Numerical analysis of subsonic laminar flow aerothermodynamics in microturbomachinery and development of a design methodology

mww&mmm

SH1EBR00KE

Faculte de genie

Departement de genie mecanique

NUMERICAL ANALYSIS OF SUBSONIC LAMINAR FLOW AEROTHERMODYNAMICS IN MICROTURBOMACHINERY AND DEVELOPMENT OF A DESIGN METHODOLOGY

Master's thesis in applied science Specialty: Mechanical engineering

ETUDE NUMERIQUE DE L'AEROTHERMODYNAMIQUE D'ECOULEMENTS LAMINAIRES SUBSONIQUES DANS LES MICROTURBINES ET DEVELOPPEMENT D'UNE

METHODOLOGIE DE CONCEPTION Memoire de maitrise en sciences appliquees

Speciality: Genie mecanique

Jury:

- Prof. Martin Brouillette, Eng., Ph.D. - Prof. Luc G. Frechette, Eng., Ph. D. - Prof. Nicolas Galanis, Eng., Ph. D.

Philippe BEAUCHESNE-MARTEL, ing. jr.

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ABSTRACT

NUMERICAL ANALYSIS OF SUBSONIC LAMINAR FLOW AEROTHERMODYNAMICS IN MICROTURBOMACHINERY AND DEVELOPMENT OF A DESIGN METHODOLOGY

This thesis presents the numerical and analytical study of the aerodynamic and heat transfer in laminar subsonic cascades along with the development of design guidelines and procedures to improve the design of microfabricated multistage radial turbines operating at low Reynolds number.

Numerical analysis was performed on 24 cascade geometries using 2D computational fluid dynamics (CFD) for over 100 flow conditions for each cascade. Two dimensional correlations were extracted from CFD for profile and mixing los'ses, deviation and heat transfer. These correlations include Reynolds number and compressibility effects, and take into account incidence and various geometrical parameters (solidity, stagger, blade angles, thickness and mean-line distribution). Adaptation of losses to account for three dimensional effects and correlation for blockage were derived from analytical relationships.

A turbomachinery simulation software based on mean-line analysis and conservation of rothalpy incorporating the developed correlations was programmed. The software can be adapted as for the physic it uses and the turbine configuration it analyses (axial, radial inward or outward, single or multi stage). The pressure profiles obtained from simulation were found to be in good agreement with experimental data for cold turbine tests.

Design guidelines and charts are provided as well as cycle analysis considering microfabrication limitations. A considerable increase in stage isentropic efficiency compared to previous devices can result from the use of slender blades, lower solidity cascades and aspect ratios of 0.5, suggesting efficiencies as high as 85% for Re > 700. The study shows that higher power density and multistage matching can be achieved through the radial outward configuration. Two designs are presented, a single stage turbine for the next generation of microturbopump prototype and a turbine configuration with four rotors and 10 stages for closed Rankine cycle providing 50.7 W of net mechanical power.

RESUME

ETUDE NUMERIQUE DE L'AEROTHERMODYNAMIQUE D'ECOULEMENTS LAMINAIRES SUBSONIQUES DANS LES MICROTURBINES ET DEVELOPMENT D'UNE METHODOLOGIE DE CONCEPTION

Ce memoire porte sur I'etude numerique et analytique de I'aerothermodynamique dans les cascades subsoniques laminaires et sur le developpement d'une methodologie de conception permettant d'augmenter I'efficacite des turbines radiales multi-etages microfabriquees operant a faible nombre de Reynolds.

Des analyses numeriques (CFD 2D) ont ete realisees sur 24 geometries de cascade differentes a I'aide d'un logiciel commercial, toutes soumises a plus de 100 regimes d'ecoulements. Des correlations 2D ont ete extraites des resultats pour les pertes de profil et de melange, la deviation et le transfer! thermique, integrant les effets du nombre de Reynolds, de la compressibility, de I'incidence et de la geometrie (solidite, calage, angles des pales, distribution de cambrure et d'epaisseur). Les effets tridimensionnels sur les pertes et la correlation pour le blocage ont ete derives de relations analytiques.

Un programme de simulation de turbomachinerie base sur I'analyse de la ligne mediane et sur la conservation de la rothalpie, incluant les relations developpees, a ete programme. II permet I'adaptation de la physique et de la configuration utilisee. Les profits de pression obtenus par le programme correspondent aux donnees experimentales obtenues.

Des analyses de cycles, des chartes et des recommandations pour le design considerant les limites de la microfabrication sont presentees. Un gain en efficacite considerable par rapport a ses predecesseurs peut etre obtenu par I'utilisation de pales minces a facteur de forme de I'ordre de 0.5 et de plus faible solidite, prevoyant une efficacite atteignable de 85%. Une plus grande densite de puissance et un meilleur couplage multistage peuvent etre obtenus par une configuration centrifuge. Deux designs sont presentes, un pour le prototype de prochaine generation et I'autre pour le cycle Rankine ferme, comportant quatre rotors et 10 etages, fournissant 50.7 W de puissance mecanique net.

TABLE OF CONTENTS

LIST OF FIGURES VII LIST OF TABLES IX NOMENCLATURE X REMERCIEMENTS I 1 INTRODUCTION , 1

1.1 BACKGROUND 1

1.2 MOTIVATION AND OBJECTIVES 1 1.3 PREVIOUS WORK AND RESEARCH ; 3

1.3.1 Background literature 3 1.3.2 Previous work done by MICROS 3

1.3.3 Work done at MIT. : , J

1.3.4 Continuity assessment. 5

1.4 OVERVIEW OF THE PROJECT 6 1.5 ORGANIZATION OF THE THESIS 7 2 MICROTURBOMACHINERY ANALYTICAL MODELING 8

2.1 FLOW MODELING CONSIDERATIONS IN A GAS TURBINE 8

2.1.1 The one dimensional analysis 8 2.1.2 The two dimensional analysis 9 2.1.3 The three dimensional analysis 9 2.1.4 Experimental and numerical adjustments to the cascade's performance 9

2.1.5 The selected model. 10

2.2 GOVERNING EQUATIONS OF THE MEAN-LINE ANALYSIS 10

2.2.1 Nomenclature 10 2.2.2 Conservation equations in compressible fluid flow 11

2.2.3 Power production and flow turning 13

2.2.4 Isentropic efficiency 13

2.3 AERODYNAMIC AND THERMODYNAMIC PHENOMENA IN A GAS TURBINE 14

2.3.1 Total pressure variation 14 2.3.2 Loss simplifications 15 2.3.3 Total temperature variation 15 2.3.4 Flow angle variation 16 2.3.5 Passage area variation 16

2.4 CHAPTER'S SUMMARY 16 3 NUMERICAL ANALYSIS OF LAMINAR CASCADES 17

3.1 NUMERICAL ANALYSIS OBJECTIVES 17 3.2 NUMERICAL ANALYSIS DOMAIN 18

3.2.1 Geometric domain 18

3.2.2 Fluidic domain v 18

3.3 PREPROCESSING 20 3.4 PROCESSING 20

3.4.1 Viscous and Compressible Effects 21

3.4.2 Incidence Effects 21 3.4.3 Rotation Effects 21 3.4.4 Thermal Transfer 21

3.5 POSTPROCESSING 22

3.5.1 Properties averaging in non-uniform flows 22

3.5.2 Position and flow properties 22

3.5.3 Data extraction 23

4 NUMERICAL RESULTS 24 4.1 VALIDITY DOMAIN : 24

4.2 FOREWORD ON INCIDENCE AND CENTRIFUGAL EFFECTS 2 4

4.2.1 THE EFFECT OF INCIDENCE 2 5

4.2.2 THE EFFECT OF ROTATION 25,

4.3 PROFILE LOSSES 2 6

4.3.1 General profile loss mechanism and the effect of geometry 26

4.3.2 The effect of the Reynolds number 30 4.3.3 The effect of compressibility 37 4.3.4 The effect of incidence 42 4.3.5 General profile loss correlation 48

4.4 MIXING LOSSES , 52

4.4.1 Mixed-out losses 52 4.4.2 Mixing loss evolution downstream of the trailing edge 54

4.5 BLOCKAGE AND DEVIATION 55

4.5.1 The blockage phenomenology. 55 4.5.2 The blade blockage vs deviation 58 4.5.3 The evaluation of deviation 60

4.6 THERMAL TRANSFER 65

4.6.1 The working fluid's Prandtl number 65 4.6.2 General heat transfer mechanism 66 4.6.3 The effect of compressibility and the recovery factor 68

4.6.4 The combined entry length correlation 75

4.6.5 General heat transfer model 77

4.7 COMPARISON OF EXPERIMENTAL DATA VERSUS ANALYTICAL MODEL 7 9

4.7.1 Loss coefficients and 3D effects 79

4.7.2 Tip leakage 81 4.7.3 Thermal transfer 82 4.7.4 Evaluation of the model accuracy 83

4.8 CHAPTER SUMMARY '. 85 5 MULTISTAGE RADIAL MICROTURBINE DESIGN 86

5.1 DESIGN GOALS .86

5.1.1 Operating point 86 5.1.2 Global integration, performance and robustness 86.

5.1.3 Resistance and reliability 87

5.2 DESIGN CONSTRAINTS 87

5.2.1 Microfabrication constraints 87

5.2.2 System constraints 88

5.3 DESIGN ANALYSIS 89

5.3.1 The flow coefficient, the loading coefficient and the degree of reaction 90

5.3.2 The ^-y/ chart 92 5.3.3. Effect of the curve on stage robustness and power output 93

5.3.4 Inward versus outward turbine configuration 94

5.4 IMPACT OF THERMAL TRANSFER ON THE HOT TURBINE CYCLE AND DESIGN 98

5.4.1 The effect of the stator's thickness and solidity 100

5.4.2 The degree of reaction 101

5.5 MULTISTAGE DESIGN 102 5.5.7 The choice of an operating point 102

5.5.2 The choice of the stage loading and the number of stages 102

5.5.3 The choice of the radial positions 104

5.6 THE CHOICE OF DETAILED BLADE GEOMETRY AND RECOMMENDATIONS 104

5.6.1 The choice of blades' height 105 5.6.2 The choice of blades' solidity 105 5.6.3 The choice of blade angles 106

5.6.4 The choice of blade profile 106 5.7 THE MICROS II DESIGN 108

5.7.1 Design specifications 108 5.7.2 Detailed design 108 5.7.3 Mean-line analysis 109 5.8 THE RECOMMENDED MULTI-SPOOL MICROS DESIGN 110

5.8.1 Assumptions and simplifications Ill

5.8.2 Design specifications Ill 5.8.3 Baseline design Ill 5.8.4 A first attempt: The impulse configuration 112

5.8.5 The reacting configuration 114

5.9 CHAPTER SUMMARY 115 6 CONCLUSION 116

6.1 SUMMARY OF THE RESEARCH 116

6.1.1 Numerical analysis 116, 6.1.2 Aerodynamic and heat transfer 116

6.1.3 Mean-line analysis software 117 6.1.4 Laminar subsonic turbine design 118 6.1.5 Detailed design for MICROS 118

6.2 CONTRIBUTIONS OF THE WORK 118

6 3 RECOMMENDATIONS FOR FUTURE WORK ....119

REFERENCES 121 APPENDIX A : DESIGN OF THE MICROS 1 TURBINE 124

APPENDIX B : CASCADE GEOMETRY AND AERODYNAMIC COEFFICIENTS 125

APPENDIX C : BLADE PROFILES AND THICKNESS DISTRIBUTION 128 APPENDIX D : PREPROCESSING - MESHING AND SOLVER SETTINGS 130

Meshing procedure 130 Mesh Validation 131 Solver settings 133 APPENDIX E : CENTRIFUGAL AND INCIDENCE EFFECTS 134

APPENDIX F : BLOCKAGE AND DEVIATION 137 APPENDIX G : DEVELOPED CORRELATIONS 141 APPENDIX H : SIMULATION SOFTWARE 143 APPENDIX I : FABRICATION PROCESS FOR A ROTOR 154

APPENDIX J : <|>-y CHARTS 155 APPENDIX K : CHOOSING THE BLADE HEIGHT 175

APPENDIX L : COMMON MICROFABRICATION MATERIAL PROPERTIES 183

APPENDIX M : DETAILED DESIGN OF THE MICROS II TURBINE 185 APPENDIX N : DESIGN OF THE MULTISPOOL RANKINE CYCLE 187 APPENDIX O : AUTORISATION POUR LA REDACTION DU MEMOIRE EN ANGLAIS 188

LIST OF FIGURES

FIGURE l -1: VELOCITY CONTOURS AND COMPARISON OF CASCADE'S BOUNDARY LAYERS WHILE SUBJECTED TO (A)

LOW RE LAMINAR FLOW AND (B) HIGH RE TURBULENT FLOW (MA = 0.75) 2

FIGURE 1-2: MICROS' FIRST GENERATION PROTOTYPE 4

FIGURE 2-1: ANALYSIS METHODOLOGIES OF THE FLOW THROUGH TURBINE CASCADE 8

FIGURE 2-2: FLOW AND CASCADE NOMENCLATURE 10 FIGURE 2-3: ABSOLUTE AND RELATIVE VELOCITY COMPONENTS 11

FIGURE 3-1: BOUNDARY ADAPTATION AROUND A BLADE WALL 20 FIGURE 4-1: MIXING AND PROFILE LOSSES ALONG A BLADE 26 FIGURE 4-2: LOCAL LOSS COEFFICIENT ALONG A BLADE 27 FIGURE 4-3: VELOCITY CONTOURS AND DIMENSIONLESS WALL SHEAR STRESS FOR TWO DIFFERENT BLADE

GEOMETRIES.... 28

FIGURE 4-4: A3K7 BLADES PROFILE LOSS COEFFICIENT AND VELOCITY CONTOURS FOR DIFFERENT THICKNESS

DISTRIBUTIONS 29 FIGURE 4-5: PROFILE LOSS COEFFICIENT FOR VARIOUS AERODYNAMIC PROFILES 30

FIGURE 4-6: PROFILE LOSS COEFFICIENT OF DIFFERENT A3K7 BLADE GEOMETRIES AS A FUNCTION OF RE 31

FIGURE 4-7: Loss PRESENTATION SHOWING PASSAGE WIDTH IMPACT 33

FIGURE 4-8: INPUT OF THE INDIVIDUAL TERMS OF THE PROFILE LOSS SERIES AT DIFFERENT REYNOLDS NUMBER ..36

FIGURE 4-9: LINEAR DEPENDENCE OF MODIFIED LOSS COEFFICIENT 36

FIGURE 4-10: PROFILE LOSS COEFFICIENT FOR DIFFERENT MACH AND REYNOLDS NUMBERS FOR AN A3K7 PROFILE

WITH A SOLIDITY OF 2 AND A TURNING OF 130° 3 7 FIGURE 4-11: VELOCITY PROFILES AT THE BLADE OUTLET FOR DIFFERENT FLOW CONDITIONS 38

FIGURE 4-12: INFLUENCE OF MACH NUMBER ON LOSS PARAMETERS 39 FIGURE 4-13: DIMENSIONLESS WALL SHEAR STRESS ON BLADE WALLS FOR VARIOUS REYNOLDS AND MACH

NUMBER 41 FIGURE 4-14: VELOCITY VECTORS AROUND THE LEADING EDGE OF A BLADE ROW FOR DIFFERENT VALUES OF

INCIDENCE 42

FIGURE 4-15: EVOLUTION OF THE PROFILE LOSSES ON THE BLADE LENGTH FOR DIFFERENT INCIDENCE VALUES ....43

FIGURE 4-16: DIMENSIONLESS WALL SHEAR STRESS ON THE BLADE WALLS FOR DIFFERENT VALUES OF INCIDENCE 44 FIGURE 4-17: VELOCITY CONTOURS AT ZERO INCIDENCE (A), 15° INCIDENCE (B) AND VELOCITY VECTORS OF THE

SEPARATION BUBBLE (c) 44 FIGURE 4-18: EFFECT OF INCIDENCE FOR VARIOUS BLADE GEOMETRIES 45

FIGURE 4-19: EFFECT OF INCIDENCE FOR VARIOUS REYNOLDS AND MACH NUMBERS 45

FIGURE 4-20: VELOCITY CONTOUR AT I = 1 5 ° OF AN A3K7 BLADE FOR A SOLIDITY OF 1.5 (A) AND 2.0 (B) 46

FIGURE 4-21: EFFECT OF INLET BLADE ANGLE ON INCIDENCE EFFECT 47 FIGURE 4-22: COMPARISON BETWEEN INCIDENCE MODEL (DASHED LINES) AND CFD RESULTS 48

FIGURE 4-23: PROFILE LOSS COEFFICIENT COMPARISON BETWEEN CFD RESULTS AND MODEL (DASHED LINES) ....51

FIGURE 4-24: INFLUENCE OF REYNOLDS AND MACH NUMBER ON MIXED-OUT LOSSES AND COMPARISON WITH

MODEL 53 FIGURE 4-25: EVOLUTION OF MIXING LOSS IN THE STREAMWISE DIRECTION DOWNSTREAM OF THE LEADING EDGE

55

FIGURE 4-26: SIMPLIFIED 3D BLOCKAGE DEVELOPMENT 55 FIGURE 4-27: ENDWALL BLOCKAGE APPROXIMATION AT DIFFERENT RADIAL POSITIONS FROM INLET FOR VARIOUS

VALUES OF CONSTANT A 5 7 FIGURE 4-28: MASS-AVERAGED VELOCITY COMPONENTS AND FLOW ANGLE AS A FRACTION OF CHORD LENGTH

DOWNSTREAM OF CASCADE 5 9 FIGURE 4-29: CARTER'S M COEFFICIENT (CUMPTSY,2004) 60

FIGURE 4-30: CFD DEVIATION DATA FOR DIFFERENT BLADE GEOMETRIES AND COMPARISON WITH MODEL 61

FIGURE 4-31: DEVIATION MODEL 62 FIGURE 4-32: VISCOUS DEVIATION MECHANISM AND COMPARISON WITH CFD 64

FIGURE 4-33: COMPARISON BETWEEN THERMAL AND AERODYNAMIC BOUNDARY LAYERS 65

FIGURE 4-35: COMPARISON BETWEEN TOTAL PRESSURE LOSS (DASHED LINE) AND THERMAL TRANSFER (FULL LINE)

EVOLUTIONS THROUGH A CASCADE.... 6 7 FIGURE 4-36: AXIAL TEMPERATURE VARIATIONS IN A TUBE AT CONSTANT SURFACE TEMPERATURE 67

FIGURE 4-37: TEMPERATURE VARIATION AS A FUNCTION OF RE AND M FOR I =0° >. 69

FIGURE 4-38: ILLUSTRATION OF THE ADIABATIC WALL TEMPERATURE 70 FIGURE 4-39: TEMPERATURE DISTRIBUTIONS ALONG A BLADE CHORD FOR ADIABATIC FLOW CONDITIONS 71

FIGURE 4-40: TEMPERATURE DIFFERENCE IN A BLADE PASSAGE HAS A FUNCTION OF RE AND M FOR I=0° 74 FIGURE 4-41: COMPARISON BETWEEN NUSSELT NUMBERS EXTRACTED FROM CFD RESULTS AND THE SIEDER-TATE

COLLERATION(M = 0.3 AND I=0°) 76 FIGURE 4-42: GENERAL ACCURACY OF THE SIEDER-TATE CORRELATION FOR ALL THE STUDIES CASES 77

FIGURE 4-43: HEAT TRANSFER IN A LAMINAR BLADE PASSAGE AS A FUNCTION OF GEOMETRY AND FLOW

CONDITIONS 78

FIGURE 4-44: IMPACT OF GEOMETRY (4-69) AND NUSSELT NUMBER (4-61) ON THERMAL TRANSFER 79

FIGURE 4-45: IMPACT OF THE BLADE'S ASPECT RATIO ON THREE DIMENSIONNAL LOSSES (A) AND COMPARISON

WITH MACRO-SCALE EXPERIMENTAL RESULTS (HORLOCK, 1966) (B) 80 FIGURE 4-46: COMBINED IMPACT OF ASPECT RATIO AND VISCOUS EFFECTS ON LOSSES 81

FIGURE 4-47: TIP CLEARANCE DUE TO DRIE CONCAVITY AT THE ROTOR'S BLADE ROOT 82

FIGURE 4-48: MODEL VALIDATION AND IMPACT OF THE VARIOUS PHENOMENA 83

FIGURE 4-49: MODEL VALIDATION AT VARIOUS OPERATING SPEEDS 84

FIGURE 5-1: TWO-DIMENSIONAL SHAPE OF THE TRAILING EDGE OF A DRIE MICROFABRICATED BLADE 88

FIGURE 5-2: GENERAL LAYOUT OF THE DEVICE AND GAPS DUE TO AIR BEARINGS 89 FIGURE 5-3: IMPACT OF THE THREE DEFINING PARAMETERS ON VELOCITY TRIANGLES ...91 FIGURE 5-4: DESIGN CHART OF A 50 PERCENT REACTION STAGE HAVING AN AVERAGED LOSS COEFFICIENT OF 0.2

AND A RATIO OF RADIAL POSITIONS BETWEEN THE OUTLET AND INLET OF THE ROTOR OF 1.1 92

FIGURE 5-5: LOADING COEFFICIENT CURVES AND CORRESPONDING POWER OUTPUT 93 FIGURE 5-6: IMPACT OF CONFIGURATION OF A SINGLE STAGE ON LOADING COEFFICIENT 94 FIGURE 5-7: IMPACT OF CONFIGURATION ON THE PRESSURE PROFILE AND AXIAL FORCE OF THE ROTOR 96

FIGURE 5-8: ALLOWABLE CORRECTED MASS FLOW PER ANNULUS AREA AS A FUNCTION OF FLOW ANGLE 97

FIGURE 5-9: EFFECT OF STATOR'S TEMPERATURE ON THE TURBINE'S TEMPERATURE PROFILE 99 FIGURE 5-10: REHEAT CYCLE OF THE TURBINE , 99

FIGURE 5-11: IMPACT OF STATOR'S REHEAT OF THE TURBINE'S THERMAL EFFICIENCY FOR A FLUID'S INLET

TEMPERATURE OF 5 0 0 K 100

FIGURE 5-12: IMPACT OF THE BLADE'S OUTLET ANGLE ON A CASCADE LOSS COEFFICIENT FOR REC=1000 103

FIGURE 5-13: SOLIDITY EFFECT ON PROFILE LOSS COEFFICIENT AND DEVIATION FOR A 130° A3K7 PROFILE 106

FIGURE 5-14: MICROS 1 PRESSURE PROFILE AND PROJECTED RESULTS OF A NEW DESIGN 107

FIGURE 5-15: POWER AND EFFICIENCY MAP OF THE DESIGN GEOMETRY 110

FIGURE 5-16: MULTI-SPOOL IMPULSE CONFIGURATION 113 FIGURE 5-17: MULTI-SPOOL REACTING TURBINES CONFIGURATION 114

FIGURE D-l: TYPICAL COMPUTATIONAL DOMAIN 130 FIGURE D-2: CASCADES CREATED BY PERIODICITY 131 FIGURE D-3: INFLUENCE OF MESH SIZE ON THE STUDIED PARAMETERS 132

FIGURE D-4: BOUNDARY ADAPTATION AROUND A BLADE WALL 132

FIGURE A-1:STATOR (UP) AND ROTOR (DOWN) 124 FIGURE A-2: INTERDIGITATION AND TIP LEAKAGE 124 FIGURE A-3: PICTURE ROTOR DISK (TOP SIDE) 124 FIGURE D-l: CENTRIFUGAL EFFECTS ON PROFILE LOSSES AND DEVIATION 134

FIGURE D-2: INCIDENCE EFFECTS ON PROFILE LOSSES AND DEVIATION 135

FIGURE D-3 INCIDENCE EFFECTS ON THERMAL TRANSFER 136 FIGURE E-l: CALCULATION PROCESS FOR THE TWO BLOCKAGE-DEVIATION FORMULATIONS ASSUMING NO MIXING

LOSSES AND INCOMPRESSIBLE FLOW 137

FIGURE E-2: FLOW ANGLE EVOLUTION FOR DIFFERENT CASES WITH RE=300 AND MA = 0.3 137

FIGURE E-3: EFFECT OF REYNOLDS AND MACH NUMBER OF DEVIATION 138

FIGURE E-4: EFFECT OF BLADE ANGLE ON DEVIATION FOR AN A3K7 PROFILE AT M A = 0 . 7 5 138

FIGURE E-5: EFFECT OF SOLIDITY AND MAXIMUM THICKNESS OF DEVIATION FOR A 130° A3K7 PROFILE AT

MA=0.75 139

FIGURE E-7: EFFECT OF MEAN-LINE AND THICKNESS DISTRIBUTION ON DEVIATION FOR A 100° BLADE OF SOLIDITY

1 . 5 A T M A = 0 . 8 0 140 FIGURE J-1: TRAILING EDGE GEOMETRY 176

FIGURE J-2: MAXIMUM BLADE ASPECT RATIO AS A FUNCTION OF THE FABRICATION TECHNIQUE'S MAXIMUM

ASPECT RATIO AND BLADE ANGLE FOR T/C = 4 % 176 FIGURE j-3: MASS CENTER SHIFTING DUE TO FABRICATION PROCESS 177

FIGURE J-4: MAXIMUM DIMENSIONLESS BLADE'S HEIGHT FOR A SINGLE STAGE USING 25% OF THE OUTER RADIUS

(4>=0.36) 179 FIGURE J-5: STRUCTURAL MODEL FOR BLADE'S STRESS ANALYSIS 180

FIGURE J-6: MAXIMUM BLADE'S HEIGHT FOR VARIOUS ROTATING SPEED : 181

FIGURE J-7: FEM ANALYSIS OF A BLADE ESTIMATING THE CENTRIFUGAL STRESS 182

LIST OF TABLES

TABLE 3-1: GEOMETRIC DOMAIN OF NUMERICAL ANALYSIS 18 TABLE 3-2: FLUIDIC DOMAIN OF NUMERICAL ANALYSIS 19

TABLE 3-4: CFD DATA TAKEN . 22

TABLE 4-1 CORRELATIONS RANGE OF VALIDITY 24 TABLE 4-2: COEFFICIENTS AND RESIDUALS OF THE LEAST SQUARES FIT ANALYSIS 35

TABLE 4-3: EFFECT OF THE MACH NUMBER ON SHEAR STRESS AND DYNAMIC PRESSURE TWO REYNOLDS NUMBERS

; 40

TABLE 4-4 EVALUATION OF "A" FOR 3 DIFFERENT CASES 57 TABLE 4-5: EFFECT OF VARIOUS PARAMETERS ON DEVIATION 61

TABLE 4-6: CARTER'S M COEFFICIENT: 63 TABLE 4-7: MEAN-LINES CHARACTERISTICS 63 TABLE 4-8: IMPACT OF GEOMETRIC AND FLOW PARAMETERS ON PRESSURE LOSSES, DEVIATION AND THERMAL

TRANSFER 85 TABLE 5-1: THE EFFECT OF GAS EXPANSION, PASSAGE AREA AND TURBINE CONFIGURATION ON THE FLOW

COEFFICIENT 95

TABLE 5-2: MAXIMUM MASS FLOW (MG/S) PER UNIT RADIUS (MM) (H=70UM AND x2=70°) 97

TABLE 5-3: MAXIMUM MASS FLOWS (MG/S) FOR INWARD AND OUTWARD CONFIGURATIONS 97 TABLE 5-4: PERFORMANCE COMPARISON BETWEEN AN IMPULSE AND A 50% REACTION STAGE 102

TABLE 5-5 : VARIOUS DATA SETS FOR DESIGN 103

TABLE 5-6: PROFILE CHARACTERISTICS AT R EC= 1000 107

TABLE 5-7: FIRST DESIGN TESTED IN THE MEAN-LINE ANALYSIS (ANGLES IN DEGREES AND LENGTH IN M) 109

TABLE 5-8: RESULTS OF THE FIRST DESIGN AT THE OPERATING POINT ....109

TABLE 5-9 : MODIFIED DESIGN TESTED IN THE MEAN-LINE ANALYSIS (ANGLES IN DEGREES AND LENGTH IN M) .. 110

TABLE 5-10 : RESULTS OF THE MODIFIED DESIGN AT THE OPERATING POINT 110

TABLE 5-11: MULTI-SPOOL IMPULSE PERFORMANCES 113 TABLE 5-12: MULTI-SPOOL REACTING TURBINES PERFORMANCES 114

TABLE D - l : BOUNDARY TYPES FOR MESHING CASCADES 131

TABLE A-l: STAGES'GEOMETRY 124 TABLE B-l: CASCADE GEOMETRY AND DEVIATION AND LOSS COEFFICIENTS 125

TABLE C-l: MEAN-LINE DISTRIBUTIONS 128 TABLE C-2: THICKNESS DISTRIBUTIONS :. 129

TABLE C-3: LEADING EDGE AND TRAILING EDGE RADII 129

TABLE F-l: CARTER'S M COEFFICIENT 142 TABLE F-2: MEAN-LINES CHARACTERISTICS 142 TABLE J-l: VALUE OF <D FOR VARIOUS CHORD LENGTH AND ROTOR RADIUS 178

TABLE L-l: RELEVANT PROPERTIES OF VARIOUS MICROFABRICATION MATERIALS 183

TABLE L-2: PROPERTIES OF FUSED SILICA (BULK) 183 TABLEL-3: PROPERTIES OF YTTRIA STABILIZED ZIRCONIA (YSZ) >. 184

NOMENCLATURE Blade Geometry

Flow - Absolute referential frame cc Absolute flow a n g l et

V Absolute flow velocity

Special angles / Incidence ( i = ax — X\ or/ = /?, - %x) S Deviation ( S = a2 - x2 &$ = A - X2) (°) (m/s) (°) O Blade geometry c Chord length d Diameter h ' Blade height/span r Radial position t Blade thickness s Blade spacing w Blade throat width <y Blade solidity <J = cls E, Stagger angle +

0 Blade camber/turning Xi Blade inlet angle f X2 Blade outlet a n g l ef

Flow - Relative referential frame

P Relative flow a n g l ef W Relative flow velocity

U Blade speed (m) (m) (m) (m) (m) (m) (m) C) D

n

(°)

n

(m/s) (m/s)

fF o r radial turbines, the angles are measured normal to the circumference while for axial turbines they are measured normal to the axial direction.

General Nomenclature a Speed of sound

Tangential Acceleration Passage area normal to the flow Effective passage area

Aspect ratio Biot number ae A A' AR Bi Ci C2

Low Reynolds number profile loss factor

High Reynolds number profile loss factor

(m/s) (m/s2)

(m2) (m2)

Cf Cp Dh E . F H h 1 k K kB K, K5 L LETR m m M M MR Nu P Pr

Q

R R Re Ro RPM S St T Friction coefficient

Specific heat at constant pressure Hydraulic diameter

Rotor thickness Force

Specific enthalpy

Averaged convection coefficient Section moment of inertia Conductivity

Blockage coefficient Boltzmann's constant Stress concentration factor

Deviation's thickness correction factor Length

Leading edge temperature ratio Mass flow Mass Mach number Torque Mixing Ratio Nusselt number Pressure Prandtl number Thermal power Ideal gas constant Degree of reaction Reynolds Number Rossby number Revolutions per minute Structural rigidity coefficient Stanton number Temperature (J/kgK) (m) (m) (N) (J/kg) (W/m2K) (m4) (W/mK) (J/K) (m) (kg/s) (kg) (Nm) (Pa) (W) (J/kgK) (rpm) (Mpa1/2) (K)

W Mechanical power (W)

x Axial position (m) X Gravity center position (m)

5H Recovery factor

8 Boundary layer thickness (m)

S* Displacement thickness (m)

7 Ratio of specific heats

A Relative thickness non-uniformities C, Dimensionless blade height

V Efficiency

K Profile loss compressibility factor

X Molecular mean free path (m)

M Dynamic viscosity (Pas)

P Density (kg/m3)

cr Structural stress (Pa)

t Shear stress (Pa)

V Kinematic viscosity (m2/s)

Flow coefficient

*

(+ = V

X

/U)

<P Ratio of rotor covered by stages

W Loading coefficient

co Loss coefficient

Q Turbine angular speed (rad/s)

Subscripts

0 Total (stagnation) properties/Initial (coefficients) 1 Stage inlet position/CFD upstream

2 Stage outlet position/CFD stage inlet position 3 Stage downstream/CFD stage outlet position 4 CFD downstream

2D Two dimensional 3D Three dimensional

c Characteristic/Chord based (Reynolds number) GC Gravity center in Stage input is Isentropic ^ 4 Mass Averaged max Maximum mix Mixing out Stage output pro Profile r Radial ref Reference s Solid te Trailing edge TV Thickness variation

w Wall/Throat width based (Reynolds number) x X axis component y Y axis component « Core flow/Upstream v Viscous component 9 Tangential Superscripts ' Relative reference frame value — Average

REMERCIEMENTS

Ce travail a ete de longue haleine et a demande un effort considerable. II n'aurait pas pu etre realise sans le soutient et I'apport de plusieurs personnes.

Je tiens d'abord a remercier mon directeur de recherche, le professeur Luc Frechette. II a su m'eclairer dans mes recherches par son experience et ses connaissances dans le domaine de la microfabrication et de la turbomachinerie, mais plus encore, il a un talent pour mettre les choses en perspectives, retourner a la base, et permettre de comprendre la nature et le fondement d'un phenomene ou d'un probleme. De plus, il a une passion contagieuse pour la science et I'ingenierie qui se manifeste par sa patience et son interet pour ses etudiants. J'ai appris enormement grace a lui, il m'a permis de developper mon savoir et mon savoir-etre. Merci beaucoup Luc.

Je tiens a etendre mes remerciements aux etudiants gradues de mon groupe de recherche. Felix et Mokthar, vous m'avez aide regulierement dans le cadre de mes recherches et vous me reveniez toujours avec la reponse a mes questions. Mathieu, nos conversations de bureau ou de laboratoire et I'energie que tu apportes ont permis de raccourcir des journees qui auraient autrement parues tongues. Mokthar, bonne chance avec ta nouvelle famille! Je remercie I'Universite de Sherbrooke, la faculte de genie et le departement de genie mecanique qui m'ont offert un milieu d'apprentissage et de vie hors pair au cours des six demieres annees et demie, tout au long de mon baccalaureat et de ma maitrise. J'y ai grandi, evolue et vecu une des plus belles periodes de ma vie.

Je tiens a remercier ma conjointe qui a ete presente pour moi lors de la redaction de mon memoire. Sa comprehension, son support et son attention m'ont permis de me concentrer sur la tache a accomplir et de continuer de I'avant lorsque la motivation semblait vouloir vaciller. Merci beaucoup Marie, je t'aime.

Finalement, je tiens a remercier specialement mes parents pour tout ce qu'ils ont pu faire pour moi. Si je suis au point ou j'en suis aujourd'hui, c'est grace a eux. Merci enormement pour I'education de grande qualite que vous m'avez offerte tout au long de ma vie, pour le support moral et financier et pour I'amour inconditionnel que vous m'avez voue. Michel et Aline, merci infiniment.

1 INTRODUCTION

This chapter explains the background, the motivation and the objectives of the current work. It then provides a brief overview of related work before outlining the thesis.

1.1 Background

In the past two decades, microelectromechanical systems (MEMS) have influenced the everyday life of most people whether they are aware of it or not. These systems implement various functions (mechanic, electric, magnetic, fluidic, thermal, optical, chemical and biological) on devices of the micrometer/millimeter scale. They can be found in car systems as accelerometers and gyroscopes for safety and control purposes, in inkjet printers where they use the piezoelectric effect in printer heads, in gas turbines and various other applications as pressure and flow sensors, in displays as micromirors such as the ones developed by Texas Instruments and in medical and biological applications as "Lab-On-Chip" (BIO-MEMS).

The constant development and improvement of portable electronics, wireless sensor arrays, small scale actuators and micro vehicles is calling for compact, high power sources. For over ten years, numerous ongoing research projects aimed at miniaturizing power plants and propulsion systems using MEMS technology. Just as their large scale counterparts, these small power systems use turbomachinery for their large power-to-weight ratio. They integrate the various features of the power plants and propulsion systems on micrometer and millimeter scale: heat exchangers, compressors and turbines, combustion chambers, nozzles, generators.,.

The current project aims at understanding the aerodynamic and thermodynamic behavior of compressible fluids in very small scale turbine cascades and at obtaining guidelines and a methodology for microturbomachinery design.

1.2 Motivation and Objectives

The group MICROS of the Universite de Sherbrooke, led by Professor Luc Frechette, has been working on a micro Rankine cycle for power generation or recuperation, either by having a specific heat source or by scavenging excess heat from a process (exhaust gas, electronic components,...).

Such devices have characteristic lengths of the order of a hundred micrometers, dimensions at which the viscous effects are far from negligible compared to macro turbines. Indeed, conventional turbines operate at Reynolds numbers of the order of 105-106 while MICROS microturbines operate at Reynolds numbers of 102-103. This shifts the highly turbulent flow to a laminar or transitional one. The flow pattern, loss mechanism and heat transfer vary greatly from turbulent to laminar flow and the use of conventional turbomachinery design offers reduced mechanical efficiency (55-70%) at microscale (Lee and Frechette, 2005), (Epstein, 2004), (Mehra, 1997). The velocity contours of Figure 1-1 shows the difference between the boundary layers of a turbulent macroscopic cascade and of a laminar microscopic cascade. The laminar cascade has much thicker boundary layers than the turbulent cascade, implying varying flow properties through the channel's width and a small core flow area.

(a) Rec = 600 (b) Rec = 700 000

Figure 1-1: Velocity contours and comparison of cascade's boundary layers while subjected to (a) low Re laminar flow and (b) high Re turbulent flow (Ma = 0.75)

Since the overall system efficiency scales directly with the turbine's efficiency, improvements in turbine aerodynamics are needed to offer a viable and high performance device. The main objective of the thesis is thus to develop design guidelines to improve the attainable efficiency of laminar microscale turbines.

The secondary objectives of the current work are:

To understand the impact of the blade geometry on its aerothermodynamics; To understand the impact of the flow properties on its aerothermodynamics; To produce charts, correlations and a model for use in microturbine design; To evaluate thermal and aerodynamic impacts on the power cycle;

To expand knowledge groundwork for microturbomachinery design; To provide a new design for the next prototype of MICROS.

1.3 Previous work and research

A microturbine, just like its large scale counterpart, is a complex machine which includes various components. The present section will only consider the work done in the aerothermodynamic field on the subject.

1.3.1 Background literature

Even though some phenomena might be of greater importance at small scale (losses) and others of lesser importance (mixing), the basic physics at small scale are still the same as for large scale turbines. This work was based on a large array of books covering turbomachinery design (Bathie, 1996), (Cumptsy, 2004), (Horlock, 1966), (Kerrebrock, 1992), (Wilson, 1984), and cascade aerodynamics (Greitzer et al., 2004), (Lakshminarayana, 1996), (Schlichting,

1979), (Shapiro, 1953).

1.3.2 Previous work done by MICROS

Most of the work done by MICROS so far has been performed in the early 2000s by Changgu Lee during his doctorate degree at Columbia University under the supervision of Dr. Luc Frechette. Lee developed the first generation prototype to demonstrate operability of a single rotor multistage radial turbine operating with air at ambient conditions (Lee, 2006).

The prototype can be seen in Figure 1-2 and detailed design of the turbine is presented in Appendix A. The main objective of the project was to test the design and to produce a viable prototype in order to obtain experimental data and compare to modeling tools for future generations of microturbines, while proving the concept of a micro Rankine device.

The multistage (four stages) radial configuration, arising from microfabrication limitations (covered later in this thesis), was a first worldwide attempt. The rotor, a 4mm diameter x 450 urn thick disk, is actually floating on gas bearings to maintain axial and radial balance. It was designed to run in open circuit to ensure proper operation and control of each individual subsystem (turbine, pump, bearings) in order to mitigate the risks of the project. Furthermore, air was used instead of superheated steam to test the turbine and the bearings. The turbine was tested extensively, showing good results up to speeds of 340 krpm, providing an estimated 0.4 W of mechanical power, with a pressure ratio of 1.3. The design speed of 1.2 Mrpm was never reached due to system failure at around 350 krpm.

Stator Rotor S e ai Center Pressure

Blades Blades j , 0| e taps

i A - B stack _ ^ ; | j . i - - ' ' K A -J il ! . .." "I.... i n ."L.-1 ,_ i Journal bearing —i

i h

IJlMPKSllXIIiB Pump Pump guide

Thrust Journal bearing bearing plenum

Figure 1-2: MICROS' first generation prototype

Directly related to the current work, Lee (Lee and Frechette, 2005) developed a correlation based on CFD (Computational Fluid Dynamics) of two blade geometries with different camber describing the total losses cotot (profile and fully mixed losses) for a given geometry

which is of the form:

co.r — _ _ !

c,

c,

_(1-1)

Here, C^ and C2 are dependent on blade geometry, mostly camber since solidity was kept constant at 2. This form of the equation was chosen in order to include two phenomena in one formula, general airfoil drag, scaling with the squared root of the Reynolds number, and the viscous flow pressure loss, scaling with the Reynolds number. Aside from Reynolds number and approximate geometric correlations for one aerodynamic profile, the model neglected the following:

1. Mach number effect; 2. Mixing process; 3. Incidence angle;

4. Wide geometric understanding and correlations;

It should also be noted that deviation was neglected because the exit flow angle was judged to be quite constant over a wide range of Reynolds numbers and that thermal transfer was not studied while designing the cold turbine.

To account for 3D effects, Lee found good agreement between his model and experimental results by multiplying the loss coefficient by a factor of 1.40. He also added a blockage coefficient based on Mehra's work at MIT (Mehra, 1997).

1.3.3 Work done at MIT

MIT was the first place to develop and test microturbomachinery (Epstein et al., 1997). Extensive work has been done on gas bearings, materials and microfabrication techniques. The aerodynamics of the turbine were not comprehensively investigated, limiting to flow separation assessment and losses magnitude analysis for specific designs.

Mehra (Mehra, 1997) used in-house CFD codes solving the Navier-Stokes equations to analyze the flow of a particular microturbine design in both 2D and 3D. The favorable pressure gradient in turbine cascade suggested relatively 2D flow field and thin endwall boundary layers compared to compressor analysis.

He concluded that a 2D design philosophy based on numerical tools is adequate for the design of the microturbine components, as long as the designer scales the efficiency to account for the additional losses generated by the endwall boundary layers. He found that 3D losses were about twice the 2D losses, resulting in a ~5% drop in mass flow.

His turbine cascade CFD analysis was done on a specific geometry range and at higher Reynolds numbers than MICROS (5000<Re<10 000 instead of 200<Re<1000), shifting the flow regime from laminar to transitional-turbulent. Insight was provided but no actual design tools or correlations were developed. General scaling laws in microturbomachinery were also

provided (Jacobson, 1998). 1.3.4 Continuity assessment

At very small scale, the molecular mean free path (A) can be of comparable size with the characteristic channel width (I), and deviation from the continuum laws can be observed. The Knudsen number, defined as

Kn = — (1-2)

/

The smaller the Knudsen number is, generally Kn«0.1, the closer to continuum. As the Knudsen number gets larger, the behavior goes through slip-flow (0<Kn<0.1), transition (0.1<Kn<10) and ballistic (Kn> 10) (Hadjiconstantinou and Simek, 2002).

The molecular mean free path is:

•3 kBT

X=

^F¥

(1

-

3)

where d is the molecule diameter (3.5E-10m) and kB is Boltzmann's constant (1.38E-23J/K). An evaluation of the Knudsen number, using realistic flow properties and blade channel width gives:

(l.38.10-

23

.//

J

K)(500£)

Kn « i '- Y = 0.0126

nj2{l00000Pa)(3.5'\0-

l0

m) (lO^m)

That means the flow exhibits similar behavior to macro scale flow in heat transfer and allows the use of standard Navier-Stokes CFD solver (Hadjiconstantinou and Simek, 2002).

1.4 Overview of the project

Most of the work so far was performed using CFD to characterize individual turbine designs. Furthermore, the MICROS turbine operates at Reynolds numbers much lower than what has been investigated by other groups. Moreover, larger scale turbines have almost negligible heat transfer compared to the energy transport, while laminar microscale turbines have considerable heat transfer due to lower mass flow and to the layout, having the cold (compressor) and hot (turbine) parts separated by less than 400pm on the same disk. Hence, very limited knowledge, correlations and tools are available for quick and accurate preliminary design of multistage laminar turbines.

In order to reduce considerably the design iterations and computational time, a broader understanding of the aerothermodynamics of laminar blade cascades is necessary. This project aims at providing such groundwork. The project aims at understanding the effect of both the geometry and flow conditions (Reynolds and Mach numbers, incidence and rotation) on the aerodynamics (losses, deviation and blockage) and thermal transfer of laminar cascades. The approach consists of performing 2D CFD calculations, using a commercial code (FLUENT), to get extensive data, allowing multiple geometries and flow conditions to be tested against each other due to low computational cost of laminar 2D CFD. Development of correlations for mean-line analysis is done and their respective impact is analyzed. These

correlations are implemented in a mean-line analysis software developed by the author and compared to the experimental results of the first generation turbine. Design guidelines and methodology are presented. Finally, these guidelines are used to obtain the geometry for a second generation, hot turbine, prototype for MICROS as well as a multispool configuration for a future and complete device.

1.5 Organization of the thesis

In the next chapter, the analytical model selected for multistage microturbomachinery is presented. Starting from perfect isentropic compressible flow, the equations of conservation, velocity triangles and power production are stated. Then, efficiency definition and aerodynamic phenomena are presented and inserted in the governing equations to represent the real turbine.

Chapter 3 covers the numerical analysis. First, the fluidic and geometric domains of investigation are defined. Then, the preprocessing procedure (meshing, boundary conditions and physical model) is presented. The processing methodology which explains how to vary each parameter individually is stated. The chapter concludes by the post-processing procedure, explaining the averaging method for flow properties and data treatment to get the required parameters.

In Chapter 4, the numerical results are analyzed in detail, providing understanding and correlations for profile and mixing losses, deviation, blockage and thermal transfer. Then, the correlations are adapted to include three dimensional effects, are included in the simulation software and validated by comparing them with experimental data from the first generation prototype.

Finally, Chapter 5 presents the microturbine design, starting from the design constraints and system requirements, and then choosing the number of stages and the detailed geometry. The chapter concludes with the design of two microturbines with different requirements, one for the next generation prototype for MICROS based on specific system-level constraints, a required power level and rotational speed, and one for the future and complete system, based on maximum power and efficiency.

2 MICROTURBOMACHINERY ANALYTICAL MODELING

This chapter presents the governing equations and the analytical model which was used to make turbomachinery calculations. First, a brief description of the flow patterns in a turbine cascade is given, before explaining the selected model and simplifications. Then, the governing equations of the model are stated. Finally, aerodynamic and thermodynamic phenomena in turbomachinery are briefly described and the approach for integrating them into the model is treated.

2.1 Flow modeling considerations in a gas turbine

There are multiple analytical methods in literature, with increasing complexity, to depict the flow in blade cascades and more generally, in the throughflow area of the gas turbine. Figure 2-1 shows the different flow pattern models used by turbine designers to compute flow properties and behavior in a turbine cascade.

Figure 2-1: Analysis methodologies of the flow through turbine cascade

2.1.1 The one dimensional analysis

Even nowadays, one dimensional analysis is widely used in the preliminary design steps due to its great simplicity and its capability to evaluate general turbine performances and to allow quick preliminary dimensioning, which may then require refinements.

Even though the flow possesses a two dimensional velocity vector, radial and tangential, it is considered as a one dimensional model because all the properties of the flow are averaged to a single value for a given position. Hence, in the case of a radial turbine, the flow

properties are considered constant (i.e. averaged) in the spanwise and tangential direction and vary only with the radial position. The value of the selected properties is generally taken at mid-span and the method is often called the mean-line (or pitchline) analysis which is based on velocity diagrams. The turbine analysis using this method is sufficiently accurate when the velocity and pressure distribution are almost constant along the blade span.

2.1.2 The two dimensional analysis

The two dimensional analysis considers the variations of the flow properties in the tangential and perpendicular (axial/radial) directions for a given spanwise (z) position. This analysis allows evaluation of the pressure and velocity distributions by solving the flow field through the cascade.

Various methods are available to solve the problem. They usually involve solving the stream function and potential function simultaneously or solving the boundary layer equations for near-wall treatment and coupling it to inviscid core-flow calculations.

2.1.3 The three dimensional analysis

A partial three dimensional analysis basically integrates a series of two-dimensional results along the blade span. Various methods are used, based on different assumptions about the spanwise velocity and pressure distributions. Amongst them are the radial equilibrium theory, the actuator disk theory and the streamline curvature method.

2.1.4 Experimental and numerical adjustments to the cascade's performance

Even the best analytical model fails to represent the flow accurately and entirely. Historically, experimental data were extensively used to correct the analytical predictions of pressure losses, deviation and, to some extent, thermal transfer and to integrate phenomena almost impossible to evaluate analytically, such as casing and blade tip effects. Nowadays, the increased computational capabilities and accuracy allow the use of extensive CFD by turbine designers in the design process. Three dimensional CFD allows simulation of all the flow features, decreasing the development costs and experimental reliance.

This was a brief overview of turbine flow modeling. All methods share one aspect; the equations of conservation of mass, momentum and energy must be satisfied at every point. More information is available in the literature (Horlock, 1966), (Cumptsy, 2004), (Lakshminarayana, 1996) & (Wilson, 1985).

2.1.5 The selected model

Due to its inherent simplicity, the mean-line analysis is the model selected for microturbine modeling. Even nowadays, preliminary design of turbines and compressors is normally carried out at the mid-span, augmented by experience and correlations (Cumptsy, 2004). 2.2 Governing equations of the mean-line analysis

The following section states the governing equations which are used in modeling the flow in the radial turbine.

2.2.1 Nomenclature

While the full geometric and fluidic nomenclature is available at the beginning of the thesis, clarifications are given here on those necessary for the mean-line analysis. Figure 2-2 shows the important flow and geometric features necessary for a stage analysis. Station 1 is at the stage's inlet while station 2 is the stage's outlet, both represented by averaged flow temperature, pressure and velocity. Station 3 is the mixed condition downstream.

Figure 2-2: Flow and cascade nomenclature

Furthermore, V and a are absolute flow velocities and angles while W and 3 are values in the relative reference frame. Velocities and angles are positive when oriented in the rotating direction.

Figure 2-3 shows the velocity components as well as how to convert velocities from the absolute to relative reference frame. The absolute velocity vector is given by:

v = w

r

+w

g

+u

(2-1)

It can be seen that for a stator, the blade has no velocity and V = W.

e

u

Figure 2-3: Absolute and relative velocity components

2.2.2 Conservation equations in compressible fluid flow

The solution of the flow with the mean-line analysis relies on the simultaneous solution of the conservation of mass, momentum, energy and on the perfect gas law. This section will cover the theory for an isentropic and adiabatic turbine, a turbine having inviscid flow with no losses or heat transfer.

Both the stator and the rotor are subject to a force created by a transfer of momentum due to the flow's deflection. Since the stator is stationary, there is no work done by that force on the stator. From the first law of thermodynamics, there is conservation of energy in the fluid. Thus, there is conservation of the absolute total enthalpy and absolute total pressure in the stator.

A similar principle applies to the axial rotors in the relative reference frame. The relative velocity in the rotor row is deflected by a stationary rotor and the relative total properties are conserved. For a radial rotor, the relative total enthalpy and relative total pressure are not

m = —jJ=J—M'cos(/?) 1 + - (AT) (2-3) conserved because of the difference in wheel speed from the inlet to the outlet. Instead, there

is conservation of what is called rothalpy (Cumptsy, 2004):

u

2

w

2

u

2

I = CT'-— = CT + - — r2-2V

P o 2 . P 2 2 v i)

In the relative reference frame of a rotor, centrifugal fields will do work on the flow. Rothalpy is essentially enthalpy corrected for this known work. Knowing a cascade's geometry (blade span and angles), mass flow and thermodynamic properties, one can solve at any point in the turbine what is known in compressible flow as the mass flow function (Shapiro, 1953):

J\/l ' nnc I l-i I I J - 1 I l\/l ' I

Here, the apostrophes are used as a superscript when referring to relative properties. Once again, these are equal to their absolute counterpart for the stator. R is the gas constant and y is the ratio of the gas specific heats.

Iterations are necessary until the adequate Mach number is found. For unchoked flow, there are two possible solutions to the equation; a subsonic (M<1) and a supersonic (M>1). In the current thesis, design is limited to subsonic flow and it holds on the designer alone to make sure it doesn't go supersonic.

Solving the Mach number, one can easily find the fluid's velocity magnitude by:

W = M'4rRT (2-4)

While the velocity components are:

W

e

=Wsm(/3)

(2

.

5)

W,=W com (2-6)

Static fluid properties are the same whether in the relative of absolute reference frame. For the total properties, the following relations are used:

T

0

' = T 1 + ^ - H M ' )

2 V 2 (2-7)

P

0

' = P\\ + ty-(Mf

y y

V-i fr„'V-i

For the rotor, the use of Eq. (2-1) to find the absolute velocity and then of Eq. (2-4) to find the Mach number allows to get the absolute total properties.

2.2.3 Power production and flow turning

Now that all of the flow's properties can be derived from the previous equations, the power output of a stage can be determined.

The conservation of energy, when applied to a flow process, can be expressed as:

Q-W^mi^-h^) (2_9)

This relation is known as the steady-flow energy equation (SFEE). For an adiabatic turbine

Q = 0 and Eq. (2-9) becomes:

W = m(h0l-h02) (2_10)

The power in rotating machinery is equal to:

Power = Rotating speed x Torque (2-11) The torque being the product of the force and its moment arm and using the conservation of

momentum to derive the force produced, Eq. (2-11) becomes:

W = nm(r2Vff2 -riVei) = m{U2Vei-UxV9l) ( 2.1 2 )

Equation (2-12) is known as Euler's turbine equation and links momentum transfer and velocity triangles to the power and the enthalpy of the fluid. More detailed developments are available in aforementioned literature.

2.2.4 Isentropic efficiency

An isentropic turbine with no heat transfer will have an isentropic efficiency of 100%. That means that the pressure drop solely comes from the work done by the turbine. In a real turbine, there are various loss mechanisms which decrease its efficiency. The isentropic efficiency is defined as the actual work from the turbine compared to the ideal work for an identical pressure drop. From Eq. (2-8) & (2-10), the isentropic efficiency of a stage, or of the whole turbine, is given by:

Vis =

(

r

p

\Wr\

r0out 1 -• p V 0/n J (2-13)

For thrust providing engines, the total-to-static efficiency is sometimes used, which replaces the outlet total pressure by the static pressure. In the case of a power plant, like the one used in the context of a Rankine cycle, the exhaust kinetic energy is not wasted and the total-to-total efficiency is the most relevant definition.

2.3 Aerodynamic and thermodynamic phenomena in a gas turbine

The previous section assumed inviscid flow with no heat transfer in a cascade where the flow follows the blade path. This section will briefly describe the various differences between that idealized cascade and the actual case.

Starting from the mass flow function:

m = -fJ=.-L- M' cos (/?) 1 + - ( M ' ) (2-3)

it can be seen that for a given mass flow, one needs the following information to find the Mach number of the fluid: the passage area, the fluid's total pressure, total temperature and the flow angle. For an ideal cascade, and especially for a stator, it was shown that the solution is fairly easy to find. The total pressure and temperature remain constant; the area can be computed from the blade's height and the radial position using Eq. (2-14) and the flow angle is given by the blade's angle.

A = 2nrh (2.i4)

In a real cascade however, none of the above assumptions are exact. 2.3.1 Total pressure variation

The total pressure in a stationary cascade is not constant. There are various sources of losses in turbomachinery (Lakshminarayana, 1996):

Profile losses: Caused by the boundary layers, flow separation, flow

nonuniformity, wake mixing and dissipation.

Shock losses: Caused by the viscous dissipation across the shock.

Clearance losses: Caused by leakage flow in the tip clearance and its mixing with the

main flow.

Secondary flow losses: Caused by the mixing and dissipation of secondary flow caused by

temperature and pressure gradients in the spanwise direction.

Endwall losses: Losses due to the casing and the hub-wall boundary layers.

It is usually very hard to distinguish between the various sources of losses. Endwall and secondary flow losses are strongly dependant due to the three dimensional nature of both, leakage flow contributes to a large extent to the secondary flow pattern and mixing, while shock and profile losses both occur in the same area. Nevertheless, they all cause the total pressure and the efficiency of the turbine to decrease.

2.3.2 Loss simplifications

Amongst the various loss mechanisms, three dimensional phenomena are not investigated in the present work. Profile and mixing losses due to drag and mixing of the wake are both investigated separately and are defined as:

P -P ' ,„ _ * 02« f 02 03Pro' p ,_p (2-15) •r02 r2 P '-P ' . _ •'02 J0 3 <*>mix ~ p ,_ p (2-16) •"02 r2

For axial configuration, P02is' can be replaced by P0i' in Eq. (2-15). Use of the compressible dynamic pressure is cpmmon to define losses in turbomachinery, while the choice of the cascade's outlet instead of inlet's dynamic pressure stands on the fact that it is the most relevant in a turbine cascade, since it usually is the greater.

Since the design focuses on subsonic flow, the shock losses are completely neglected. Due to the spanwise uniformity assumption of the flow, secondary losses are neglected as well.

Inefficiency due to tip leakage is not considered as pressure losses but as reduced work by the turbine and will be treated in Chapter 4.

Finally, the endwall losses are a three dimensional phenomenon, hence won't be investigated either. Even though they need to be considered, it was shown that they can increase the profile losses by 50-125% (Mehra, 1997) & (Lee and Frechette, 2005). A simple analytical correlation is applied to the 2D profile and mixing losses, which multiplies the 2D losses by the ratio of total area (3D) to the blade area (2D). More details are provided in Chapter 4. 2.3.3 Total temperature variation

In large gas turbines, such as in jet engines and power plants, the heat transfer is usually very low compared to the total energy carried by the flow and they are usually considered adiabatic. Sometimes, secondary mass flow used for blade cooling is added to the main flow, but then again, the total temperature of the flow will only slightly decrease.

In the configuration and flow regime studied here, there is important thermal transfer between the flow and the blades, making the total temperature vary greatly in a blade row. Equation (2-9) must be used instead of Eq. (2-10), and evaluation of the heat transfer Q will therefore be done in Chapter 4 and considered in Chapter 5 during design.

2.3.4 Flow angle variation

The flow does not always have the necessary loading needed to follow the camber line and to balance the centrifugal effects. Furthermore, viscous effects increase the deviation of the flow (Cumptsy, 2004). That means that the flow angle, /?, is actually different from the blade angle, % • Tn© deviation is defined as:

$ = P2-Xi ' (2-17)

2.3.5 Passage area variation

One of the most critical viscous effects in turbomachinery is blockage which is created by the growth of the boundary layers. The displacement thickness diminishes the "effective" area. The blockage is defined as:

K - (effective flow area)/(geometric flow area) (2-18)

And the correct area to be used in the mass flow function is:

A* = KA (2-19)

So the mass flow function becomes:

^^K^cos{p)[\

+r

-^{Mi

VvV*

_{\ ^ J}

(2-20)

2.4 Chapter's summary

The present chapter presented the basic flow models used in conventional turbomachinery design. The governing equations for compressible flow and for the mean-line analysis were given. It was seen how momentum is transferred from the flow to the rotor and how power is generated. Finally, phenomena driving the turbine away from the idealized case were briefly explained as well as how to account for them. Extended developments on every one of them will be done in Chapter 4. This development was done to present the physics and the simplifications integrated within the turbine simulation and design software which will be used through this thesis.

3 NUMERICAL ANALYSIS OF LAMINAR CASCADES

In order to use the mean-line analysis, one needs to know how the airfoils will behave in laminar flow. Numerical analysis using computational fluid dynamics (CFD) was done to capture the impact of the various geometric and fluidic parameters on the following phenomena:

• Aerodynamic losses; • Deviation;

• Blockage; • Thermal transfer.

The numerical analysis was performed using FLUENT V6.1, while the GAMBIT module provided with FLUENT was chosen to create the meshes. Details of preprocessing, processing and postprocessing methodology follow in the next sections. FLUENT was chosen because of its proven reliability when simulating cascade flows (Papa et al., 2007) and because it was and is still the program used by the group MICROS to perform CFD (Lee etal.,2004).

3.1 Numerical analysis objectives

Simply put, the main objective of the CFD analysis is to answer the following question:

"What should the bladings of a subsonic laminar cascade look like and how will they react under given flow conditions?"

In order to answer that question, the impacts of the following parameters were investigated: Airfoil aeometrv

Solidity (cr) Camber (9) Stagger (£)

Maximum thickness (t/c)max Thickness distribution Mean-line distribution

Flow conditions Reynolds number (Rec)

Mach number (M) Incidence (i) Rossby number (Ro) Stanton number (St)

Three dimensional effects are not investigated and thus three dimensional CFD was not undertaken since it is very time consuming and is not primordial to the understanding of the main geometric and flow features which impact on performances.

3.2 Numerical analysis domain

3.2.1 Geometric domain

The analysis needed to cover a wide range of cascade geometries in order to develop correlations based on an extensive data base, allowing for sufficient freedom during turbine design. Table 3-1 shows the various geometric parameters which were tested.

Table 3-1: Geometric domain of numerical analysis

e

75 100 130 145 a 1.0 1.5 2.0 2.5

5

10 20 30 40 (t/c)max 10% 15% 20% Thickness distributions A3K7 C4 NACA747A015 Mean-line distributions A3K7 NACA 65

The choice of the blade cambers comes from typical turning of turbine blades, while the choice of the solidities covers the usual turbomachinery range, which is typically between 1.0 and 2.0. The variation of stagger angle for a given camber allows varying the cascade's inlet and outlet angles. The chosen thickness and mean-line distributions offer good flow behavior in conventional turbomachinery (Horlock, 1966) & (Cumptsy, 2004).

Testing all the combinations possible would create 1152 geometries. Instead, 24 different geometries were tested which could isolate the effect of each parameter. All the blade geometries are available in Appendix B while information about the aerodynamic profiles can be found in Appendix C.

3.2.2 Fluidic domain

The study concerns subsonic laminar flows (M<1). Transition of the flow was shown to occur at a Reynolds number between 1800 and 2000 (Agostini et al., 2004). Varying the blade chord and the flow velocity allows variations to the Reynolds and Mach numbers.

Incidence at the design point is usually 0° for a well designed turbine, while it can vary during off-design operation by approximately ±15°. Changing the incoming flow angle for a given cascade allows variation of the incidence.

The Rossby number,

Ro = -

w.

ref

w.

ref _(3-1)

Qr U

compares the relative flow inertia to the Coriolis force and usually stands between 0,1 and 1. It is varied by increasing the blade speed.

The Stanton number, a ratio of the heat transfer rate at the wall to that advected by the flow,

k

eff

(d(T-TJ)

w

St =

-is varied by changing the blade's wall temperature.

(3-2)

Table 3-2 shows the various tests that were done on each blade geometry in order to understand the effect of the Mach number, Reynolds number, incidence, Stanton number and Rossby number on losses, deviation, thermal transfer and blockage.

Table 3-2: Fluidic domain of numerical analysis Viscous and compressibility effects

Settings: Stationnary blades Parameters Reynolds Mach Range 100-1500 0.1-1.0 Incidence effect

Settings: Stationnary blades Parameters Incidence Reynolds Mach Range -10" to 15° 300 & 800 0.3 & 0.9 Rotation effect Settings: 0 incidence Parameters Rotating speed Reynolds Mach Range 0 - 2.4 MRPM 300 & 800 0.3 & 0.9 Thermal transfer

Settings: Stationnary blades Parameters Wall temperature Reynolds Mach Range 400 K- 600 K 300 & 800 0.3 & 0.9 Adiabatic blades Step -150 - 0 . 2 Subtotal Adiabatic blades Step 5° 500 0.6 Subtotal Adiabatic blades Step 400 MRPM 500 0.6 Subtotal 0 incidence Step 50K 500 0.6 Subtotal Total 0 incidence Number of tests 9 5 45 Number of tests 6 2 2 24 Number of tests 6 2 2 24 Number of tests 5 2 2 20 113

These 113 tests, applied to 24 different meshes, give over 2700 CFD results. Furthermore, 4 meshes were chosen in order to test the incidence effect on thermal transfer. Rotation effects on thermal transfer were not investigated, and experience showed it was judicious, as discussed later in section 4.2.

3.3 Preprocessing

This section briefly covers the meshing and solver setting steps of the CFD process. A detailed description of those steps is available in Appendix D.

GAMBIT was used for creating the meshes. The blade's geometry was specified by importing profile points (~70) and linking them together using splines. The computational domain, a single blade passage with periodic boundaries to recreate blade interaction, extends from 75% radial chord upstream to 125% radial chord downstream. The refined meshes have between 16 000 and 25 000 triangular elements, depending on the blade angles and solidity. To increase accuracy boundary adaptation on the first 10 cell rows of each wall is done (Figure 3-1). This increases the thermal transfer accuracy at low computational cost (-20% more cells).

(a) Before adaptation (b) After adaptation

Figure 3-1: Boundary adaptation around a blade wall

The settings of the physics done in FLUENT, once importing the mesh, are also available in Appendix D.

3.4 Processing

Journals were made for automatic processing of the 2700+ cases computed. Choosing a pressure relaxation factor of 0.15 for the first 200 iterations and then increasing it to 0.3 greatly helps convergence and stability. The following settings and boundary conditions were used for different points of the analysis described in Table 3-2.

3.4.1 Viscous and Compressible Effects

The walls velocities are set to zero relative to the reference plane (stationary walls) and thermal transfer is set to zero Watts. The pressure outlet is set to zero relative pressure. The velocity vectors at the pressure inlet are set to zero incidence and the fluid's temperature to 600 K. Finally, the Reynolds and Mach numbers are varied independently by varying the inlet total pressure and the scale of the model.

3.4.2 Incidence Effects

Cases from the previous step are taken and the velocity vectors are simply varied to change the flow incidence.

3.4.3 Rotation Effects

Cases from the first step are taken. The fluid's motion type is set to moving reference frame, with rotating speed which progressively increased. Hence, for a given relative flow speed, the Rossby number varies.

3.4.4 Thermal Transfer

Cases from the first step are taken. The wall thermal formulation is changed to constant wall temperature and is varied from 600 K to 400 K to increase thermal transfer. The assumption of constant wall temperature comes from the Biot number.

Bi = Ji-£- (3-3)

K

The characteristic length to be taken is the one most likely to be subject to high temperature gradients (Incropera and Dewitt, 2001). In turbine blading without internal cooling or bleed flow, the greatest temperature gradient will occur along the blade span, from the root to the tip. Using the silicon's conductivity and a conservative convection coefficient:

Bi*2500(W/m2 • K) 1 0 0 e~6 w =0 . 0 0 0 1 7 « 0 . 1

Since the Biot number is much smaller than unity, almost no temperature gradient is found along the blade, thus justifying the assumption of constant wall temperature over the whole blade. Even with great improvements in material properties, several orders of magnitude are necessary to invalidate this assumption.