flow analysis and

further investigations regarding the blade geometry design. Its

overall dimensions, thermodynamic design conditions and design

performance are given below.

Turbine dimensions

Ao = 1702.41 mm^ : Scroll inlet area

Ro = 80.53 mm ; Distance from the rotor axis to the

centroid

Di 1.37 D2 : Vaneless duct inlet diameter

D2 83.58 mm : Rotor inlet mean diameter

B2 17.99 mm : Rotor inlet blade height

p2b 20.0 deg. : Mean blade angle at rotor inlet

52 40.0 deg. ; Rotor inlet cone angle

Z2 = 12 : Number of rotor blades

DS3 = 78.65 mm Exducer tip diameter

DH3 = 27.07 mm Exducer hub diameter

79

D3 = D2/1.40 Exducer root mean square diameter

P3 = -52.00 deg. Blade angle at the exducer root mean

diameter

Design conditions

Po- 2.91 bar

To- = 923.0 degrees

Nd = 98000. rpm

m = 0.414 kg/s

UC = 0.61

Predicted Efficiency

^TS ~ 0.73

T i j j = 0.84

The turbine off-design performance characteristics

are shown in figures 3.18 to 3.23. Figures 3.18 and 3.19 show the

total-to-stat ic efficiency curves, for three different rotational

speeds, plotted against the velocity ratio and pressure ratio

respectively. It can be noticed from figure 3.18 that the peak

efficiency is located near the design velocity ratio (0.61) and

remains fairly high over a wide range of pressure ratios (figure

3.19) for the design rotational speed (98000 rpm). Figure 3.20 on

the other hand shows the pressure ratio variation against the non

dimensional mass flow rate . The swallowing capacity of Holset

turbine X I 7 0 3 on the matching line is plotted on the same figure.

A comparison between the X17Q3 swallowing capacity (which is a

80

"matching line" for engine running conditions) and that predicted

for the design is presented in figure 3.21. Finally , the absolute

flow angle and incidence flow angle variations against pressure

ratio are presented respectively in figures 3.22 and 3.23.

81

TABLE 3.1 EFFECT OF DIAHETER RATIO D1/D2

D1/D2 1 .30 1.37 1.45

D2 83.58 83.58 83.58

B2B 20.0 20.00 20.00

DELTA 40.0 40.00 40.00

D2/D3 1.40 1.40 1.40

ALFA2 15.00 15.00 15.00

AO 1645.32 1702.41 1801.78

RO 77.21 80.53 84.54

B2 18.00 17.99 17.94

D2S 94.34 94.33 94.31

D3H 26.89 27.07 27 .57

D3S 78.70 78.65 78.51

BETA3 -52.08 -52.00 -51.78

TABLE 3.2 EFFECT OF VOLUTE EXIT ABSOLUTE FLOW ANGLE

D1/D2 1 .37 1 .37 1.37

D2 83.58 83.58 83.58

B2B 20.00 20.00 20.00

DELTA 40.00 40.00 40.00

D2/D3 1.40 1 .40 1.40

ALFA2 10.0 15.00 20.00

AO 1702.41 1702.41 1735.07

RO 80.53 80.53 80.75

B2 27.33 17.99 13.42

D2S 99.27 94.33 91.75

D3H 28.40 27.07 25.16

D3S 78.27 78.65 79.14

BETA3 -51 .39 -52.00 -52.77

82

TABLE 3.3 EFFECT OF BLADE AND CONE ANGLE AT ROTOR INLET

D1/D2 1.37 1.37 1.37

D2 83.58 83.58 83.58

B2B

DELTA

0.00

0.00

10.00

20.00

20.00

40.00

D2/D3 1.40 1.40 1.40

ALFA2 15.00 15.00 15.00

AO 1868.82 1801.32 1702.41

RO 81.64 81.19 80.53

B2 18.30 18.06 17.99

D2S 83.58 89.52 94.33

D3H 36.80 33.43 27.07

DBS 75.24 76.59 78.65

BETA3 -45.58 -48.30 -52.00

TABLE 3.4 EFFECT OF DIAMETER RATIO D2/D3

D1/D2 1 1.37 J.3f

D2 83.58 83.58 83.58

B2B 20.00 20.00 20.00

DELTA 40.00 40.00 40.00

D2/D3 1.30 1.40 1.50

ALFA2 15.00 15.00 15.00

AO 1702.41 1702.41 1702.41

RO 80.53 80.53 80.53

B2 17.99 17.99 17.99

D2S 94.33 94.33 94.33

D3H 36.29 27.07 12.31

D3S

BETA3

82.46

—54 .64

78.65

-52.00

75.47

-49.58

83

Fig.3.1 Velocity Triangle at Rotor Inlet

Figure 3.2 Mixed Flow Turbine Overall Dimensions

84

Fig.3.3 Velocity Triangle at Rotor Exit

Fig.3.4 Expansion Process In a Mixed Flow Turbine

85

Wi SIN pi

a) Positive incidence pi pf > 0

Wi SIN Pi

b) Negative incidence pi pf > 0

Wi SIN Pi

c) Negative incidence Pi Pf < 0

Wf SIN pf d) Velocity triangle just

downstream of rotor inlet

Fig.3.5 Velocity Triangles at rotor inlet ( NASA Model for incidence loss calculation ) - a), b), c) Just upstream of

rotor inlet - d) Just downstream of

rotor inlet

"Ti* « Tf*

Pi* Pf*

f*s

a) Constant Pressure Model b) NASA Model

Fig.3.6 Incidence Loss Model (Ref.30)

86

BTAKT ^

INPUT DATA

Mfl =0.0 wrz = MTorO

MT = (Mjri-t-iiyat/a.j

I SCROLL IWLBT gAKAMBTERa ]

OD>0.0

UPSTREAM ROTOR INLET PARAMETERS

IMCIDEMCB LOSSES AND OOniSTREAM ROTOR IMLBT

PARAMETERS

ISENTROPIC PARAMETERS

STAGNATION AT EXDUCER

STAGNATION PARAMETERS AT EXDUCER

30+>P

W 2 = MF

MF2 = MF

MF2 = MF

- * ©

•* ®

- ©

©

30+/P4>PRc

ROTOR LOS8K HIGH 7

EXIT VELOCITY M3

CONVERGENC REACHED FOR

«3 7

CLCDLATB MASS FLOW

MFOMF

CALCULATES OTHER LOSSES

RESULTS

STOP

1 » ®

Fig.3.7 OFF-DESIGN PERFORMANCE PREDICTION

00

Figure 3.8 Expansion process

88

F i g u r e 3 . 9 CWARBONGmEENIOSlREDWCCOLFm TOTAL TO

STATE EFFCENCir OF H H20 X17Q3 TU»C

80J]

m-

60i)-

5011

40i)

M

m

• •

* PR = 16

• PR = IB

I PR = ZO

I PR = 12

I I I I M M I I M I I I I I I I I I I I I I I I I I I I I

Of ti) 20i) 30D 40i) 50i) 60D 70i)

REDUCED SPED N/fTo*

89

0 .8

0 . 7 -

0.6 -

0 . 5

0 . 4 -

o D1/D2 = 1.30 D1/D2 = 1.37

0 D1/D2 = 1.45

1 . 4

T 1.8

a)

2 .2 2 . 6

P r e s s u r e R o t i o

3 . 4 3 . E

D Q;

c 0 z

D . 2 B -

0 . 2 7 -

P r e s s u r e R o t i o

3 . 4 3 . 8

Fig.3.10 Effect of Diameter Ratio D1/D2 : Turbine Characteristics at 98000 rpm, a) Efficiency b) Non Dimensional Flow Rate

90

> o z w

L. w y < K W o *-

z o

0 . 7 5

0 . 7 4 8

0 . 7 4 6

0 . 7 4 4

0 . 7 4 2

0 . 7 4

0 . 7 3 8

0 . 7 3 6

0 . 7 3 4

0 . 7 3 2

0 . 7 3

0 . 7 2 8

0 . 7 2 6

0 . 7 2 4

0 . 7 2 2

0 . 7 2

o D1/D2 = 1.30 + D1/D2 = 1.37 0 D1/D2 = 1.45

r 1 . 2

a)

T I I T I r 1 . 4 1 . 6 1 . 8 2 2 . 2

P R E S S U R E RATIO

2 . 4 2 . 6 2 e

u < o: 5 o

< z o w z u 5 o z o z

0 . 4

0 . 3 9

0 . 3 8

0 . 3 7

0 . 3 6

0 . 3 5

0 . 3 4

0 . 3 3

0 . 3 2

0 . 3 1

0 . 3

0 . 2 9

0 .28

0 . 2 7

0.26

0 . 2 5

0 . 2 4

0 . 2 3

0.22

0.21

0.2

/ / V / ^ ////

•f // /V D D1/D2 = 1.30

g// + D1/D2 = 1. 37 / J 0 D1/D2 = 1.45 A A X17Q3

• T i l l 1 . 2

b)

1 . 4 1 . 6 1 . 8 2 2 . 2


2 . 4 2 . 6 2 . 8

Fig.3.11 Effect of Diameter Ratio D1/D2 : Turbine Characteristics at X17Q3 Turbine Running Conditions a) Efficiency b) Non Dimensional Flow Rate

91

z w u h.

y <

K 1/1 o t-<

0 . 7 5

0 . 7 -

0 . 6 5 -

0 . 6 -

0 . 5 5 -

0 . 5 -

0 . 4 5 -

0 . 4 -

0 . 3 5

2 . 2 2 . 6 3

PRESSURE ratio

0 . 3 E

< K $ O

< z o to z w 5 5 z o z

b)

2 . 6 3

PRESSURE ratio

3 . 4 3 . 8

a) Efficiency b) Non Dimensional Flow Rate

92

z w u L. L. W O k-< t-(/) o K

o

0 . 7 5

0 . 7 4 8 -

0 . 7 4 6 -

0 . 7 4 4

0 . 7 4 2 -

0 . 7 4 -

0 . 7 3 8 -

0 . 7 3 6 -

0 . 7 3 4 -

0 . 7 3 2

0 . 7 3 H

0 . 7 2 8

0 . 7 2 6 H

0 . 7 2 4

0 . 7 2 2 H

0 . 7 2

1

O ALFA2 = 10. + ALFA2 = 15. 0 ALFA2 = 20.

1 . 2

a)

r 1 . 4 1 . 6 1 . 8 2


2.2 2 . 4 2 . 6 2 5

< Q: g o

_I < 2 O i/t 2 w 5 D

2 O 2

0 . 4

0 3 9 -

0 3 5 -

0 . 3 7 -

0 . 3 5 -

0 . 3 5 -

0 . 3 4 -

0 . 3 3 -

0 . 3 2 -

0 . 3 1 -

0 . 3 -

0 . 2 9 -

0 . 2 S -

0 . 2 7 -

0 . 2 6 -

0 . 2 5 -

0 . 2 4 -

0 . 2 3 -

0 . 2 2 -

0 . 2 1

0 . 2

-0 . 2 1

0 . 2

0 ALFA2 + ALFA2 « ALFA2 * X17Q3

10. 15. 2 0 .

—I— 1 . 2

—I r

1 4

—I— 1 . 6

—I— 1 . 8 2 . 2

—I— 2 . 4

I

2 . 6 2 . B


b)

Fig.3.13 Effect of Volute Exit Absolute Flow Angle Turbine characteristics at X17Q3 Turbine Running Conditions a) Efficiency b) Non Dimensional Flow Rate

93

y u z w

L. U. w

<

o

<

0 . 7 4

0 . 7 2

0 . 7

0.68

0.6€

0 . 6 4

0 . 5 2

0.6

0 . 5 8

0 . 5 5

0 . 5 4

0 . 5 2

0 . 5

0 . 4 8

0 . 4 5

0 . 4 4

0 . 4 2

0 . 4

1 . 4

BETA2B = 0. BETA2B = 10. BETA2B = 2 0 .

DELTTA2 DELTTA2 DELTTA2

2 . 2 2 . 6 3

PRESSURE r a t i o

< Q:

5 o

< z o (/) z w 5 D Z o z

b) PRESSURE r a t i o

Fig.3.14 Effect of Blade and Cone Angles at Rotor Inlet Turbine Characteristics at 98000 rpm. a) Efficiency b) Non Dimensional Flow Rate

34

y u 2 W

w o t-g

o

z o

0 . 7 5

0 . 7 4 —

0 , 7 3 -

0 . 7 2 -

0 . 7 1 -

0 . 7 -

0 . 5 9

0 .68

0 . 6 7 -

f BETA2B + BETA2B 0 BETA2B

0. DELTTA2 = 0. 10. DELTTA2 = 20. 20. DELTTA2 = 40.

0 . 6 6 I I I r 1 . 2 1 . 4

a)

1 I I r 1 6 l . B 2


2 . 2

T r 2 6 2 E

< Q;

5 o

< z o 1/1 z w z o z o z

0 4

0 . 3 9 -

0 . 3 8 -

0 . 3 7 -

0 . 3 6 -

0 . 3 5 -

0 . 3 4 -

0 . 3 3 -

0 . 3 2 -

0 . 3 1 -

0 . 2 E

0 . 2 7 H

0 . 2

-—

y y y T

CI / D BETA2B = 0. DELTTA2 = 0. / + BETA2B = 10. DELTTA2 = 20.

0 BETA2B = 20. DELTTA2 = 40. * X17Q3

- 1 1 1 1 1

1 . 2 1 . 4 1 . 6 1 . 8 2


2 . 2 2 . 4 2 .6 2 . 8

b)

Fia 3 15 Effect of Blade and Cone Angles at Rotor Inlet : Fig.3.ir> characteristics at X17Q3 Turbine

Running Conditions a) Efficiency b) Non Dimensional Flow Rate

95

>-u z

L. W

O

B O K-z o

0 . 7 5

0 . 7 -

0 , 6 5 -

0 6

0 . 5 5 -

0 . 5 -

0 4 5

0 . 4

o D2/D3 = 1.30 * D2/D3 = 1.40 4 D2/D3 = 1.50

0 . 3 5

1 . 4 1 . 8

a)

T I I I

2.2 2 .6 I I I r 3 3 . 4 3 . 8

PRESSURE ratio

0 3S

< Q:

5 o

< z o in z w 5 Q Z o z

0 26 -

b)

2 . 5 3

PRESSURE ratio

Fig.3.16 Effect of Diameter Ratio D2/D3 : Turbine Characteristics at 98000 rpm. a) Efficiency b) Non Dimensional Flow Rate

96

>-o z w u

w y

< t-(/I o t-<

0 . 7 4 8

0 . 7 4 6

0 . 7 4 4

0 . 7 4 2

0 . 7 3 8

0 . 7 3 5

0 . 7 3 4

0 . 7 3 2

0 . 7 2 8 D D2/D3 = 1.30 + D2/D3 = 1.40 0 D2/D3 = 1.50

0 . 7 2 6

0 . 7 2 4

0 . 7 2 2


w < o: 3 o

< z o z Ul 2 O z o z

0 4

0 . 3 9 -

0 . 3 8 -

0 . 3 7 -

0 . 3 5 -

0 . 3 5 -

0 . 3 4 -

0 . 3 3 -

0 . 3 2 -

0 . 3 1 -

0 . 3 -

0 . 2 9 -

0 . 2 8 -

0 . 2 7 -

0 . 2 6 -

0 . 2 5 -

0 . 2 4 -

0 . 2 3 -

0 .22 -

0 . 2 1 -

0 . 2 - -

1

D D2/D3 — 1 • 30 4 D2/D3 = 1. 40 0 D2/D3 = 1. 50 * X17Q3

1 . 2

b)

1 . 6 1 . 8 2


2 . 2

—I— 2 . 4 2 . 6 2 . 6

Turbine

Running Conditions a) Efficiency b) Non Dimensional Flow Rate

97

I H U •H •p <0 a I B I

ID

I

0.8

0.6

0.4'

0.2 0.4

A 50000. rpm

o 75000. rpm

• 98000. rpm

0 .6 —I— 0.8 1.0

Velocity Ratio UC

Fig.3.18 Total-to-static Efficiency vs Velocity Ratio UC

ir S u IM

H

a I 0 •u I

0.8'

0.7 •

0.6-

0.5

t

0.3'

0.2 1.0

D 50000. rpm

• 75000. rpm

B 98000. rpm

—I— 1.5

—I— 2.0

— I 1 2.5 3.0 3.5

Pressure Ratio PR

Fig.3.19 Total-to-static Efficiency vs Pressure Ratio PR

98

3.0

g

en

2 5 n 0)

6

2.5

2.0-

1.5-

1.0 0.1

50000. rpm

75000. rpm

98000. rpm

X17Q3

/

—I— 0.2

—I— 0.3

Non Dimensional Mass Flow Rate

0.4

Fig.3.20 Mass Flow Rate Characteristics (Design) and Holset Turbine X17Q3 Swallowing Capacity

3.0

4J

2 9 U n

I

2.5-

2.0-

1.5-

1.0

X17Q3

74600

58350 \ Design

96000 rpm

89242

81750

Fig.3.21

0.25 0.30 0.35 0.40

Non Dimensional Mass Flow Rate

Comparison Between the Design and Holset Turbine X17Q3. Swallowing Capacity

99

20.0

Of

< 17.5

B -fa o

0

1

is.oH

12.5 H

10.0

O 50000. rpm

• 75000. rpm

• 980,00. rpm

Fig.3.22

3.0 3.5

Pressure Ratio PR

Absolute Flow Angle at Rotor Inlet

CT> 01 Q

0) iH tr c <

0) u c 01 •o -H u c H

° 50000. rpm

• 75000. rpm

98000. rpm

2.5 3.0 3.5

Pressure Ratio PR

Fig.3.23 Incidence Flow Angle at Rotor Inlet

100

Chapter 4

4. BLADE GEOMETRY

The one dimensional model described previously

makes it possible to set up the overall dimensions of a mixed flow

turbine and predict its off-design performance. At this stage of the

design procedure, the influence of the blade geometry is not taken

into account. To complete the design, an analysis of the flow

pattern inside the rotor is necessary. This is generally achieved by

a two or three dimensional flow analysis whenever this is possible.

A combined method for radial turbines, developed by

Katsanis (25), for calculating the flow distribution in the

meridional plane and on a blade to blade surface is used for the

flow analysis. This method has been modified in order to cope with

the flow calculation for a mixed flow turbine. It enables the

velocity and pressure distribution inside the rotor channel to be

obtained.

The first step consists then of setting the rotor and

blade geometry . The design is then checked and whenever it is

found unsatisfactory, the blade shape has to be modified and the

design reanalyzed until an optimum geometry is reached. Because

the blades have radial fibres, the projection of the mean blade

surface on a surface of reference is unique (camberline). Thus the

blade geometry is completely defined by the hub and shroud

projection on a meridional plane and the camberline. The task of

defining analytically the blade geometry is carried out by using

Bezier polynomials. The choice of this method has been dictated by

101

its simplicity and flexibility in the selection of the form of the

curves by varying the degree of the polynomial and the position of

the points describing it.

4.1. Bezier Polynomial

The parametric representation of a curve in the (X.Y)

plane is written in the following form;

X = fx (U)

y = f y (u) [4.1]

One way of expressing f ^ (u) and fy (u) is by using Bezier

polynomials. A set of points including the end points of the curve

described by equations 4.1 is used to define polynomials f^ and fy.

If (n+1) points (Po,...Pi,...P^) are used, any point of the curve (figure

4,1) is defined by the following n ^ degree polynomial.

n

OP (u) = (u) OR [4.2] i = 0

In this equation, the Bezier curve defined is a type of weighted

average of the polygon points (PQ,...Pj,...Pn) and the weighting

functions are the Bernstein polynomial B, (u).

B"(u) = ( " ) u' (l-u)""' [4.3]

The coordinates { x(u), y(u) } of P are derived from equation 4.2

n

X (U) = (U)

i = 0

n

y (u) = (u) Yj [4.4] i = 0

A 3^^ degree Bezier polynomial (figure 4.2) is used to illustrate the

proprieties of a Bezier curve.

102

X - (1-u) XQ + 3 u (1-u)^ + 3 (1-u) x + Xg

y = (1-u) yQ + 3 u (1-u)^ yj + 3 (1-u) y + y [4.5]

From equation 4.5, it can be shown that the end points of the

polygon points representing the Bezier curve are the end points of

the curve.

{x(0), y(0)} = (xo.yo) for u = 0

a n d { x (1 ) . y (1 ) } = (X3.y3) for" u = 1

The slopes at the two ends of the curve are given by the following

expressions:

It.. yo

X, -Xo

and

dv

dx U = 1

3 -^2 [4.6] ^ -X2

The second point of the polygon points from one end is therefore

located on the tangent ( PQP^ or PgPg) to the curve at this point (Pg

or P3). By varying the position of P along the straight line PQP.,

and/or Pg along PgPg , one can obtain different curves satisfying

the same end conditions. In many cases, a curve is divided into

several sections which are in turn defined by Bezier polynomials. It

is then necessary to ensure continuity between two sections at the

joining point up to a certain degree of derivatives. Continuity of the

second derivative is generally needed as it defines the curvature of

the curve. It can be shown that the second derivative at an end

point is defined by that end point and the two points nearest to it.

In the general case, it is obtained from equation 4.7.

103

d \

du^ dy du^

dx 2 dx fdx

du l ^ .

d \ _ , __ 7 T ~ • — — • — T [4.7] dx

and d^y/dx^ is a function of the coordinates of the three first

points for u = 0 and of the three last points for u = 1.

4.2. Blade Geometry Generation

The method outlined above is used to complete the

rotor design by setting the three dimensional geometry of the

blade. Rotor tip and hub diameters as well as the mean blade angle

at both rotor inlet and outlet have already been defined by the one

dimensional design method. The blade angle distribution along the

leading edge, blade curvature and the rotor length remain to be

fixed and their influence on the turbine performance analysed.

A fourth degree Bezier polynomial is used for

generating the hub, shroud and camberline profiles.

OP(u) = (1-u)^ OPQ + 4 u (1-u)3 OPi + 6 (l-u)^ OP^

+ 4 u3 (1-u) OPg + OP3 [4.8]

Where : u is a real number in the range ( 0 - 1 ) corresponding to

the point P of the curve.

PQ corresponds to the rotor inlet ( u=0 )

P3 corresponds to the rotor exit ( u=1 )

PI is a point of the tangent to the curve in PQ,

defined as PQPI = P P Q ^ ( 0<P <"1 )•

P2 is a point of the tangent to the curve in P3,

104

defined as P3P2 = q P3 C ( 0<q <1 ).

C is the intersection between the straight lines

( P QP I ) a n d ( P 3 P 2 ) .

OP, OPO, 0P1 , 0 P 2 and 0P3 are the posit ion vectors of

the points defining the curve.

4.2.1. Hub and Shroud Profile Generation

From figure 4.3, the hub or shroud profile is described

by the following parametric equations;

r = (1-u)4 TQ + 4 u (1-u)3 r + 6 u^ (l-u)? r

+ 4 u^ (1-u) rg + u' rg

X = (1-u)4 XQ + 4 U (1-u)3 X + 6 u^ (1-u)2 Xg

+ 4 u^ (1-u) Xg + u' X3 [4.9]

From figures 4.3 and 4.4, the end conditions necessary to define the

position of P^, Pc and P2 are as follows:

a Hub. At inlet ;

Xn = 0

Ty = 0.5 (Dj-b, SIN 62 )

dr

dx = TAN ( f - §2 )

and at exit

105

= xl

3 ^3H

2

dr

dx = 0 [4.10]

Shroud. At inlet:

X "0

0

f -

= COS 62

— 0.5 (D_ + b~ SIN 5 )

= T A N ( % - 62)

and at exit

= xl

_

2

= 0 [4.11]

coordinates of P.,, Pc and Pg for both hub and shroud profiles are

given by the following relations :

= Xy + ( ) TAN 5

Xj = Xy + p ( ) TAN 5, I", = - P ( To - )

x^ = X3 + q ( Xy - X3 + [r , - r^] TAN 5^ ) [4.12]

Varying the position of P and Pg along P^P^ and PgPg respectively

as well as the length of the rotor (Xg -XQ)|^^^ = xl , gives a family

of hub and shroud profiles which satisfy the constraints imposed by

106

the rotor geometr ica l character is t ics def ined in the one

dimensional design.

4.2.2. Blade Curvature

4.2.2.1. Radial Fibre Blade

Figure 4.5 represents an element of the blade. Because

it has radial fibres, its projection on a reference cylinder of radius

r g, is represented by one unique line called the camberline. The

properties of such a blade are outlined below.

The blade angle (3 is expressed as follows;

TAN p = COS Y [4.13]

where

P is the blade angle ( TAN p = r d9 / dm ).

Y is the cone angle with TAN y = - dr / dx

( r, X, 0 ) are the coordinates corresponding to the mean blade

surface.

The slope of the analytical curve representing the camberline is

given by equation 4.14.

r , d6 TAN (p., = [4.14]

" dx

where

r^ef is the radius of a reference cylinder where the mean

blade surface is projected. Because the blade fibres

are radial, this projection is a single line called

camberl ine.

By simply considering the slope dG/dx , relation 4.15 is derived

from equation 4.13.

107

d e 1 TAN B ^ - 7 [4.15] " ' cos Y

4.2.2.2. Camberline Generation

The camberline is defined by two curves consisting of

the leading edge up to the blade tip and the projection of the shroud

line of the blade on the (x,6) plane enabling the leading edge to

be modelled separately. The leading edge curve is defined first

allowing for the conditions related to the end point of figure 4.6

to be set and the blade angle distribution along the leading edge to

be predefined as explained in the following section.

a : Leading Edge

The mean blade angle at the rotor inlet was set for a

flow approaching the rotor at an optimum incidence angle by

analogy with radial turbines, for which maximum efficiency has

been recorded at incidence angles around -20. degrees. As far as

mixed flow turbines are concerned, the radius along the leading

edge varies from hub to shroud which implies the same for the flow

angle. In the first instance, it is tempting to use the same

reasoning for optimum incidence angle along the leading edge of a

mixed flow turbine rotor. Therefore the blade angle p2b along the

leading edge is obtained from the computation of the flow angle j i j

with the optimum incidence angle being fixed.

P2b = P] " 'Popt

The relative flow angle along the leading edge is computed by

108

assuming a uniform flow approaching the rotor and the free vortex

law valid upstream of the rotor. The meridional velocity component

is calculated from continuity equation as follows :

m = — — [4.17]

nD^ P2

While the tangential component at radius r is obtained from the

free vortex equation.

Wg = VQ - 0) r with r Vg = Cte

The density appearing in the R.H.S of equation 4.17 is in turn a

function of V^.

P? = PF 1 e m

7-1

[4.18]

An iterative calculation combining equations 4.17 and 4.18 enables

P2 and to be found.

The flow angle Pg 's then calculated as ;

P2 = A T A N ( WQ / V m )

and the blade angle distribution along the leading edge pgbci

constant incidence angle is obtained by equation 4.16.

Another option consists of considering a constant

blade angle distribution ( ) ^'ong the leading edge, Pgbo being

the blade angle calculated at the mean rotor inlet diameter.

These two distributions referred to as pgbci (constant

109

incidence angle distribution along the leading edge) and Pgbo

(constant blade angle distribution along the leading edge) are used

to define the blade angle distribution along the leading edge in the

following form ;

P2b - C ' C) Pzbci + C P2b0 [4.19]

where

P2bci is the blade angle distribution along the leading edge

and corresponding to a constant incidence angle.

P2bo is the blade angle at the mean diameter.

^ is a weighting factor between 0 and 1.

The slope of the camberline d9/dx along the leading

edge is obtained from relation 4.15 by replacing p by Pgb ^nd the

cone angle y by 69 - ^^2 . Thus

d8 ] TANP,^

d x r S I N 6 [4.20]

9

The (x,e ) coordinates along APg are calculated by the following

expression;

X

6 = e , + — ! f TAN &. — 14.211 SINS, J " f

where r = r^^ + ( x - Xop, ) TAN Sg

b : Shroud curve

The second curve P0P3 (figure 4.6) is generated by a

fourth degree Bezier polynomial. The end points coordinates (x,6 )

1 1 0

are :

(x,e)o = ( bg C O S 62, 0 )

(x,9 jg = ( xl , 6 3 )

while the slope at PQ and P3 are ;

and

de

dx

de

dx

TAN p, 2 b s

R«. SIN 5;

= ? TAN p j

d7~" [4.22]

Where

p2bs and RQ5 are respectively the blade angle and the radius at the

tip rotor inlet .

P3 and D3 are the blade angle and diameter at the rotor exit

mean root diameter.

Straight lines PQC and P3C are the tangents to the camberline

respectively in PQ and P3 thus:

Xc = (83- 80+ 83X3 - aoXo )/ (83 - ao )

8c ~ 80 ^0 ( Xc " XQ ) [4.23]

where a^ and a3 are the slopes PoC and PsC. defined by equation

4.22 and

X 1 = X Q + P ( XJ; " XQ )

81 = 8Q + p ( 9^ - 60 ) ( 0<p<1 ) [4.24]

111

Coordinates of point B (figure 4.6) are found by using a continuity

of the second derivatives of section APQ and PpPg at P^ and zero

curvature at P3. Thus B, Pg and P3 are situated on the same tangent

to the curve at P3.

d ^ 8 d ^ e

2 9 [ 4 . 2 5 ]

dx dx xO (APO) xO ( P 0 P 3 )

The L.H.S of equation 4.25 is defined for the section APq w/hile the

R.H.S is obtained from equation 4.7. The position of B being defined,

point Pg is obtained as follows ;

X2 - X3 + q ( Xg - X3 )

62 = 83 + q ( 6g - 83) ( 0<q<1 ) [4.26]

The coordinates (x,8 ) of any point of the camberline ( curve P^Pg )

are defined by equation 4.27.

X = (1-U)4 XQ + 4 U (1-U)3 X + 6 U^ (1-U)2 Xg

+ 4 u^ (1-u) Xg + u" Xg

8 = (1-u)4 80 + 4 u (1-u)3 81 + 6 u2 (1-u)2 Sg

+ 4 u3 ( l -u) 82 + u^ 83 [4.27]

4.2.3 Examples of Bezier Polynomial applications.

The method presented in this chapter has been devised

and programmed for the blade design. By varying the position of the

interior points of the polygon points defining the Bezier curve, the

method enables the geometry of the blade to be modified throughout

112

the design process. Also by varying the coefficient ^ in equation

4.19, several blade geometries can be obtained.

An example of camberlines generated by this method

is presented in figure 4.7. The curve %=0.0" corresponds to a blade

camberline of a rotor having a constant blade angle at inlet and the

curve "^=1.0" describes the blade camberline of a rotor having a

constant incidence flow angle at inlet for design conditions. The

rotor length XL and "blade curvature" 63 are two other parameters

which are used to modify the blade geometry. The effect of XL

and 03 on the rotor performance will be analysed in chapter 6 which

is aimed at completing the turbine design.

113

Y(u)

u-0.

Polygon

Bezier Curve

X(u)

Fig.4.1 Nth degree Bezier polynomial.

Fig.4.2 3rd degree Bezier ploynomial.

114

Fig.4.3 Hub and shroud profiles generation by a Bezier polynomial.

Fig.4.4 Mixed f l o w r o t o r : O v e r a l l dimensions

115

Fig.4.5 Radial fibres blade element.

e

Fig.4.6 Camberline generation. APo : Leading edge part of the camberline. P0P3 : Shroud part of the camberline generated

by a Bezier polynomial.

116

0) +> Id c •H •d M o o u

0) g

t*

r-rH rH

Axial coordinate

Fig.4.7 Examples of Camberline generation by a Bezier polynomial.

Chapter 5

5. FLOWFIELD ANALYSIS

In the one dimensional design analysis, many

parameters concerning the rotor geometry remain undefined. The

rotor design can be completed either by experimentally testing the

prototype or simply by numerical analysis of the selected design.

The first option is quite laborious and too expensive as it involves

in each case of analysis the manufacturing of the prototype while

the second needs only the input geometrical parameters to be

modified. It is the second option which is generally adopted by

turbomachine designers at this stage of the design process as a

result of the remarkable progress in both the state-of-the-art of

numerical methods in computational fluid dynamics and computers

capabi l i t ies.

Inviscid methods used for turbomachine flow

calculations can be subdivided in different classes such as :

- Potential flow method

- Stream function method

- Streamline curvature method

- Solution of Euler equations

Potential flow method assume that the velocity

components derive from a scalar function cp. In a cylindrical system

of coordinates, they are written in the following form :

V = • V = ; V„ =AE [5.1] ' 6r ' 8 r68 ^ 6z

The substitution of these expressions of velocity components into

118

the continuity equation leads to a second order differential

equation of the scalar cp. Efficient techniques exist for solving such

equation for both two and three dimensional flows. However the

solution is limited to isentropic and irrotational flows. Subsonic

and shock free transonic solutions can be obtained.

Stream function methods are based on a similar

approach to that of potential flow methods in that the governing

equations of the flow are reformulated in terms of a scalar

function \|/ (stream function). The mass flow components pVr and

pVe or pVr and pVz are written in the following form :

(r -9) plane : p V = — ; - n v = — ' b rSQ 6 b gr

(r-z) plane ; p V = - — ; - p V = — [ 5 . 2 ] ' b / b

where b is the local thickness of the stream sheet of the

calculat ion domain. This reformulation of the fundamental

equations results in a second order differential equation for which

ef f ic ient re laxat ion methods exist. The st ream function

formulation of the flow retains the generality of the Euler

equations but is limited to two dimensional or axisymmetric flows.

Although transonic solutions have been obtained for particular

applications, the method is generally restricted to subsonic flows

because the density is a double valued function of the stream

function \\i corresponding to subsonic and supersonic solutions.

The streamline curvature method and the time

dependent solution of the Euler equations are presented in detail in

the following sections. The streamline curvature method was used

here for the rotor geometry design while the solution of the Euler

equations has not been used in the design process because of the

119

difficulty of getting a converged solution with this method when it

was applied to a mixed flow rotor.

5.1. Streamline Curvature Method

The streamline curvature method is two dimensional

and is used to solve the equation of motion in a grid mesh formed

by streamlines and orthogonals or quasi-orthogonals (figure 5.1).

Bindon and Carmichael (24) solved the momentum equation written

as a first order ordinary differential equation for the velocity

gradient along lines normal to the streamlines. This condition of

orthogonality makes Bindon's method less preferable than that used

by Katsanis (25), (26) and Wilkinson (27).

Katsanis in his method, which was used for the rotor

geometry design, used a grid mesh for the flow calculation

generated by streamlines and lines not necessarily normal to the

streamlines and referred to as quasi-orthogonals (Q.O). A Q.O in the

meridional surface is a straightline which goes from hub to shroud

(figure 5.1a) and in a blade-to-blade surface, it is a portion of

circle which goes from pressure surface to suction surface (figure

5.1b). The differential equation giving the velocity gradient along

an arbitrary Q.O is given by the following equation :

dq dq dq dq W

dh, dX

dq dq [5.3]

where A, B and C are coefficients obtained from the blade angle,

the streamline slope and curvature and the velocity distribution on

the surface of the flow calculation. The fourth term on the R.H.S of

equation (5.3) is a function of the stagnation enthalpy and prewhirl

gradients along the Q.O.

120

W c o s ^ P + cosy W sin^ B ^ dW A = —— + sin y cos P - 2 co sin P

C

g _ W c o / 3 s i n r

dm

C = W sin Y cos P sin P + r cos p —— + 2 0) sin Y dm

Details of these terms and the calculation procedure are given in

reference 25 for the meridional surface calculation and in

reference 26 for the blade-to-blade calculation.

The velocity distribution along each Q.O is calculated

by equation 5.3 and adjusted so that the mass flow crossing each

section must equal the specified mass flow. In addition to the

specification of mass flow, stagnation conditions and prewhirl rVe

at inlet boundary as well as the rotational speed are specified.

After each iteration, the streamline positions are updated so that

the mass flow is equally distributed between the streamsheets.

This streamline geometry and the velocity distribution are used to

update the coefficients of equation 5.3 in the next iteration until

convergence is reached. The effect of viscosity is approximated by

the introduction of a stagnation pressure loss proportionally

distributed from inlet to outlet so that the exit pressure is

approximately equal to that defined in the one dimensional

calculat ion.

5.1.1. Meridional Surface Calculation

121

The flow calculation is performed on a stream

surface which coincides with the mean blade surface except near to

the rotor inlet. In this region of the rotor, the stream surface

deviates from the blade surface allowing for the incidence to be

taken into account. A method to model the flow incidence for radial

turbines is presented in reference 25. A similar method which is

presented in this section have been used for mixed flow turbines.

At rotor inlet, the stream surface deviates from the

blade surface and reduces from inlet to a certain Q.O on which the

two surfaces join together. The distance from inlet to this Q.O is

arbitrarily taken to be 25% of the rotor channel length.

The relative flow angle P2 at the leading edge is

computed in the same way as it is done in section 4.2.2 and the

blade angle pgb is defined by the blade geometry at rotor inlet. The

incidence angle, not being necessarily constant along the leading

edge, is computed on each point at inlet.

For each streamline, the incidence angle i(3 is

empirically defined as i(3 = ip(sm), where sm is the distance along

the meridional streamline. Let s,, denotes the distance at which no

incidence occurs and ip1 the incidence at rotor inlet. Incidence at

any position is given as follow:

ip = ip1 (1 - sm/s,)) " when s < So

ip = 0 s > So

and the relative flow angle:

p = + ip where Pb denotes the blade angle

122

The change dG of the streamsurface angular coordinate 6 along a

streamline is due to the blade shape d6b and the flow incidence

dQjnc-

d6 = d6|3 + dGjpiQ

For radial blade surface,

tan pb = r cos y (SOb/Sx)

and for the stream surface,

tanp = r sin Y (6ejnc/8r) + r cos y [ (68b/6x) + (50inc/Sx)]

If it is assumed that for a radial turbine, the deviation from the

blade surface due to the direction of the flow coming into the rotor

is independent of the axial position but varies in the radial

direction, then

59jnc/S^ = 0

A similar analysis for an axial turbine, leads to

59jp|Q/5r = 0

For a mixed flow rotor, it is logical to say that the flow deviation

is dependent on both radial and axial positions. One can assume a

d e p e n d e n c e between (86 inc /6r ) , (88 inc /8x ) dBjnc

following form:

123

(56jnQ/5r) dr = dGjnc sin^ y

3nd (SGjnc^Sx) dx = dGjpiQ cos^ y

and finally S e j n c / S r = tan y ( g 8 i n c / 6 x )

The combination of these equations leads to

8fi siny — = - 7 - ( tan (3 - tan ) and

_ c o ^ ( tan P + tan^y tan P ) [5.4] ox r "

Parameters (56/5r) and (60/5x) are used to update the terms A

and B in the right hand side of equation 5.3 which gives velocity

gradient in a quasi-orthogonal.

5.1.2. Blade-to-Blade Calculation

On the same principle, Katsanis applied the method to

solve the flow on a blade to blade surface . The stream sheet

surface on which the flow is calculated is delimited by two

surfaces of revolution generated by two successive streamlines

resulting from the meridional plane solution. In this case, the Q.O

are portions of circles going from suction surface to pressure

surface. The intersection of streamlines and Q.O in the blade to

blade surface, generates a grid for the flow calculation. By solving

the flow in different blade to blade surfaces from hub to shroud, a

picture of the flow inside the rotor can be obtained. One of the

difficulties encountered when using a blade to blade calculation is

to decide on the shape of the stagnation streamline upstream and

downstream of the rotor. Although Katsanis (26) and Wilkinson (27)

estimate the stagnation streamline shape, this can be a source of

124

errors as it Is demonstrated by Wilkinson and Allsop (28). The

method was used only in a domaine restricted to the blade channel.

5.2. Finite Volume Method

Time marching solutions of Euler equations by a finite

volume method has been extensively used for the flow calculation

in turbomachines during the last years (1), (10), (14) and (19). This

method consists of marching the solution of the time dependent

Euler equation from an arbitrary solution until a converged steady

state solution is reached. The physical domain of the flow

calculation is subdivided into elementary volumes on which the

Euler equations written in a conservative form are integrated. The

advantage of the method is its conservative character and the

facility of discretizing complex flow domain geometries without

resorting to a coordinate transformation and therefore giving it a

character closer to the physics of the problem. A long calculation

time is needed before convergence is reached, a high storage

memory and the inviscid nature of the solution are the

disadvantages of the method.

5.2.1 Governing Equations

For an adiabatic, inviscid flow, equations governing

the flow and applied to a small volume can be written in a general

form as follows;

- Continuity equation

+ p div V = 0 dt

- Momentum equation [5-5]

125

dV p -gp + Grad P = 0

Energy equation

dE p —— + P div V = 0

dt

It is more convenient to write these equations in

cylindrical coordinates when applied to turbomachinery. In this

case, the system of equations written in a quasi-conservative form

becomes :

Where o, F, G, H and B are given by the following equations ;

o =

p p w ^ e

p \ \ pw^ + P p w w ^ e '

p w ^ e F = p w w

^ r e G = pw^ + P ^ e

p W W ^ r X

PE p ( E + p/p ) P Wg (E + P/p )

H =

pw^

p w w r X r

P W W ^ x e

P w" + P

p w ( E + P/p

B =

- P

( + Q r )"

w w r e

P — - — + 2 p Q

0

0

[5.7]

This system of five equations with six unknowns (p,

126

W|-, W0, WX, P and E) is completed by the equation of state which,

after some combinations, results in the following form:

P = ( y - 1 ) p ( E - — + ) [5.8]

where E = Cv T + W^/2 - U^/2 is the relative energy.

5.2.2. Grid Generation

The volume which is bounded by two successive

blades, hub and shroud surfaces and rotor inlet and exit surfaces is

divided into elementary volumes used in the finite volume

discretisation. These are generated by three types of surfaces

namely a blade to blade surface, a meridional surface and a quasi-

orthogonal surface. The domain constituted by the channel between

two blades is extended upstream and downstream of the rotor

where inlet and exit boundary conditions are applied (fig.5.1).

The control volume used in the numerical procedure is

in fact the sum of two streamwise volumes in the meridional

direction. Nodes, at which the unknown parameters are computed,

are situated at the centres of the faces generated by the quasi-

orthogonal surface (Fig.5.4)

5.2.3. Finite Volume Discretisation

The integration of equations [5.6] in the control

volume AV already defined and for a time interval At can be simply

expressed by equation [5.9], assuming that the fluxes are constant

in each face. In equation [5.9], a supplementary damping term is

127

added.

At ^ Aa = - — A + G Ag + H A - At B + Aa - Aa

rrl

[5.9]

where A|-, AQ and Ay are respectively the projection of a face over

plans (x, 8), (r,x) and ( r , d ). F, G and H are values calculated in the

centre of each face while B is the value at the node considered. A a

represents the change of property o during the time step At, Ao and

Aa* are defined below in equation (5.11).

Equation (5.9) can simply be expressed in the form:

A(property) 6

AV + ^ FLUX^ + SOURCE = 0 [5.10]

5.2A. Corrected Viscosity Scheme

This scheme was first introduced by Mc Donald to

discretise the time dependent term of equation [5.6]. It was largely

used in two-dimensional and three-dimensional flow calculations

with finite volume. According to Arts (1) and Van Hove (19), the

scheme has shown very good convergence, stability and accuracy

properties for different types of control volumes employed. The

scheme applied to a three-dimensional flow calculation leads to

the following discretized equations:

128

- O;: i, (Net transport by convection )' - At B ' »j ' .J ^ V

+ <,j.k + + V i +

- 6 + i.ij,k + i,j-i.k + ®i.j+u + V - i + ®io.k.r6<jjc )

[5.11]

The fourth term in the right hand side of equation

5.11, equivalent to a second derivative of a, is used to speed up the

convergence in the time marching method. The introduction of this

important numerical viscosity-like results in a poor accuracy. The

last term is introduced as a correction to the numerical viscosity

term. Terms superscripted by an asterisk remain constant during Nv

iterations after each updating (Nv=15~20) . (i is a numerical

coefficient calculated as a function of the local density gradient

and a numerical constant u (0<u<1)

^ = D ( 1 - IPj.] J + Pj+i J J; + Pi,j.],k + Pi,j+l,k Pi,j.k-1 Pi,j.k+1

- 6 p _ j J / 6 [5.12]

It permits the retention of some viscosity and hence the capture of

shock waves. The time increment At has to satisfy the CFL

stability condition as defined in section 5.2.7. Since no time

history for the calculation is needed, the maximum stable time

step can be used for each individual element to obtain the fastest

convergence of the solution.

129

5.2.5 Boundary Conditions

Additional conditions are needed when evaluating

terms in the right hand side of equation (5.9) applied to an element

volume adjacent to the boundaries of the computational domain.

No mass crosses the solid boundaries constituted by

the hub, shroud, pressure and suction surfaces. Mass and energy

fluxes are set equal to zero when computed in the solid boundary

faces, while flux terms in the momentum equations reduce to a

pressure force normal to the face considered. The pressure on this

face is obtained by extrapolation from the interior nodes of the

domain.

In one of the periodic boundaries, which are surfaces

extending the mean blade surface upstream and downstream of the

rotor, flow parameters (p, W, E and P) are set to be equal to those

of the opposite face on the other boundary. Thereby, fluxes on these

two faces are equal in absolute value but opposite in sign.

In the inlet plane, total pressure, total temperature

and the flow direction are specified. These parameters are assumed

to be constant along the tangential direction while their

distribution from hub to shroud is specified as input. Solving one of

Euler equations, generally one of the momentum equations, enables

all the unknown parameters at inlet to be computed. The choice of

the momentum equation to be solved depends on the turbine type.

The axial momentum equation is used for axial turbine and the

radial momentum equation is used for radial inflow turbines.

Experience has shown that the use of continuity or energy equation

at inlet leads to the instability of the solution process (1). For

130

mixed flow turbines, a combination of radial and axial momentum

equations is used to compute the meridional velocity (figure 5.3) at

the inlet boundary (equation 5.13)

A ( p ) = cos Y A( p V J + sin y A( p ) [5.13]

In the outlet plane, it is generally the static pressure

which is specified (uniform in the tangential direction). Although

the radial pressure distribution can be specified, it is generally

accepted that it is defined at the hub and the radial distribution is

deduced from the radial momentum equilibrium. The remaining

conditions are obtained by the continuity equation and the three

momentum equations.

Viscosity terms intervening in equation (5.9) as it is

written do not apply to control volumes adjacent to the boundaries.

Except for periodic boundaries, an excentred viscosity scheme is

applied instead of the centered scheme used in the general case.

5.2.6 Initial Conditions

In a time marching method, the solution process is

started assuming an initial distribution of the flow parameters. As

the initial solution is considered as a perturbation of the steady

state solution, its choice is made arbitrarily providing it is

compatible with the flow conditions. Convergence towards the

steady state solution occurs regardless of the initial solution

which is of little importance when no history of the flow

calculation is needed.

13.

5 .2 .7 S tab i l i ty

Time marching methods are subject to stability

criterion known as the Courant-Friedrichs-Lewy (CFL) condition.

The mathematical formulation of the CFL condition is given by

equation (5.14)

A1 T T w '5,141

where At is the time step, Al the minimal distance from the node

on which the computation is made, to the faces of the domain of

numerical dependence (fig.5.4), a is the local speed of sound and W

is the local velocity (positive). This means that the pressure waves

may not propagate beyond the domain defined by the surrounding

nodesduring the time interval. The CFL condition is a necessary

condition for the stability of the computing scheme and has to be

observed at each node. For a steady state solution only, the time

step is maximized at each node by equation (5.14) in order to

reduce the computing time.

5.2.8 Control Volume and Surface Calculation

Control volumes on which the Euler equations (5.6)

are integrated consist of polyhedra of irregular shape and therefore

the evaluation of the measure of the control volume and the areas

are not straightforward. In order to overcome this difficulty, the

control volume in the (r,e,x) space is transformed into a cube with

unit length in a new space (^,ti,0- The coordinates of any point of

the control volume AV (figure 5.5) is expressed by the following

132

re lat ions:

'm ; G = ^ X, m-l m=l m=l

The coefficients Km are given as follows :

[5 .15]

K i = ( 1 - C ) (1-Ti) ( l - S )

K2 = C (1-Tl) (1-S)

K3 . ( n (1-S)

K4 = (1-0 T1 (1-S)

K5 = (1-0 (1-Tl) %

K6 = ; (1-Tl) ^

K7 = ; Ti ^

KB = (14^ n S [5.16]

and the measure of the control volume is given by the following

re lat ion;

11 1

AV = J J J r dr d9 dx = J J J r J d^ drj d^ [5.17]

0 0 0

J is the Jacobian of the transformation for the control volume of

which spanwise and streamwise surfaces are surfaces of

revolution.

J =

5r M

5x

5C

0 M 0 5n

5r

5^

M 5^

5x

[5.18]

The area of each face of the control volume are

calculated in a similar way to that used for the measure of the

control volume calculation. The surface defined by the four points

133

1, 2, 3 and 4 ( figure 5.6) is defined as follows :

^ ; 8 = 2 . K „ 6 „ ; x = 2 . K „ x „

m = l rn=l m = l

where

Ki = (1-a) (1-p) K3 = a p

K2 = a (1-p) K4 = (1-a) p

The pair ( a, p) being ( ri). ( n, ^) or ( Q depending

on the face of the control volume considered. The area S is then

calculated as follows :

and

S = Sr + Se + S)

' JJ 6a 6p 6a 6p 0 0

=ij 0 0

1 1

S = F F , ^ , D A D P [5.19] JJ 6a 6B 6a 6B 0 0

6a 6p 6a 6p

The full developments of equations 5.17 and 5.19 are given in

reference [1].

134

Periodic Boundary

Streamline

1 < •H X w

a) Meridional Surface

Streamline

— I —

Periodic Boundary

b) Blade-To-

Fig.5.1 Rotor Channel Discretisation

135

i +1, Jt3

( i - i . j . k )

r 4

'*•» j. k-lJ

. k )

J i. . j . k • 1 J

( i + l . j . k )

/ C i . j - l . k D

k :

e

a: Blade Surface b: Quasi—orthogonal Surface

^i.j,k+i)

fi-1,j,k)

J f »• » j , k -i)

e *

r*

c : S t r e a m w i s e S u r f a c e d ; C o n t r o l V o l u m e

Fig.5.2 Volume Discretisation

136

c

Fig.5.3 Velocity Triangle in the Meridional Plane

( V - 1 , j » k ) \ \

^ ' j , k+i)

Fig.5.4 Domain of Numerical Dependence ( CFL Condition )

137

b)

Fig.5.5 Control Volume Transformation a) Real Control Volume b) Transformed Unit Volume

b)

a)

Fig.5.6 Surface Transformation a) Real Surface b) Transformed Unit Surface

138

Chapter 6

6. TURBINE DESIGN

6.1 Casing Design

Some features of the casing geometry have already

been defined through the one dimensional design procedure and

concern the inlet scroll and vaneless geometry given in section 3.6.

The full geometry of the casing is still to be defined, bearing in

mind that the flow has been assumed to satisfy the free vortex law

upstream of the rotor. The casing configuration analysis is done

without considering the friction and other losses.

The mass flow entering the rotor is assumed to be

uniform and therefore at each scroll section, one can write the

expression of the mass flow m\|/ with respect to the .azimuth angle

\|/ as follows (see figure 6.1) :

m\j/ = m ( 1. - ) [6.1]

The continuity equation applied at the cross section corresponding

to the azimuth angle v (figure 6.1) leads to the following relation

between the scroll inlet parameters and those at the section

considered :

A p V G = ( 1. - ) Ao po Vo [ 6 . 2 ]

and the free vortex law ( ro Vo = r Ve ) combined with the energy

equation leads to the following expression for the density ratio :

139

IT -1

_G_ =

Pn

0 *

r cosa 1 -

[6.3]

Substituting equation 6.3 into 6.2 leads to the relation 6.4 between

the area A, radius r, the azimuth angle \\f and the flow angle a .

A = A (r,\|/,a) [6.4]

Equation 6.4 can be reduced to the form

A = A (y) [6.5]

by considering a relationship between radius r and azimuth angle y

along the line containing the centroid of the cross section. The flow

angle a is then calculated at each cross section centre from the

following relation:

TAN a = dr

r d\|/ [6.6]

Equation 6.6 must satisfy the end condition at xy = 360° which is set

by calculating the flow angle at the vaneless inlet (radius r j in

the same way as that used in the off-design conditions analysis for

calculating the flow parameters at rotor inlet. So

TAN = dr

d y 360"

Where r = D^/2 is the radius at the vaneless inlet (figure 6.2). The

scroll section area is then defined at any azimuth angle from

140

equation 6.5. The selection of the cross section configuration

shown in figure 6.2 has been made according to the following

considerations :

- The distance between the bearing housing and the back of the

rotor is limited so that possible rotor vibrations, which can be

damaging to the machine, can be avoided.

- The need to have a straight vaneless duct upstream of the rotor

which directs the flow at the desired angle (62) in the meridional

direct ion.

- Simplicity of the casing geometry so that it can be easily

manufactured.

The full details of the volute cross section geometry

are given in table 6.1.

6.2. Rotor Design

The streamline curvature method outlined in section

5.1 has been used to analyse the flow in the rotor channel. The

calculation has been carried out in the meridional surface for the

design operating conditions. The blade surface velocity distribution

in three surfaces (hub, mean and shroud blade-to-blade surfaces)

has been obtained from the meridional surface calculation by an

approximation method which is presented in reference (25) and

which assumes absolute irrotational flow and linear velocity

distribution between pressure and suction surfaces.

From the turbine design dimensions already defined by

the one dimensional design and presented in section 3.6, several

blade geometries have been analysed. The effect of three factors

influencing the blade geometry has been investigated and

141

consisting of ;

1) The blade camberline of the leading edge ( of equation 4.19) .

Figure 6.3a shows three camberlines corresponding to ^ = 0.0, 0.5

and 1.0.

2) The rotor length XI ( figures 6.3b and 6.3d).

3) The blade curvature 93 ( figure 6.3c).

The method presented in section 4.2 has been used to

generate the hub, shroud and camberline profiles for each case to

be analysed. Spline curve fit routines have been used to find the

streamline derivatives necessary for the calculation of the

coefficients in equation 5.3.

6.2.1. Influence of the leading edge shape

The leading edge shape effect on the rotor

performance is presented in terms of the blade surface velocity

distribution for three different blade-to-blade surfaces as shown

in figure 6.4. Three rotors whose blade camberlines are presented

in figure 6.3a have been analysed. The rotor length XI and the

tangential coordinate 63 at the blade trailing edge for the three

rotors have been kept identical while the leading edge blade

camberline differs from each other as follows :

- Curve A1 represents the camberline of a blade having a constant

blade angle along the leading edge ( ^ = 0.0 in equation 4.19 ).

- Curve A3 represents the blade camberline of a rotor designed to

have a constant incidence flow angle along the leading edge at the

design operating conditions ( ^ = 1.0 ).

- Curve A2 represents the blade camberline of an intermediate

design where the blade angle distribution along the leading edge is

the averaged blade angle of A1 and A2 obtained for ^ = 0.5.

142

From the blade velocity distribution of figure 6.4, the

large loading at rotor inlet can be noticed which varies with the

leading edge shape. At the shroud, the flow is generally

accelerating in the most part of the rotors corresponding to A1 and

A2 except near the leading edge where the suction surface velocity

is decelerating. Rotor A3 on the other hand shows a decelerating

flow region on both pressure and suction surfaces. Negative

velocities computed on the pressure surface for each case, which in

addition to decelerating flows in some parts of the rotor channel

are likely to be the cause of high losses due to the mixing flow and

flow separation in these regions of the rotor channel.

6.2.2. Influence of the Rotor Length

Two sets of blade camberlines are analysed to show

the influence of the rotor length XI on the flow distribution along

the rotor channel. Curves of figures 6.3b and 6.3d represent the

blade camberlines of rotor having the following characteristics :

1) Curves B ( figure 6.3b )

Constant blade angle along the leading edge ( ^ = 0.0 ) and 03 = -25.°

for the three camberlines ( B1, B2 and B3 ), while the rotor lengths

are as follows :

B1 : XI = 35. mm

B2 : XI = 40. mm

B3 : XI = 45. mm

2) Curves D ( figure 6.3d)

Constant incidence angle along the leading edge ( ^ = 0.0 ) and

143

93 = -25.° for the three camberlines ( D1, D2 and D3 ), while the

rotor lengths are as follows :

D1

D2

D3

XI = 32.5 mm

XI = 36.0 mm

XI = 40.0 mm

Figures 6.5 and 6.6 show the blade surface velocity

diagrams corresponding to the family of camberlines of figures

6.3b and 6.3d respectively. The blade surface velocity distribution

at the shroud (figure 6.5) corresponding to the set of camberlines B

(figure 6.3b) seems to improve by increasing the rotor length.

Figure 6.6 also shows the influence of the rotor length (figure 6.3d)

on the blade surface velocity. The main feature of increasing the

rotor length is an increase of the blade leading edge loading

accompanied by a velocity deceleration inside the rotor channel for

both the pressure and suction surfaces. This suggests the use of a

short rotor when the blade geometry is designed in order to have a

constant incidence angle along the leading edge.

6.2.3. Influence of Blade Curvature ( 93 )

The blades represented by the camberlines of figures

6.3c have the same leading edge shape ( constant blade angle along

the leading edge ) and the same axial length ( XI = 40. mm ). They

differ only by their tangential coordinates of the trailing edge 93 .

CI : 93 = -20.°

C2 93 = -25.°

C3 93 = -30.°

The blade surface velocity diagrams corresponding to

this set of camberlines (figure 6.3c) are presented in figure 6.7.

144

The effect of 03 is less apparent than in the previous cases but it

can be seen from the shroud velocity profiles that increasing 03

leads to a sharper flow acceleration in the pressure surface and a

larger region of decelerating flow in the suction surface near the

rotor exit.

6.2.4. Selection of Prototype

Two rotors, which will be referred to as "rotor A" and

"rotor B", have been selected from the set of designs analysed in

this section. The two rotors are defined as follows (table 6.2);

Table 6.2 : Characteristics of rotor A and B

Rotor A Rotor B

Leading Edge Constant Blade angle

C = 0.0

Constant Incidence

Angle ^ = 1.0

Rotor Length: XI

( mm )

40.0 32.5

Tangential Coordi-

nate of Trailing

Edge : 03

( degrees )

-25.0 J

-25.0

1

In order to investigate the effect of blade angle

distribution at rotor inlet on the turbine performance, the exducers

of rotor A and rotor B have been kept approximately identical. The

145

two rotors differ only by their lengths and leading edge shapes.

Details of the two rotor geometries as well as a flow

analysis by the streamline curvature method are presented in

section 6.3. Further experimental analysis of the two prototypes

will be carried out as the final step in the mixed flow turbine

design process.

6.3. Analysis of Rotor A and Rotor B

6.3.1. Blade Geometry

Blade coordinates of rotor A and rotor B are presented

in tables 6.3 and 6.4 respectively. The blade thickness of the two

rotors obtained from the blade stress analysis by Holset is also

given in tables 6.5 and 6.6. The blade thickness distribution used

during the design process is slightly different and was arbitrarily

chosen . Camberlines and meridional blade surface projection of the

two rotors are shown in figures 6.8 and 6.9. Figure 6.10 on the

other hand shows the blade angle distribution along different

streamlines for both rotors A and B while figure 6.11 shows the

streamline projections on a (r0,r) plane. Finally, transverse

projections of the two rotors are presented in figure 6.12.

6.3.2. Flow analysis of Rotor A and Rotor B

The two rotors were analysed at different rotational

speeds by a streamline curvature method calculating the flow in a

meridional surface. The velocity ratio DC and stagnation

temperature TO* were maintained constants for every case { UC =

0.61 and TO* = 923. K ). For each rotor, four values of the rotational

146

speed were used for the flow analysis ( No = 50000, 70000, 98000

and 110000 rpm). The parameters needed for the calculation such

as the stagnation pressure/density, the prewhirl at inlet boundary

(rVe) and the mass flow were provided by the one dimensional off-

design code analysis.

The results for turbine A consisting of the velocity,

pressure and Mach Number distribution along hub, mean and shroud

streamlines for each rotational speed are presented in figures 6.13

to 6.16. The blade surface velocity distribution (loading diagram) in

three streamsurfaces ( hub, mean and shroud) corresponding to each

case are shown in figures 6.17 to 6.20.

Similar results for turbine B are also presented in

figures 6.21 to 6.28.

Both rotors seems to have similar loading diagrams at

the exducer part of the rotor while near to the inlet, the flow in the

shroud suction surface is decelerating for turbine A and

accelerating for turbine B (at high speed). This is probably due to

the difference on the shroud blade angle between the two rotors

(figure 6.10).

147

TABLE 6.1; VOLUTE CROSS SECTION DIMENSIONS

17.99

Y r A X Y A/r Y/2 (DEG.) (nun) (mm2) (mm) (mm) (mm) (mm)

0. 80.530 1702.410 48.848 41.388 21.140 20.694 10. 80.505 1629.253 47.991 40.238 20.238 20.119 20. 80.429 1557.574 47.134 39.088 19.366 19.544 30. 80.303 1487.374 46.276 37.939 18.522 18.969 40. 80.128 1418.651 45.419 36.789 17.705 18.394 50. 79 . 903 1351.407 44.562 35.639 16.913 17.820 60. 79.630 1285.641 43.705 34.490 16.145 17.245 70. 79.311 1221.353 42.848 33.340 15.400 16.670 80. 78 . 945 1158.543 41.991 32.190 14.675 16.095 90. 78.534 1097.211 41.133 31.041 13.971 15.520 100. 78.080 1037.358 40.276 29.891 13.286 14.945 110. 77.584 978.982 39.419 28.741 12.618 14.371 120. 77.049 922.085 38.562 27.592 11.968 13.796 130. 76.475 8 66.666 37.705 26.442 11.333 13.221 140. 75.864 812.725 36.848 25.292 10.713 12.646 150. 75.219 760.262 35.990 24.143 10.107 12.071 160. 74.542 709.278 35.133 22.993 9.515 11.497 170. 73.834 659.771 34.276 21.843 8.936 10.922 180. 73.098 611.743 33.419 20.694 8.369 10.347 190. 72.336 565.193 32.562 19.544 7.813 9.772 200. 71.550 520.121 31.705 18.394 7.269 9.197 210. 70.741 476.527 30.847 17.245 6.736 8.622 220. 69.912 434.411 29.990 16.095 6.214 8.048 230. 69.066 393.774 29.133 14.945 5.701 7.473 240. 68.203 354.615 28.276 13.796 5.199 6.898 250. 67.326 316.934 27.419 12.646 4.707 6.323 260. 66.437 280.731 26.562 11.497 4.225 5.748 270. 65.538 246.006 25.704 10.347 3.754 5.173 280. 64.630 212.759 24.847 9.197 3.292 4.599 290. 63.716 180.991 23.990 8.048 2.841 4.024 300. 62.796 150.700 23.133 6.898 2.400 3.449 310. 61.873 121.888 22.276 5.748 1.970 2.874 320. 60.947 94.554 21.419 4.599 1.551 2.299 330. 60.020 68.699 20.561 3.449 1.145 1.724 340. 59.094 44.321 19.704 2.299 0.750 1.150 350. 58.170 21.421 18.847 1.150 0. 368 0.575 360. 57.249 0.000 17.990 0.000 0. 000 0.000

148

Table 6.3 : Blade coordinates of rotor A

XH RH XS RS XT TETA

0. 000 36.006 13.781 47.569 0.000 -10.768

1.238 34.532 14.523 46.692 1.238 -9.670

2.481 33.062 15.284 45.831 2.481 -8.596

3.735 31.601 16.079 45.002 3.735 -7.542

5.003 30.153 16.919 44.219 5.003 -6.505

6.290 28.722 17.810 43.494 6.290 -5.480

7.602 27.313 18.751 42.836 7.602 -4.463

8 . 942 25.931 19.738 42.248 8 . 942 -3.451

10.316 24.583 20.765 41.733 10.316 -2.439

11.728 23 . 275 21.824 41.288 11.728 -1.427

13 . 182 22.014 22.907 40.907 13.182 -0.411

14.684 20.809 24.010 40.584 14.684 0.609

16.235 19.670 25.127 40.314 16.235 1.581

17.839 18.607 26.253 40.090 17.839 2 . 399

19.498 17.629 27.387 39.905 19.498 2.950

21.209 16.749 28.526 39.755 21.209 3 .128

22 .971 15.974 29.669 39.635 22.971 2.830

24.778 15.311 30.814 39.541 24.778 1.975

26.623 14.764 31.960 39.469 26.623 0.505

28.498 14.332 33.108 39.415 28.498 -1.603

30.396 14.008 34.256 39.377 30.396 -4.342

32.307 13.783 35.405 39.352 32.307 -7.666

34.227 13.642 36.553 39.336 34.227 -11.502

36.150 13.567 37.702 39.328 36.150 -15.752

38.075 13.539 38.851 39.325 38.075 -20.298

40.000 13.535 40.000 39.325 40.000 -25.000

149

Table 6.4 : Blade coordinates of rotor B

XH RH XS RS XT TETA

0.000 36.006 13.781 47.569 0.000 -13.123

1. 058 34.745 14.335 46.912 1.058 -10.373

2.119 33.486 14.893 46.258 2.119 -7.972

3 . 183 32.231 15.461 45.613 3.183 -5.956

4.253 30.981 16.044 44.981 4.253 -4.246

5.332 29.738 16.644 44.365 5.332 -2.852

6.421 28.504 17.265 43.771 6.421 -1.723

7.522 27.280 17.911 43.204 7.522 -0.864

8 . 638 26 . 071 18.582 42.667 8 . 638 -0.231

9.771 24.877 19.280 42.165 9.771 0.161

10.925 23.703 20.005 41.703 10.925 0.356

12.101 22.553 20.755 41.284 12.101 0.335

13.305 21.430 21.528 40.909 13.305 0.126

14.538 20.341 22.323 40.581 14.538 -0.252

15.807 19.292 23.134 40.298 15.807 -0.771

17.114 18.292 23.960 40.059 17.114 -1.444

18.465 17.352 24.797 39.862 18.465 -2.302

19.863 16.484 25.642 39.704 19.863 -3.383

21.312 15.705 26.492 39.580 21.312 -4.733

22.812 15.029 27.347 39.486 22.812 -6.403

24.361 14.474 28.204 39.419 24.361 -8.450

25.952 14.053 29.062 39.373 25.952 -10.922

27.572 13.768 29.921 39.346 27.572 -13.852

29.210 13.608 30.781 39.331 29.210 -17.234

30.854 13.545 31.640 39.326 30.854 -21.000

32.500 13.535 32.500 39.325 32.500 -25.000

150

Table 6.5 ; Rotor A blade thickness

x\r 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. 0. 3.64 3.41 3.17 2.93 2.69 2.45 2.20 1.96 1.72 1.48 2. 3.69 3.47 3.24 3.01 2.77 2.54 2.30 2.07 1.84 1.61 4. 3.73 3.51 3.29 3.07 2.84 2.61 2.39 2.16 1.93 1.71 6. 3.75 3.54 3.32 3.11 2.89 2.67 2.45 2.23 2.01 1.79 8. 3.75 3.55 3.34 3.13 2.92 2.70 2.49 2.28 2.06 1.86 10. 3.74 3.54 3.34 3.13 2.93 2.72 2.51 2.31 2.10 1.90 12. 3.72 3.52 3.32 3.12 2.92 2.72 2.52 2.32 2.12 1.92 14. 3.68 3.49 3.30 3.10 2.91 2.71 2.51 2.32 2.12 1.94 16. 3.64 3.45 3.26 3.07 2.88 2.69 2.49 2.30 2.12 1.93 18. 3.58 3.40 3.21 3.03 2.84 2.65 2.46 2.28 2.10 1.92 20. 3.52 3.34 3.16 2.97 2.79 2.61 2.43 2.25 2.07 1.90 22. 3.45 3.27 3.10 2.92 2.74 2.56 2.38 2.21 2.04 1.87 24. 3.37 3.20 3.03 2.85 2.68 2.51 2.33 2.16 1.99 1.83 26. 3.29 3.12 2.95 2.78 2.61 2.44 2.28 2.11 1.94 1.78 28. 3.20 3.03 2.87 2.70 2.54 2.37 2.21 2.05 1.89 1.73 30. 3.10 2.94 2.78 2.62 2.46 2.30 2.14 1.98 1.82 1.67 32. 2.98 2.83 2.67 2.52 2.36 2.21 2.05 1.90 1.75 1.60 34. 2.86 2.71 2.56 2.40 2.25 2.11 1.96 1.81 1.66 1.52 36. 2.72 2.57 2.42 2.28 2.13 1.99 1.84 1.70 1.56 1.43 38. 2.55 2.41 2.27 2.13 1.99 1.85 1.71 1.58 1.44 1.31 40. 2.37 2.23 2.10 1.96 1.83 1.69 1.56 1.43 1.30 1.18

x\r 32. 34. 36. 38. 40. 42. 44. 46. 48. 0. 1.25 1.02 0.80 0.59 0.39 0.21 0.04 0.01 0.01 2. 1.38 1.16 0.95 0.74 0.54 0.36 0.20 0.04 0.01 4. 1.49 1.28 1.07 0.87 0.68 0.50 0.34 0.19 0.05 6. 1.58 1.38 1.18 0.99 0.81 0.63 0.47 0.32 0.19 8. 1.65 1.46 1.27 1.09 0.92 0.75 0.59 0.45 0.31 10. 1.70 1.51 1.33 1.16 1.00 0.85 0.70 0.56 0.43

12. 1.74 1.55 1.38 1.22 1.06 0.92 0.79 0.66 0.53

14. 1.75 1.57 1.41 1.25 1.11 0.98 0.86 0.74 0.62 16. 1.75 1.58 1.42 1.27 1.13 1.01 0.90 0.81 0.72 18. 1.74 1.58 1.42 1.27 1.13 1.02 0.94 0.87 0.79

20. 1.72 1.56 1.40 1.26 1.13 1.03 0.96 0.89 0.83

22. 1.70 1.54 1.38 1.24 1.12 1.03 0.96 0.89 0.84

24. 1.66 1.51 1.36 1.22 1.10 1.02 0.94 0.88 0.82

26. 1.62 1.47 1.32 1.19 1.08 0.99 0.91 0.84 0.78

28. 1.58 1.43 1.28 1.15 1.04 0.96 0.88 0.80 0.73

30. 1.52 1.37 1.23 1.10 1.00 0.91 0.83 0.74 0.67

32. 1.46 1.31 1.18 1.05 0.95 0.86 0.77 0.68 0.60

34. 1.38 1.24 1.11 0.99 0.89 0.79 0.70 0.61 0.52

36. 1.29 1.16 1.03 0.91 0.81 0.72 0.63 0.53 0.44

38. 1.18 1.06 0.93 0.82 0.72 0.63 0.54 0.45 0.36

40. 1.06 0.94 0.82 0.71 0.62 0.53 0.45 0.36 0.27

151

Table 6.6 ; Rotor B blade thickness

x\r 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. 0. 3.90 3.65 3.40 3.14 2.89 2.63 2.37 2.10 1.84 1.58 2. 3.95 3.71 3.47 3.23 2.98 2.73 2.48 2.23 1.97 1.72 4. 3.98 3.76 3.52 3.29 3.05 2.81 2.57 2.33 2.09 1.84 6. 3.99 3.77 3.55 3.33 3.10 2.87 2.64 2.41 2.17 1.94 8. 3.97 3.76 3.55 3.34 3.12 2.90 2.68 2.45 2.23 2.01 10. 3.93 3.73 3.52 3.32 3.11 2.90 2.69 2.47 2.26 2.05 12. 3.86 3.67 3.48 3.28 3.08 2.88 2.67 2.47 2.27 2.07 14. 3.78 3.59 3.41 3.22 3.02 2.83 2.63 2.44 2.25 2.05 16. 3.67 3.49 3.32 3.13 2.95 2.76 2.58 2.39 2.21 2.02 18. 3.55 3.38 3.21 3.03 2.86 2.68 2.50 2.32 2.15 1.97 20. 3.42 3.26 3.09 2.92 2.75 2.58 2.41 2.24 2.07 1.90 22. 3.28 3.12 2.96 2.80 2.63 2.47 2.31 2.14 1.98 1.82 24. 3.14 2.98 2.82 2.67 2.51 2.35 2.19 2.04 1.88 1.73 26. 2.98 2.83 2.68 2.52 2.37 2.22 2.07 1.92 1.77 1.63 28. 2.82 2.67 2.52 2.37 2.23 2.08 1.94 1.79 1.65 1.51 30. 2.64 2.49 2.31 2.21 2.07 1.93 1.79 1.65 1.51 1.38 32. 2.43 2.29 2.15 2.02 1.88 1.75 1.61 1.48 1.35 1.22 34. 2.21 2.08 1.94 1.81 1.68 1.55 1.42 1.29 1.17 1.05

x\r 32. 34. 36. 38. 40. 42. 44. 46. 48. 0. 1.31 1.06 0.80 0.56 0.32 0.11 0.01 0.01 0.01 2. 1.47 1.22 0.97 0.73 0.50 0.29 0.09 0.01 0.01 4. 1.60 1.36 1.13 0.90 0.67 0.46 0.26 0.07 0.01 6. 1.71 1.48 1.26 1.04 0.83 0.62 0.42 0.24 0.07 8. 1.79 1.57 1.36 1.16 0.96 0.76 0.57 0.39 0.22 10. 1.84 1.64 1.44 1.25 1.07 0.89 0.79 0.53 0.36 12. 1.87 1.68 1.49 1.31 1.14 0.98 0.82 0.66 0.49 14. 1.87 1.69 1.51 1.34 1.19 1.04 0.91 0.77 0.62 16. 1.84 1.67 1.50 1.35 1.20 1.07 0.96 0.86 0.74 18. 1.80 1.63 1.47 1.32 1.19 1.07 0.99 0.91 0.83 20. 1.74 1.58 1.42 1.28 1.15 1.06 0.99 0.92 0.83 22. 1.66 1.51 1.36 1.22 1.10 1.02 0.95 0.88 0.83 24. 1.58 1.43 1.29 1.15 1.05 0.96 0.88 0.81 0.75 26. 1.48 1.34 1.20 1.08 0.98 0.89 0.80 0.72 0.65 28. 1.37 1.24 1.11 0.99 0.89 0.80 0.71 0.62 0.54 30. 1.25 1.12 1.00 0.88 0.79 0.70 0.61 0.52 0.43 32. 1.10 0.98 0.86 0.75 0.66 0.57 0.48 0.39 0.31 34. 0.93 0.81 0.70 0.60 0.51 0.42 0.34 0.25 0.17

152

k

Fig.6.1 Scroll Channel

153

v i 7

\

Fig.6.2 Casing Design

154

0) +J (d C •H •a o o u Ul

0) iH 0) Id k •H tr •P 0) C T3 0) tn c

! • • •

- ^ i i r

Case B : Effect of Rotor Length

## #

! • • • ##4

Case C : Effect of Tangential Coordinate Gj

4t«« ##.# l i^

•ii.0

i

Case A Effect of the Leading Edge Shape

Axial Coordinate (nun) !•••

!#.# »o.#

-wa

-M.I Case D Effect of Rotor Length

in in

Fig.6.3 Blade Camberlines

4 .

10 e

>1 •p •H o o r4

>

0) > •H +J Id iH 0) PH

9^

«.#

#.#

4M.«

tOQ.t

4M.t

#.#

A3

Shroud

Mean

vo in

Hub

Axial Distance (mm)

Fig.6.4 Blade Surface Velocity Distribution : Effect of the Leading Edge Shape ( Camberlines A )

Ui

8

>1 •P •H O o rH >

> -H •P n) I—t a) PH

n.a 4# # M-a

B2

Shroud

• tH «•••

Mean

t-in rH

Hub

a<o

Wa B3

Axial Distance (mm)

Fig.6.5 Blade Surface Velocity Distribution : Effect of Rotor Length ( Camberlines B ) ( Constant Blade Angle Along the Leading Edge )

in

>1 •p •H o o iH >

0) > •H +J 10 I—I 0) PH

##.# M-a

Shroud

»#.« ## #

4M4

D1

• a.t 40 wa

D2

Axial Distance (mm)

Fig.6.6 Blade Surface Velocity Distribution : Effect of Rotor Length ( Camberlines D ) ( Constant Incidence Angle Along the Leading Edge )

Mean

CO in

Hub

D3

4

U) M # ## # 4#.# * #

Shroud

>1 •P •H o o iH 0) >

0) > -H •P (d iH 0) K M-9 4t4 W-i !#.# M.i te-a <«.• M.a

Mean

104

m in iH

Hub

Axial Distance (mm)

Fig.6.7 Blade Surface Velocity Distribution : Effect of the Tangential Coordinate of the Trailing Edge ( Camberlines C ). Constant Blade Angle at Inlet

CAMBERLINE ROTOR

It-O

X (MM)

40 . •

#0

•04

40.0

m-Q

o VD rH

10.0

404

X (MM)

Fig.6.8 TURBINE A ( Constant Blade Angle at Rotor Inlet) Blade Geometry : Camberline and Meridional Blade

Surface Projection

CAMBERLINE ROTOR

X (MM)

40.0

n JZ

vo

X (MM)

Fig.6.9 TURBINE B ( Constant Incidence Angle at Rotor Inlet) Blade Geometry ; Camberline and Meridional Blade

Surface Projection

TURBINE A TURBINE B

o Ul a

ci) z cc

o oc CO

Shroud

-to^

•4a.o

O lU a

d z tr

Q ac m

«* # M 4 I

Shroud

-M-O

-M.O

a U) H

1*0 a

MERIDIONAL DISTANCE : MERIDIONAL DISTANCE i

Fig.6.10 Blade Angle Along Streamlines

TURBINE A TURBINE B

a cc tc.

m

<r t— lU X H-* oc

Shroud

o cr a:

w x: • a:

- I I

- I *

Hub

1

Hub

. . . . . - -V

Hub

. - j . - j

Shroud

i

10

RADIAL DISTANCE RADIAL DISTANCE

Fig.6.11 streamline Projection on a ( r, r© ) Plane

Rotor A

Rotor B

k

Fig.6.12 Blade Surface Projection on a ( r, e ) Plane

164

oc cc CD

OC ZD (O CO LU o: a .

t*4

l.t

! • ! -

1.0

0.#

1 1 1

!

1 : !

1 1 1 1

1 1

i 1 1

'

h f

1

T

I

i 1 1 i 1

0.1

0.4

f .O : 0 . 0 lO.O to .o t 6 . 0 #0.0 ## O h 40.0

QC lU CD c ZD z

(_) cc c •fi.

-—

! t

r 1

i

i

L_.

—i -

—

!

10.0 1#.0 to.o M . 0 M . 0 M . 0 40.0

to

CJ o UJ >

w

UJ cc

100.0

400.0

N0.0

no.o

100.0

AXIAL DISTANCE (MM)

(B) PRESSURE

'•8

i

, . . . J . . . .

1

— — 1

I i 1 !

i 1 i 1 :

1 i - "

-—-"""I t

i , i , , , , i , , ,

1 i ! : i

AXIAL DISTANCE (MM)

(C) MRCH NUMBER IT) VD

•H

Fig.6.13 Turbine A : Meridional Surface Flow Calculation (S.L.C) ND = 50000. rpm UC = 0.61 TO* = 923. K

1.0 10.0 It.O to.o K.O M.O Ik.O 40.0

AXIAL DISTANCE (MM) IR) RELATIVE VELOCITY

j

a: cc CO

a: ZJ tn 0 3 UJ cc Q.

OJ

(_) o

Ul >

cr _j UJ oc

100.0

1.0 10.D 11.0 fO.O t l . O M.O W.O 40.0

AXIAL DISTANCE (MM)

(B) PRESSURE

400.0

#00.0

too.o

a: UJ (D c

CJ en 1.0 10.0 li.o to.o M.o M.| M.0 40.0

AXIAL DISTANCE (MM)

(C) MACH NUMBER

vo VD

Fig.6.14 Turbine A : Meridional Surface Flow Calculation (S.L.C) ND = 75000. rpm UC = 0.61 TO* = 923. K

40.0

AXIAL DISTANCE (MM) (A) RELATIVE VELOCITY

a: cc m

on 3 1/3 to UJ ce Q_

i.t

>41

I . T

1 . 4

I . I

! N

! 1 1 !

X i >v 1

:

1 ! 1 1 "f 1 i 1

. . . . . . . , J , . ! i i

Ql LU CO i : 3

<_) cr

t.o 1 0 0 11.0 10.0 40.0 * 0 10.0 ll.o to.e H.e n.i N4 4 0 . 0

(O c

CJ C3

lU >• UJ >

cr

UJ oc

AXIAL DISTANCE (MM)

(B) PRESSURE

400 .0

MM.O

no.o

AXIAL DISTANCE (MM)

(C) MACH NUMBER r -vo

Fig.6.15 Turbine A : Meridional Surface Flow Calculation (S.L.C) ND UC TO*

= 98000. = 0.61 = 923. K

rpm

11.0 M . O WO 40 .0

AXIAL DISTANCE (MM) (H) RELATIVE VELOCITY

oc t r CQ

W OH =) (O to

1.0

0.1

.0 H.O #0.0 MO 40.0

a: LU (D c

n CJ t r ac M.a M4I

(O V 2=

400.0

MM.0

CJ o toa.o bj >

UJ > 100.0

cr _i UJ a:

AXIAL DISTANCE (MM)

(B) PRESSURE

' • 8

. . . .

/ ' '

L

— . . . — . . . — y/'"- _

...

1 1

AXIAL DISTANCE (MM)

(C) MACH NUMBER 00 VD

Fig.6.16 Turbine A ; Meridional Surface Flow Calculation (S.L.C) ND = 110000. rpm UC =0.61 TO* — 923. K

t . O 10.0 l i . O M O M O M.O w o 40.0

AXIAL DISTANCE (MM) CR) RELATIVE VELOCITY

700.0

MO.O

• l O O . Q

••0 10.# 11.0 t o . o M .0

AXIAL DISTANCE (MM)

(B) MEAN

400.0

000.0

w 100.0

- 1 0 0 . 0

1 . 0 10.0 l l . O tO.Q M . O

AXIAL DISTANCE (MM)

in) HUB

M.O M .O 40.0

CO

r

CJ CD

700.0

MO.O

#00.0

400.0

MO.O

t o o . o

LU 100.0 UJ >

cc bJ or

0 .0

-100.0

•too . J

1 1

- —

!

j

1 .

1

r \ !

1

i l l !

-

I 1 I

1

„ T 1

; :

1 . 0 10.0 11.0 fiO.O *# .o

AXIAL DISTANCE (MM)

(C) SHROUD

• 0 . 0 M . 0 40 .0

V£)

Fig.6.17 Turbine A : Blade Surface Velocity (S.L.C) ND = 50000. rpra UC =0.61 TO* = 923. K

700

•00.0 I

100.0 CO

• 1 —

•00.0 #00 0

w t o o . o o o

o 100.0 100.0

>

Ul > (JU 0.0 0.0

^ -100.0

0 0 II .0 10.0 to.o to.o t l . O M . O 40 .0 ( .0 10.0 I t . O M . O 0.0 10.0 I I . 0 to.o M . O

AXIAL DISTANCE (MM)

(0) MEAN

't.O t.O 10.0 11.0 10.6 MO W.0 M.0 40.0

AXIAL DISTANCE (MM)

(C) SHROUD

400.0

100.0

o r-

Fig.6.18 Turbine A ; Blade Surface Velocity (S.L.C) ND = 75000. rpm UC = 0.61 TO* = 923. K

i.O 10.0 l l .o to.o M.O

AXIAL DISTANCE (MM)

( A ) HUB

M.O M . O 40.0

(O xr

o o

UJ >

cr LU oc

400.0

100.0

•too

700.0

(O

<_) O

lU >

UJ on

to.e If.o to.o M.o M.O M.O

AXIAL DISTANCE (MM)

(B) MEAN

AXIAL DISTANCE (MM)

CRl HUB

40.0

#00,0

400.0

100.0

40 .0

#00.0

(O

o o UJ

UJ >

cc UJ 01

400.0

#00.0

too.o

100.0

10.0 i#.e to.o to 0 #o.# 40 .0

AXIAL DISTANCE (MM)

(C) SHROUD rH

rH

Fig.6.19 Turbine A : Blade Surface Velocity (S.L.C) ND = 98000. rpm UC = 0.61 TO* = 923. K

to

o o

>• UJ

CE _L LLI Q£

700*0

100.0

400.0

MOO

100.0

1.0 10.e 11.0 MO H.O

AXIAL DISTANCE (MM)

(8) MEAN

to

o

>

UJ >•

UJ OS.

AXIAL DISTANCE (MM)

( A ) HUB

40 .0

100.0

400 .0

100.0

100.0

•0 40 .0

tn

n

w o

UJ >

CE _ J UJ a;

700.0

MO.O

WM.O

400.0

(00.0

t o o . o

t o o . o

100.0

0.0 10.0 ii.o to.o M.o 00.0

AXIAL DISTANCE (MM)

(C) SHROUD

00.0 40 .0

CM

iH

Fig.6.20 Turbine A : Blade Surface Velocity (S.L.C) ND = 110000. rpm UG = 0.61 TO* = 923. K

a: cr CO

u tn

in CO cc 0.

1.4

l.t

1.1

1.1

1.0

O.f

I ! 1

— i i : 1 1

1 1

i • \ ; \

i i r [ ! : 1 1 ! : t i

1.0 10.0 11.0 10.0 M.o M.O II.U • 40 .0 40.0

to c

CJ o lU >

UJ

cr —I lij cc

KM.0

400.0

100.0

100-0

AXIAL DISTANCE (MM)

(B) PRESSURE

—

•

—

1

1

1 1 i '

1

, 1 i i . . . . i . . . . 1 . . . . 1 . . . . 1 . . . . i . . . . i . . . .

AXIAL DISTANCE (MM)

(C) MACH NUMBER

Fig.6.21 Turbine B ; Meridional Surface Flow Calculation (S.L.C) ND = 50000. rpm UC = 0 . 6 1 TO* = 923. K

1.0 10.0 11.0 to.o M.O M.O WO 40.0

RXIflL DISTANCE (MM) CR) RELATIVE VELOCITY

1.0

|.#

W ct: cc m 141

UJ a: =) 1.0 in tn tu cc a.

1 ! I 1 1

1 ^

1 1 t 1 i 1 1 i 1 ^

\ 1 i

! i 1 I

1 1 .

#0041

AXIAL DISTANCE (MM)

(B) PRESSURE

1.0

1 .0 10.0 l i .O ta.O M.O M.O M.O ^ 40.0

400.0

M.O

AXIAL DISTANCE (MM) (R) RELATIVE VELOCITY

w o 40.0

a: UJ (D H Z3

(_) tr z: 10.0 li.O M O n . i M.O M.0 4041

AXIAL DISTANCE (MM)

(C) MACH NUMBER

Fig.6.22 Turbine B : Meridional Surface Flow Calculation (S.L.C) ND = 75000. rpm UC = 0 . 6 1 TO* = 923. K

a: cr (DO

UJ a: Z3 <o to

* f . O 4 0 . 0

Qi lU CD i:

x: (_) cr c

0.1

0 . 4

0.1

II .0 4t.O

to

c

u o UJ >

UJ >

w Q:

uo

400.0

#00.0

no.0

100.0

AXIAL DISTANCE (MM)

(B) PRESSURE

AXIAL DISTANCE (MM)

(C) MACH NUMBER IT)

Fig.6.23 Turbine B : Meridional Surface Flow Calculation (S.L.C) ND = 98000. rpm UC = 0 . 6 1 TO* = 923. K

II.O 40.0

AXIAL DISTANCE (MM)

(A) RELATIVE VELOCITY

4.0

I . I

1.0

I . I

cn GC m 1.0

^ ..I to (n UJ ? »•

" " 1 I 1

I 1 1 n , K I I 1 1 1 1 5 ^ : 1 1 !

i )

I i M —

t ! i

- !

1 ! 1 1 i 1 1 1.0 10.0 wo to.o MO M.O »*0 ' 40.0

ai UJ CO c

n (_) en JC H.0 41.0

(O "S

100.0

100.0

400.0

100.0

u o _l Ul 100.0 >

lU M 100.0

K '-8

AXIAL DISTANCE (MM)

(B) PRESSURE

—

. . . . . . . . ' ' ' . . . .

1 /

^ !

i i

c C

/T ^ / ! /

1

r •

M M ! i . . . . A . . . . 1 . . . . A . . . . 1 . . . .

AXIAL DISTANCE (MM)

(C) MRCH NUMBER vo iH

Fig.6.24 Turbine B ; Meridional Surface Flow Calculation (S.L.C) ND = 110000. rpm UC = 0 . 6 1 TO* = 923. K

( .0 10.0 11.0 to.o MO #0.0 40.0

AXIAL DISTANCE (MM) (A) RELATIVE VELOCITY

700.0

#00.0

400.0

100.0

100.0

700.0

t0*0 WO AXIAL DISTANCE (MM)

(B) MEAN

CO "S

000.0

000.0

400.0

100.0

100.0

1 . 0 10.0 l l . o tO.Q w o

AXIAL DISTANCE (MM)

( R ) HUB

M.O M.O 40 .0

in X. XL

CJ o UJ >

UJ >

cr

oc

•00.0

400.0

Mn.o

roo.o

100 .0

100.0

-MO.I 8.0 1 . 0 10 .0 10.0 10 .0 M.O 1 0 . 0 M.O 40 .0

AXIAL DISTANCE (MM)

(C) SHROUD r-r-iH

Fig.6.25 Turbine B : Blade Surface Velocity (S.L.C) ND = 50000. rpm UC = 0 . 6 1 TO* = 923. K

N0.0

•oa.a

MW.O

Ui tM.O

-100.0

ai 1.0 10.0 11.0 tfl.o

AXIAL DISTANCE (MM

(81 MEAN

M.O #0.0

to

700.0

000.0

•00.0

400.0

>_ 100.0

f-c j n o . o o _J UJ 100.0 >•

^ 0.0

t -100.0

"""'S.O 1.0 10.0 ll.b to.o M O

AXIAL DISTANCE (MM)

( A ) HUB

M.O M.O

40.0

. . . . J . . . . i ^

J , , , , , , ,,

1 1 i

— _ 1 i

i i !

I

- — i i

:

^ !_ j i . 1 . 1 . 1 1 1 1 1 1 . ^ 1 . . . . 1 . . . . 1 . . . . A 1 . . . .

tn x :

o

lij >

CE _i lU oc

700 .0

MO.O

HO.O

400.0

MO.O

n o . o

100.0

100.0

'f.O 1.0 10.0 II .0 to.o M.O 10.0 M.« 40.»

AXIAL DISTANCE (MM)

(C) SHROUD 00

Fig.6.26 Turbine B ; Blade Surface Velocity (S.L.C) ND = 75000. rpm UC = 0 . 6 1 TO* = 923. K

CO

n

CJ o

UJ >

UJ oc

100-0

000.0

iOO.O

400.0

>00.0

too.o

100.0

0 .0

•100.0

-tOO.D

(O x:

CJ o UJ

UJ >

cc UJ a:

100.0

000.0

•00.0

400.0

000.0

100.0

100.0

0.0

- 1 0 0 0

-too.j 1.0 10.0 11.0 to.o H.O

AXIAL DISTANCE (MM) C R ) H U B

M.0 M.0

. . . . 1 . . . . ' j "

r " j x ;

1 1

; 1 i

1 ! •0 loO 10*

AXIAL

0 11.0 to.o n

DISTANCE (MM)

0 M 0 M 0 40.

(B) MEAN

1 i

; _

1

i

:

-1 r

i

1 1 ! 1

I

1 i 40.0

<n V. c

u o UJ >

UJ >

cr UJ a:

100.0

000.0

MO.O

400.0

•00.0

100.0

100.0

-100.0

f . o i.o 10.0 110 t0.0 M.0 to.o M.0 404

AXIAL DISTANCE (MM)

(C) SHROUD 3,

Fig.6.27 Turbine B : Blade Surface Velocity (S.L.C) ND = 9.8000. rpm UC = 0 . 6 1 TO* = 923. K

M0.0

•00.0

•100.0

K -W fl.'i 1.0 10.0 11.0 to.o M.O

AXIAL DISTANCE (MM)

(B) MEAN

(O V.

700.0

#00.0

no.o

400.0

>- 100.0

cj no.o o _J

W 10041

^ 0.0

-100.0

1.0 10.0 li.O to.o tf.o

AXIAL DISTANCE (MM)

( R ) HUB

IQ.Q M.O

to.0

, , ,

1 i M 1 —

1 1 1 1 1 — 1

1 i i J 1 I i — T i ^ I I ^ 1 i 1 1 i 1 4*.0

CO

n

u o

100.0

#00.0

100.0

400.0

MO.O

too.o

UJ 100.0

lU > 0.0

S -loo.o

too.I w on

1 • ' ' '

- — — — — —

/ 1

! I j A . U .

1 1 i 1.0 10.0 ll.o w.@ M.O #0.0

AXIAL DISTANCE (MM)

(C) SHROUD

1.0 40.0

o CO

Fig.6.28 Turbine B : Blade Surface Velocity (S.L.C) ND UC TO*

110000, 0.61 9 2 3 . K

rpiti

Chapter 7

7. EXPERIMENTAL INVESTIGATION OF T W O MIXED FLOW

TURBINES

7.1. Description of the Test Rig

A mixed flow turbine has been designed and

manufactured to meet the design conditions presented in paragraph

3.1. Two rotors ( A and B ) have been selected for experimental

testing. These two rotors present an identical exducer geometry

but differ by their leading edge shape. Rotor A, whose geometry is

given in table 6.3, has been designed so that the blade angle along

the leading edge is constant ( Pgb = 20 ° ). Rotor B ( table 6.4 ), on

the other hand, has been designed so that the incidence angle along

the leading edge ( at the design conditions ) is constant.

The experimental investigation of the mixed flow

turbine consists of the measurement of the turbine performance

(efficiency, mass flow rate and torque ), the shroud pressure

distribution and the flow field at the turbine exhaust duct

(temperature, pressure, flow angle, etc...). It has been carried out

on an existing turbocharger test rig at Imperial College. The test

rig, which was modified so that it can house the research mixed

flow turbine, consists of the following components (figure 7.1) :

Mixed flow turbine prototype (figure 6.2). The rotor has been

manufactured from aluminium by Holset using a five axis milling

machine while the turbine housing has been made at Imperial

College BY K . A . AWAN ( 7 5 ) .

181

Turbine air supply : The pressurized air is supplied by two

Howden screw type compressors which can deliver up to 0.5 kg/s

each at an absolute pressure of 5.0 bar. The air is supplied through

a 4 inch diameter pipe in which is placed a 59.84 mm orifice plate

used to measure the air mass flow according to the British

standard BS1042.

Exhaust duct : Consisting of a straight 86.76 mm diameter pipe

supporting the traverse probe system which is used to measure the

flow parameter distribution ( pressure, temperature and flow

angle) in a cross section of the exhaust turbine duct.

A radial compressor acting as a power absorber and a 3 inch

diameter pipe for the atmospheric air intake. The compressor air

mass flow is also measured by a 59.84 mm diameter orifice plate

according to the British standard BS1042.

- The compressor exhaust duct contains a remotely controlled

butterfly valve which enables the compressor air mass flow and

thus the power absorbed to be varied.

- The bearing housing and the lubricating system.

- The control panel and data acquisition hardware system.

- The instrumentation necessary to make the desired measurement

( speed pick up sensor, thermocouples, pressure tappings, pitot

tube, oil flow meter, etc...).

A complete description of the rig and instrumentation

is given in reference 73.

7.2 Performance Measurement

The experimental determination of the mixed flow

turbine overall performance is made by measuring the mass flow

crossing the turbine, the specific work and the ideal work ( work

done by the fluid for an isentropic expansion across the turbine )

182

for ranges of rotational speeds and pressure ratios. These

parameters enable the torque and efficiency to be determined. The

conditions at which the test is conducted and the measurement of

different parameters are presented in the following section.

7.2.1. Test Rig Conditions

Temperatures and rotational speeds at which these

tests were conducted were much lower than those encountered in

real applications. The utilization of air at a low temperature

enables conventional instrumentation to be used and therefore

limits the rig equipment costs.

The test is done at the same pressure ratios as those

of the actual turbine operating conditions but at lower

temperatures and speeds. The similitude in Mach number and

velocity diagrams ( characterized by the U/C parameter ) are

respected while the dissimilarity in the Reynolds number is

encountered. However , the effect of the Reynolds number on the

turbine performance is generally considered only of the second

order of importance (ref.71) especially for large turbines for which

the friction losses are low compared to the other losses as a

result of the small boundary layer to the flow passage width ratio.

The similarity in Mach number between the cold test

(total temperature T*c) and the conditions at which the turbine is

actually operated (total temperature T*h) leads to the following

expression which gives the cold test mass flow (mc) as a function

of the actual operating condition (or hot test) mass flow (mh) :

mc = mh [7 .1 ]

183

In addition to that, the similarity in velocity diagrams leads to the

relation giving the equivalent rotational speed :

Nc = Nh [ 7 . 2 ]

The parameters at which the mixed flow turbine has been designed

and the equivalent design parameters for the cold test ( obtained

by equations 7.1 and 7.2 ) are given in table 7.1. These parameters

are given for three values of the total inlet temperature used

during the test.

Table 7.1 Equivalent design conditions

Design

Conditions

Equivalent Design Conditions

Total Inlet

Temperature 923. 334. 338. 344.

Mass Flow 0.414 0.688 0.684 0.678

Rotational speed 98000. 58952. 59304. 59828.

Pressure Ratio 2.91

Velocity Ratio 0.61

The total temperature at the turbine inlet has been

kept as small as possible provided that no condensation occurs at

184

the turbine exit. The temperature used during the test varies

from 61 ° C to 71 ° C and it was only at high pressure ratios that

the exit temperature approached 0 ° C . This reduction in inlet

temperature led to a reduction in the equivalent design speed and a

safer running of the research turbine whose rotor is made of

aluminium and therefore less able to stand high stresses due to the

higher speed.

The rotational speeds for which the test has been

conducted cover a range from 50% to 100% equivalent design speed.

The speed was measured by a speed pick up sensor fixed on the

bearing housing close to the shaft.

7.2.2. Mass Flow Measurement

The air mass flows crossing the compressor or the

turbine are measured by means of orifice plates according to the

British standard BS1042. The orifice plate is placed in a straight

part of the pipe , far from any device ( elbow, valve, etc... ) which

can perturb the flow and affect the measurement. Static pressures

upstream (Pu) and downstream (Pd) of the orifice plate as well as

the upstream temperature are measured. The mass flow is obtained

from the following expressions ;

m = a E C Ay ^ 2 p ( Pu - Pd) [ 7.3

where a is the discharge coefficient and is a function of the

orifice to pipe diameter ratio ( p ) and the Reynolds number Re.

185

in^

a = 0.5959 +0.0312 '^-0.184(3% 0.0029

+ 0.039 / (1- p" ) - 0.015839 P C is the approach velocity factor C = 1 / ^ / l - p *

E is the compressibility factor

E = 1 - ( 0.41 + 0.35 p ) Z^L-Z l for P u _ P ^ < 0.3 yPu Pu

AQ is the orifice plate area and p is the density.

The oil mass flow is measured by means of a turbine

flow meter. The mass flow has been found to be dependant on both

the speed of rotation and the oil viscosity and hence temperature

(ref .72).

7 .2 .3 Turbine Per formance Character is t ics

In order to assess the turbine performance such as the

total to static efficiency, the non dimensional mass flow rate and

the torque with respect to either the pressure ratio or the velocity

ratio, it is necessary first to assess the terms intervening in the

equation defining the turbine performance ( equations 3.38, 3.40

and 3.41 ).

The turbine total to static efficiency is defined as :

ri,g = Net Turbine Output Power/Gross Turbine Output Power [7.4]

The gross turbine output power ( W|g )is the power resulting from

an isentropic expansion from the turbine inlet stagnation condition

( defined by PQ* and Jq* ) to the exhaust turbine static pressure

186

^EX-

- 1 y

Wis = " 4 Cp V n - ( P E x / P o . ) ^ ] [7 .5]

Hence, total temperature TQ* and total pressure PQ* at the turbine

inlet as well as the exhaust turbine static pressure have to be

measured. The total pressure PQ* is measured by a pitot tube

placed at the cross section centre and facing the flow. The total

pressure measured in this way does not represent the averaged

stagnation pressure at turbine inlet as it is measured at the cross

section centre only and therefore it is not used for the turbine

performance calculation but serves as a means of maintaining a

constant turbine inlet stagnation pressure during the test. The

total pressure (PQ-). which is instead used to define the pressure

ratio and to calculate the turbine performance, is obtained from

the experimental measurement of the mass flow (m), the static

pressure (Pg) and the total temperature (TQ.) and is given by the

following expression from references 69 and 70.

Ps 0.5 + 0.5 I l. + 2 . X z L i ( _ ^ ^ ) 2 _ ^ _ l 2 1 y 0 cos a

[ 7 . 6 ]

The static pressure is the averaged value provided by four tappings

on a common cross section. Static pressure measurements are

made at both the turbine inlet and the turbine exhaust duct. The

different pressure tappings are connected to a 24 channels

scanivalve. Low pressure and a high pressure transducers

(previously calibrated) are used to measure the pressure.

187

E-Type thermocouples are used to measure the

temperatures. These are fed to a BBC computer through an Inlab

interface unit. The measured value (Tm) is lower than the total

temperature therefore a recovery factor (r), combined with the

continuity equation and the measurement of the static pressure are

used to calculate the static (Ts) and total (T*) temperatures. The

recovery factor is defined as follows :

r = ( Tm - Ts ) / ( T* - Ts ) [7.7]

Two different ways have been used to assess the net

output power. The first method which was used for the complete

test condition range consists of measuring the power absorbed by

the compressor and the shaft friction losses. Thus

Turbine net power = compressor work + bearing losses

W J = m C P ( T*EX - T*IN ) Q Q M P + NN C P ( T E X - TIN ) Q||_ [ 7 . 8 ]

The measured temperatures and static pressures at

both the compressor inlet and exit enable the total temperatures to

be found in the same way as for the turbine inlet. The total

temperature at the compressor inlet was found to be greater than

that at the orifice plate ( which is used for the compressor mass

flow measurement ) especially for low mass flow and high speed.

It was thought that the rise of the compressor exit temperature

led to the compressor housing being heated. A heat transfer from

the housing to the compressor inlet duct led to the rise in the

compressor inlet temperature and therefore the measured inlet

temperature is different from the actual one. The total

temperature at the orifice plate is therefore used instead for the

188

compressor work calculation. The bearing losses are obtained from

the measurement of the oil temperatures at both the inlet and exit

of the bearing housing.

The second method used to evaluate the net turbine

output power is based on the flow field measurement at the turbine

exhaust duct. This method provides a valuable way of checking the

accuracy of the first method. Details of this method are given in

section 7.3.

The turbine mass flow is usually illustrated by curves

of pressure ratio variation with respect to the non dimensional

mass flow rate for a constant speed. The non dimensional mass

flow rate is given by the following equation :

Mfr - R T Q./Y / AQ [7.9]

where Ao is the turbine inlet area.

The torque is defined as follows :

X = r\\s Wis / 0) [7.10]

where co is the turbine rotational frequency.

7.2.4, Shroud Pressure Measurement

In addition to the turbine performance measurement,

provision for the pressure measurement along the shroud profile

has been made. A series of tappings are used to measure the

pressure distribution along the shroud profile. The disposition of

the tappings is shown in figure 7.2.

189

7.3. Exhaust Turbine Flow Measurement

The measurement of the distribution of fluid

parameters ( pressure, temperature and flow angle) in a cross

section of the turbine exhaust duct provides an additional way of

measuring the turbine efficiency. The measurement is made by

means of a traversing mechanism (figure 7.3) supporting a probe

which traverses the exhaust duct from one wall to another in the

radial direction. The traversing wedge probe is rotated until the

pitot tube is facing the flow and the reading at the left side

tapping and the right side tapping are equal ( the probe is said to be

nulled ). A K-Type thermocouple measuring the temperature is

placed at 3.66mm from the pitot tapping centre ( figure 7.3) and

therefore the temperature at the measuring point is obtained by

extrapolation. The pressure and probe angle readings are taken at

10 positions equally spaced while the temperatures are measured

at 11 positions. These measurements are done on two perpendicular

lines across the duct for each test case. The traverse readings at

each position consists of :

GP

Pf - Pr

Pf - PR

Pf - PL

Tm

Tr, Pr

The probe angle

Front pressure reading

Right tapping reading

Left tapping reading

Measured temperature

Reference temperature and reference

static pressure measured at one cross

section upstream of the cross section

for the traverse measurements.

190

7.3.1 . Cal ibrat ion Factors

a - Thermocouple recovery factor

The K-Type thermocouple used at the turbine exhaust

gives a temperature value between the static and total

temperatures. The recovery factor for the K-Type has been found to

be dependent on the angle of flow onto the probe ( ref.72).

r(6) = ( Tm - Ts ) / ( T* - Ts ) [7.11]

where 9 is the difference between the flow angle and the probe

angle. T. and Ts are respectively the total and static temperatures

at the probe position of measurement.

b - Total pressure recovery factor

During the test, the pitot tapping of the traversing

wedge probe is pointed to a near flow direction. The calibration of

the probe has shown that the measured pressure is smaller than

the total pressure as a result of the non isentropic deceleration of

the flow approaching the pitot tube as well as the non coincidence

between the flow and the probe directions. The total pressure can

be obtained from the measurement by means of a total pressure

recovery factor which is defined as follows;

p(e) = ( Pf - Ps ) / ( P* - Ps ) [7.12]

P . and Ps are respectively the total and static pressures at the

probe position of measurement.

191

c - Gas flow factor

The parameter 0 which expresses the difference

between the flow angle a and the probe direction 8p is dependent

on the pressure differential between the two side tappings of the

probe. This pressure difference is expressed by means of the gas

flow factor q.

q = ( PL - PR ) / ( P* - Ps ) [7.13]

The determination of the factor q enables the parameter 9 = 0 (q)

to be found and therefore the absolute flow angle a .

a = 0p + e [7.14]

d- Left and right pressure factors

One of these factors is used to calculate the static

pressure at the measuring point . The pressure factors are defined

as follows :

X = ( P< - PL ) / ( P* - Ps ) ; Left pressure factor

Y = ( Pf - PR ) / ( P* - Ps ) ; Right pressure factor

It was found that these factors are dependent on both the Mach

number and the 0 parameter. Thus

X = X ( 0 , M ) a n d Y = Y ( 0 . M ) [7.15]

192

7.3.2 . Flow Parameter Calculat ions

The procedure to determine the flow parameters from

the traverse reading as well as the relevant calibration factors are

presented in detail in reference (72). Once the total pressure P*,

the static pressure Pg and 9 have been calculated , the flow angle

is calculated by means of equation 7.14.

The total temperature is given by equation 7.16

T. = T„ [7,161 2 + r ( Y - 1 ) M

The test conditions drifted slightly during the test and therefore a

thermocouple measuring the reference temperature Tr (at a fixed

position upstream of the cross section for the traverse

measurements) was used to monitor the change in flow conditions.

The difference between the averaged reference temperature (at 11

positions of the probe) and the reference temperature Tr is

subtracted from the measured probe temperature at each position,

thus allowing the drift in flow conditions to be taken into account.

7.3.3. Performance Calculat ion

An iterative calculation combining the parameters

defined in section 7.3.2, as shown in figure 7.4, enables the flow

parameters at the turbine exhaust duct to be found.

The mass flow is measured by means of an orifice

plate placed in the pipe feeding the turbine. This mass flow is

compared with the mass flow obtained from the turbine exhaust

duct parameters ( equation 7.17 ).

193

= iTcjp r dr [7.17]

0

The mean total and static pressures are averaged with respect to

the exit area ( equation 7.18 ) while the mean total and static

temperatures are mass averaged ( equation 7.19)

J P r dr

P = [7.18]

J r dr 0

rw

i T ^ J x p Y ^ r d r

T = [7.19]

The turbine total to static and total to total efficiencies are

obtained by the following expressions ;

Tits = ( C P T . i n - C P T*EX ) / ( C P T . i n - C P TISEX ) [ 7 . 2 0 ]

ritt = ( C P T . i n - C P TIS*EX) / ( C P T . i n - C P TISEX ) [7.21]

The procedure outlined in section 7.3 for the traversing probe

performance measurement is carried out in the horizontal plane

first and then in the vertical plane by rotating the traversing

mechanism 90° around the rotor axis for the same conditions.

7.4 Experimental Results

The two mixed flow turbine prototypes ( A and B )

1 9 4

have been experimentally tested following the procedure explained

in the previous sections of this chapter. The results are presented

below and concern ;

- The overall turbine performance characteristics : The total to

static efficiency assessment makes use of the measured

compressor work and the bearing losses ( figures 7.5 to 7.15 ).

- The turbine exit duct flow survey ( figures 7.16 to 7.28 ).

- The shroud pressure distribution ( figures 7.29 to 7.32 ).

7.4.1 Overall Performance

The two mixed flow rotors are identical in their

overall dimensions but differ mainly in the rotor inlet, which is a

constant blade angle in one case ( rotor A ), and a notionally

constant incidence angle in the other case ( rotor B ). They also

differ by their rotor length, which is 40mm for rotor A and 32.5mm

for rotor B. The same housing was used for both rotors and

therefore the static radial clearance in turbine B case (0.52mm) is

slightly larger than in the case of turbine A (0.40mm).

Turbine A is showing significantly higher efficiencies

than its counterpart B across the operating range (fig.7.5). Total to

static efficiency characteristics for the lines of constant speeds

ranging from 50% to 100% equivalent design speed are shown in

figures 7.6 (with respect to U/C) and 7.7 (with respect to pressure

ratio). The range of pressure ratios at which the test was

conducted for each speed was limited by the capability of the

compressor to absorb the turbine power for the highest pressure

ratio and the compressor surge line for the lowest pressure ratio.

High peak efficiencies at low U/C were observed for

195

turbine A. Both the peak efficiency and the optimum U/C vary with

the rotational speed. Turbine B efficiency characteristics show the

same trends as those of turbine A but with values lower by about

0.07 .

The mass flow characteristics (fig.7.8), presented in

terms of pressure ratio as a function of the non dimensional mass

flow rate for lines of constant speeds, show that turbine B is

flowing more air mass flow than turbine A. Figure 7.9, on the other

hand shows the two turbine torque characteristics. They are

presented in terms of torque versus the non dimensional mass flow

rate.

A comparison between measured and calculated total

to static efficiency and non dimensional mass flow rate along with

the computed incidence angle at the mean diameter are shown in

figures 7.10 to 7.15 ( each figure corresponds to a constant speed

ranging from 50% to 100% equivalent design speed). The one

dimensional model used to predict the turbine performances does

not differentiate between turbine A and turbine B as all the

parameters used in the calculation process are those at the mean

area of the station considered. The computed results are in

reasonable agreement with the experimental ones over the overall

operating range especially in the case of turbine A.

The volute which was used in the experimental tests

differs at the inlet part from the one which dimensions are given

in table 6.1. The tip of the tongue is situated at -40° azimuth angle.

The area AQ and the radius of the centroid RQ at the volute inlet are

respectively 2150. mm^ and 82.50 mm. The volute dimensions

between 0° and 320° azimuth angles are those given in table 6.1.

196

The values adopted for the coefficient Kgp used in the calculation

of the scroll loss coefficient and the coefficient CR of equation

3.31 are respectively 6. and 0.2.

7 .4.2 Traverse Measurements

The turbine exit duct flow survey by a traversing

probe was carried out at four different turbine speeds (50%, 70%,

90% and 100% equivalent design speed). The number of conditions

at which the test was conducted for each speed was limited to

three pressure ratios selected as follows:

- Pressure ratio 1 : corresponds to the maximum power absorbed by

the compressor at this speed. For 100% equivalent design speed, it

was not possible to test the two turbines at this pressure ratio

because of high fluctuations in the reading of the flow parameters

at the turbine exhaust duct.

- Pressure ratio 2 : was selected so that it is near to the pressure

ratio for which the peak efficiency was observed at this speed.

- Pressure ratio 3 : corresponds to the minimum power condition at

the onset of compressor surge.

The test procedure and performance calculation are

explained in section 7.3 while the details of each test conditions

are given in table 7.2. Results from the traverse probe

measurements are presented in figures 7.16 to 7.20 for the turbine

performance and in figures 7.21 to 7.28 for the exit duct flow

surveys of turbine A and B.

The total to static efficiencies (calculated from the

exhaust duct flow measurements and referred to as the second

method for the performance measurement) plotted along with those

197

of figures 7.7 ( for which the turbine net output power is obtained

from the compressor work and the bearing losses and referred to

as the first method for the performance measurement) are

presented in figures 7.16 and 7.18 for respectively turbine A and

turbine B. Figure 7.16 shows that the efficiency measured by the

second method is lower than that measured by the first one in the

case of turbine A for the operating range. Figure 7.18 on the other

hand shows that the efficiencies of turbine B measured by the two

methods are of comparable values.

The mass flow (used to plot the characteristics of

figure 7.8) and those calculated from the exhaust duct flow

measurement are plotted against pressure ratio in figures 7.17

(turbine A) and 7.19 (turbine B). The two measurements are in good

agreement in the case of turbine B (fig. 7.19) but present a small

difference in the case of turbine A (fig. 7.17).

The limited number of points of probe measurement at

the turbine exhaust duct (10 position in the horizontal plane and 10

others in the vertical plane for each test condition) seems to be

insufficient to calculate accurately the parameters involved in the

performance assessment by the second method. This is illustrated

by the difference between the measurements in the horizontal and

vertical planes on the one hand and the results obtained by the

first method on the other hand.

Figure 7.20 confirms clearly the results obtained by

the first method ( use of compressor work ) of assessing the

turbine efficiency which showed that turbine A is performing

better than turbine B across the whole range of operating

condi t ions. The two turbine total to total ef f ic iency

198

characteristics of the same figure suggests that rotor A is better

designed than rotor B.

The exhaust duct flow surveys (figure 7.21 to 7.28 ) of

each turbine are presented in terms of total and static pressures,

absolute flow angle, swirl velocity, Mach number and axial velocity

for the test conditions already described and which are given in

table 7.2. A similar pattern of the flow is observed for both

turbines. In the case of high pressure ratio (curve 1), the flow

presents the features of the flow in a straight pipe as it is the

case for the exit duct. At low pressure ratio (curve 3) very large

changes in flow angle and swirl velocity at the duct centre are

presumably caused by the rotor hub core. This phenomenon becomes

more accentuated with increasing speeds. A more uniform flow

near the peak efficiency conditions is observed although at low

speeds, the peak efficiency does not correspond to the lowest

swirl velocity. At the near design pressure ratio and equivalent

design speed, the swirl velocity and absolute flow angle are close

to zero ( the turbine has been designed for zero exit swirl

veloci ty) .

The absolute flow angle variation along the rotor exit

calculated by the streamline curvature method presented in

chapter 5 is plotted in figures 7.33 (50% speed) and 7.34 (90%

speed) along with that measured by a traversing probe at a

downstream station. The distance between the rotor exit and the

exhaust duct cross section at which the traverse measurements

were made is 2f2.G mm in the case of turbine A and 280 mm in the

case of turbine B. The area ratio of the two cross sections (rotor

exit area / duct cross section area ) is about 0.68 and therefore a

reduction in the axial velocity of the same order and an increase in

199

the absolute flow angle magnitude are expected. This trend is

observed in most of the cases shown in figures 7.33 and 7.34 which

compare the absolute flow angle computed at the rotor exit and

that measured downstream of the rotor. The predicted smooth

distr ibution at the rotor exit is not observed across the

downstream duct cross section where the measured irregular flow

variation is probably the result of the fluid mixing processes and

the viscous nature of the fluid especially near the solid boundary

of the duct. However a shifting from negative towards positive

values with increasing velocity ratio U/C (for constant speed) is

observed for both predicted and measured flow angles.

7.4.3 Shroud Pressure

The pressure distribution along the shroud profile has

been measured for each test condition between the horizontal and

vertical traverse measurements. The results of such measurements

are presented in figures 7.29 to 7.32. The conditions for each test

are given in table 7.2. At all speeds and pressure ratios, the

expansion process is taking place across all the rotor shroud

profile as far as rotor A is concerned. This type of pressure

distribution is characteristic of accelerating flow and therefore

minimal losses. Rotor B by contrast shows that the flow is

overexpanding and the pressure recovery observed just upstream of

the trailing edge is likely to result in flow separation especially

on the suction surface. The consequence of such phenomenon is

higher rotor losses.

It appears from this analysis, that the rotor design

and therefore the flow behaviour in the rotor channel is the main

factor which has made turbine A more efficient than turbine B.

2 0 0

A comparisorl between measured and computed

pressure distribution along the shroud profile of turbine A (100%

speed, pressure ratio = 2.55) is presented in figures 7.35. The 2-D

flow calculation was performed using the streamline curvature

method of chapter 5 ( in the meridional plane) while the 3-D

inviscid calculation was achieved by the method developed by Hua

Chen (74). The results from the two methods are in good agreement

across the whole shroud profile except near the inlet and exit of

the rotor where small differences are encountered. However, the

measured pressures are lower than those obtained by the 2-D and

3-D calculation methods. The only measurements of the flow made

upstream of the rotor concern the stagnation temperature and the

static pressure at the turbine inlet and the mass flow. The

stagnation pressure was obtained from these measurements by

equation 7.6 while the rotor inlet conditions which were used in

the 2-D calculations, were the result of a one dimensional

calculation based on the method developed in chapter 3. These

conditions are :

- Flow angle distribution along the leading edge.

- Swirl velocity coefficient ( r Ve ).

- Stagnation temperature and pressure.

The flow behaviour at a mixed flow turbine rotor inlet needs to be

fully investigated so that more precise inlet conditions can be

applied for the 2-D or 3-D calculations. These limitations in the

flow modelling added to those inherent to the method itself are

sufficient to result in the difference between the measured and

predicted pressure distribution along the shroud profile shown in

figure 7.35.

201

7.4.4. Incidence Angle at Rotor Inlet

An approximation of the incidence angle variation

along the rotor inlet for the test cases of table 7.2 is presented in

f igures 7.36 and 7.37. The calculations were made on the

assumption of a uniform meridional velocity distribution at rotor

inlet and a free vortex flow in the volute. The main features of this

analysis can be summarized as follows:

- The incidence angle decreases with the velocity ratio U/C and

the rotational speed. The measurements of the flow field at the

rotor inlet of a twin-entry vaneless radial turbine under equal

admission made by Baines and Yeo (28) showed also a decrease in

incidence angle with increasing U/C and that the incidence angle at

the design point has a mean value of -30°.

- The relative flow angle varies from hub to shroud as a result of

the changes in radius along the rotor of a mixed flow turbine. In the

case of turbine A (constant blade angle along the leading edge), the

incidence angle decreases from hub to shroud by about 40° at 50 %

speed and 60° at 100 % speed. In the case of turbine B whose rotor

was designed for a constant incidence angle along the rotor inlet,

the incidence angle distribution varies with speed. An increase in

incidence angle from hub to shroud is predicted at low speed (50%)

while at higher speeds, its variation along the leading edge

becomes more uniform.

- The incidence angle distribution corresponding to the near design

conditions (100% speed and U/C = 0.61) are represented by curves

(2) which show a variation in incidence angle from approximately

+ 6° (hub) to -50° (shroud) in the case of turbine A and a quasi-

uniform variation with a mean value of -30° in the case of turbine

B.

202

7.5. Conclusion From the Experimental Investigation

Efficiencies measured by two methods, which differ

in the way the turbine net output power is assessed (from the

compressor work in one case and the turbine exhaust duct

f lowfield in the other case), although of slightly different

magnitudes, showed that turbine A is performing better than its

counterpart B. The difference in the two turbine peak efficiencies

of about 0.07, across the range of speeds at which the tests were

made, shown in figures 7.38 is mainly due to the difference in the

two rotor blade shapes ( see figures 6.8 to 6.11 ). This is

illustrated by the pressure distribution along the shroud profiles

presented in figures 7.29 to 7.32 which showed that the fluid is

expanding across the whole blade channel of rotor A. In the case of

rotor B which is shorter than rotor A, the fluid is overexpanding

and therefore the pressure recovery occurring at the rotor exit

results in greater losses.

An increase in the peak efficiency and the optimum

velocity ratio U/C with respect to the rotational speed is observed

in figure 7.6. The optimum velocity U/C^pj and the peak efficiency

plotted against the rotational speed for the two mixed flow

turbines in figures 7.38, showed a linear variation across the

operating range. For radial turbines, the optimum velocity ratio

U/C is around 0.7 and remains of the same order across the

operating range of speeds while for mixed flow turbines, it is much

lower and varies with speed. Curves (2) of figures 7.36 and 7.37

are the incidence angle distr ibution along the rotor inlet

corresponding to the near optimum conditions at each speed. The

increase in peak efficiencies with speed (figure 7.38) can be

explained by the displacement from positive incidences at low

203

speeds towards negative ones at higher speeds. At 100 % speed for

instance, the mean incidence angle for both turbines is about -30°

which corresponds to the optimum incidence angle for radial

turbines according to Baines and Yeo (28). The mixed flow turbine

for which the variation in the incidence angle along the rotor inlet

depends on the blade geometry as it is the case for rotor A

(constant blade angle at inlet) and rotor B (constant incidence

angle at inlet), it will be more appropriate to consider an optimum

incidence angle distribution at rotor inlet instead of an optimum

incidence angle (radial turbine).

The total to static efficiency of the two turbines at

the design conditions (U/C = 0.61 and 100 % speed) versus the

specific speed are plotted in figure 7.39 along with those of high

specific speed mixed flow turbines and the curves of maximum

efficiency for radial turbines by Rohlik (48). Turbine A is clearly

showing higher efficiency than the others but at lower specific

speed.

204

T a b l e 7 . 2 T u r b i n e f low condi t ions for t r ave r se m e a s u r e m e n t s

5 0 % speed

T u r b i n e A T u r b i n e B

C u r v e I 2 3 1 2 3

IJ/C .452 .544 .578 .452 . 530 . 6 3 7

TO* 3 3 4 . 0 3 3 4 . 0 3 3 4 . 0 3 3 4 . 0 3 3 5 . 0 3 3 4 . 0

PO* 1.611 1.379 1.326 1 .593 1 .399 1 . 2 7 4

Pex 1.034 1.022 1.018 1.021 1 .020 1 .027

M F .344 .246 J i l 9 .361 .271 . 2 0 4

7 0 % s peed


C u r v e 1 2 3 1 2 3

U/C .510 .587 .652 .508 . 591 .666

TO* 3 3 4 . 0 3 3 5 . 0 3 3 4 . 0 3 3 4 . 0 3 3 5 . 0 3 3 5 . 0

PO* 2 . 0 9 1.725 1.567 2 . 0 4 5 1 .660 1 .496

Pex 1.023 1.023 1.032 0 . 9 9 8 0 . 9 9 4 1 . 0 0 5

M F .525 J 7 9 .303 .542 J 8 8 . 3 1 2

9 0 % ipeed


C u r v e 1 2 3 1 2 3

U/C 5 6 8 .633 .713 .562 . 614 . 6 9 1

TO* 3 4 1 . 0 3 3 & 0 3 3 9 ^ 3 4 0 . 0 3 3 9 . 0 3 3 9 . 0

PU* 2 . 7 6 5 2 J W 6 1.878 2 .801 2 . 3 3 1 1 .923

Pex 1 .022 1.036 1 .033 1 .010 1 .011 1 .013

M F .741 .560 .411 J 8 4 . 620 .464

1 0 0 % speed


C u r v e 1 2 3 1 2 3

U / C .619 .702 . 624 7 2 7

TO* 3 4 4 . 0 3 3 & 0 3 4 5 . 0 3 4 4 . 0

PO* 2 . 8 7 7 2 . 2 5 9 2 . 7 7 9 2 . 0 7 3

Pex 1 .030 1.029 1 .014 1 .018

M F .760 .551 . 754 5 0 2

2 0 5

Air Inlet to Compressor

(Atmospheric Conditions)

.Xl

E %

u

I Valve I

[ T u r b i n e Inlet (Plot, P»t, T t o t ) j

Shroud Static Pressure

[Control Valve Turbine Exhaust Duct

(Pit, T»i, a)

[Safety Valve]

(L

J Q

1

Q

Compressor Bearing

Housing

Turbine

O 5

vo o CJ

u

E

0.6B

Shroud ring pressure tapping coordinates

Tapping X ( U ) Y (deg.) theU (deg.)

1 14.6 40.0 0.0

2 16.4 5 ^ ^ 5.0

3 19.6 60.0 10.0

4 23. 1 7 ^ ^ 15.0

5 26.1 7 ^ ^ 20.0

6 30.6 90.0 25.0

7 34.1 90.0 30.0

8 )#.l 90.0 15.0

$ 47.6 90.0 40.0

10 56.6 90.0 45.0

FIGURE 7.2 SHROUD PRESSURE TAPPING No 1: Rotor Inlet No 10 : Turbine Exit

207

EXHAUST DUCT

X

6

e .

PROBE ROTATION

TRAVERSING

CROSS SECTION

# 4 . 0 0 m m

TRAVERSING WEDGE PROBE

VERTICAL PLANE

FIGURE 7 J EXHAUST TURBINE DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM

2 0 8

Estimate P* - Ps

Calculate X(e,M) or Y(e,M)

Assess P* - Ps

Calculate q(9)

Calculate 9

Calculate p(0)

Calculate Ps

Calculate P*

Calculate M

9 Converged ?

N

Calculate : T* and Ts

Absolute Velocity

Absolute Flow Angle

Relative Velocity

Relative Flow Angle

Turbine Performance

FIGURE 7.4 TRAVERSING PROBE :

FLOW PARMETERS AND PERFORMANCE CALCULATION

209

31

TuraneA

Turane B

I I I I ' I I I I ' I M T l rm I I I I I I I ' I I I I 0.70

VELOCIIY RATIO; U/C

FIGURE I S MIXED FLOW TURBINE TOTAL TO STATIC EFnCIENCY vs

VELOCITY RATIO U/C

210

(Tll»£A)

ago

OfO

I i 0.70

IN.

•

0.40 050 m 0.70 OfO U/C

(TUatB)

LEGOCS

•50XSPEED

6 0.70

I 1 1 I ' I I 1 I • I 1 1 I

0.70 0 0

U/C

FIGURE 7.6 MIXED FLOW TURBINE TOTAL TO STATIC EFFICIENCY vs

VELOCITY RATIO U/C and ROTATIONAL SPEED

211

[HfOEA]

IfGDCS

•50XSPEED

PRESSIHRMD

aso

080

0L7O

060

(TUfaCB)

1 1 1 1 f 1 1 i 1 1 1 1 1 * 1 1 i 1 1 1

1 1 1 1 1 1

•

n

^ t / 1 * 1

• 1 1 1 i 1 i 1 )

lOQ 150 200 250

PRESSURE RAID

100 150

nCURE 7.7 MIXED FLOW TURBINE TOTAL TO STATIC EFnCIENCY TS

PRESSURE RATIO and ROTATIONAL SPEED

212

(TUOCA]

IIGQOS

•50ZSPEED O60X 07DX

BOX •gox •IX} X

e m

(TURBICB)

g zso

NGN I 'J M

025 OJO

#LWkSS FLOW RATE

FIGURE 7.8 NON DIMENSIONAL MASS FLOW RATE CHARACTERISTICS

213

[HIBCA]

•SOXSPED

5

NONDtCOmyi FLOW RATE

(TUaCB)

12J)0

too

m

I ? 6i»

^ 4i)0

2JI)

m

•

/

n / p

•

L - -

• ^

0.5 020 02 030

NON OICNSIQNALMSSaOW RATE

035

FIGURE 7.9 TORQUE AS A FUNCTION OF THE NON DIMENSIONAL MASS FLOW RATE and ROTATIONAL SPEED

214

m

an

QfiO

1140

020

aoo

f ^ A

/ EFFCENCr

/ / /

/

o o o MDIUJI

I I I I 1 do do 140 170

40i)

M

2Qi)

no

OD

-NO PRESSURE RATIO

FIGURE 7.10 COMPARISON BETWEEN THE MEASURED AND PREDICTED PERFORMANCE

OF TURBINE A AND B (50 % EQUIVALENT SPEED )

too

0L8O

060

&40

020

BETA

EFFtENCr

• o KWlfJl

• • TurtineB

O o Turbine A

51 I I I I I I I I I I I I I I I I I I I I 170 IH) 19)

40D

300

200

no ^

oo I

-no

-200

-300 PRESSURE RATIO


OF TURBINE A AND B ( 60 % EQUIVALENT SPEED )

215

too

an

060

(140

020

BETA « o o

EFFcecr

KDiLFJl

• • Turbine B

O O TurtineA

aio

200

CO

Oi)

-m

-200

OOO 1 1 1 1 1 1 1 1 1 -300 PRESSURE RATIO 140 150 160 170 IBO 190 100 IB 120 2J0


OF TURBINE A AND B ( 7 0 % EQUIVALENT SPEED )

EFFcecr

HDMFR.

l)o ' ite ' ik) ' 2J)0 ' IB ' 2i!0 ' 2^ ' Z40 ' 2io PRESSURE RATIO


OF TURBINE A AND B ( 80 % EQUIVALENT SPEED )

216

too ETA QD

QiS

0L6O

0.40

o o o ^ o, • • •—• ^ ^ • . —1_

/ / /

/ EFFcecir

/ /

o- % ^ o" & o — ^ * 0 * o KDil/Jl

020 :

m

• # Tiftine B

0 0 Turbine A

-to

-m

-30i)

-m

lAo ' ztio ' I'l) ' 2^0 ' L I ) ' 2.40 ' 2io ' 2io ' 2 . ^ ' zio ' 2io ' sio ^ PRESSURE RATIO


OF TURBII>iE A AND B ( 90 % EQUIVALENT SPEED )

100

oao

0l60

0.40

020

000

BETA

^ o O o o o o • • • • • •

/ / / /

/ EJTim

-200

-30J3

--m

— % 9—§ * o — ^ RDiLFJl

/ / /

fi ' 2I0 ' 2!I ' 212' 2!3 ' 2!4 ' 2!5 ' 2!6 ' 2!? ' 2J ' 2!9 ' 3!O ' i r 12 ' 3I3 ' 14

-500

-GOO PRESSURE RATIO


OF TURBINE A AND B (100 % EQUIVALENT SPEED )

217

HOR VER SPEED e )K O • 90X o e 70% • # 50%

m 150 PRESSURE RTIO

FIGURE 7.16 COMPARISON BETWEEN THE TOTAL TO STATIC EFnCIENCY OBTAINED

FROM THE EXHAUST DUCT TRAVERSING MEASUREMENT AND THAT

OBTAINED FROM THE COMPRESSOR WORK MEASUREMENT (TURBINE A )

s HQR VEH SPED ^ ^ 1)0%

• 90% # 70%

50%

100 250 PRESSLKERMB

FIGURE 7.17 COMPARISON BETWEEN THE MASS FLOW OBTAINED FROM THE EXHAUST

DUCT TRAVERSING MEASUREMENT AND THAT MEASURED BY MEANS OF

AN ORIFICE PLATE ( TURBINE A )

218

ago

080

HOR. VER. SPEED e 1 0 % o • 90% o • 70% o • 50%

zoo 250 PRESSURE WTO

FIGURE 7.18 COMPARISON BETWEEN THE TOTAL TO STATIC EFFICIENCY OBTAINED

FROM THE EXHAUST DUCT TRAVERSING MEASUREMENT AND THAT

OBTAINED FROM THE COMPRESSOR WORK MEASUREMENT ( TURBINE B )

S

I HOR e

o o •

VER SPEED ^ mx

• 90%

70% 50%

m 250 PRESSURE RATO

FIGURE 7.19 COMPARISON BETWEEN THE MASS FLOW OBTAINED FROM THE EXHAUST

DUCT TRAVERSING MEASUREMENT AND THAT MEASURED BY MEANS OF

AN ORIFICE PLATE (TURBINE B )

219

TRWBGEIDSUOEm mTDSUTCEFFCENDr

OJD

an

(LTD

m

T . A T . B SPEED

e X

o • n x

O • 7DX

0 • SOX

T . A T . B SPEED

e X

o • n x

O • 7DX

0 • SOX

T . A T . B SPEED

e X

o • n x

O • 7DX

0 • SOX #

1 0 o 0 Q- — -— •

< 3 0 — —1

' " ^ N \ • "\i f n

&

i 1

uo 150 m PRESSURE WTB

ZSO m

TRAVERSE iCASUROOn TOTAL TO TOTAL ETTTENCr

QfiO tDO

T.A T.B SPEED e n x o • gox o • • 7DX a . • SOX

-I—[ I ' I 150 zm

PRESSIFERATD

-I—I—'—I—I—I—r 250 m

FIGURE 7 JO TRAVERSING MEASUREMENT :

- TOTAL TO STATIC EFHCIENCY

- TOTAL TO TOTAL EFnCIENCY

220

TOTAL PRESSURE STATIC PRESSURE

I " f iM

t & IA3

X —"K

\ N

X

•40 -SO - I D 10

mm. m\mi (*m)

t : /

N s

-« -X -i« le E M

FLOW ANGLE SWIRL VELOCITY

J I " §

? .0

I

' / \ \ /

\ /

V 1 L 0 y / Sfc V

V J X so -so -30

ABSOLUTE MACH NUMBER

3 :

• ^ 5

-so -30 -»0 10

AXIAL VELOCITY

» CUM 3 * CUM 7

HGURE 7 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBINE A ( 5 0 % Speed )

221


tji

lit

IJ7

I

i -!" 6

\

/ > \ / D- \ - A

/

\ / V,

/ \ /

/

s -N

J s s V

% -10 10

MHOClMCt (mm) -30 -10 10

MMLKVKt (MO


2 M

\ \ / / \

\ 1 1 ' -V

/ /

\ 1 / \ \

s >

/ /

-50 - » - 1 0 10 X se


-M -» -10 10 30 iO

AXIAL VELOCITY

/ \

X %

\ \ J

-50 -30

0 curvE I

-10 10

• OJM. i A CUM 2 •

HGURE 7J2 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECII/VNISM . TURBINE A (70 * Speed )

222


WW. BEMCI BWC (MO


e »

\ 1 /

X V

s N —f"

r

V

r~

\ / N / / \ 1 / \ \/ /

\ Y V N \ A

—V V - U -10 10 30 iC - 1 0 10

ABSOLUTE MACH NUMBER AXIAL VELOCmr

/ \ \,

s.

—

\ J >

\ / / V

—

^ 100

i " i

/ \

IT" "

A s

\ / J

-JD

0 WW I

•>10

• CUIM i 4 CUMC 2

W

FIGURE 7J3 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBINE A { 90 % Speed )

223


% / \ \

/ \ / /

i / 'V /

\, \ / \ / \ / V /

y / V

S l-R

g HI

/ \ /

/ A r

/

/ /

\ \ /

/

\ / V

-so -3C -10 to

m . OBWKf (mm)

-to 10 wm. oeiMKt w


\

\ f \ \ \

/

/ W

V \ / i

W -30

ABSOLUTE MACH NUMBER AXIAL VELOCITY

\ \

"

\ / \ \ V

-SO -30 -10 10 30 50 -50 -30 -to 10 30 M

0 CUPVY 3 6 CURrt 7

nCURE 7 J4 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBINE A (100 % Speed )

2 2 4

TOTAL PRESSURE

-60 -30 -10 10

WML DGTAHa (mm)

/

\ I / y \

N / > /

> /' X

V

# \tt

STATIC PRESSURE

/ /

- r " i

' %

\

? -

-M -30 -10 10

m. OCTANQ (liO

FLOW ANGLE

-30 -10 30 SO

SWIRL VELOCmr

\

A

-50 -30 -10 10 30 so


-8—

/ \ \ \ \

(K \. >< / /

, ,

V ]

-50 -JO -10 10 » »

s s «

AXIAL VELOCITY

/

-50 -30

0 CURVC I

\

-10 10 u so

0 CUM i t CUM 1

FIGURE 7as TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBINE B (50 % Speed )

225


A \ / \ /

/ \ \ / /

-"-y i—, A, 1 / V \ / V /

i <M

/ /

\ / , r -

X / / / \ y

V

-50 - » -10

m m . O C U M t (mm)

-40 -30 -10 10 iO 50

IMWi DSlANa (MO


-50 -30 -10 10

ABSOLUTE MACH NUMBER AXIAL VELOCITY

2

i 07

i

/ /

/ \

\ \

\ / V

-50 -30 -10 10 30 50

• M M I 0 n*vt} t CUM 2

FIGURE 726 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING PROBE MECHANISM . TURBINE B ( 70 » Speed )

226


-M -30

J \ 1 B

/ / " A 7

/ V /

\ / \ y

/

V - 1 0 10

W.06UMCS (im) 30 50

/ /

/ /

/

f

/

/ 4 /

\ / \

V /

- 1 0 10

WMLDGIWd (MM)


ABSOLUTE MACH NUMBER AXIAL VELOCmr

-50 -30 -to 10

/ , r * - \

/ \ \/

/ V

A \ /

0 / \ k \

V

, / \. '

/ \ / \

\ / V

-so -30 - 1 0 10 30 M

• CUBVC I 0 CURVE 3 t CUM 7

FIGURE 7 J7 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBI7VE B(90% Speed )

227


i

; I DS

>

-7^ f— \

—

/

7 ^ 7^

d - V

7 ^

I " <

^ 10

• 1 0 10

MM. DCUMX (mn)

F L O W A N G L E

r-—• N

/ 1 1 /

1 /

V I

MM. 061AMX (IM)

SWIRL VELOCmr

-30 -10 10 30 50

ABSOLUTE VACH NUMBER

1 0.3 2

-10 10

AXIAL VELOCITY

\ \ £

/ \ > f

\ / V —

-30 -10

e OtNl 3

10 A CUM 7

FIGURE 7 J8 TURBINE EXHAUST DUCT FLOW MEASUREMENT BY A TRAVERSING

PROBE MECHANISM . TURBINE B (100 * Speed )

228

SHOD PimiE DSnOUnON [niBE A] 50 XEqiiviM Speed

120

IB

III

105

too

: ( i ) \

•

(3 )<y

\ Tf 1

N O S L I

1 : ^

1 1 1 1 1 1 M

) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 J

20 30 40 50 1 1 1 1 1 1 1 1

6

AXKLOSTANCEM

SWOliD PRESSIK OETRBJTIGN [nJOE B] 50 X EquhnM Speed

(1)

30 musinairn]

FIGURE 7 J9 MIXED FLOW TURBINE SHROUD PRESSURE MEASUREMENT

(50 % Equivalent Design Speed )

229

SWOll) PIESSURE DETRBinQN CniOE A] 7D X Eq i iv iM Speed

? 120

GO AXKLOSIANCEM

SHOiD PfmJE OETimiON fnJOE B]

70 X Eqiivdent Speed

m-

AXKLDETANCEM

nCURE 7 JO MIXED FLOW TURBINE SHROUD PRESSURE MEASUREMENT

( 70 % EquiTOlent Design Speed )

230

smom PiEssuE osTiBjnQN [nisK A] 90 XEqutvoM Speed

AXKLOCTANCEH

S H ^ P i m i E aSTRBirUN [TIFBI£ B] 90 X Equvoient Speed

I 60

AXW.DGTANCEW

nGURE 7 J1 MIXED FLOW TURBINE SHROUD PRESSURE MEASUREMENT

( 90 % Equivalent Design Speed )

231

SHOE FiGSlJE lOBinQN tniOE A] I D X EfJtvM Speed

IBO

IfiO

S" 140

120

too

m

AXM.DETANCE[nn]

SWGU) PESSIJRE DETRBJnON nUOE B] I D X Equivdent Speed

IfiO:

OA)

a x m l d e t a n c e m

FIGURE 7 J2 MIXED FLOW TURBINE SHROUD PRESSURE MEASUREMENT

(100 % Equivalent Design Si>eed )

232

Exit f l o w angle : Turbine A Exit f l o w ongle : Turbine B

f i

200

15J3

00

5i)

OD

-M

-OD

- 6 D

-200 -5011

; 50Xspe<^d ;

' U /C = 0.45 '

0| Rotor exit J S i X i j <

» Exhoust duct (meosured]

i

;

:

-3011 - C D CD

Radiol distonce [mm]

300 SOD

I Ji

200

BO

CO

5 0

OO

-5J)

-CO

-150

-200 -500

1 50Xspe<jd

' U /C = 0 .45

] f 0 M o f exit ( s i x )

> Exhoust duct (measure! ] f ;

— y

-300 - C O tt.D

Radiol distonce [mm]

300 50

i

GOO

500

, 4 0 0

. 3 0 0

200

CO

00

0

50 % spa

U / C = OJ

- m a r e a "

Exhoust du

r — — —

>4

rt (measure* y

-500 -300 -CO CO

Rodiol distonce [mm]

300 500

50 % s p e ^

U/C = 0.W

my aR-pcq 0 Exhoust duct [meosured]

r - . 400

C O

-300 -0 .0 CO

Rodiol distonce (mm]

GOO

50 % speed

U / C = 0.56

[?ator exit (5.1.^7 0 Koior ex* » Exhoust i ud (meosut^

r r r r m

-co Rodiol distance [mm]

rrrrrrrrr CO 300

GOO

500

^ 400

- S . 3 0 0

I 200

J •< 0 0

OO

-co -500 -30.0

50 X s p ^

U / C = 0.64

0 Rotor exit (S.l .CJ

0 Exhoust iuct (meo

- 0 0 CO

Rodiol distonce (mm]

300 50

F i p n i e 7 . 3 3 C o m p a r i s o n b e t w e e n t h e a b s o l u t e f l o w a n g l e a t r o t o r e x i t

a n d a t t h e e x h a u s t d u c t c r o s s s e c t i o n ( 5 0 % s p e e d )

2 3 3

Exit flow angle : Turbine A Exit flow angle : Turbine 0

20.0

10D

0.0

-10,0

- 2 0 . 0

-30.0 -50.0 -30.0

~1 — — " "

90 % s p e ^ ^

u / u = 0 . 5 /

0 Rotor exit (S.L.C)

0 Exhoust duct [meosur

20i)

10.0

0.0

-10.0

-20.0

:d)

-10.0 10,0

Radial distance [ m m ]

30.0 50.0 -30.0

20.0

10.0

0.0

.s

-10.0

-20,0

-30,0 h -50,0

1

1 1 1 1

1 1 1 1

i 90 % speed

, U/C = d . i b

• 0 Rotor e x i [S.L.C]

.0 Exhaust duct [meo

-30.0 -10,0 10,0 30,0

Radial distance [ m m ]

50

40.0

30.0

20.0

0.0

-10,0

-20.0 -50,0

90 % speed

U/C = g.G3

0 Rotor e x i [S,L,C)

0 Exhaust duct [meosur

-30.0 • -10.0 10,0

Radiol distance [ m m ]

30.0 50,0

40i)

30J]

201)

10J3

0.0

-10.0

:d)

-20,0

40D

3 0 i )

1* 20.0

E-I 10.0

a i J 0.0 «c

-10.0

-20.0 -50,0

/ 90 % s p e e S ^

uyc = g .6 i

0 Rotor e x i [S.L.C]

0 Exhoust duct [meo

-30.0 -10.0 10,0


30.0 50

60.0

50.0

40.0

— 30.0

% 20.0 a

I 100 I

0.0

-10.0

-20.0

;

1

90 % speed

U/C = q .7 i

0 Rotor ex'5 [S .L .C j

0 Exhaust duct [meosur

eon

50i]

40X1

3011

201)

10.0

0,0

-10,0

d] -20,0

-50,0 -30,0 -10,0 10.0


30,0 50.0

60.0

50.0

4 0 D

i ^ 30.0 g "

» 20.0

I m «c

0.0

-10,0

-20,0

y \ 90 % speed

U/C = q,69

0 Rotor ex^ [ S l . C j

0 Exhaust duct [meo

-50,0 -30,0 -10,0 10,0


30.0 50

i i s i i r e 7 .34 C o m p a r i s o n b e t w e e n t h e a b s o l u t e f l o w a n g l e a t r o t o r e x i t

and a t t h e e x h a u s t d u c t c r o s s s e c t i o n ( 9 0 % s p e e d )

2 3 4

Axial Distance [m]

0.00 oj)i om WB iMw oa wm (W7

2.1)

190

170

150

130

IB

0.90

• meosurejnent

0 3-Oinviiiii BOX Speed. R?

0 2-OsicJn

0.00 0.01 0.02 0.03 OiM OJB 0i)6 m

I i futx- 7 . 3 5 ( . o m p a r i s o n b e t w e e n m e a s u r e d and c o m p u t e d s h r o u d

p r e s s u r e d i s t r i b u t i o n ( T u r b i n e A )

3 - D inv i sc id c a l c u l a t i o n (74)

2-D s t r e a m l i n e c u r v a t u r e m e t h o d (25)

235

SOXEOJVALENTSPED

(1)

TurtineA

Turbine B

I I I I I I I M I I M M • M I I I M

» 30 DGTANCE ALONG TKLEAONGEDGEtX)

70 XEQUVALENT SPEED

40 TurtineA

- 2 0

-40

-60

TurbneB

DISTANCE ALONG THE LEADWG EDGE ( X )

r i y i i r e 7 . 3 6 C o m p u t e d i n c i d e n c e a n g l e a l o n g t h e r o t o r in l e t

236

go XEQJVALENT SPEED

TurhneA o.

I 1 I I I I

30

H

-t)

-50

I I I I ' I I I I I I I I I ' I I I I I I I I I ' I I I M I I I I I I I I I I I I I I

30 50 70 90 -70

DISTANCE ALONG DElEAOWGEDGEfX)

IX) % EQUVALENT SPEED

Turbine A ^ ^

TuftineB

DISTANCE ALONG TT€ LEADWG EDGE ( X )

F i p u r e 7 . 3 7 C o m p u t e d i n c i d e n c e a n g l e a l o n g t h e r o t o r in l e t

237

Ifixed flow turbine: Curve of optimum U/C

0.70

0£5

0.60

055

050

1

k 1

1 f

_ i J

>

L _ J

\ 1

J

1

r

0 •

_

Turbine A TifbineB

40 50 60 70 80

X of equivoknt design speed 90 CO m

kCxed flow turbine: Curve of maximum efficiency

0.90

0.85

g- 0.80

i

I 0.75

^ 0.70

0.65

0.60

>

0 • TiffcineA Turbine B

40 50 60 70 80 90 no It) % of equ volent design speed

F i g u r e 7 . 3 8 .Mixed f low t u r b i n e o p t i m u m v e l o c i t y r a t i o a n d t o t a l t o

s t a t i c e lT ic i ency a s a f u n c t i o n of r o t o r s p e e d .

238

100

OJB 0 ^ a40 OfO I I I I I I I I I I I I I

Specific speed Ns[ 9 Units]

OA)

O i O -

0.60

I 0.40

0.20

100 120 I I I I I t i l l

140

Totol to totd efficiency corresponding 1 tocurve of rradrnumstotic efficiency j

too

- m

OiO

Dive of maximum totd b static efficiency'

_ m ( 4 8 )

» Garrett (35)

a Baines(2)

0 Yamoguchi (34)

0 TurtineA

• TurtineB

0.40

Mixed flow turtines 0.20

0-00 f I I I I I I I I I I I I M I I I I I I I M I I I

0 20 40 60 BO 120 I I I I I I I

140 BO

OiB

BO

Spedfic speed Ks

Fitnii-o 7 . 3 9 C o m p a r i s o n b e t w e e n high s p e c i f i c s p e e d m i x e d f l o w

t u r b i n e s a n d r a d i a l t u r b i n e s o p e r a t i n g a t m a x i m u m

e f f i c i e n c y (4SJ

239

Chapter 8

8. Conclus ion

This chapter gives a summary of the work

presented in this thesis as well as the most important results of

the experimental investigation concerning two mixed flow

turbines. Suggestions of future work to be made in order to achieve

more understanding of the mixed flow turbine are presented.

8.1. Summary of the Design Model

The design process of a new generation of highly

loaded mixed flow turbines has been developed and is presented in

this thesis. It can be summarized as follows :

a - 1-D Design Model

A 1-D design model has been developed to define

the turbine overall dimensions and to predict its performance at

the off-design conditions. A series of designs were produced and

then analysed by the 1-D off-design performance prediction code. A

final design was selected for further analysis.

b - Rotor Geometry Design

An analytical method was developed so that the

blade geometry can be generated and modified easily. The flow

inside the blade channel is then analysed by a quasi-three

dimensional streamline curvature method for the flow calculation.

240

The effects of several parameters such as the rotor inlet blade

geometry, the rotor length and the blade curvature were

investigated. Two prototypes were selected for the experimental

investigation and a volute which is used for the two rotors was

also designed and manufactured. The two rotors differ only in their

blade leading edge shapes and rotor lengths.

8.2. Exper imental Ana lys is

The experimental analysis of the two mixed flow

turbine prototypes consisted of the determination of the overall

performance characteristics, the flow field at the exhaust duct and

the pressure distribution along the rotor shroud profile. The

turbine performance characteristics were assessed by two

methods which differ in how the measurement of the turbine net

output power is made. The turbine net output power was obtained

by measuring the work absorbed by the compressor in one case and

by measuring the flow field at the turbine exhaust using a

traversing probe and then calculating the drop in total enthalpy

across the turbine in the other.

8.3. Results of Exper imental Analys is

The experimental investigation of the two mixed

flow turbines designed by the method presented in this thesis has

raised high expectations concerning the use of this type of turbine

in the turbocharger applications as a result of the significantly

high efficiencies obtained for a wide range of operating conditions.

Turbine A and turbine B efficiency characteristics are of similar

trend but differ in their magnitudes by about 0.07. Peak

efficiencies and the corresponding velocity ratios U/C, (riTs,U/C)op,,

241

showed a variation with the variation of speed. The corresponding

( t i jg ,U/C)op^ for different equivalent design speeds are shown in

figures 7.38.

The variation of optimum velocity ratio with speed

seems to be a particular characteristic of the mixed flow turbine.

The optimum U/C for radial turbines for instance is constant and

has a typical value of 0.7 while for the mixed flow turbines A and

B, it varies from 0.55 at 50 % equivalent design to about 0.65 at

100 % equivalent design speed.

The pressure distribution measured along the shroud

profile indicates that higher losses are occurring inside the rotor

channel of turbine B than in the case of turbine A. This confirms

the results obtained by the two methods used to measure the

turbine efficiency characteristics, that is turbine A is more

efficient than turbine B. Therefore the blade geometry design is of

high importance as it has been demonstrated by the behaviour of

the two rotors which have identical overall dimensions but differ

only by the blade camberline and the rotor length. The constant

rotor inlet blade angle ( rotor A ) appears to be better suited for

future design of high efficiency mixed flow rotors.

8.4. Future Work

a - The work presented in this thesis constitutes a useful tool for

the design of mixed flow turbines. Although the effects of

different geometrical parameters were investigated, only two

rotors were manufactured and experimental ly tested. The

experimental study has shown interesting results and that the

efficiencies of the two rotors, thought to be both good designs.

242

were significantly different as a result of the differences in some

geometrical features ( blade camberline and rotor length ). In order

to widen the understanding of the effects of other geometrical

parameters affecting the mixed flow turbine design, more rotor

prototypes have to be built and experimentally tested following the

procedure adopted for the two rotors ( A and B ) already studied.

This can be achieved for instance by having different designs

similar to rotor A (which showed high efficiencies) but with

different cone angles and rotor inlet blade angles.

b - The testing of an equivalent radial inflow turbine ( whose

performances are already known ) on the same test rig will provide

a good comparison with the two mixed flow turbines. It can also be

used as a calibrating device for the whole rig and the results

obtained will serve to correct the mixed flow turbine

character ist ics.

c - The mixed flow turbine is destined to be part of a turbocharger

where the flow in the real application ( turbocharged engine ) is of

a highly pulsating nature. There fore its per formance

characteristics under these unsteady conditions are even more

important than those obtained during the steady condition tests

and need to be measured. A similar turbine to turbine A but with

scaled down dimensions ( due a limitation in the power which can

be absorbed by the dynamometer ) will be tested at Imperial

College on an existing dynamometer test rig for the determination

of the pulsation flow performance characteristics. Further testing

of the turbocharger, which is made up with this mixed flow turbine

and a compressor designed at Bath University, on the engine will

also be undertaken.

243

APPENDIX A

O p t i m u m Ve loc i ty Rat io

The express ion of max imum to ta l - to -s ta t i c e f f ic iency

(zero tangent ial component of absolute velocity at the rotor exit) is

as fol lows :

IA.11

From figure 3.3 and for an isentropic expansion in the volute, V2U can

be written in the following form :

^2U = U, + Vg, sin [^2]

where pp is the degree of reaction, ocg is the absolute flow angle and

|32b is the rotor inlet blade angle and also the relative flow angle in

this analysis. Replacing V2U in equation A.I by its expression given

in equation A.2, the following relation between DC = Ug/Vg., Pgb'

(Xg. Pd snd rijg, can be obtained.

UC^ + UC sin a ^ 1 - P ^ tan [3 ^ - = 0 [A.3]

The maximum efficiency r i js theoretically achievable (ideally) is

equal to unity (the exit kinetic energy is totally recovered and the

expansion process in the turbine is isentropic). The degree of

reaction for radial turbines is assumed to be 0.5. Thus equation A.3

becomes

UC + UC sin (X ' tan — 0 [A.4]

244

Solutions of equation (A.4) are presented in figure A.I which shows

that increasing the blade angle at rotor inlet leads to a decrease of

the optimum velocity ratio at which optimum efficiency is obtained

(for a given rotational speed, stagnation conditions and rotor inlet

mean diameter). Equation (3.46) on the other hand is used to

represent the variation of pressure ratio Pr with respect to UC

(figure A.2) for a given rotational speed and stagnation temperature

at the turbine inlet.

245

a 0 u 3

o Alfa2 = 10. deg. -f 1 5 . d e g . ^ 2 0 . degr.

0 . 7 9 -

0.76 -

0 . 7 5 -

0 . 7 3 -

0 . 7 1 -

0 . 5 8 -

o.ee -0 . 6 5 -

0 . 6 4 -

0 . 6 3 -

- 1 0 0

Ro to r Inlet B lade Ang le ( DEC. )

Fig.A.l Variation of Optimum Velocity Ratio UCopt with Rotor Inlet Blade Angle for Mixed Flow Turbines.

L

0 . 5 6 0 . 6 4 0 . 6 8

Ve loc i t y Ro t io ( U / C )

0 . 7 2 0 . 7 6

Fig.A.2 Pressure Ratio vs Velocity Ratio ( Equation 3.46 )

246

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250

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251

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252

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256

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257

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Design and manufacture of complex surfac es

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258

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