APPLICABILITY OF SPECIFIC SPEED AND ZWEIFEL COEFFICIENT
RECOMMENDATIONS TO LOW HEAD AXIAL HYDRAULIC TURBINES
By
Zachary Hoyle

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of
Mechanical Engineering â Master of Science
2017

ABSTRACT
APPLICABILITY OF SPECIFIC SPEED AND ZWEIFEL COEFFICIENT
RECOMMENDATIONS TO LOW HEAD AXIAL HYDRAULIC TURBINES
By
Zachary Hoyle
This thesis work investigates the applicability of literature recommendations regarding
specific speed and Zweifel coefficient for low head axial hydraulic turbines with constant blade
thickness circular arc blade profiles, and no inlet guide vane. Highest efficiency was observed in
Computational Fluid Dynamics (CFD) investigations for the designs running at speeds 12-33%
that of the speed recommended by the Cordier line. Designs that followed the Zweifel coefficient
recommendations of the literature in range of 0.8-1.1 demonstrated highest efficiency in CFD
investigations. To test the applicability of the Cordier line recommendations for machines of this
type, designs at differing specifications were tested at six different rotational speeds. Peak
efficiency was measured at specific speeds between 1.4 and 1.9, compared to the Cordierrecommended value of 5.5 to 6.1. Next designs with differing values of Zweifel coefficient were
simulated at the rotational speed at which highest efficiency was measured for each set of
specifications. Zweifel coefficient was altered by changing the axial blade length. It was found
that highest efficiency was measured at Zweifel coefficients between 0.8 and 1.1. The designs
with higher axial blade lengths had lower Zweifel coefficient, and experienced greater friction
losses. Decreasing blade length and increased Zweifel coefficients were correlated with more
severe velocity gradients at the leading edge, increased flow deviation and areas of low pressure
where cavitation could occur. Designs with the shortest blades (Zwefiel coefficient above 1.2)
experienced greater velocity gradients and turbulence at the trailing edge and decreased
efficiency.
ii

ACKNOWLEDGEMENTS

I greatly appreciate the support of those who have helped me through the Mechanical
Engineering Masterâs program at Michigan State University. First, I thank my family for the love
and support they have always provided, my father Randy, my mother Jill, my brother Randall,
and my sister Jacqui. Iâd like to thank my committee, Dr. Xinran Xiao, Dr. Abraham Engeda,
and Dr. Norbert Mueller for their assistance, patience, and help to complete this work. A special
thank you to Dr. Norbert Mueller, who for the past 5 years has provided enormously valuable
advice and opportunities which have helped me gain experience in engineering and
turbomachinery. All of the experience which I now have on my resume is the result of
opportunities which Dr. Mueller has provided. I would like to thank my friends and colleagues
who have provided very valuable guidance which helped me complete this thesis, in particular
Blake Gower and Thomas Qualman. Thank you to Tim Hinds and Andrew Mccolm for
providing the financial support which has allowed me to complete the program.

iii

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... vi
LIST OF FIGURES ...................................................................................................................... vii
KEY TO SYMBOLS ..................................................................................................................... xi
CHAPTER 1. INTRODUCTION AND BACKGROUND ............................................................ 1
1.1. Introduction to Hydroelectric Power .................................................................................... 1
1.2. Hydroelectric Power Challenges .......................................................................................... 2
1.3. Hydroelectric Dam Retrofits ................................................................................................ 3
1.4. Selection of Type of Water Turbine ..................................................................................... 6
1.5. Woven Wheel Hydro Turbine .............................................................................................. 7
1.5.1. Lightweight, Modular Design........................................................................................ 9
1.5.2. Low-cost Additive Manufacturing ................................................................................ 9
1.5.3. Rapid Customization and Manufacturing .................................................................... 10
1.5.4. Outer Shroud Eliminates Tip Leakage and Reduces Fish Mortality ........................... 11
1.6. Definition of Turbine Type Investigated ............................................................................ 12
1.7. Conservation of Energy...................................................................................................... 13
1.8. Definitions of Geometry and Design Parameters ............................................................... 17
1.9. Turbine Cascade Forces ..................................................................................................... 25
1.10. Dimensionless Coefficients .............................................................................................. 28
1.10.1. Flow and Capacity Coefficients ................................................................................ 28
1.10.2. Blade Loading and Head Coefficients ....................................................................... 28
1.10.3. Specific Speed ........................................................................................................... 29
1.10.4. Specific Diameter ...................................................................................................... 30
1.11. Cordier Diagram and Line................................................................................................ 30
1.11.1. Effect of Flow, Loading Coefficients on Cordier Diagram Position ......................... 33
1.11.2. Using Cordier Diagram as a Design Tool ................................................................. 35
1.11.3. Operating below the Cordier Line ............................................................................. 36
1.12. Literature Recommendations on Blade Loading and Flow Coefficients ......................... 37
1.13. Cost Advantages of Designs With High Flow and High Loading Coefficients .............. 39
1.14. Zweifel Lift Coefficient ................................................................................................... 40
1.14.1. Recommended Values of Zweifel Coefficient .......................................................... 45
1.14.2. Industry Trend - Higher Zweifel Coefficients ........................................................... 45
1.14.3. Zweifel Analysis ........................................................................................................ 47
1.15. Cavitation ......................................................................................................................... 49
1.15.1. Cavitation for Composites ......................................................................................... 49
1.16. Discussion of Loss Mechanisms ...................................................................................... 50
1.17. Goals of This Thesis and Description of Work ................................................................ 52
CHAPTER 2. ANALYSIS DESCRIPTION AND METHODS................................................... 55
2.1. Description of Setup ........................................................................................................... 55
2.2. Analysis Description, Description of Specifications.......................................................... 56
2.3. Estimation of Friction Losses ............................................................................................. 57
iv

2.4. CFD Methodology and Assumptions ................................................................................. 64
2.4.1. CFD Overview ............................................................................................................. 64
2.4.2. CFD Methodology and Assumptions .......................................................................... 67
2.5. Description and Inputs for Speed Study............................................................................. 69
2.6. Description and Inputs for Blade Length Study ................................................................. 76
2.7. Mesh Independence Study ................................................................................................. 80
CHAPTER 3. CFD RESULTS AND DISCUSSION ................................................................... 84
3.1. Speed Study CFD Results .................................................................................................. 84
3.2. Discussion of Cordier Recommendations and Location on Diagram ................................ 86
3.3. Discussion of Speed Study Results .................................................................................... 89
3.3.1. Overall Summary......................................................................................................... 89
3.3.2. Flow Visualization and Vectors .................................................................................. 91
3.3.3 Trailing Edge Effects .................................................................................................... 95
3.3.4. Blade Loading.............................................................................................................. 97
3.3.5. Hub and Shroud Effects ............................................................................................. 103
3.3.6. Work Extraction ........................................................................................................ 107
3.3.7. Cavitation................................................................................................................... 110
3.4. Second Study: Blade Length Study.................................................................................. 113
3.5. Blade Length Study Results Discussion........................................................................... 115
3.5.1. Overall Summary....................................................................................................... 115
3.5.2. Flow Visualization and Vectors ................................................................................ 118
3.5.3. Trailing Edge Effects ................................................................................................. 123
3.5.4. Blade Loading............................................................................................................ 126
3.5.5. Hub and Shroud Effects ............................................................................................. 131
3.5.6. Work Extraction ........................................................................................................ 134
3.5.7. Cavitation .................................................................................................................. 139
CHAPTER 4. CONCLUSIONS ................................................................................................. 142
4.1. Speed Study and Cordier Line ......................................................................................... 142
4.2. Blade Length Study and Zweifel Coefficient ................................................................... 143
4.3. Next Steps of the Project .................................................................................................. 144
REFERENCES ........................................................................................................................... 147

v

LIST OF TABLES

Table 2.1. Case 1 Geometric Parameters and Flow Angles .......................................................... 72
Table 2.2. Case 1 Speed Study Inputs........................................................................................... 74
Table 2.3. Case 2 Speed Study Inputs........................................................................................... 75
Table 2.4. Case 3 Speed Study Inputs........................................................................................... 75
Table 2.5. Case 4 Speed Study Inputs........................................................................................... 75
Table 2.6. Case 5 Speed Study Inputs........................................................................................... 75
Table 2.7. Case 6 Speed Study Inputs........................................................................................... 76
Table 2.8. Blade Length Study Runs and Inputs .......................................................................... 80
Table 3.1. Case 1 Speed Study CFD Results ................................................................................ 84
Table 3.2. Case 2 Speed Study CFD Results ................................................................................ 84
Table 3.3. Case 3 Speed Study CFD Results ................................................................................ 85
Table 3.4. Case 4 Speed Study CFD Results ................................................................................ 85
Table 3.5. Case 5 Speed Study CFD Results ................................................................................ 85
Table 3.6. Case 6 Speed Study CFD Results ................................................................................ 86
Table 3.7. Performance of Cordier Recommendations of Cases in Speed Study ......................... 88
Table 3.8. Case 1 Blade Length Study Results ........................................................................... 113
Table 3.9. Case 2 Blade Length Study Results ........................................................................... 113
Table 3.10. Case 3 Blade Length Study Results ......................................................................... 114
Table 3.11. Case 4 Blade Length Study Results ......................................................................... 114
Table 3.12. Case 5 Blade Length Study Results ......................................................................... 115
Table 3.13. Case 6 Blade Length Study Results ......................................................................... 115

vi

LIST OF FIGURES

Figure 1.1. Hydroelectric Dam Schematic ...................................................................................... 2
Figure 1.2. Bellaire Dam, Antrim County, Michigan ..................................................................... 5
Figure 1.3. Spillway at the Bellaire Dam, Antrim County, Michigan ............................................ 5
Figure 1.4. Hydroelectric dam retrofit schematic ........................................................................... 6
Figure 1.5. (a) 3-D model of Woven Wheel winding scheme (b) Woven Wheel in mandrel (c)
Woven wheel removed from mandrel............................................................................................. 8
Figure 1.6. Woven Wheel tidal turbine being tested inside the tow tank ....................................... 8
Figure 1.7. Modular Woven Wheel compressor unit design .......................................................... 9
Figure 1.8. (a) 3-D printed mandrel. (b) carbon fiber winding. (c) mandrel post-winding after
being dipped into epoxy/resin and cured. (d) Woven Wheel after removal from mandrel. ......... 10
Figure 1.9. Turbomachine control volume ................................................................................... 14
Figure 1.10. Example velocity triangle ......................................................................................... 17
Figure 1.11. Axial turbine geometry parameters .......................................................................... 19
Figure 1.12. Turbine blade cascade with velocity triangles.......................................................... 22
Figure 1.13. Pressure distribution across a turbine stage .............................................................. 25
Figure 1.14. Example turbine blade cascade with blade forces .................................................... 26
Figure 1.15. Example efficiency curve for turbine for specific speed đđ  [15] ............................ 30
Figure 1.16. đˇđ  â đşđ  diagram with Cordier line and typical machine type ............................... 32
Figure 1.17. đˇđ  â đşđ  diagram with Voith Hydro turbines with Cordier line ............................. 33
Figure 1.18. đˇđ  â đşđ  diagram with Cordier line, lines of constant flow coefficient đ ............... 34
Figure 1.19 đˇđ  â đşđ  diagram with Cordier line, lines of constant blade loading coefficient đ .. 34
Figure 1.20. Smith loading vs. Flow coefficient diagram [22] ..................................................... 37
Figure 1.21. Blade loading and flow coefficients for different types of hydraulic turbines [15] . 39
Figure 1.22. Profile loss coefficient vs. s/b ratio resulting from separation and friction [14] ...... 41
Figure 1.23. Idealized pressure distribution with actual pressure distribution for Zweifel
coefficient calculation ................................................................................................................... 43
Figure 1.24. Blade length vs. Zweifel coefficient ......................................................................... 47
Figure 1.25. Number of blades vs. Zweifel coefficient ................................................................ 48
Figure 1.26. Tip speed vs. Zweifel coefficient at differing blade lengths .................................... 48
Figure 1.27. Trailing edge of blade with separated boundary layer [32] ...................................... 50
Figure 2.1. Hydroelectric dam retrofit schematic with height positions....................................... 55
Figure 2.2. Design 1j5 generated in BladeGen ............................................................................. 67
Figure 2.3. Example turbine geometry with entry and exit annuli ............................................... 67
Figure 2.4. Process flow chart for Speed Study ............................................................................ 71
Figure 2.5. Velocity triangles for leading and trailing edges of design 1j of the Speed Study (50%
span) .............................................................................................................................................. 73
Figure 2.6. Velocity triangles for leading and trailing edges of design 1CORD of the Speed
Study (50% span) .......................................................................................................................... 74
Figure 2.7. Blade Length Study process flow chart ...................................................................... 77
Figure 2.8. Velocity triangles for leading and trailing edges of design 2l of the Blade Length
Study (50% span) .......................................................................................................................... 78
Figure 2.9. Velocity triangles for leading and trailing edges of design 2l8 of the Blade Length
Study (50% span) .......................................................................................................................... 79
vii

Figure 2.10. Mesh cell count vs. Power for case 1 mesh Study ................................................... 81
Figure 2.11. Mesh cell count vs. Power for case 2 mesh Study ................................................... 81
Figure 2.12. Mesh cell count vs. Power for case 3 mesh Study ................................................... 82
Figure 2.13. Mesh cell count vs. Power for case 4 mesh Study ................................................... 82
Figure 2.14. Mesh cell count vs. Power for case 5 mesh Study ................................................... 82
Figure 2.15. Mesh cell count vs. Power for case 6 mesh Study ................................................... 83
Figure 3.1 Specific speed đşđ  Vs. Efficiency of Speed Study designs ......................................... 87
Figure 3.2. Location of peak efficiency points for each case on đşđ vs đˇđ  diagram with Cordier
line................................................................................................................................................. 88
Figure 3.3 Flow coefficient vs. Efficiency of Speed Study designs ............................................. 90
Figure 3.4. Blade loading coefficient vs. Efficiency of Speed Study designs .............................. 90
Figure 3.5. Relative velocity vectors for case 1: Span 50% ......................................................... 91
Figure 3.6. Relative velocity vectors for case 2: Span 50% ......................................................... 91
Figure 3.7. Relative velocity vectors for case 3: Span 50% ......................................................... 92
Figure 3.8. Relative velocity vectors for case 4: Span 50% ......................................................... 92
Figure 3.9. Relative velocity vectors for case 5: Span 50% ......................................................... 92
Figure 3.10. Relative velocity vectors for case 6: Span 50% ....................................................... 93
Figure 3.11. Relative velocity vectors at leading edge for case 2: 50% span ............................... 93
Figure 3.12. Relative velocity vectors at leading edge for case 1: Span 50% .............................. 94
Figure 3.13. Relative velocity vectors at leading edge for case 3: Span 50% .............................. 94
Figure 3.14. Relative velocity vectors at trailing edge for case 2: 50% span ............................... 94
Figure 3.15. Relative velocity vectors at trailing edge for case 1: 50% span ............................... 95
Figure 3.16. Relative velocity vectors at trailing edge for case 3: Span 50% .............................. 95
Figure 3.17. Turbulence kinetic energy contours for case 3: Span 50% ...................................... 96
Figure 3.18. Turbulence kinetic energy contours for case 1: Span 50% ...................................... 96
Figure 3.19. Relative velocity of case 1 Speed Study designs near the trailing edge: Span 50% 97
Figure 3.20. Blade loading diagrams for Speed Study case 2: Span 50% .................................... 99
Figure 3.21. Blade loading diagrams for Speed Study case 1: Span 50% .................................... 99
Figure 3.22. Static pressure contours for Speed Study case 2: Span 50% .................................. 100
Figure 3.23. Static pressure contours for Speed Study case 1: Span 50% .................................. 100
Figure 3.24. Static pressure contours for Speed Study case 3: Span 50% .................................. 101
Figure 3.25. Velocity triangles with relative velocity contours for design 1CORD of the Speed
Study: (Span 50%) ...................................................................................................................... 102
Figure 3.26. Velocity triangles with relative velocity contours for design 1j of the Speed Study:
(Span 50%).................................................................................................................................. 103
Figure 3.27. Relative velocity of Case 2 Speed Study designs at the trailing edge along span.. 104
Figure 3.28. Relative velocity of Case 1 Speed Study designs at the trailing edge along span.. 104
Figure 3.29. Relative velocity of highest efficiency Speed Study designs at the leading edge
along span ................................................................................................................................... 105
Figure 3.30. Relative velocity of highest efficiency Speed Study designs at the trailing edge
along span ................................................................................................................................... 106
Figure 3.31. Turbulence kinetic energy of highest efficiency Speed Study designs at the trailing
edge along span ........................................................................................................................... 107
Figure 3.32. Absolute circumferential flow velocity for designs in case 2 of Speed Study at the
trailing edge along span .............................................................................................................. 108

viii

Figure 3.33. Absolute Circumferential flow velocity for designs in Case 3 of Speed Study at the
trailing edge along span .............................................................................................................. 108
Figure 3.34. Average absolute circumferential flow velocity from inlet to outlet for designs case
1 of Speed Study ......................................................................................................................... 109
Figure 3.35. Absolute circumferential flow velocity for highest efficiency designs in Speed Study
at the trailing edge along span .................................................................................................... 110
Figure 3.36. Areas of low static pressure for case 1 of Speed Study: 50% Span ....................... 111
Figure 3.37. Areas of low static pressure for case 2 of Speed Study: 50% Span ....................... 111
Figure 3.38. Areas of low static pressure for case 3 of Speed Study: 50% Span ....................... 111
Figure 3.39. Areas of low static pressure for case 4 of Speed Study: 50% Span ....................... 112
Figure 3.40. Areas of low static pressure for case 5 of Speed Study: 50% Span ....................... 112
Figure 3.41. Areas of low static pressure for case 6 of Speed Study: 50% Span ....................... 112
Figure 3.42. Zweifel coefficient vs. Efficiency for Blade Length Study .................................... 116
Figure 3.43. Zweifel Coefficient vs. Flow deviation at trailing edge for Blade Length Study .. 117
Figure 3.44. Difference of the peak efficiency and efficiency of the design vs. the Zweifel
coefficient ................................................................................................................................... 118
Figure 3.45. Relative velocity vectors for case 1 of Blade Length Study: Span 50% ................ 119
Figure 3.46. Relative velocity vectors for case 2 of Blade Length Study: Span 50% ................ 119
Figure 3.47. Relative velocity vectors for case 3 of Blade Length Study: Span 50% ................ 120
Figure 3.48. Relative velocity vectors for case 4 of Blade Length Study: Span 50% ................ 120
Figure 3.39. Relative velocity vectors for case 5 of Blade Length Study: Span 50% ................ 120
Figure 3.40. Relative velocity vectors for case 6 of Blade Length Study: Span 50% ................ 121
Figure 3.41. Relative velocity vectors at leading edge for case 5 of Blade Length Study (Span
50%) ............................................................................................................................................ 121
Figure 3.42. Relative velocity vectors at leading edge for case 3 of Blade Length Study (Span
50%) ............................................................................................................................................ 122
Figure 3.43. Relative velocity vectors at leading edge for case 1 of Blade Length Study (Span
50%) ............................................................................................................................................ 122
Figure 3.44. Relative velocity vectors at trailing edge for case 5 of Blade Length Study (Span
50%) ............................................................................................................................................ 122
Figure 3.45. Relative velocity vectors at trailing edge for case 3 of Blade Length Study (Span
50%) ............................................................................................................................................ 123
Figure 3.46. Relative velocity vectors at trailing edge for case 1 of Blade Length Study (Span
50%) ............................................................................................................................................ 123
Figure 3.47. Relative velocity near trailing edge of case 3 and 5 of Blade Length Study.......... 124
Figure 3.48. Turbulence kinetic energy contour for cases 1j5 and 1j8: Span 50% .................... 125
Figure 3.49. Turbulence kinetic energy contour for designs 3n and 3n6: Span 50% ................. 125
Figure 3.50. Turbulence kinetic energy contour for designs 5n and 5n6: Span 50% ................. 125
Figure 3.51. Zweifel Coefficeint vs Exit Loss Coefficient for Blade Length Study .................. 126
Figure 3.52. Blade loading diagrams for case 3 of Blade Length Study: Span 50% .................. 128
Figure 3.53. Static Pressure Distributions for Case 3 of Blade Length Study (Span 50%) ........ 128
Figure 3.54. Blade loading diagrams for case 5 of Blade Length Study: Span 50% .................. 129
Figure 3.55. Static Pressure Distributions for Case 5 of Blade Length Study (Span 50%) ........ 129
Figure 3.56.Velocity triangles superimposed over velocity vector contours for design 2l of the
Blade Length Study at leading and trailing edge (span 50%) ..................................................... 130

ix

Figure 3.57.Velocity triangles superimposed over velocity vector contours for design 2l8 of the
Blade Length Study at leading and trailing edges (span 50%) ................................................... 131
Figure 3.58. Relative velocity of case 6 for Blade Length Study at the leading edge along span
..................................................................................................................................................... 132
Figure 3.59. Relative velocity of case 6 for Blade Length Study at the trailing edge along span
..................................................................................................................................................... 133
Figure 3.60. Zweifel coefficeint vs Profile loss coefficient for Blade Length Study ................. 134
Figure 3.61. Average relative velocity from inlet to outlet for case 1 for Blade Length Study . 135
Figure 3.62. Average relative velocity from inlet to outlet for case 6 for Blade Length Study . 135
Figure 3.63. Absolute circumferential velocity for case 6 of the Blade Length Study at the
trailing edge along span .............................................................................................................. 136
Figure 3.64. Absolute circumferential velocity for case 1 of the Blade Length Study at the
trailing edge along span .............................................................................................................. 137
Figure 3.65. Average Absolute circumferential velocity from inlet to outlet of case 6 of Blade
Length Study ............................................................................................................................... 138
Figure 3.66. Average Absolute circumferential velocity from inlet to outlet of case 1 of Blade
Length Study ............................................................................................................................... 138
Figure 3.67. đđđśđ˘ for case 1 of the Blade Length Study at the trailing edge across span ........ 139
Figure 3.68. Areas of low static pressure for case 1 of Blade Length Study (Span 50%) .......... 140
Figure 3.69. Areas of low static pressure for case 2 of Blade Length Study (Span 50%) .......... 140
Figure 3.70. Areas of low static pressure for case 3 of Blade Length Study (Span 50%) .......... 140
Figure 3.71. Areas of low static pressure for case 4 of Blade Length Study (Span 50%) .......... 141
Figure 3.72. Areas of low static pressure for case 5 of Blade Length Study (Span 50%) .......... 141
Figure 3.73. Areas of low static pressure for case 6 of Blade Length Study (Span 50%) .......... 141

x

KEY TO SYMBOLS

đ´đ = meridonial cross-sectional area (đ2 )
đź = Absolute flow angle (đđđđđđđ )
đ = Axial blade length (đ)
đ˝ = Relative flow angle(đđđđđđđ )
đ˝đ = Blade angle (đđđđđđđ )
đ˝đ = Mean relative flow angle (đđđđđđđ )
đ

đś = Absolute flow speed ( đ  )
âââ = Absolute flow speed (vector form) (đ)
đś
đ 
đ = Chord length (đ)
đ

đśđ = Absolute flow speed in meridional direction ( đ  )
đ

đśđ˘ = Absolute flow speed in tangential direction ( đ  )
đśđ = Tangential force coefficient
đśđ =Drag coefficient
đśđĽ =Axial force coefficient
đśđ = Lift force coefficient
đˇ = Outside diameter (đ)
đˇđ  = Specific Diameter
đ2

đĚ = Mass-specific shaft work output ( đ 2 )
đ2

đ = Total mass-specific work done on the fluid from leading edge to trailing edge ( đ 2 )
đ = friction factor
xi

đšđĽ = Axial force (
đšđ = Drag force (
đšđ = Lift force (

đđ đ
đ 2

đđ đ
đ 2

đđ đ
đ 2

)

)

)
đ2

đšđ = friction work lost per unit mass ( đ 2 )
đ

đ = gravitational acceleration constant (đ 2 )
âđ = Blade height (đ)
đ2

â = Enthalpy ( đ 2 )
đ2

âđ = Stagnation enthalpy( đ 2 )
đť = total head (đ)
đ â˛ = Inlet incidence (đđđđđđđ )
đ2

đ = Turbulence kinetic energy ( đ 2 )
đž = Friction loss coefficient from 90 degree turn
đżđĄ = length of penstock pipe (đ)
đđ

đĚ =Mass flow rate ( đ  )
đđ

âđĚ = Mass flow rate through a single blade passage ( đ  )
đ = Rotational speed (

đđđĄ
đ 

)

đđ  = Specific Speed (đđđĄ)
đđđđ = rotational speed (đđđĄ/đđđ)
đđ = number of blades
đđ

đ = Static pressure (đđ 2 )
xii

đđ

đđĄ = Total Pressure (đđ 2 )
đđ

đđđĄđ = Atmospheric Pressure (đđ 2 )
2

đ đđ
đĚ = Heat input ( đ 3 )

đđ = Reynolds Number
đâ = Inside (hub) radius (đ)
đđ  = Outside (shroud) radius (đ)
đđ = Distance from centerline in tangential direction (đ)
đđ = Mean radius (đ)
đđ = Radius of the penstock pipe (đ)
đđ

đ = Density (đ3 )
đ  = Blade spacing (đ)
đĄ = Blade thickness (đ)
đ = Local temperature (đśđđđđđ˘đ )
đ

đ˘ = Local flow velocity ( đ  )
đ

đ = Rotor tip speed( đ  )
đ
âââ
đ =Absolute rotor velocity (vector form) ( đ  )
đ

đđ = Absolute speed of rotor at position along span ( )
đ 

3

đ
đĚ = Volume flow rate ( đ  )

đ = Angular velocity (

đđđ
đ 

)
đ

đ = Relative flow speed ( đ  )

xiii

đ
âââââ
đ = Relative flow speed (vector form) ( đ  )
2

đ đđ
đĚ = Shaft power output ( đ 3 )
đ

đđđđđ = Mean relative velocity ( đ  )
đ

đđ = Relative flow speed in meridional direction( đ  )
đ

đđ˘ = Relative flow speed in tangential direction ( đ  )
đ = Tangential blade force (

đđ đ

đđđ = Idealized blade force (

đ 2

)

đđ đ
đ 2

)

đ = Zweifel Coefficient
đ§ = Position above datum (đ)
đ = Loss coefficient
đż â˛ = Outlet deviation (đđđđđđđ )
đđ = surface roughness(đ)
đ = Flow Coefficient
đ = Blade loading coefficient
đâ˛ = Head coefficient
đ â˛ = Capacity Coefficient
đ = Kinematic viscosity
đđĄ = Kinematic turbulence viscosity
đżđđ = Kroneker delta
ÎŠđ  = Specific Speed (đđđ)
đž = Hub-tip ratio

xiv

đ = Isentropic turbine efficiency
đđĄ = Total turbine efficiency
đđ

đđđ â˛ = Reynolds stresses(đđ 2 )
đđ = Time rate of change of momentum about a
đđ

đ = Shear stress(đđ 2 )

Subscripts
đđđĽ = At 100% efficiency, idealized
1 = Turbine leading edge
2 = Turbine trailing edge

a = Position a: water surface/inlet of penstock pipe
b = Position b: outlet of penstock pipe/inlet of turbine entry annulus
1= Position 1: Turbine leading edge
2 = Position 2: Turbine trailing edge
c = Position c: Outlet of turbine exit annulus, water level

đĽ = axial direction
â = hub
s = shroud
i = ideal property (assuming 100% efficiency)
k = index notation
j = index notation
đ = Tangential direction

xv

CHAPTER 1. INTRODUCTION AND BACKGROUND
1.1. Introduction to Hydroelectric Power
Hydroelectricity is known as a renewable, sustainable, clean energy source with large
potential for new development in the United States [1]. Hydroelectric power plants convert the
kinetic and potential energy of water into electric power. To accomplish this, hydroelectric
power plants use a turbine which converts the energy of the water into mechanical work, which
is then transferred to a generator converting that mechanical work into electric power. Hydraulic
(hydro) turbines usually have a stator which directs the water flow, as well as a runner or rotor.
The stator directs the flow in a desired direction, using vanes or a nozzle. The rotor has blades
which change the angular momentum of the flow, exerting a torque on the rotor inducing rotation
[2].
The two main parameters which govern the amount of power which can be generated by
a hydraulic turbine are the total available head đť and the volume flow rate đĚ . Head is defined as
the height difference between two water surfaces. Most hydropower plants use a dam to generate
the head necessary for hydroelectric power generation. These dams hold back water, creating a
reservoir with a high water level on one side of the dam. The potential energy of the water
behind the dam is used to extract energy. Hydroelectric dams often use an intake pipe, called a
penstock, to feed water into the turbine. After the water flows through the turbine, it flows
through an outlet tube called a draft tube which has an expanding cross-section. The draft tube is
designed to decelerate the water. The use of a draft tube also allows for the turbine to be placed
above the lower water level and extract the full potential energy of the water. Therefore the draft
tube ensures that the head of the water below the tail race level [3].

1

An example of this is shown below in Figure 1.1, a diagram of a dam with a hydroelectric
system.

Figure 1.1. Hydroelectric Dam Schematic

The amount of electric power generated by a hydroelectric system (đĚ ) depends on the
total efficiency of the system (đđĄ ), the total head đť, the gravitational acceleration constant đ, and
the volume flow rate đĚ , and is calculated using equation 1-1 [4]:
đĚ = đđĚ đťđđĄ

(1-1)

The total efficiency of the system depends on the isentropic efficiency of the turbine
rotor, the amount of friction losses in the penstock and draft tube, as well as the efficiency of the
generator, among other factors.
1.2. Hydroelectric Power Challenges
Hydroelectric sites can be categorized by the potential for power output. They are divided
into Large, Medium, Small, Mini, and Micro categories. This thesis focuses primarily on Mini
hydro (between 100 and 1000 kW) and Micro hydro (up to 100 kW) turbine designs and
development. Up to this point, constraints and problems related to the development of
mini/micro hydropower include [5]:
1. High capital cost of hydropower generation equipment
2

2. High cost of civil works typically associated with mini/micro hydropower
3. High operations/maintenance cost
The US Department of Energy has suggested pre-packaged, pre-assembled modular low
cost hydroelectric units are needed to solve this problem [1]. It has been suggested that a
reduction in the capital cost of mini/micro hydroelectric units could lead to additional growth of
hydropower generation in the United States [1]. This thesis focuses on the first problem,
specifically the high capital cost of the turbine equipment.
1.3. Hydroelectric Dam Retrofits
In order to achieve the goal of expanding hydropower use, it is advantageous to look for
potential sites with the lowest potential cost of installation and civil works. âHydroelectric dam
retrofitsâ are considered to be a cost-effective hydropower development strategy. Hydroelectric
dam retrofits involve installing hydropower generation equipment into existing dams that have
the necessary water volume flow and head for potential power generation [6]. The power
generated could either be used to supplement the grid or offset electricity costs for municipal
dam owners.
In a report prepared for the US Department of Energy, The Oak Ridge National
Laboratory stated that the primary advantage of hydroelectric dam retrofits compared to new
hydropower development is that âmany of the costs and environmental impacts of dam
construction have already been incurred at NPDs (Non-Powered-Dams) and may not be
significantly increased by the incorporation of new energy production facilities. Thus, the
development of some NPDâs for energy purposes is assumed to be achievable with lower
installed cost, lower levelized cost-of-energy, fewer barriers to development, less technological
and business risk, and in a shorter time frame than development requiring new dam constructionâ

3

[6]. The Oak Ridge National Laboratory has estimated that a potential additional 12.6 gigawatts
of power could be generated from hydroelectric dam retrofits in the United States. It is
considered an abundant potential source of renewable energy in the US. There are over 80,000
non-powered dams in the United States, and it is estimated 54,000 of these have the water flow
and head necessary for considerable power output if a hydroelectric system was installed [6].
Civil works contribute around 40% of the total cost of conventional small hydro projects
[7]. The civil work projects that need to be done for a hydroelectric dam retrofit include the
intake, penstock, and power house construction. Civil works cost also include any temporary
infrastructure needed during installation and construction. Typically when installing new
hydropower equipment into an existing dam, the dam must be de-watered and a new temporary
dam must be constructed. The cost of the civil works projects associated with a hydroelectric
dam retrofit are highly dependent on the design of the hydroelectric system selected.
It is advantageous to design the hydroelectric dam retrofit system such that extensive
modification of the existing dam structure is not required. Many non-powered dams in the
United States have spillways from which water flows. Most spillways have gates which control
the rate at which water flows through the dam. Figure 1.2 below shows the Bellaire Dam in
Antrim County, Michigan, and Figure 1.3 shows a spillway gate at the dam.

4

Figure 1.2. Bellaire Dam, Antrim County, Michigan

Figure 1.3. Spillway at the Bellaire Dam, Antrim County, Michigan

The analytical model in this work was set up to obtain CFD boundary conditions which
were applicable for a proposed application of a hydraulic turbine for use as a retrofit turbine unit
to fit on the front of dam spillways. The setup analyzed in this work is illustrated in Figure 1.4
below.

5

Spillway

Turbine

Figure 1.4. Hydroelectric dam retrofit schematic

1.4. Selection of Type of Water Turbine
When selecting a turbine system to be used for a potential site for hydroelectric plant
installation, a number of different factors are considered. The first being the volume flow rate,
head, and power obtainable given the geographic constraints of the hydropower site. The degree
of complexity involved to build/install the hydroelectric system, as well as maintenance
requirements of the turbine technology being evaluated are also considered. The portability and
shipping cost of the system are factors as well as the degree of modularity of the turbine being
considered for selection and potential environmental impacts. Modular turbine systems are of
value as it allows for the turbine to be broken up into smaller components for easy maintenance
and replacement in the field [2]. Before going forward on a hydropower project, the capital and
operating costs of the system are weighed against the value of the energy the plant will generate.
Engineers often pick the type of turbine to be used for a particular application by observing the
designed flow rate and head, and picking the type of turbine based off of recommendations of
literature.

6

1.5. Woven Wheel Hydro Turbine
A technology invented by Dr. Norbert Mueller at Michigan State University known as the
Woven Wheel offers promise to reduce the capital cost and complexity of hydraulic turbines.
The Woven Wheel is a manufacturing process which involves manufacturing turbomachinery
blades by winding the wheel out of continuous composite fiber strands. The wheels are designed
such that the fibers are wound in tension. For the Woven Wheel manufacturing technique, the A
technology invented by Dr. Norbert Mueller at Michigan State University known as the Woven
Wheel offers promise to reduce the capital cost and complexity of hydraulic turbines. The
Woven Wheel is a manufacturing process which involves manufacturing turbomachinery blades
by winding the wheel out of continuous composite fiber strands. The wheels are designed such
that the fibers are wound in tension. For the Woven Wheel manufacturing technique, the
continuous fiber is wound around a low-cost mandrel which can be 3-D printed. The technology
allows for motor and generator components to be integrated directly into the hub or shroud,
decreasing the cost of producing modular turbines or compressors which use the Woven Wheel
as the rotor [8]. The winding of these wheels can be done by hand or using a commercially
available computer controlled winding machine. The Woven Wheel design includes an outer
shroud [9]. This winding process allows for a large number of different winding patterns to be
employed. Figure 1.5(a) below shows a computer rendering of one such winding pattern. Figure
1.5(b) shows the wound turbomachine wheel inside of the mandrel, and Figure 1.5(c) shows the
wheel after removal from the mandrel.

7

Figure 1.5. (a) 3-D model of Woven Wheel winding scheme (b) Woven Wheel in mandrel (c) Woven wheel
removed from mandrel

This technology developed at Michigan State University allows for lower cost
manufacturing of axial water turbines with integrated generator components. Inclusion of an
integrated generator along the shroud of the rotor can decrease the part count and required
maintenance as well. A prototype Woven Wheel water turbine was successfully tested for a tidal
turbine application in a tow tank at the Marine Hydrodynamic Lab, at the University of
Michigan, shown below in Figure 1.6.

Figure 1.6. Woven Wheel tidal turbine being tested inside the tow tank

Integrated motor components have been demonstrated in the Woven Wheelâs application
as a compressor; however, the integration of the generator components in the Woven Wheelâs
application as a water turbine has yet to be prototyped. The turbine rotor tested, manufactured
using the Woven Wheel manufacturing method was wound from Kevlar-49. Kevlar fiber has a
specific strength three times higher than titanium alloys [10]. The advantages of using this
Woven Wheel technology as a water turbine are discussed further below.
8

1.5.1. Lightweight, Modular Design

The Woven Wheel modular turbine or compressor units can be assembled in a row to
form a modular multistage unit. Kevlar is a high strength, low cost, low weight fiber material
which can be considered a good choice of composite material for winding a Woven Wheel for
use as a water turbine. Figure 1.7 below shows a rendering of one such modular design. A recent
Woven Wheel prototype weighed approximately 50% of a wheel of the similar geometry
machined from 6061 T6 aluminum.

Figure 1.7. Modular Woven Wheel compressor unit design
1.5.2. Low-cost Additive Manufacturing

The Woven Wheel can be manufactured by winding a continuous carbon fiber bundle
around a 3-D printed mandrel to form the shape of the blades. The wheel with mandrel is then
infused with epoxy and cured, after which the wheel is removed from the mandrel. Preliminary
prototyping of manufacturing Woven Wheels using this method has taken place at Michigan
State University. The cost of manufacturing traditional turbine blades with an integrated shroud
with traditional turbine manufacturing methods is high as it would require more raw material,
and machining processes. The weight of the turbine would inevitably increase as well. Figure
1.8 below shows a 3-D printed mandrel and the winding process. Although the process requires

9

further maturation, prototypes have demonstrated it costs less to purchase the materials and
produce compared to traditional rotors machined out of metal.

Figure 1.8. (a) 3-D printed mandrel. (b) carbon fiber winding. (c) mandrel post-winding after being
dipped into epoxy/resin and cured. (d) Woven Wheel after removal from mandrel.
1.5.3. Rapid Customization and Manufacturing

Although this manufacturing process is still being refined and perfected, this additive
manufacturing process allows for the time between design and production to be minimal. New
designs derived from analytical models can be tested in CFD, a mandrel can be manufactured
and wound in a matter of hours. This allows for faster, lower cost production and rapid
customization. This added versatility lends itself well to the application of the technology for use
as a water turbine. Individual hydroelectric applications are unique and are often custom

10

projects, and the ability to adjust the Woven Wheel manufacturing technique to fit any required
size is advantageous.
Currently only blade profiles with constant thickness can be manufactured using the Woven
Wheel method. The turbine designs investigated in this thesis all have constant blade thickness
from hub to shroud, and leading to trailing edge.
1.5.4. Outer Shroud Eliminates Tip Leakage and Reduces Fish Mortality

Fishes can suffer injuries and death passing through hydraulic turbines. This is
considered a significant environmental concern when considering any new hydroelectric
development [11]. Fish injuries and death while passing through hydraulic turbines is a result of
rapid pressure changes along flow path, cavitation, narrow gaps between rotating parts and
stationary structures, and fish collision with structures including turbine runner blades and guide
vanes.
Traditional axial-flow turbines (such as Kaplan turbines) are not shrouded, there is a gap
between the turbine runner and outer housing which is the most common source of the mortality
of fish passing through hydraulic turbines [12]. Turbine manufactures like Voith have attempted
to solve this problem by implementing design changes which reduce the factors listed above.
Voith has developed a âMinimum Gap Runnerâ technology which features specially contoured
and machined runner blades to minimize the gaps between the turbine runner blades and the hub,
as well as the stationary outer housing. The minimization of the gap between turbine runner
blades and the housing also reduces tip leakage, improving performance [13]. Computational
Fluid Dynamics (CFD) is often used to ensure that cavitation, shear stresses, and rapid pressure
changes are kept to a minimum.
Gaps between the turbine rotor blades and outer housing can lead to fish death.
Companies like Voith have developed technologies for axial hydraulic turbines which reduce the
11

size of this gap, however almost all axial hydraulic turbines do not have this feature [13]. Using
the Woven Wheel as a hydraulic turbine reduces the risk of fish death as a result of the gap
between rotor blades and the outer housing as the shroud is integrated into the design of the
blades. This also reduces losses associated with tip leakage.
1.6. Definition of Turbine Type Investigated
This work explores the viability of using an axial turbine with simplified geometry for
low head hydroelectric applications. Constraints were placed on the design to keep
manufacturing costs as low as possible, and to allow the Woven Wheel method to be employed
for manufacturing the turbine rotor. The turbines analyzed in this work have the following
features:
ďˇ

Axial flow

ďˇ

Circular arc blade profile

ďˇ

Constant thickness blade profile

ďˇ

Constant OD/ID

ďˇ

Shrouded rotor

ďˇ

No inlet or outlet guide vane

ďˇ

No-pre swirl, đśđ˘1 = 0

The performance of the hydro turbine designs tested in CFD could be improved by
employing blade profiles, adjusting the OD/ID along the flow path axially, but these were not
included in the designs, this was done to keep manufacturing production cost low and to make
the designs applicable to the Woven Wheel manufacturing method. In this work turbines of this
type referred to as CTPAT, or Constant Thickness Profile Axial Turbine.

12

The work in this thesis applies for turbine designs with shrouded rotors manufactured
from metal as well as Woven Wheels. The CTPAT scheme is applicable with the modular
Woven Wheel turbine concept which has been proposed. The use of constant thickness blade
profiles makes the CTPAT scheme different from those turbines most commonly employed in
the field. The analytical model described in section 2.3 was set up to obtain CFD boundary
conditions which were applicable for a proposed application of a CTPAT turbine for use as a
retrofit turbine unit to fit on the front of dam spillways.
1.7. Conservation of Energy
The first law of thermodynamics is applied through a control volume to obtain an
equation for the steady flow energy balance for a turbine. A control volume representing a
turbine is illustrated in Figure 1.9 below. For the analyses in this work, â1â represents the
properties at the leading edge of the turbine, and â2â represents the properties at the trailing edge
of the turbine. Equation 1-2 below shows the energy balance, where đĚ is the heat transfer from
the surroundings to the control volume measured in Joules per second, â is mass-specific
enthalpy, đś is absolute flow speed, đ§ represents the elevation above the datum, and đĚ represents
the power that is transferred from the fluid to the blades of the turbomachine via the shaft (shaft
power).

13

Figure 1.9. Turbomachine control volume

1
đĚ â đĚ = đĚ [(â2 â â1 ) + (đś22 â đś12 ) + đ(đ§2 â đ§1 )]
2

(1-2)

The heat transfer from the surroundings to the control volume đĚ is negligible for water
turbines, and is assumed to be equal to zero. To derive an expression for the isentropic efficiency
of the turbine, equation 1-2 is rewritten in differential form below in equation 1-3 [4]:
1
đđĚ = đĚ [đâ + đ(đ 2 ) + đđđ§)]
2

(1-3)

Ě is the total energy extraction from the fluid per second, and is the maximum
đđđđĽ
possible power output of the turbine. For an isentropic process, đâ =

đđ
đ

Ě can then be
[4]. đđđđĽ

expressed using equation 1-4:
Ě = đĚ [
đđđđĽ

(đ2 â đ1 ) 1 2
+ (đś2 â đś12 ) + đ(đ§2 â đ§1 )]
đ
2

(1-4)

Ě can
For incompressible flow, the total energy extraction from the fluid per second đđđđĽ
be rewritten in terms of the total available head đť, which takes into account gravitational
potential energy, kinetic energy, and enthalpy difference, seen in equation 1-3, applied to the
energy equation in equation 1-5 below [4]:

14

đđť =

đ 1 2
+ đś + đđ§
đ 2

(1-5)

Ě , equation
The expression for the total energy extraction from the fluid per second, đđđđĽ
() can then be rewritten in terms of đť, shown below in equation 1-6:
Ě = đĚđ(đť1 â đť2 )
đđđđĽ

(1-6)

The isentropic turbine efficiency đ is defined as the ratio of the shaft power output đĚ to
Ě , and defined below in equation 1-7
the total energy extraction from the fluid per second đđđđĽ
[4]. Isentropic turbine efficiency is referred to as âefficiencyâ in this work.
đ=

đĚ
đĚ
=
Ě
đĚđ(đť1 â đť2 )
đđđđĽ

(1-7)

The mass-specific form of the shaft power output đĚ is denoted by đĚ , measured in meters
squared per seconds squared, đĚ is defined in equation 1-8 below:
đĚ =

đĚ
đĚ

(1-8)

Using equation 1-8, the expression for efficiency in equation 1-7 can then be rewritten in
terms of the mass-specific shaft work đĚ , shown below in equation 1-9:
đ=

đĚ
đ(đť1 â đť2 )

(1-9)

Ě . đ denotes the mass-specific
A variable is defined, đ, the mass-specific form of đđđđĽ
work extracted from the fluid from the leading to trailing edge, which includes losses in the
turbine, and is defined below in equation 1-10:
đ=

Ě
đđđđĽ
đĚ
= = đ(đť1 â đť2 )
đĚ
đ

15

(1-10)

Work extraction from a hydraulic turbine can be expressed in terms of the total
(stagnation) pressure, đđĄ . đđĄ can be expressed in terms of static pressure, dynamic pressure, and
the gravitational head, shown in equation 1-11 below [4]:
đđĄ =

đ 1 2
+ đś + đđ§
đ 2

(1-11)

The mass-specific energy extracted from the fluid from leading to trailing edge, đ, is
expressed in terms of the total pressure đđĄ at leading and trailing edges in equation 1-12 below
for incompressible flow for a hydraulic turbine:
đđĄ2 = đđĄ1 â đđ

(1-12)

A portion of the total pressure drop across a turbine stage (đđĄ1 â đđĄ2 ) is converted into
useful shaft work (đĚ đ). The portion of the pressure drop that is not converted into shaft work are
considered losses. The losses can be expressed using a reordered version of equation 1-12, in
terms of the total loss coefficient đ and the absolute velocity at the trailing edge, đś2 , shown
below in equation 1-13.
1
đđĄ1 â đđĄ2 = đĚ đ + đđđś22
2

(1-13)

Total loss coefficient đ can be expressed in terms of its components, a sum of the sources
of loss in a turbine stage, where đđđđđđđđ represents losses in the blade row, đđđĽđđĄ represents losses
between the trailing edge and the outlet, đđđđĄđđŚ represents losses between the turbine inlet and
leading edge, expressed below in equation 1-14:
đ = đđđđĄđđŚ + đđđđđđđđ + đđđĽđđĄ

(1-14)

The shaft power output of the turbine đĚ can be calculated using the mass-specific shaft
work đĚ and the mass flow rate đĚ, shown below in equation 1-15:

16

đĚ = đĚ đĚ

(1-15)

1.8. Definitions of Geometry and Design Parameters

Figure 1.10. Example velocity triangle

The variables used to generate a turbine rotor design for CFD analysis are defined in this
â . The absolute velocity of the fluid is đś , and
section. Absolute velocity of the turbine rotor is đ
âââ , defined below in equation 1-16.
the velocity of the fluid relative to the rotor is đ
âââ = đś â đ
â
đ

(1-16)

âââ are often expressed in terms of their meridonial and tangential
The vectors đś and đ
(wheel rotation direction) components. đśđ is the absolute velocity of the fluid in the meridonial
direction, and đđ is the flow speed in the meridional direction relative to the wheel. đśđ˘ is the
absolute velocity of the fluid in the direction of the rotation of the wheel, and đđ˘ is the flow
speed in the rotation direction of the wheel relative to the wheel. Velocity triangles help to
visualize how the velocities relate to each other. Figure 1.10 above shows an example velocity

17

triangle. The magnitude of the absolute flow velocity, đś, is calculated using the meridonial and
tangential velocity components shown below in equation 1-17:
2
đś 2 = đśđ˘2 + đśđ

(1-17)

Similarly the magnitude of the relative flow velocity đ can be calculated using the
meridional and tangential relative velocity components, seen in equation 1-18:
đ 2 = đđ˘2 + đđ2

(1-18)

â vector, and
Absolute flow angle đź is defined as the angle between the đś vector and the đ
can be calculated using the components of the đś vector, expressed below in equation 1-19:
đź = cos â1

đśđ˘
đśđ
= = sinâ1 ( )
đś
đś

(1-19)

âââ vector and the
Similarly the relative flow angle, đ˝, is defined as the angle between the đ
â vector, and can be calculated using the components of the đ
âââ vector, expressed below in
đ
equation 1-20:
Î˛ = sinâ1

đđ
đđ˘
= cos â1( )
đ
đ

(1-20)

Figure 1.11 below shows an example CTPAT axial turbine geometry, where âđ is the
blade height, đâ is the inner (hub) radius, and đđ  is the outside (shroud) radius of the turbine rotor,
and đ is the angular velocity of the rotor. Position 1 represents the leading edge, position 2
represents the trailing edge.

18

Figure 1.11. Axial turbine geometry parameters

For the analysis in this work, the outside (shroud) radius of the rotor, đđ  , and the
rotational speed of the rotor in RPM are all input. The hub-tip ratio đž is input as well. The inner
(hub) radius đâ can be calculated using the shroud radius and the hub-tip ratio, shown in equation
1-21 below:
đâ = đđ  đž

(1-21)

Angular velocity đ can be calculated using the rotational speed given in rotations per
minute đđđđ , which was input for this analysis, using equation 1-22,then the tip speed of the
wheel (đ) can then be calculated using equation 1-23.
đ = 2đ

đđđđ
60

đ = đđ  đ

(1-22)
(1-23)

The tangential speed of the rotor for a position along the span is denoted by đđ , where
âđâ represents a position along the span, calculated using equation 1-24 below, where đđ is the
radial distance from the centerline to the position along the span being considered. This is

19

different than the tip speed đ in that tip speed uses outside radius. For example, the tangential
speed of the rotor at the inner radius/hub is đâ = đâ đ, calculated using the inner radius, đâ .
đđ = đđ đ

(1-24)

The inner radius and outside radius of each turbine in this analysis were kept constant
form leading to trailing edges. Thus the tip speed of the wheel is constant from the leading to
trailing edges, so đ = đ1 = đ2 . For this analysis the thickness of the blades (đĄ) and the number
of blades (đđ ) were also input. The thickness of the blades is constant from leading to trailing
edge, and is also constant across the blade span. Blade height âđ is calculated for the IGV and
rotor geometries using equation 1-25:
âđ = đđ  â đâ

(1-25)

Meridional cross-sectional area đ´đ is then calculated for the axial rotor using equation 1-26:
đ´đ = đ(đđ 2 â đâ2 ) â đĄâđ đđ

(1-26)

Mass flow rate đĚ can be calculated using the meridional cross-sectional area đ´đ , density
đ, and absolute meridonial flow speed đśđ , shown below in equation 1-27:
đĚ = đśđ đ´đ đ

(1-27)

Equation 1-28 below is referred to as the equation of continuity, and states that mass flow
rate, đĚ, stays constant from inlet to outlet of the turbomachine control volume. đĚ stays constant
from leading to trailing edge, and from inlet to outlet as there is no accumulation of water within
the control volume. Equation 1-29 is a reordered version of equation 1-28, using the definition of
mass flow rate in equation 1-27:
đĚ 1 = đĚ 2

(1-28)

đśđ1 đ´đ1 đ1 = đśđ2 đ´đ2 đ2

(1-29)

20

The mass flow rate of water flow (đĚ) as well as the density of the water (đ) are input.
Using these input properties and known geometry, meridional absolute flow speeds along the
flow path can be calculated using equation 1-30 below, after rearranging the definition of mass
flow rate expressed in equation 1-27:
đśđ =

đĚ
đ´đ đ

(1-30)

Flow speed in the direction of the wheel relative to the wheel đđ˘ is then calculated using
âââ vector equation 1-16 ,in the tangential direction, seen below in equation
the definition of the đ
1-31:
đđ˘ = đ â đśđ˘

(1-31)

Similarly by applying equation 1-16 in the meridional direction đđ is calculated using
equation 1-32 below:
đđ = đśđ

(1-32)

Figure 1.12 below illustrates an example CTPAT axial turbine blade cascade with
velocity triangles at leading and trailing edges. Geometric parameters are defined, where đ is the
axial blade length, đ  is the spacing between blades, đ is the chord length, and đĄ is the blade
thickness. Circular arc blade profiles are used for the analytical model and the CFD simulations.
For this analysis geometric parameters đĄ, đđ , đđ , đđ , đ are each provided as an input. For the
analyses in this work, â1â represents the properties at the leading edge, and â2â represents the
properties at the trailing edge of the turbine.

21

Figure 1.12. Turbine blade cascade with velocity triangles

Blade spacing đ  can be calculated using equation 1-34 below, which uses known
geometric inputs and mean radius đđ , which is calculated using equation 1-33 below:
(đ 2 + đđ 2 )
đđ = â â
2
đ =

2đđđ
đđ

22

(1-32)

(1-33)

Flow does not follow the blade path perfectly, the difference between the ideal trailing
edge relative flow angle if flow followed the blades perfectly and the actual trailing edge relative
flow angle is referred to as the outlet deviation, đż â˛ , defined in equation 1-34 below, where Î˛b2 is
the angle between the blade at the trailing edge and the âââ
đ vector, and Î˛2 is the relative flow
angle at the trailing edge:
đż â˛ = Î˛2 â Î˛b2

(1-34)

Similarly the incidence is the difference between the leading edge flow angle and the
blade leading edge angle, shown below in equation 1-35:
đ â˛ = Î˛1 â Î˛b1

(1-35)

The mass-specific shaft work đĚ can be related to the rotor speed and absolute tangential
velocity using the Euler turbomachinery work equation, derived using conservation of angular
momentum. Conservation of momentum equates the sum of external forces acting on a fluid to
the rate of change of momentum. Applied to turbomachinery, the force applied by a fluid onto
the blades is caused by the acceleration of the fluids passing through the blades [4]. Equation 136 below applies conservation of momentum in the x-direction for a control volume where fluid
enters with uniform velocity đśđĽ1 in the x direction and leaves with uniform velocity đśđĽ2 in the x
direction, đ represents the mass of a fluid element, and đšđĽ represents forces acted on the control
volume in the x-direction [4]:
â đšđĽ =

đ
(đđśđĽ ) = đĚ(đśđĽ2 â đśđĽ1 )
đđĄ

(1-36)

Conservation of momentum can be applied to relate change in tangential flow speed to
the change in angular momentum and shaft power. The sum of all moments of all forces acting
on a system about an axis is equal to the time rate of change of angular momentum about that
axis [4]. This is seen below in equation 1-37, where đđ is the time rate of change of angular
23

momentum about axis A, đ is the distance between the center of mass to the rotation axis A and
đśđ is the component of velocity perpendicular to both the axis A and the radius vector:
đđ = đ

đ
(đđśđ )
đđĄ

(1-37)

Equation 1-37 can then be applied for an arbitrary turbomachine, with radius at leading
edge đ1, radius at trailing edge đ2 , with tangential velocity at the leading and trailing edges
đśđ˘1 and đśđ˘2 respectively for one dimensional steady flow, shown below in equation 1-38 [4]:
đđ = đĚ(đ2 đśđ˘2 â đ1 đśđ˘1 )

(1-38)
đ

Applying the definition of tip speed đ by plugging in đ = đ to equation 1-38, equation 139 is formed:
đđ đ = đĚ(đ2 đśđ˘2 â đ1 đśđ˘1 )

(1-39)

The product đđ đ is equal the shaft power output of the turbine đĚ . Using equation 1-8,
equation 1-39 is rearranged in terms of the mass-specific shaft work đĚ , shown below in equation
1-40:
đĚ = đ2 đśđ˘2 â đ1 đśđ˘1

(1-40)

The above is a form of the Euler turbomachinery work equation, which is valid for
steady, adiabatic flow [4]. For the analysis in this work, tip radius stays constant from leading to
trailing edge such that đ1 = đ2 . Equation 1-40 can then be simplified to form equation 1-41:
đĚ = đ(đśđ˘2 â đśđ˘1 )

(1-41)

By rearranging equation 1-41 and applying equation 1-10, đĚ can then be expressed in
terms of the total head đť difference from leading to trailing edge, shown below in equation 1-42:
đ(đśđ˘2 â đśđ˘1 ) = đđ(đť1 â đť2 )

24

(1-42)

1.9. Turbine Cascade Forces
An example pressure distribution across a turbine stage is shown below in Figure 1.13,
where đđ is the pressure curve on the pressure side of the blade, đđ  is the pressure curve on the
suction side.

Figure 1.13. Pressure distribution across a turbine stage

The area between the đđ and đđ  curves in Figure 1.13 is equal to the tangential force đ
acting on the flow imparted by a single blade, calculated by integrating the pressure difference
from the leading edge to the trailing edge, expressed below in equation 1-43.
1

đĽ
đ = đ âŤ (đđ â đđ  ) đ ( )
đ
0

(1-43)

To derive an alternate form of equation 1-43, conservation of angular momentum is applied in
the tangential direction, the tangential force can be expressed using equation 1-45 assuming
Ě is calculated, which represents the mass flow rate through a
constant axial velocity. First âđ
single blade passage, using equation 1-44 [4]:

25

Ě = đśđ đ âđ đ
âđ

(1-44)

Ě (đđ˘2 â đđ˘1 )
đ = âđ

(1-45)

Force in the axial direction đšđĽ can be derived using conservation of momentum, written
below in equation 1-46, assuming axial velocity stays constant from leading to trailing edge:
đšđĽ = (đ1 â đ2 )đ âđ

(1-46)

These forces can be used to calculate lift and drag forces đšđ and đšđ on the blades. Figure
1.14 below shows an example turbine blade cascade with blade forces.

Figure 1.14. Example turbine blade cascade with blade forces

26

First mean flow angle đ˝đ is calculated using equation 1-47, necessary for the calculation
of lift and drag forces.
1
tan đ˝đ = (tan đ˝1 + tan đ˝2 )
2

(1-47)

Mean relative velocity đđđđđ can then be calculated using the mean relative flow angle
đ˝đ and the relative meridional velocity đđ , shown below in equation 1-48:
đđđđđ = đđ / sin đ˝đ

(1-48)

Lift and drag forces đšđ and đšđ can be calculated using equation 1-49 and equation 1-50
respectively, below:
đšđ = đšđĽ cos đ˝đ + đ sin đ˝đ

(1-49)

đšđ = Ycosđ˝đ â đšđĽ sin đ˝đ

(1-50)

Tangential force coefficient đśđ , axial force coefficient đśđĽ , lift coefficient đśđ , and drag
coefficient are expressed in equations 1-51, 1-52, 1-53, and 1-54 respectively. They are useful
non-dimensional coefficients which are a measure of the blade forces defined in equations 1-45,
đ

2
1-46, 1-49, and 1-50 respectively, compared to the mean dynamic pressure , 2 đđđđđ
, multiplied

by the area on which it acts, đ âđ .
đ
đśđ = đ
2
2 đđđđđ đ âđ

(1-51)

đšđĽ
đśđĽ = đ
2
đ
đ â
2 đđđđ đ

(1-52)

đšđ
đśđ = đ
đ 2 đ â
2 đđđđ đ

(1-53)

đšđ
đśđ = đ
2
đ
đ â
2 đđđđ đ

(1-54)

27

1.10. Dimensionless Coefficients
1.10.1. Flow and Capacity Coefficients

The flow coefficient, đ, is a non-dimensional coefficient that depends on flow rate and
rotational speed. The flow coefficient is used to compare different types of turbines. It relates the
axial flow velocity to the speed of the blade, and is defined below in equation 1-55 [15]:
đ=

đśđ
đ

(1-55)

The capacity coefficient, đ â˛ , similar to the flow coefficient, is a non-dimensional
parameter which is effected by the rotational speed, rotor diameter, and flow rate. The capacity
coefficient is defined below in equation 1-56:
đâ˛ =

đĚ
đđˇ3

(1-56)

1.10.2. Blade Loading and Head Coefficients

The blade loading coefficient đ provides a measure of the work extraction for a turbine
stage. The blade loading coefficient relates the mass-specific shaft work đĚ to the blade tip speed
đ. is defined below in equation 1-57: [4].
đ=

đĚ
đ2

(1-57)

The equation is rearranged for an adiabatic axial water turbine with constant shroud
radius, plugging in equation 1-10 into equation 1-57 to obtain equation 1-58 below: [4]
đ=

đĚ
đđđť
=
đ 2 đ 2 đ 2 đđ 2

(1-58)

The head coefficient đâ˛ is similar to the blade loading coefficient in that it relates the
work extraction to the rotational speed. However, blade loading coefficient đ includes isentropic
efficiency in the expression, while head coefficient đâ˛ uses total work extraction from the fluid,

28

đ, in the expression. The head coefficient đâ˛ is defined below for an axial water turbine with
constant shroud radius using equation 1-59:
đâ˛ =

đ
đđť
=
2
(đđˇ)
(đđˇ)2

(1-59)

Turbines which extract more work at a low tip speed have higher blade loading and head
coefficients.
1.10.3. Specific Speed

Specific speed is a non-dimensional quantity that is used to describe and categorize
turbomachinery. Specific speed, also called shape parameter, is discussed by Horlock [15] as a
useful tool to help select the type of machine that will give highest efficiency for a given
application. The specific speed was derived by raising the head and capacity coefficients to a
power, eliminating diameter from the expression [16]. The specific speed takes into account
rotational speed, flow rate, and total mass-specific work extracted from the fluid, where đ is in
units of rotations/second. For a water turbine with incompressible flow, the mass-specific work
done on the fluid from leading to trailing edge, đ, can be expressed in terms of total available
head đť. Specific speed đđ  is defined below in equation 1-60 [15]:
1

đđ  =

1
đâ˛2
3
đâ˛4

=

đĚ 2
( 3)
đđˇ
3
4

đ
(
)
(đđˇ)2

1

=

đđĚ 2
3
đ4

1

=

đđĚ 2
3
đđť 4

(1-60)

It should be mentioned the head in this case is the total designed head of the turbine
system, including losses. This form of specific speed is in units of rotations [4] . Another form of
specific speed, ÎŠđ  , uses the angular speed of the turbine measured in radians per second instead
of rotations per second [4], defined below for incompressable flow in equation 1-61:

29

1

ÎŠđ  =

đđĚ 2

(1-61)

3
(đđť)4

Horlock stated there is an optimal value of the specific speed for a given type of machine,
independent of size, at which efficiency is highest [15]. Figure 1.15 illustrates this concept,
showing there is a specific speed at which peak efficiency is reached.

Figure 1.15. Example efficiency curve for turbine for specific speed đđ  [15]
1.10.4. Specific Diameter

Specific diameter, đˇđ  , is also used to describe and categorize turbines. The specific
diameter was defined by using the head and capacity coefficients, similar to the specific speed.
The head and capacity coefficients can be raised to a power to eliminate rotational speed from
the equation [4]. This is shown below in equation 1-62 [4]:
1

đˇđ  =

đâ˛4
1
đâ˛2

1

=

đˇ(đđť)4
1

(1-62)

đĚ 2

1.11. Cordier Diagram and Line
Dixon stated that once specific speed is determined; the ideal machine type can be
selected using the Cordier diagram [4]. The Cordier diagram is also often used to select a
30

specific speed or specific diameter for a given type of machine where efficiency is predicted to
be highest. Otto Cordier carried out an experimental analysis of high efficiency turbomachines
during the 1950s, and placed their data on a plot showing specific speed ÎŠs against specific
diameter đˇđ  , and defined a trend line on a diagram showing where the highest efficiency
turbomachines lie [17]. This curve is referred to as the âCordier lineâ.
Each type of machine has a range of specific speeds in which they perform with highest
efficiency [15]. Machines with high head and low flow are on the right side of the Cordier
diagram, while low head, high flow machines are on the left. The Cordier diagram with Cordier
line is shown below in Figure 1.16. Circles are added to indicate where different types of
machines typically lie on the diagram. Lines from Wright are placed above and below the
Cordier line which indicate the accuracy/margin [18]. Wright divided the Cordier diagram into
six regions [18]. The first region A, with specific speed between 6 and 10 and specific diameter
between .95 and 1.25, propeller-type machines are typically used. Region B has specific speed
between 3 and 6, specific diameter between 1.25 and 1.65. Wright mentioned this region has
primarily axial turbomachines, like axial fans, axial pumps, and shrouded propellers. Region C
contains machines with speed between 1.8 and 3, and specific diameter between 1.65 and 2.2.
Ducted axial machines or multistage axial machines typically operate in this region. Region D
includes machines with specific speeds between 1.0 and 1.8, and specific diameter between 2.2
and 2.8. Mixed flow pumps, blowers, mixed flow hydraulic turbines operate in this region.
Region E includes machines with specific speed between .7 and 1, and specific diameter between
2.8 and 4. Centrifugal fans, pumps, heavy duty blowers, compressors operate in this range.
Region F has specific speeds below .7 and specific diameters above 4. High pressure blowers,
centrifugal compressors, high head pumps operate in this region.

31

ÎŠs

Axials

Mixed
flow

Radial

Dđ 
Figure 1.16. đˇđ  â đşđ  diagram with Cordier line and typical machine type

Balje compiled test data and specifications from 92 water turbines built between 1940
and 1974 and placed them on the Ds â ÎŠs chart, superimposed over the Cordier line,. The water
turbines included in that test data have specific speeds between 2 and 6. The water turbines are
close to the line originally defined by Cordier [15].
The below Figure 1.17 shows the Cordier line alongside eight modern axial hydro
turbines manufactured by Voith Hydro, the specifications of which were obtained on the
company website [20].

32

Specific Speed ÎŠđ  vs. Specific Diameter
14.00
12.00

Sp. Speed ÎŠđ 

10.00
8.00
Voith Hydro

6.00

Cordier

4.00
2.00
0.00
0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

3.0000

Specific Diameter Ds
Figure 1.17. đˇđ  âđşđ  diagram with Voith Hydro turbines with Cordier line
1.11.1. Effect of Flow, Loading Coefficients on Cordier Diagram Position

A turbine designâs position on the Ds âÎŠs diagram relative to the Cordier line can be fully
defined by two dimensionless coefficients, Flow coefficient đ and blade loading coefficient đ.
The below figures show the Cordier diagram with lines of constant đ (Figure 1.18) and đ
(Figure 1.19) superimposed.

33

đˇđ  vs. ÎŠđ 
10
Cordier

Sp. Speed ÎŠđ 

FC = 0.005
FC = 0.01
FC=0.05

1

FC=0.1

1

10

FC = 0.25
FC=0.5
FC=1
FC=1.5
FC=3

0.1

Specific Diameter
Figure 1.18. đˇđ  âđşđ  diagram with Cordier line, lines of constant flow coefficient đ

đˇđ  vs. ÎŠđ 
10
Cordier
LC=2

Sp. Speed ÎŠđ 

LC=1
LC=0.5
LC=0.25

1

1

10

LC=0.1
LC=0.05
LC=3

0.1

Specific Diameter Ds

Figure 1.19 đˇđ  âđşđ  diagram with Cordier line, lines of constant blade loading coefficient đ

Turbine designs with high blade loading and flow coefficients have both low specific
speed and specific diameter, are located on the lower left side of the Ds âÎŠs diagram, below and
to the left of the Cordier line. Designs with low loading and flow coefficients are located above
and to the right of the Cordier line.
34

1.11.2. Using Cordier Diagram as a Design Tool

Wright recommended using the Cordier line as a design tool to help decide what type of
machine to use for a particular application, or to pick optimal values of either specific diameter
or specific speed given one of the two. In Fluid Machinery, Performance and analysis, Wright
stated: âWhen it is necessary to meet certain requirements in volume flow rate and pressure rise,
one can follow the overall trend prediction of the Cordier diagram to help decide what type of
machine will perform the job. For an initial selection of speed and known fluid density, one can
estimate the specific speed. Enter the diagram at ÎŠđ  , and determine a workable value for the
machine size by choosing a value near or on the âCordier lineâ. This initial procedure will
indicate both the size and type of machine one should be considering and the highest value of
efficiency one can reasonably expect to achieveâ. [18] Wright stated that by using designs on the
Cordier line highest efficiency can be achieved: âThe efficiency band represents the best total
efficiency that can reasonably be expectedâ [18]. Balje stated that efficient turbomachines are
located close to the Cordier line [19]. Balje applied a curve fit to the data to derive an equation to
pick optimal specific speed given specific diameter, which place the design on the Cordier line,
the equations are shown below in equations 1-63 and 1-64 below [19]:
đşđ  â 9.0đˇđ â2.103 for đˇđ  < 2.8

(1-63)

đşđ  â 3.25đˇđ â1.126 for đˇđ  âĽ 2.8

(1-64)

Adhikari et. al. described using the Cordier line as a starting point to help pick axial water
turbine diameter given specific speed [21]. Given a flow rate, head, and desired diameter, the
Cordier line can be used to pick an optimal specific speed, which then defines the RPM of the
turbine. For example, for a turbine with a rotor diameter of .4 m, head of 2 meters, and flow rate
of .46

đ3
đ 

, the specific diameter is 1.22. Then the Cordier line recommendation is used to select

35

the specific speed of 5.9. This gives a rotational speed of 737 RPM. Doing this in reverse gives a
recommended diameter given rotational speed, head, and volume flow rate. For example, a 170
RPM turbine with head of 2 meters, a volume flow rate of .46

đ3
đ 

, specific speed is 1.37. Using

the Cordier diagram to pick optimal specific diameter of 2.2. Using equation 1-62 and solving
for diameter gives an outside diameter of .7 meters.
1.11.3. Operating below the Cordier Line

The Cordier line can be used to pick a recommended specific diameter or specific speed
given one of the two, however using the Cordier line for CTPAT turbines for this purpose results
in recommendations of specific speed that may be larger than necessary for the application.
Although there are correlations available within literature for predicting efficiency of specific
types of hydraulic turbines at points off of the Cordier line, there will be no literature available
for turbines with a new design, so it can be advantageous to test the effect of specific speed on
efficiency when considering a new type of machine, like CTPAT.
Although literature suggests that the Cordier line provides a reasonable estimation of the
specific speed at which a given machine will perform at peak efficiency. Korpela noted that
turbines with low specific speed are better suited for radial turbines, and as specific speed
increases axial turbines perform with higher efficiency [16]. Dixon mentioned that the Cordier
line is a mean curve based upon data from a large number of machines Cordier compiled, and âit
is possible to diverge from the line and still obtain high performance pumps, fans and
compressorsâ [4]. However Cordier and Baljeâs recommendations suggest that turbomachines
should perform with higher efficiency when they lie on the Cordier line. Therefore it makes
sense to test whether this applies for CTPAT turbines.

36

This work investigates the viability of the CTPAT turbine scheme for low head hydro, at
specific speeds above and below that recommended by the Cordier line. This work investigates
whether designs of this type within the range of specifications tested below the Cordier line
(lower than recommended specific speed) perform any worse than designs on the Cordier line.
1.12. Literature Recommendations on Blade Loading and Flow Coefficients
Water turbines are often categorized by their blade loading coefficient đ and flow
coefficient đ. The Smith efficiency chart, shown below in Figure 1.20, was compiled using a
large set of gas turbine test data to correlate head coefficient đâ˛ and flow coefficient đ to
efficiency for a turbines with constant axial velocity [22]. Each turbine was tested at different
operating points to find its point of highest efficiency, and the flow coefficient and loading
coefficient were plotted at that point. Although the original diagram was published in 1965, the
Smith diagram is still used today for preliminary turbine design, as more modern turbine designs
follow the same efficiency vs. flow and head coefficient trends [23].

Figure 1.20. Smith loading vs. Flow coefficient diagram [22]

37

Smithâs work suggests that as flow coefficient đ and head coefficient đâ˛ are increased,
efficiency tends to decrease. The same relationship applies for capacity coefficient đâ˛ and the
blade loading coefficient đ. Turbines with higher flow coefficients have higher flow velocities
relative to blade speed, this tends to increase losses as there is a smaller acceleration through the
blade path, that acceleration is usually beneficial as it minimizes boundary layer growth or
secondary flows [24]. When head coefficient is increased, the change in flow tangential velocity
becomes larger, which tends to increase loss. [4]
The Smith diagram is often used to choose optimum values of flow coefficient and blade
loading coefficient given one of the two coefficients. Subsequent tests by Kacker and Okapuu
confirmed the chart is useful for preliminary turbine design [25]. The diagram recommends
values of flow coefficient and head coefficient that can offer higher performance, less flow
separation and losses. However this does not mean that values of flow coefficient and head
coefficient have to lie within recommended ranges in order to be considered acceptable designs
[24]. The specifications of the axial turbines simulated lie both inside and outside of the
recommendations of the Smith diagram and the Cordier line.
Smithâs diagram was derived using gas turbine data and compressible flow, however the
same trend applies for water turbines and incompressible flow, increasing flow coefficient and
loading coefficient is correlated with decreased efficiency [15]. Figure 1.21 shows the range of
typical flow coefficient đ and and blade loading coefficients đ for different types of hydraulic
turbines from Horlock. Flow coefficients for Kaplan and Francis turbines rarely reach above .6,
and blade loading coefficients rarely reach above 2 [15].

38

Figure 1.21. Blade loading and flow coefficients for different types of hydraulic turbines [15]

1.13. Cost Advantages of Designs With High Flow and High Loading Coefficients
Losses tend to increase with increasing blade loading and flow coefficients, however
turbines with higher blade loading and flow coefficients can be produced for lower cost.
Turbines with high flow coefficient generally have a smaller cross-sectional area compared to
those with low flow coefficients, so turbines with lower đ tend to have increased turbine
machining costs and weight [23]. Turbines with higher loading coefficient require fewer stages
to achieve the same work extraction, and can extract more head for the same number of stages.
Reducing the number of stages required for a turbine decreases part count, reducing weight,
capital cost and operating cost. Industrial gas turbines are usually designed to maximize
efficiency, so they generally have low flow and loading coefficients. Aerospace engines are
usually designed for higher blade loading and high flow coefficient as the engine weight is of
highest importance [23]. For aero engines, optimum value of đ is between 1.5 to 2.5, and đ
39

ranges from 0.8 to 1.2. This contrasts with industrial gas turbines which typically have đ
between 1.0 to 1.5, and đ from 0.4 to 0.8 [23].
Smith, in his paper defining his diagram, stated that in practice the designer should use
the highest possible feasible values of blade loading and flow coefficient, and has to âstrike a
compromise between the conflicting requirements of efficiency versus weight and costâ [22]. A
critical part of turbine selection is to weigh turbine efficiency with capital cost, and designs
which are less costly to produce, but have lower efficiencies can be considered acceptable for
many applications. This is especially true for low-head hydro applications, the revenue generated
from power is often not enough to justify large capital expenses for the most efficient turbines.
1.14. Zweifel Lift Coefficient
The Zweifel coefficient, also referred to as the Zweifel lift coefficient or Zweifel
tangential force coefficient, đ, is often used to find the optimal blade spacing where losses are
đ 

predicted to be lowest. The blade solidity, , is the ratio of the blade spacing đ  to the axial blade
đ

length đ. The solidity and has a large effect on the friction and losses resulting from deceleration
on the suction side (diffusion) in the blade row. The Zweifel coefficient recommendations in
literature can be used to pick a value of

đ 
đ

at which it is predicted that minimum friction losses

and losses resulting from diffusion will occur [4]. If the spacing between blades is too small, the
flow follows the blade with low deviation, but losses from friction are higher. If the spacing
between blades is too high however, friction losses are low but the load distributed per blade
increases, and losses resulting from diffusion increase [4]. Diffusion is undesirable in a turbine
blade row, this is because the adverse pressure gradient, especially for large fluid deflections,
makes boundary layer separation more probable [4].

40

The losses in the turbine from leading to trailing edge can be expressed in terms of the
profile loss coefficient, đđđđđđđđ . Profile losses are a function of losses as the result of friction and
diffusion, such that đđđđđđđđ â đđđđđđĄđđđ + đđđđđđ˘đ đđđ . Schoberi stated that for every turbine
design there is a value of

đ 
đ

where overall profile loss coefficient đđđđđđđđ is minimum [14].

Figure 1.22 below shows profile loss coefficient at differing blade spacing/chord length ratios, as
the result of diffusion and friction.

Figure 1.22. Profile loss coefficient vs. s/b ratio resulting from separation and friction [14]

Zweifel stated that âthe ratio of an âactualâ to an âidealâ tangential blade loading has an
approximately constant value for minimum lossesâ [26]. đ denotes the actual tangential force
per blade, previously defined in equation 1-43, and đđđ is the ideal tangential force per blade. The
Zweifel coefficient đ is defined using the below equation 1-65:
đ
đđđ

đ=

(1-65)

The tangential force imparted per blade, đ, is equal to the area between the pressure and
suction curves in Figure 1.13, and can be calculated using equation 1-43. đ can also be expressed
using conservation of momentum, assuming axial velocity stays constant from the leading to
trailing edge. Equation 1-66 below shows đ expressed using conservation of momentum, in
41

Ě and the change in relative tangential velocities from
terms of mass flow rate per blade âđ
trailing edge to leading edge, shown below in equation 1-66 [27]:
Ě (đđ˘2 â đđ˘1 )
đ = âđ

(1-66)

The ideal tangential force per blade đđđ is determined using an idealized pressure
distribution from leading to trailing edge across a blade, multiplied by the area on which it acts.
Zweifel defined this idealized pressure distribution using the assumption that the total pressure at
the leading edge đđĄ1 acts over the entire pressure side of the blade, meaning that velocity is
assumed to be zero [14]. For this idealized pressure difference used for derivation of the Zweifel
coefficient, it is assumed that the pressure is equal to đ2 on the entire suction side of the blade,
and it is assumed the pressure is constant across the surface with no diffusion. The pressure
distribution across the blade is thus assumed to be a rectangular shape [14]. đđđ is calculated by
multiplying this pressure difference by the surface area on which this force acts, the product of
the axial blade length and the blade height, đâđ , applied below in equation 1-67.
đđđ = (đđĄ1 â đ2 )đâđ

(1-67)

Figure 1.23 below shows an example pressure distribution across a blade with idealized
pressure distribution for calculation of Zweifel coefficient. The Zweifel coefficient is a measure
of how close the area of the real pressure distribution on the blades is to the area of this idealized
pressure distribution.

42

Figure 1.23. Idealized pressure distribution with actual pressure distribution for Zweifel coefficient
calculation

For incompressible flow the idealized pressure difference đđĄ1 â đ2 is equal to the
dynamic pressure at the trailing edge,

đđ22
2

[14]. Plugging into equation 1-67, the expression for

idealized tangential force is seen below in equation 1-68:
đđđ =

đâđ đđ22
2

(1-68)

It is assumed that meridional velocity đđ stays constant from leading edge to trailing
edge, thus đđ1 = đđ2. By plugging in the definition of the mass flow rate through one blade
path, âđĚ, equation 1-44, into the definition the actual tangential blade force đ (equation 1-66), đ
can then be calculated, below in equation 1-69.
đ = đâđ đ đđ (đđ˘2 â đđ˘1 )

(1-69)

Equations 1-68 and 1-69 can be plugged into the definition of đ (equation 1-65). Using
âââ (equation 1-16), the definition of đ can be rearranged
the definition of relative flow velocity đ
to be expressed in terms of relative flow angles, seen below in equation 1-70 [27]:
43

đ=

Ě (đđ˘2 â đđ˘1 )
đ
âđ
đđ2 đ (đđ˘2 â đđ˘1 )
=
=
1 2
(đđĄ1 â đ2 )đâđ
đđđ
2 đ2 đ

(1-70)

The Zweifel coefficient is proportional to the ratio of the blade spacing to the axial blade
đ 

length, đ. Zweifel coefficient is similar to the tangential force coefficient đśđŚ , originally defined
đ 

in equation 1-51, simplified in equation 1-71 below. Unlike Z, đśđŚ is not dependent upon đ, and
đśđŚ uses mean relative velocity to normalize the tangential force. A set of designs with the same
relative flow velocity components can have different values of Z depending on blade spacing and
axial blade length, however the value of đśđŚ will stay constant across each design.
đśđŚ =

đ
1
2
2 đđđđđđ đ âđ

=

đđ2 (đđ˘2 â đđ˘1 )
1 2
2 đđđđđ

(1-71)

The Zweifel coefficient recommendations from literature can be used to find the
đ 

đ 

recommended value of đ. After calculating the ideal đ, the ideal đ can then be calculated using a
đ 

known value of đ . Equation 1-70 is rearranged to solve for đ in equation 1-72 below:
đ 
đđ22
=.
đ
2đđ2 (đđ˘2 â đđ˘1 )

(1-72)

The Zweifel coefficient depends on the number of blades, axial blade length, blade
spacing, and the blade angles, which are all interconnected. Blades which are closer together
have lower Z, and results in higher weight, increased cost, reduced blade loading per blade, and
increased friction losses due to increased surface area. Blades which are farther apart have higher
Z, lower cost, lower weight, more tangential force distributed per blade, but lower friction losses,
due to the decreased blade surface area [22].
Farther apart blades, therefore higher loading per blade can increase losses resulting
from diffusion and adverse pressure gradients, while offering the benefit of lower friction. As
44

Zweifel coefficient is raised further the risk of flow separation increases [28] [4] [23]. Literature
suggests that for the designs with higher Zweifel coefficient, losses via friction from the blade
surfaces, hub and shroud are reduced, however the losses resulting from diffusion and trailing
edge losses are higher [24] [28].
1.14.1. Recommended Values of Zweifel Coefficient

Zweifel suggested that for turbine cascades with incompressible flow at low Mach
numbers, minimum profile losses occur and maximum efficiency can be reached when đ = .8
[26]. However since Zweifelâs original paper, additional work has been done to determine
optimal values of đ, after Zweifels initial work Pfiel demonstrated the optimal value of the
coefficient varies from đ =.75 to 1.15 depending on flow deflection [30]. The experimental data
used to determine the optimal value of Zweifel coefficient is based on dated turbine blade
profiles, with moderate loading [4].
E.Dick noted the recommended value of the Zweifel coefficient does not change
depending on the Mach number, and works for both incompressible and compressible fluids. He
mentions that the optimal value of Zweifel coefficient is between 1.0 and 1.2, with separation
beginning with values between 1.4 and 1.6 [27]. Although there is no agreement on a Zweifel
coefficient that results in the highest efficiency, the literature suggests that for a given turbine
design, the peak efficiency will be measured at a Zweifel coefficient somewhere between .75 and
1.2.
1.14.2. Industry Trend - Higher Zweifel Coefficients

A major trend in the turbine industry is the design of blades with ever increasing Zweifel
coefficients, usually achieved by decreasing the number of blades. The desire to use turbine
designs with higher Zweifel coefficients is primarily motivated by the desire to decrease total
cost of ownership of turbine equipment. Blade profile design with high Zweifel coefficients is
45

an area of active research, motivated primarily by the need to reduce part count and engine
weight in commercial gas turbines, especially jet engines. [31] Reducing part count reduces
capital cost, and service cost, reducing total cost of ownership. Decreasing the number of blades
increases the tangential force đ imparted per blade, this increases Zweifel coefficient. The
definition of the blade spacing, equation 1-33, and equation 1-72 show that reducing the number
of blades increases blade spacing đ , in turn increasing Zweifel coefficient. Reducing the number
of blades reduces the number of parts and lowers material costs, which in turn reduces
manufacturing and maintenance costs [31]. This is considered important, especially in the
aviation industry where engine weight is critical.
Zweifel coefficient can also be increased by reducing the axial length of the blades, đ.
Reducing the axial blade length has a similar effect on the tangential loading to reducing the
number of blades, this is because Zweifel coefficient is a measure of the tangential loading
experienced per blade, weighed against the axial length. Reducing axial blade length is
advantageous from a cost perspective as well, as blades with lower axial length require less
material, cost less to manufacture, and weigh less.
Tests involving the highest Zweifel coefficient blade profiles demonstrated (in a cascade)
were conducted by Praisner et. al, who tested airfoils with Zweifel coefficients between 1.6 and
1.82 [32]. However there has been less success in demonstrating airfoils with very high Zweifel
coefficients in actual commercial gas turbines [28]. Schmitz et. al. developed a turbine stage
called the Notre Dame Highly Loaded Turbine 01 (ND-HiLT01), with a rotor Zweifel coefficient
of 1.35, with a blade loading coefficient of 2.8, and measured a stage efficiency of 90.6% [31].
The ND-HiLT01 was developed to demonstrate a reduction in stage count for a gas turbine
engine and the part count of an individual airfoil row.

46

1.14.3. Zweifel Analysis

Reducing the number of blades and reducing the axial blade length both increase Zweifel
coefficient for a design. The below Figure 1.24 shows how changing axial blade length đ affects
Zweifel coefficient, for a turbine with same power and flow rate as the first simulation case in
this work. The designs in the Blade Length Study in this work had their Zweifel coefficients
adjusted in this way by changing đ.

b vs Z
Zweifel Coefficient

2.5
2
1.5
1
0.5
0
0

0.05

0.1

0.15

0.2

Blade length b (m)
Figure 1.24. Blade length vs. Zweifel coefficient

Zweifel coefficient can also be changed by changing the number of blades of the design,
which effects the blade spacing đ . Reducing the number of blades increases the tangential force
đ which is distributed per blade, thus increasing đ. The below Figure 1.25 shows this effect for a
turbine design.

47

Zweifel Coefficient

Number of blades vs. Z
2.5
2
1.5
1
0.5
0
0

5

10

15

Number of blades Nb
Figure 1.25. Number of blades vs. Zweifel coefficient

Zweifel coefficient can be changed while keeping relative flow angles, mass flow rate,
and power constant by changing the number of blades or the blade length. Zweifel coefficient
can be changed by changing the tip speed đ, however the relative flow angles will change. The
tip speed đ has a significant effect on Zweifel coefficient, designs with larger tip speeds have
lower Zweifel coefficients. The below Figure 1.26 shows how changing đ effects đ for different
turbine designs with the same power output, with three different blade lengths.

Zweifel Coefficient

Tip Speed U vs Z
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

b=0.06
b=0.02
b=0.1

0

5

10

15

20

Tip Speed U (m/s)
Figure 1.26. Tip speed vs. Zweifel coefficient at differing blade lengths

The above figure illustrates that designs with high tip speeds have low values of Zweifel
coefficient. As tip speed is increased for a given design, to keep Zweifel coefficient near typical
48

levels between ..75 and 1.2, blade length has to be decreased. This was demonstrated in the
simulations done in the Speed Study, in particular the designs with the tip speed recommended
by the Cordier diagram.
1.15. Cavitation
Cavitation occurs when the static pressure of the flow drops below the vapor pressure.
Vapor bubbles are created by this effect, and these bubbles can collapse suddenly if the pressure
rises later in the fluid stream. These collapsing bubbles can severely damage water turbines
blades if the bubble collapse occurs on the turbine blade. The collapsing bubbles produce shock
waves and microjets which produce high pressures and temperatures in a short period of time.
Over time this effect causes fatigue in the blade material causing erosion [33]. Erosion resulting
from cavitation occurs in pumps, water turbines, propellers, and valves. Cavitation is often
generated on the suction surface of hydraulic turbine blades.
1.15.1. Cavitation for Composites
Composites are considered an attractive material choice for turbine blades, as the high
specific strength and stiffness of these materials are superior to that of metal blades [34]. The
cost of composite materials is also becoming cheaper, leading turbomachinery manufacturers to
move to composite blades in many applications. The Woven Wheel technology, which is
constructed out of continuous composite fiber, could be employed in a hydraulic turbine
application using a CTPAT setup, is useful to investigate literature to see how composites can
perform under cavitation conditions.
Studies by Yamatogi et al. investigated the cavitation resistance of composite propellers
and the mechanism by which composite impellers are damaged by cavitation. Samples of three
types of composites using reinforced fibers were tested under cavitation conditions, carbon fiber,

49

glass fiber, and aramid fibers. Specimens made of epoxy resin and aluminum bronze molded
NAB (CAC703) were also tested. The study demonstrated the aramid fiber reinforced composite
materials exhibited less erosion than carbon or glass fibers under cavitation. It was found the
resistance to cavitation erosion was superior in situations where the adhesion between fiber and
resin was stronger [34].
A study by M. ÄosiÄ, M. DojÄinoviÄ & Z. AÄimoviÄ-PavloviÄ measured the cavitation
resistance of aluminum matrix composite with silicon carbide reinforcement particles, and found
the mass loss as the result of cavitation erosion was close to the mass lost for the same test
performed for CA6NM stainless steel (a 13Crâ4Ni soft martensitic stainless steel). CA6NM is
known for good corrosion and cavitation resistance, and is commonly used in hydraulic
machinery [33]. These studies suggest that most composite blades are more susceptible to
corrosion via cavitation than metal, and careful choice of fiber and resin is required to use
composites for hydro propellers.
1.16. Discussion of Loss Mechanisms

Figure 1.27. Trailing edge of blade with separated boundary layer [32]

50

Denton described the mechanisms surrounding viscous friction and entropy production in
turbomachines [35]. Viscous friction is the result of viscous shear, which can occur in either
boundary layers or mixing processes. A large portion of entropy generation in turbomachines is
due to viscous shear, when the fluid undergoes a rate of shear strain. Velocity gradients which
cause viscous shear are experienced at the boundary layers, the leading and trailing edges, and
anywhere where flow separation is seen. Viscous shear stress is defined below in equation 1-73,
where đ˘ is the local flow velocity, đ is the local shear stress, and đŚ is position along a boundary
layer [35]:
đ(đŚ) =

đđđ˘
đđŚ

(1-73)

The rate of entropy generation per unit surface volume in a boundary layer is shown
below in equation 1-74, where đ is the local temperature [35]:
đĚ =

1 đđ˘
đ
đ đđŚ

(1-74)

Entropy production in the boundary layer is proportional to the velocity cubed. This is
why the entropy is generated more rapidly on the suction side of turbine blades than on the
pressure side, where velocity is higher. Most boundary layers have velocity changing the most
rapidly in the inner part of the boundary layer, especially for turbulent flow, the inner layer is
where most entropy is generated within a boundary layer [32].
The wake left by the trailing edge of a turbine blade is considered a major source of
losses. When two fluid streams at different velocity, pressure, or temperature mix together,
entropy is generated. High shearing rates occur in wakes left after the trailing edge. Figure 1.27
above shows an example turbine blade trailing edge with separated boundary layer. Denton
estimated the loss that can be attributed to the trailing edge can be estimated to be about 32% of

51

the boundary layer losses or 21% of the total losses in a subsonic turbine blade row. Thicker
blades are correlated with additional losses at the trailing edge [35].
Due to the adverse pressure gradient arising from flow diffusion on the suction surface
downstream of the minimum pressure, boundary layer separation near the trailing edge can
occur, and is considered a major source of blade profile losses [35]. As flow moves from leading
to trailing edge along the suction side and decelerates, the adverse pressure gradient can lead to
backflow in the boundary layer. Flow separation can occur when there is backflow in the
boundary layer, where local change velocity along the boundary layer

đđ˘
đđŚ

becomes negative. This

can lead to a distinct wake region, and for blades with high Zweifel coefficient, flow separation
is of added concern [29]. With higher Zweifel coefficients, pressure on the suction side is
reduced and pressure on the pressure side is increased, the resulting increased difference in
velocity from the suction side to the pressure side can lead to additional shear stress, entropy
generation, and losses at the trailing edge. Turbine designs with higher Zweifel coefficients have
a higher degree of flow deceleration on the suction side of the blades downstream the point of
minimum pressure [24]. Another major source of loss is referred to as âsecondary lossâ, and
includes friction losses resulting from the hub and shroud surfaces, as well as losses resulting
from secondary flows near the hub and shroud [35]. Denton estimated that for turbines, the
secondary losses are considered a major source of loss, contributing typically 1/3 of the overall
loss [35].
1.17. Goals of This Thesis and Description of Work
This work explores the viability of using the CTPAT turbine design scheme for low-head
hydroelectric applications, at speeds inside and outside of the recommendations of the Cordier
line. This work also aims to evaluate the applicability of the literature recommendations of the
52

value of Zweifel coefficient for these types of machines, and to see how varying Zweifel
coefficient by varying blade length effects performance.
Operating with highest Zweifel coefficients requires careful design of the blade profile,
as turbines with higher Zweifel loading coefficient are correlated with adverse pressure gradients
on the suction side of the blades, and have an added risk of separation. In this work, the
performance of the constant thickness blades in an axial turbine of the CTPAT type under
varying Zweifel coefficients are tested using CFD. In the gas turbine industry, blade profiles are
carefully designed using CFD in an effort to maximize performance at higher Zweifel
coefficients. However detailed design of blade profiles can be computationally expensive and
detailed precise blade profiles can be more expensive to machine. This work approaches this
problem from a different perspective, using a simple blade profile and observing how it performs
under differing conditions in simulation, with differing specific speeds and Zweifel coefficients.
The first hypothesis tested is that designs of the type simulated with specific speed
recommended by the Cordier line will perform with lower efficiency than designs with specific
speed lower than that recommended by the Cordier line. This work investigated whether designs
below the Cordier line (lower than recommended specific speed) perform with lower efficiency
than designs of similar geometry, on the Cordier line. The second hypothesis tested is that peak
efficiency will be obtained in CFD for designs with Zweifel coefficients in the range between
0.75 and 1.2, and decreased efficiency will be experienced at Zweifel coefficients above and
below that range.
To evaluate the applicability of the CTPAT design scheme to a low head hydro
application and to evaluate these hypotheses, turbine designs were generated which defined the
inputs and geometry for the CFD simulations. Losses resulting from a penstock pipe for a

53

hydroelectric dam retrofit application were estimated to estimate the total inlet pressure. Six sets
of turbine specifications were investigated in CFD. First for the âSpeed Studyâ, designs at each
set of specifications at different rotational speeds were developed to test the performance of
designs inside and outside of the Cordier recommendations. The highest efficiency designs from
each case in the Speed Study were then simulated in the âBlade Length Studyâ, with varying
axial blade lengths to investigate the effect of varying Zweifel coefficient on performance. The
effect of varying specific speed and Zweifel coefficient on efficiency, trailing edge losses,
friction losses from the blades and hub and shroud surfaces, flow deviation, and cavitation are
investigated. Constraints were imposed on the geometry to ensure the designs fit the geometric
and manufacturing constraints of the CTPAT design scheme and the Woven Wheel
manufacturing process. The analytical model was set up to obtain CFD boundary conditions
which were applicable for a proposed application of a CTPAT turbine for use as a retrofit unit to
fit in front of dam spillways. Chapter 2 focuses on the analysis setup, the results of the CFD
simulations are discussed in Chapter 3.

54

CHAPTER 2. ANALYSIS DESCRIPTION AND METHODS
2.1. Description of Setup
An example application of a CTPAT turbine in a hydroelectric application is for
hydroelectric dam retrofits, discussed in Chapter 1 of this work. The analytical model was set up
to obtain CFD boundary conditions which were applicable for a proposed application of a
CTPAT turbine for use as a retrofit turbine unit to fit on the front of dam spillways. Figure 2.1
below shows a simplified schematic of the setup.

Turbine

Figure 2.1. Hydroelectric dam retrofit schematic with height positions

The heights defined by đ§a , đ§b , đ§1 , đ§2 , đ§c are input for this analysis. Each position is defined below.
ďˇ

Position a: water surface/inlet of penstock pipe

ďˇ

Position b: outlet of penstock pipe/inlet of turbine entry annulus

ďˇ

Position 1: Turbine leading edge/outlet of turbine entry annulus

ďˇ

Position 2: Turbine trailing edge /inlet of turbine exit annulus

ďˇ

Position c: Outlet of turbine exit annulus, water level

55

2.2. Analysis Description, Description of Specifications
An analytical model was constructed in Microsoft Excel to define the inputs and
boundary conditions needed for the CFD analysis. To simulate the designs in CFD, the below
inputs are required. The purpose of the analytical model was to obtain these inputs.
ďˇ

Total pressure at inlet of the turbine entry annulus, đđĄđ , accounting for estimated
friction losses in penstock pipe

ďˇ

Mass flow rate đĚ

ďˇ

Tip speed of rotor đ1 = đ2

ďˇ

Relative blade angles at leading and trailing edges, đ˝đ1 and đ˝đ2 respectively

ďˇ

Density of water đ

ďˇ

Geometry:
o Inside and outside radius of turbine rotor đâ , đđ  respectively, equal to radii
of entry and exit annuli
o Blade thickness đĄ
o Number of blades đđ

This model takes turbine and penstock geometry, head, and flow rates as inputs and
calculates values of absolute and relative flow speeds/angles, estimated total pressure at the
turbine entry annulus, as well as blade angles for CFD analysis. The analytical model was made
to take account for losses in a penstock pipe, including friction resulting from the pipe surfaces,
and the friction loss associated with a 90 degree turn of the penstock. Six different sets of
specifications relevant to low-head hydro applications were investigated. Three size turbines
were considered, at two different values of total available head, đť. Each of the six sets of
specifications had its own input volume flow rate. Designs at each set of specifications, referred
56

to as a âcaseâ, were generated at differing rotational speeds for the Speed Study. After finding
the RPM where efficiency was measured highest through CFD, a new set of designs were
generated for each case, each with a different axial blade length and corresponding Zweifel
coefficient. This second set of designs were simulated during the Blade Length Study.
đ3

ďˇ

Case 1: đť = 2đ, .2 m OD, .12m ID, đĚ = .46

ďˇ

Case 2: đť = 2đ, .3 m OD, .18 m ID, đĚ = 1.05

ďˇ

Case 3: đť = 2đ, .5 m OD, .3 m ID, đĚ = 2.85

ďˇ

Case 4: đť = 4đ, .2 m OD, .12m ID, đĚ = .6

ďˇ

Case 5: đť = 4đ, .3 m OD, .18 m ID, đĚ = 1.5

ďˇ

Case 6: đť = 4đ, .5 m OD, .3 m ID, đĚ = 4.1

đ 
đ3
đ 

đ3
đ 

đ3
đ 
đ3
đ 

đ3
đ 

The surface roughness of 314 stainless steel was used for the penstock pipe friction
calculations. The temperature of the water flow was assumed to be 20 degrees Celsius for the
calculations. This temperature was needed to find the kinematic viscosity and vapor pressure of
the water flow. The analysis and CFD simulations did not include a draft tube, an important and
necessary part of a turbine installation which can be used to decelerate the flow. For use in a real
application a draft tube will need to be included as a part of the design.
2.3. Estimation of Friction Losses
The purpose of this section is to demonstrate how the inputs for CFD described above
were determined for the designs later simulated in CFD. For this analysis, Bernoulliâs equation
was used to estimate the total work extraction from the fluid, đ, which was needed to define the
relative and absolute flow angles needed to generated designs for CFD analysis, as well as the
total pressure at the inlet of the entry annulus, đđĄđ . Bernoulliâs equation is a modified version of
57

the energy conservation equation, and is defined below in equation 2-1, Where đ is static gage
pressure, đś is absolute flow speed, đ§ is vertical distance from the datum, water level, đ is the
gravitational acceleration constant, đšđ is the friction work done per unit mass of a fluid element
while moving from positions đ and đ along a streamline in the direction of flow, and đ represents
total mass-specific energy extraction from positions đ to đ [4]:
đđ đśđ2
đđ đśđ2
+
+ đđ§đ = +
+ đđ§đ + đšđ + đ
đ
2
đ
2

(2-1)

For the analysis in this work, đ was estimated by carrying out Bernoulliâs equation
between each system position, a through c, then combining the equations to obtain an expression
for đ. This begins with points a and b, from the water surface to the outlet of the penstock pipe,
shown below in equation 2-2.:
đđ đśđ2
đđ đśđ2
+
+ đđ§đ = +
+ đđ§đ + đšđđđ
đ
2
đ
2

(2-2)

For this analysis it is assumed that the velocity at the water surface is zero, and the gage
static pressure at the water surface is zero. Thus đśđ , đđ = 0. It is also assumed that the motion of
the water at the outlet of the penstock pipe is purely axial in direction. The velocity of the water
at the outlet of the penstock pipe can be calculated using equation 2-3 below, using the
continuity equation.
đśđ =

đĚ
đđ´đđ

(2-3)

Mass-specific friction loss from positions a to b đšđđđ is calculated using the friction loss
equation for flow in a pipe added to the friction loss equation due to a 90 degree bend, where đđ
is the friction loss coefficient for the penstock pipe, and đž is the friction loss coefficient resulting
from the 90 degree turn, shown below in equation 2-4:
58

đšđđđ =

đđ đśđ2 đžđśđ2
+
2
2

(2-4)

đđ is found using the below equation 2-5, a re-ordered version of the Darcy-Weisbach equation,
where đżđĄ is the length of the penstock pipe, đđđ is the friction factor for the penstock pipe, and đđ
is the radius of the penstock pipe [36]:
đđ =

đđđ đżđĄ
2đđ đ

(2-5)

There are many equations which can be used to estimate this friction factor for a fullflowing circular pipe, for this analysis the Haaland equation is used, valid for turbulent flow. The
accuracy of the friction factor calculated using this equation is within Âą2% for Reynolds
numbers above 3000 [36]. First đđđ is calculated using equation 2-6 below, the Reynolds
number at position b, where đ is the kinematic viscosity of water [4]:
đđđ =

đśđ đđ
đ

(2-6)

Then the Haaland equation is applied below in equation 2-7, đđ is the surface roughness
[36].
đđđ =

1
2

đđ 1.11
6.9
2đđ
(â1.8 log (đđ + ( 3.7 ) ))
đ

(2-7)

The penstock pipe modeled as a part of this analysis turns 90 degrees, before the flow
reaches the turbine leading edge. The losses resulting from turning the flow 90 degrees were
predicted. The friction loss coefficient resulting from the 90 degree turn, đž, is predicted using
correlations from literature, specifically a chart for predicting đž for 90 degree bends of uniform
diameter obtained from the Hydraulic Instituteâs Pipe Friction Manual [37]. After obtaining the
predicted friction factors, Equation 2-4 was then simplified and re-arranged to form equation 2-8
below:
59

đđ
đśđ2
= đ(đ§đ â đ§đ ) â
(1 + đđ + đž)
đ
2

(2-8)

Then Bernoulliâs equation for positions b to 1 (from the inlet of the entry annulus to the
leading edge of the turbine) is applied in equation 2-9 below.
đđ đśđ2
đ1 đś12
+
+ đđ§đ = +
+ đđ§1 + đšđđ1
đ
2
đ
2

(2-9)

The region between the inlet of the entry annulus to the leading edge of the turbine are
simulated as a part of the CFD analysis. As such the friction loss from positions b to 1, đšđđ1 , is
accounted for in the CFD simulation. The expression for

đđ
đ

(equation 2-8) is then plugged into

equation 2-9 to obtain equation 2-10 below:
đ(đ§đ â đ§đ ) â

đśđ2
đśđ2
đ1 đś12
+ đ(đ§đ â đ§1 ) = +
(1 + đđ + đž) +
2
2
đ
2

(2-10)

Equation 2-10 is then simplified to obtain equation 2-11 below:
đ(đ§đ â đ§1 ) â

đđ đśđ2 đžđśđ2 đ1 đś12
â
= +
2
2
đ
2

Rearranging equation 2-11 to solve for

đ1
đ

(2-11)

, equation 2-12 is formed:

đđ đśđ2 đžđśđ2 đś12
đ1
= đ(đ§đ â đ§1 ) â
â
â
đ
2
2
2

(2-12)

Bernoulliâs equation is then applied for positions 1 to 2 (turbine leading edge to trailing
edge) in equation 2-13 below. The total work extracted from the fluid by the turbine is đ.
đ1 đśđ2
đś22 đ2
+
+ đđ§1 =
+ + đđ§2 â đ
đ
2
2
đ

(2-13)

Plugging equation 2-12 into equation 2-13 and rearranging, equation 2-14 below is
formed,
đ(đ§đ â đ§1 ) â

đđ đśđ2 đžđśđ2 đś22 đ2
â
=
+ + đ(đ§2 â đ§1 ) + đ
2
2
2
đ

60

(2-14)

Bernoulliâs equation is then applied for positions 2 to c, from the turbine trailing edge to
the outlet of the exit annulus, seen below in equation 2-15.
đ2 đś22
đđ đśđ2
+
+ đđ§2 = +
+ đđ§đ + đšđ2đ
đ
2
đ
2

(2-15)

The region between the trailing edge of the turbine to the outlet of the exit annulus are
simulated as a part of the CFD analysis. The friction loss from positions 2 to c, đšđđđ , is accounted
for in the CFD simulations. As position c is located at the datum, water level, đđ = 0.
Rearranging equation 2-15 and simplifying, equation 2-16 below is formed:
đ2 đśđ2
đś22
=
+ đ(đ§đ â đ§2 ) â
đ
2
2

(2-16)

Plugging equation 2-16 into equation 2-14 and solving for đ gives equation 2-17 below:
đ=â

đđ đśđ2 đžđśđ2
đśđ2
+ đ(đ§đ â đ§đ ) â
â
2
2
2

(2-17)

With mass flow rate and geometry as inputs, absolute meridional flow speed at the
leading edge đśđ1 can be calculated using the continuity equation, applied below in equation 218:
đśđ1 =

đĚ
đ´đ đ

(2-18)

The turbine is designed so đśđ stays constant from leading to trailing edge, such
that đśđ1 = đśđ2 . Flow angle at the leading edge đź1 is also input, allowing đśđ˘1 to be calculated
using equation 1-19, applied below in equation 2-19:
đśđ˘1 =

đśđ1
tan đź1

(2-19)

Equation 2-17 is then rearranged to be expressed in terms of variables with known
values. To simplify the estimation of đ, đśđ , the velocity of the flow at the outlet of the exit
annulus, is assumed to be equal to đś2 , the velocity at the trailing edge. đś2 is then expressed in

61

terms of the mass-specific turbine shaft work đĚ , by first expressing đś2 in terms of its meridional
and tangential velocity components, seen below in equation 2-20:
2
2
đś22 = đśđ2
+ đśđ˘2

(2-20)

Eulerâs equation of turbomachinery, expressed in equation 1-40, is then applied between
positions 1 and 2 (leading edge to trailing edge of turbine rotor), and then solved for đśđ˘2 , shown
below in equation 2-21:
đśđ˘2 =

đĚ + đśđ˘1 đ1
đ2

(2-21)

đĚ

Using đ = đ, where đ is isentropic turbine efficiency, equation 2-21 is expressed in
equation 2-22:
đśđ˘2 =

đđ + đśđ˘1 đ1
đ2

(2-22)

For the designs considered in this analysis, hub and shroud radii stayed constant from
leading to trailing edge, such that đ1 = đ2 . For this analysis it was assumed that đśđ stays
constant from leading to trailing edges, such that đśđ1 = đśđ2 . Equation 2-22 above is then
plugged into equation 2-20 to express đś2 in terms of known variables, shown below in equation
2-23:
đś22

=

2
đśđ1

đđ + đśđ˘1 đ1 2
+(
)
đ2

(2-23)

Equation 2-23 is then plugged into equation 2-17 to express đ in terms of known
variables, shown below in equation 2-24:
2
đśđ1
+(

đ=â

đđ + đśđ˘1 đ1 2
)
đđ đśđ2 đžđśđ2
đ2
+ đ(đ§đ â đ§đ ) â
â
2
2
2

62

(2-24)

To determine the absolute and relative flow speeds and angles, a value of đ = 100%
(ideal) was assumed, which allows for preliminary calculation of relative flow angles at the
trailing edge that are necessary for generating designs for CFD. The quadratic equation is used to
solve equation 2-24 for đ. After đ was calculated, đśđ˘2 needed to be determined in order to obtain
the relative and absolute flow angles necessary for defining the turbine blade geometry. As the
efficiency of the machine is not yet known, accurate values of đśđ˘2 , đĚ , and đ˝2 cannot yet be
determined, and for this analysis are obtained via CFD simulations.
The CFD simulations require inputting blade angles, đ˝đ1 and đ˝đ2 . For an ideal turbine, đĚ
is equal to đ, and đ˝2 is then đ˝đ2 = đ˝2. However, the turbine will not perform at 100% efficiency,
and the flow does not follow the blades perfectly, so the trailing edge relative flow angle đ˝2 was
reached in CFD by specifying an trailing edge blade angle đ˝đ2 and adjusting it until the total
mass-specific work extraction achieved through CFD was equal to đ.
An initial value of đ˝đ2 was required as an input to conduct the first simulation, this is
defined as đ˝đ2đ . Flow speeds and angles were calculated by first inputting an efficiency of 100%
into equation 2-22 to obtain an initial, idealized value of absolute circumferential trailing edge
velocity đśđ˘2đ . Then initial, idealized values of đ2 , đđ˘2 , đś2 , đź2 , đ˝1 were calculated, denoted in
this work as đ2đ , đđ˘2đ , đś2đ , đź2đ , đ˝1đ . Equations 1-18,1-31,1-17,1-19, and 1-20 were used,
respectively, to calculate these initial values. Finally, equation 1-20 was used to obtain đ˝đ2đ ,
substituting đ˝đ2đ for đ˝2 in the expression. An initial value of đ˝đ1 = đ˝1đ was used for the first
simulation. After conducting a first simulation with đ˝đ2 = đ˝đ2đ , đ˝đ2 was reduced iteratively until
the desired value of đ was reached in CFD. After obtaining the desired work extraction through
CFD, đĚ , đ˝2, đśđ˘2 , and efficiency đ were logged.

63

The calculations to obtain the flow speeds and angles were carried out at three positions
along the blade: at the hub, shroud, and 50% span. Relative flow and blade angles were
calculated such that the product of the rotor tangential speed multiplied by the ideal absolute
tangential flow speed đđ đśđ˘đ is constant across the blade span at the trailing edge.
The total pressure at the inlet of the turbine entry annulus, đđĄđ was then calculated, this
value was used as a total pressure boundary condition for the annulus inlet for the CFD
simulations. To simplify the analysis the gravitational potential energy of the turbine đđ(đ§đ â
đ§đ ) was applied in the form of total pressure at the annulus inlet for CFD simulations đđĄđ . This is
shown below in equation 2-25, where đšđđđ is the sum of the friction losses considered above per
unit mass from points a to c.
đđĄđ = đđ(đ§đ â đ§đ ) + đđđĄđ â đđšđđđ

đđ đśđ2
đžđśđ2
= đđ(đ§đ â đ§đ ) + đđđĄđ â đ
âđ
2
2

(2-25)

2.4. CFD Methodology and Assumptions
2.4.1. CFD Overview

The analytical model was used to obtain the boundary conditions and inputs for
computational fluid dynamics (CFD) software. The software package used was ANSYS. The
CFD analysis was carried out in ANSYS CFX. Computational Fluid Dynamics (CFD) software
solves modified forms of the Navier-Stokes equations, as well as continuity. The Navier-Stokes
equations describe how the flow velocity, pressure, temperature, and density of a viscous fluid
are related. The Navier-Stokes equations in the x, y, and z direction are listed below for
incompressible flow with constant viscosity, where đ˘ represents the local velocity, equation 2-26
is in the x direction, equation 2-27 in the y direction, and equation 2-28 is in the z direction:
đ(

đđ˘đĽ
đđ˘đĽ
đđ˘đĽ
đđ˘đĽ
đđ
đ 2 đ˘đĽ đ 2 đ˘đĽ đ 2 đ˘đĽ
+ đ˘đĽ
+ đ˘đŚ
+ đ˘đ§
)=â
+ đđđĽ + đ( 2 +
+
)
đđĄ
đđĽ
đđŚ
đđ§
đđĽ
đđĽ
đđŚ 2
đđ§ 2

64

(2-26)

đđ˘đŚ
đđ˘đŚ
đđ˘đŚ
đđ˘đŚ
đ 2 đ˘đŚ đ 2 đ˘đŚ đ 2 đ˘đŚ
đđ
đ(
+ đ˘đĽ
+ đ˘đŚ
+ đ˘đ§
)=â
+ đđđŚ + đ( 2 +
+
)
đđĄ
đđĽ
đđŚ
đđ§
đđŚ
đđĽ
đđŚ 2
đđ§ 2
đ(

đđ˘đ§
đđ˘đ§
đđ˘đ§
đđ˘đ§
đđ
đ 2 đ˘đ§ đ 2 đ˘đ§ đ 2 đ˘đ§
+ đ˘đĽ
+ đ˘đŚ
+ đ˘đ§
)=â
+ đđđ§ + đ( 2 +
+
)
đđĄ
đđĽ
đđŚ
đđ§
đđ§
đđĽ
đđŚ 2
đđ§ 2

(2-27)

(2-28)

The continuity equation is below in equation 2-29 for incompressible flow:
đđ˘đĽ đđ˘đŚ đđ˘đ§
+
+
=0
đđĽ
đđŚ
đđ§

(2-29)

The four equations are all coupled. Time dependent solutions of the Navier-Sokes
equations are too computationally expensive to be feasible for most situations. Reynoldsaveraging is used to simplify the process of solving the equations, in a way such that small scale
turbulent fluctuations do not need to be simulated. The Reynolds averaged Navier-Stokes
equations are time-averaged versions of the Navier-Stokes equations. For Reynolds averaging,
the velocity and scalar variables of the Navier-Stokes equations are converted into mean (time
averaged) and fluctuating components, shown below in equation 2-30, where đ˘đ are the mean
velocity components (j=1,2,3) and đ˘đ â˛ are the fluctuating velocity components:
đ˘đ = đ˘đ + đ˘đ â˛

(2-30)

Pressure and the other scalar quantities in the Navier-Stokes equations are converted into
mean and fluctuating components as well. The variables in this form are substituted into the
Navier-Stokes continuity and momentum equations to derive the Reynolds averaged NavierStokes (RANS) equations, shown below in index notation in equation 2-32, where đżđđ is the
Kroneker delta, the definition of which is listed in equation 2-31 [36].
0 đđđ đ â  đ
đżđđ = {
1 đđđ đ = đ

(2-31)

đđ˘đ
đđ˘đ 1 đ
đđ˘đ
+ đ˘đ
=
(âđđżđđ + đ
â đđ˘đâ˛ đ˘đâ˛ )
đđĄ
đđĽđ đ đđĽđ
đđĽđ

(2-32)

65

â˛
The last term represents the Reynolds stresses, đđđ
, defined below in equation 2-33.
â˛
đđđ
= âđđ˘đâ˛ đ˘đâ˛

(2-33)

The nonlinearity of the Navier-Stokes equations has the result that the velocity
fluctuations still appear in the RANS equations in the Reynold stress term. To obtain forms of
the RANS equations which contain only the mean velocity and pressure, the Reynolds stress
term needs to be modeled in terms of mean flow, removing references to the fluctuating
component of the velocity. The Reynolds stresses are modeled using the Boussinesq hypothesis,
which relates the Reynolds stresses to mean velocity gradients [38]. Joseph Boussinesq
introduced the concept of eddy viscosity, and proposed relating the Reynolds stresses to the
mean velocity gradients. đđĄ is the turbulence eddy viscosity, đ is the turbulence kinetic energy,
1

defined as đ = 2 đ˘đâ˛ đ˘đâ˛ . Equation 2-34 below describes this relationship [38]:
đ˘â˛đ đ˘â˛đ = đđĄ (

đđ˘đ đđ˘đ
đđ˘đ
2
+
)đż
) â (đđ + đđĄ
đđĽđ đđĽđ
3
đđĽđ đđ

(2-34)

The Boussinesq hypothesis was used for the đ â đ and đ â đ turbulence models used in
the simulations. The đ â đ and đ â đ turbulence models use two additional transport equations
to compute đđĄ as a function of đ, đ and đ. đ is the turbulence dissipation rate and đ is the
specific dissipation rate. The đ â đ turbulence model is considered most useful for free-shear
flows with smaller pressure gradients. The đ â đ turbulence model aims to model near-wall
flow features more accurately than the đ â đ turbulence model. For the CFD simulations
conducted as a part of this work, the Shear Stress Transport (SST) RANS turbulence model was
used. The SST turbulence model combines the đ â đ and đ â đ turbulence models, where đ â đ
is used in the inner region of the boundary layer, and đ â đ is used in the free shear flow.

66

2.4.2. CFD Methodology and Assumptions

For all CFD simulations in this study, a single axial turbine stage was simulated. Each
design simulated included an annulus entry and exit section, each with an axial length of 0.3
meters. The blade profiles for each CFD run was of a constant thickness with rounded leading
and trailing edges. Blade designs were generated using BladeGen. Figure 2.2 below shows
example turbine geometry, and Figure 2.3 below shows the example turbine geometry in
Bladegen.

Figure 2.2. Design 1j5 generated in BladeGen

Figure 2.3. Example turbine geometry with entry and exit annuli

An outer shroud was included in the simulations, eliminating tip leakage from the
simulation. Meshes were generated using ANSYS TurboGrid. All simulations were steady state,
67

and were run until convergence was reached. To track convergence the total pressure at the
annulus outlet and turbine efficiency were monitored until their values stabilized. All CFD runs
were tested using the boundary condition set Total Pressure Inlet/Mass Flow Rate Outlet. Total
pressure at the annulus inlet was input, taking into account the friction losses from the penstock
pipe. Figure 2.2 shows renderings of one of the designs generated in BladeGen. The turbines
simulated ranged in power output from 3KW to 137KW, with heads ranging from 2 to 4 meters.
A number of different flow characteristics and performance metrics were measured for each
CFD run. All data was recorded using CFX-Post.
For the simulations within each case, the total work extraction from the fluid (counting
both shaft work and losses) was kept constant between each design. This was achieved by
changing the trailing edge blade angles đ˝đ2until the total pressure at the outlet of the exit annulus
đđĄđ reaches the desired value, atmospheric pressure. This allows for the efficiency and
performance of each turbine to be compared with each other on equal ground. Blade thickness
was also kept constant within each case.
CFD simulations are not perfect recreations of a physical system, some assumptions need
to be made to simplify the modeling, to reduce computation time. Gravity was not modeled in
the simulations, instead the pressure as the result of gravitational force was applied to the inlet of
the turbine entry annulus. The designs tested in CFD do not include an inlet guide vane. The
CFD simulations assume a uniform flow velocity across the inlet which was assumed to have no
pre-swirl, such that đśđ˘1 = 0, đź1 = 90Â°. A single flow path with blade was modeled in the CFX
simulations, the outputs were dependent upon an assumption of symmetry across each flow path.
Only water was simulated in the blade path, cavitation was not modeled, although areas of low
pressure are tracked using ANSYS CFD-POST. The simulations assumed an inlet water

68

temperature of 20Â°C. The simulations did not take account for flow disturbances and turbulence
resulting from the penstock pipe bend, friction losses were taken into account to obtain a total
inlet pressure for CFD, as described in section 2.2. When employed in a physical application, a
flow-guiding nose cone at the inlet and outlet of the annulus sections will be required, something
the simulations in this work do not include. These simulations did not take surface roughness of
the blades and walls into account, and assume no-slip wall conditions.
Efficiency was obtained from the CFD results, in the form of isentropic efficiency. đĚ is
the shaft power output, and was calculated in CFD by first multiplying the forces imparted on the
blades in x,y, and z axis directions by the absolute flow velocities in those directions, then
summing up the components and multiplying that by the number of blades. This is expressed
below in equation 2-35:
đđ

đĚ = â(đšđĽ đśđĽ + đšđŚ đśđŚ + đšđ§ đśđ§ )

(2-35)

1

Isentropic efficiency was calculated in CFD-Post using the below equation 2-36, where đť
is the total available head.
đ=

đĚ
đĚ đťđđ

(2-36)

2.5. Description and Inputs for Speed Study
In this work, each set of specifications described in section 2.2 were referred to as a
âCaseâ. Designs were generated and simulated at least 6 different rotational speeds for each case,
in addition to the rotational speed recommended by the Cordier diagram. Idealized relative flow
angles at the leading and trailing edges were calculated and used to define the initial blade shape.
First a value of Z = .8 was used to determine the axial blade length at each rotational speed

69

tested. Each RPM had its own corresponding axial blade length đ which was adjusted to keep
Zweifel coefficient at .8.
In this analysis, preliminary geometries were first generated in the analytical model, with
leading edge and trailing edge flow beta angles specified. For each rotational speed tested, new
leading edge and trailing edge relative flow angles were calculated using the analytical model.
For the first simulations, the leading edge blade angle đ˝đ2 was kept the same as the ideal flow
beta angle đ˝2đ . The total pressure at the annulus outlet will not reach the desired value, so for
each design the trailing edge blade beta angle was reduced until the total pressure at the annulus
outlet đđĄđ reached within 200 pA of the desired value, atmospheric pressure. When the total
pressure at the annulus outlet reached the desired value, the data was recorded. For the Speed
Study, Zweifel coefficient was kept at the original recommendation by Zweifel, Z = 0.8. This
was achieved by using equation 1-72 to calculate axial blade length đ, using a value of 0.8 for Z.
Each of the six cases had a rotational speed at which the highest efficiency and power
output was measured.

The cases at which these geometries produced the highest

power/efficiency were considered for the next analysis, the Blade Length Study. A flow chart
describing the CFD simulation process for the Speed Study is shown in Figure 2.4 below. The
below process was repeated for all six cases.

70

Figure 2.4. Process flow chart for Speed Study

Table 2.1 below shows the geometric parameters, idealized flow speeds, flow angles, and
ideal absolute circumferential flow speed at the trailing edge đśđ˘2đ which was used to obtain đ˝2đ
for the first simulation, for two designs within Case 1 of the Speed Study, designs 1j and
1CORD. The table also shows the final values of đ˝đ2 that were used to obtain the results. For the
simulations of each design, đ˝đ1 was kept equal to đ˝1đ , as first simulations showed incidence
đ â˛ was close to zero for every design simulated. Table 2.1 below shows how the higher speed
designs in the Speed Study, in this case 1CORD, require a smaller axial blade length to keep Z at

71

.8. The designs were set up such that the product of the ideal circumferential flow speed and the
circumferential rotor velocity, đđ đśđ˘đ was kept constant at all positions across the blade span.

đĄ

đđ

đ
0.005
0.005
0.005
0.005

đ
0.200
0.160
0.165
0.120

đĄ

đđ

đ
0.005
0.005
0.005
0.005

đ
0.200
0.160
0.165
0.120

Table 2.1. Case 1 Geometric Parameters and Flow Angles
1j
đ˝1đ đ˝1đ
đ˝2đ
đđ
đśđ˘1 đśđ˘2đ đđ đśđ˘đ
đź2đ
đ˝đ2
2
đ
đđđ đđđ đđđ đđđ đđđ
đ/đ  đ/đ  đ/đ 
đ 2
3.6
0.0
-5.1
-18.0 129.9 59.1 59.1 34.6 26.8
2.8
0.0
-6.3
-18.0 136.1 64.4 64.4 33.0 25.2
2.9
0.0
-6.1
-18.0 135.3 63.8 63.8 33.3 25.5
2.1
0.0
-8.4
-18.0 144.0 70.3 70.3 29.6 21.8
1cord
đđ
đśđ˘1 đśđ˘2đ đđ đśđ˘đ
đź2đ
đ˝1đ đ˝1đ
đ˝2đ
đ˝đ2
đ2
đđđ đđđ đđđ đđđ đđđ
đ/đ  đ/đ  đ/đ 
đ 2
15.4 0.0
-1.2
-18.1 101.0 21.1 21.1 19.7 14.4
12.3 0.0
-1.5
-18.1 103.6 25.8 25.8 23.3 18.0
12.7 0.0
-1.4
-18.1 103.2 25.1 25.1 22.8 17.5
9.3
0.0
-2.0
-18.1 107.9 32.7 32.7 28.0 22.7

đ
đ
0.098
0.098
0.098
0.098
đ
đ
0.022
0.022
0.022
0.022

Z

0.8
0.8
0.8
0.8
Z

0.8
0.8
0.8
0.8

Velocity triangles of the designs were produced, Figures 2.5 and 2.6 show velocity
triangles at leading and trailing edges for design 1j 1Cord of the Speed Study respectively,
superimposed over the blade profile design produced using BladeGen. The higher speed designs,
including 1Cord, have lower relative flow angles at leading and trailing edges, this is due to the
higher rotational speed of the rotor for these designs.

72

Figure 2.5. Velocity triangles for leading and trailing edges of design 1j of the Speed Study (50% span)

73

Figure 2.6. Velocity triangles for leading and trailing edges of design 1CORD of the Speed Study (50%
span)

The below tables show the rotational speeds, axial blade length, and non-dimensional
coefficients of the designs tested in CFD for each case for the Speed Study.

Case 1
đ3
đĚ = 0.46
đ 

đâ = .12 m
đđ  = .2 m

Table 2.2. Case 1 Speed Study Inputs
Run Total Head (m) đ (m) RPM
đ
ÎŠđ 
b
1.88 0.045
350
0.79
2.80
d
1.86 0.056
300
0.92
2.41
f
h
j
l

1.86
1.84
1.85
1.88

0.070
0.086
0.098
0.106

250
200
170
140

74

1.12
1.39
1.63
1.98

2.01
1.62
1.37
1.12

đˇđ 
1.22
1.22
1.22
1.22
1.22
1.22

đ
0.34
0.46
0.66
1.03
1.43
2.14

Case 2
đ3
đĚ = 1.050
đ 

đâ = .2 m
đđ  = .3 m

Case 3
đ3
đĚ = 2.85
đ 

đâ = .3 m
đđ  = .5 m

Case 4
đ3
đĚ = 0.6
đ 

đâ = .12 m
đđ  = .2 m

Case 5
đ3
đĚ = 1.5
đ 

đâ = .2 m
đđ  = .3 m

Table 2.3. Case 2 Speed Study Inputs
Run Total Head (m)
đ (m) RPM
đ
ÎŠđ 
b
1.90 0.031
350
0.53
4.19
d
1.85 0.044
300
0.62
3.67
f
1.88 0.060
250
0.75
3.01
h
1.88 0.086
200
0.94
2.42
j
1.88 0.105
170
1.10
2.05
l
1.85 0.126
140
1.34
1.71
n
1.87 0.142
120
1.56
1.45
Table 2.4. Case 3 Speed Study Inputs
Run Total Head (m) đ (m) RPM
đ
ÎŠđ 
f
1.90 0.034
250
0.45
4.92
h
1.89 0.061
200
0.56
3.96
j
1.92 0.104
150
0.74
2.94
l
1.91 0.148
120
0.93
2.36
n
1.88 0.199
90
1.24
1.79
p
1.88 0.260
60
1.86
1.19
Table 2.5. Case 4 Speed Study Inputs
Run Total Head (m) đ (m) RPM
đ
ÎŠđ 
b
3.73
0.05
450
0.80
2.45
d
3.72
0.06
400
0.90
2.19
f
3.68
0.07
350
1.03
1.93
h
3.70
0.08
300
1.20
1.64
j
3.70
0.09
250
1.45
1.37
l
3.70
0.10
220
1.64
1.21
Table 2.6. Case 5 Speed Study Inputs
Run Total Head (m) đ (m) RPM
đ
ÎŠđ 
b
3.73
0.04
450
0.59
3.88
d
3.67
0.05
400
0.67
3.49
f
3.71
0.06
350
0.76
3.03
h
3.70
0.08
300
0.89
2.60
j
3.72
0.10
250
1.07
2.16
l
3.72
0.11
220
1.21
1.90
n
3.70
0.13
190
1.41
1.65
p
3.72
0.15
160
1.67
1.38

75

đˇđ 
1.22
1.21
1.21
1.21
1.21
1.21
1.21

đ

đˇđ 
1.23
1.23
1.23
1.23
1.23
1.23

đ

đˇđ 
1.27
1.27
1.27
1.27
1.27
1.27

đ

đˇđ 
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20

đ

0.15
0.20
0.30
0.47
0.65
0.94
1.29

0.11
0.17
0.31
0.47
0.83
1.87

0.41
0.52
0.67
0.92
1.33
1.71

0.18
0.23
0.30
0.41
0.59
0.76
1.02
1.44

Case 6
đ3
đĚ = 4.1
đ 

đâ = .3 m
đđ  = .5 m

Table 2.7. Case 6 Speed Study Inputs
Run Total Head (m) đ (m)
RPM
đ
ÎŠđ 
b
3.79
0.11
200
0.79
2.82
d
3.79
0.14
170
0.93
2.39
f
3.78
0.18
140
1.13
1.98
h
3.78
0.22
110
1.43
1.55
j
3.79
0.26
80
1.97
1.13
l
3.75
0.08
250
0.63
3.55

đˇđ 
1.22
1.22
1.22
1.22
1.22
1.22

đ
0.34
0.47
0.69
1.12
2.12
0.21

2.6. Description and Inputs for Blade Length Study
The Blade Length Study was conducted to test the effect of changing Zweifel coefficient
on turbine performance. For each turbine geometry and specified rotational speed, the axial blade
length đ was systematically changed to alter the Zweifel coefficient. Designs with both short
blades (large Zweifel coefficient) and long blades (small Zweifel coefficient) were simulated.
Similar to the Blade Length Study, as blade length was changed đ˝đ2 was changed and
simulations were run until đđĄđ was within 200 Pa of đđđĄđ . Figure 2.7 below is a flow chart which
describes the steps required to obtain the desired data in CFD.

76

Figure 2.7. Blade Length Study process flow chart

Velocity triangles of the designs were produced, Figure 2.8 and 2.9 shows velocity
triangles at leading and trailing edges for designs 2l and 2l8 of the Speed Study respectively,
superimposed over the blade profile design produced using BladeGen.

77

2l Z = 0.42
đ = 82.9%

Figure 2.8. Velocity triangles for leading and trailing edges of design 2l of the Blade Length Study (50%
span)

78

2l8 Z = 0.84
đ = 85.5%

Figure 2.9. Velocity triangles for leading and trailing edges of design 2l8 of the Blade Length Study (50%
span)

At least eight designs with differing blade lengths were simulated for each case. Designs
with blade lengths corresponding to Zweifel coefficients from .36 to 1.49 were simulated for the
Blade Length Study. The below Table 2.7 shows each simulation which was run for the Blade
Length Study, showing the blade length and Zweifel coefficient for each design.

79

Case
1j
1j
1j
1j
1j
1j
1j
1j
Case
4h
4h
4h
4h
4h
4h
4h
4h
4h
-

Run
1
2
3
4
5
6
7
8
Run
1
2
3
4
5
6
7
8
9
-

đ (đ)
0.162
0.122
0.090
0.070
0.050
0.190
0.080
0.096
đ (đ)
0.16
0.14
0.18
0.11
0.08
0.06
0.09
0.05
0.07
-

Table 2.8. Blade Length Study Runs and Inputs
Z
Case Run đ (đ) Z
Case Run
0.48 2l
1
0.24
0.42 3n
1
0.63 2l
2
0.19
0.53 3n
2
0.85 2l
3
0.15
0.67 3n
3
1.09 2l
4
0.11
0.91 3n
4
1.49 2l
5
0.08
1.24 3n
5
0.41 2l
6
0.10
1.05 3n
6
0.96 2l
7
0.13
0.78 3n
7
0.80 2l
8
0.12
0.84 3n
8
9
3n
Z
Case Run đ (đ) Z
Case
Run
0.40 5n
1
0.24
0.44
1
6h
0.46 5n
2
0.27
0.39
2
6h
0.36 5n
3
0.20
0.52
3
6h
0.58 5n
4
0.16
0.65
4
6h
0.80 5n
5
0.12
0.87
5
6h
1.06 5n
6
0.10
1.03
6
6h
0.71 5n
7
0.08
1.28
7
6h
1.26 5n
8
0.11
0.94
8
6h
0.91 5n
9
0.11
0.80
9
6h
10
6h
11
6h
12
6h

đ (đ)
0.40
0.35
0.30
0.25
0.20
0.15
0.13
0.11
0.18
đ (đ)
0.40
0.44
0.47
0.50
0.37
0.33
0.29
0.25
0.21
0.16
0.13
0.20

Z
0.40
0.46
0.53
0.64
0.79
1.05
1.20
1.42
0.90
Z
0.44
0.41
0.38
0.36
0.48
0.54
0.61
0.70
0.83
1.08
1.31
0.90

2.7. Mesh Independence Study
A mesh independence study was conducted first to ensure a fine enough mesh was used
for each simulation to capture the flow features necessary to resolve the power, efficiency, and
pressure difference accurately. Simulations with a finer mesh can resolve more flow details, but
are more difficult to obtain convergence. A design was simulated for each case from the Speed
Study, logging the performance characteristics, then the mesh size factor in ANSYS TurboGrid
was changed systematically to see how mesh count affected power output. A base mesh count
was picked at a point where increasing the mesh count further does not have a significant effect
on power output. As each design has its own axial length, to ensure similarity across each design

80

simulated, mesh density was kept constant across each design, such that the cell count per meter
stays constant across each design per case. Figure 2.10 below shows the power output for a
single design measured with different cell counts in case 1, Figure 2.11 shows this for case 2,
Figure 2.12 shows this for case 3, Figure 2.13 shows this for case 4, Figure 2.14 shows this for

Power

case 5, Figure 2.15 shows this for case 6.
7000
6950
6900
6850
6800
6750
6700
6650
6600
0

200000

400000

600000

800000

1000000

1200000

Cell Count
Figure 2.10. Mesh cell count vs. Power for case 1 mesh Study

Base mesh count of 550000 was used for the case 1 simulations, corresponding to an
axial blade length of .14 meters. For case 1, simulations used a cell count of 3.9x10^6 cells/m.
17800

Power

17600
17400
17200
17000
16800
0

500000

1000000

1500000

Cell Count
Figure 2.11. Mesh cell count vs. Power for case 2 mesh Study

A base mesh count of 800000 was used for the case 2 simulations, corresponding to an
axial blade length of .16 meters. For case 2, simulations used a cell count of 5.0x10^6 cells/m.

81

47500

Power

47000
46500
46000
45500
45000
44500
0

500000

1000000

1500000

Cell Count
Figure 2.12. Mesh cell count vs. Power for case 3 mesh Study

A base mesh count of 1000000 was used for the case 3 simulations, corresponding to an
axial blade length of .3 meters. For case 3, simulations used a cell count of 3.3x10^6 cells/m.
18200

Power

18000
17800
17600
17400
17200
17000
0

500000

1000000

1500000

2000000

2500000

3000000

Cell Count
Figure 2.13. Mesh cell count vs. Power for case 4 mesh Study

A base mesh count of 1000000 was used for the case 4 simulations, corresponding to an
axial blade length of .18 meters. For case 4, simulations used a cell count of 5.5x10^6 cells/m.
34000

Power

33500
33000
32500
32000
31500
0

500000

1000000

1500000

Cell Count
Figure 2.14. Mesh cell count vs. Power for case 5 mesh Study

82

A base mesh count of 600000 was used for the case 5 simulations, corresponding to an
axial blade length of .2 meters. For case 5, simulations used a cell count of 3x10^6 cells/m.
24500

Power

24000
23500
23000
22500
22000
0

200000

400000

600000

800000

Cell Count
Figure 2.15. Mesh cell count vs. Power for case 6 mesh Study

A base mesh count of 600000 was used for the case 6 simulations, corresponding to an
axial blade length of .25 meters. For case 6, simulations used a cell count of 2.4x10^6 cells/m.

83

CHAPTER 3. CFD RESULTS AND DISCUSSION
3.1. Speed Study CFD Results
The Speed Study tested designs with similar geometry at varying rotational speeds. Axial
blade length was changed for each design such that đ = .8 for each design tested in the Speed
Study. The below tables show how varying rotational speed effected efficiency for the different
designs.

Case 1
đĚ =
đ3
đ 

0.46

đâ = .12 m
đđ  = .2 m

Table 3.1. Case 1 Speed Study CFD Results
Efficiency
đ
đ2
đĚ ( đ 2 ) đ
đť(m) đ(m)
RPM
(m/s)
1.88 0.045
14.38
77.09
350
7.33
1.86 0.056
14.60
79.04
300
6.28

Case Run
1 b
1 d
1
1
1
1
1

f
h
j
l
CORD.

1.86
1.84
1.85
1.88
2.07

0.070
0.086
0.098
0.106
0.022

14.69
14.69
14.77
14.45
6.45

79.66
80.47
80.58
77.53
34.87

250
200
170
140
737

5.24
4.19
3.56
2.93
15.44

đ
0.79
0.92

ÎŠđ 
2.80
2.41

đĚ
(kW)
đ
0.27
6.62
0.37
6.72

1.12
1.39
1.63
1.98
0.38

2.01
1.62
1.37
1.12
5.47

0.54
0.84
1.17
1.68
0.03

6.76
6.76
6.79
6.65
2.97

Peak efficiency was observed at a specific speed of 1.4, RPM 170. Decreased efficiency
was measured at rotational speeds above 300 RPM.
Case 2
đ3
đĚ = 1.050 đ  Case
đâ = .2 m
2
đđ  = .3 m
2
2
2
2
2
2
2

Run
b
d
f
h
j
l
n
CORD.

Table 3.2. Case 2 Speed Study CFD Results
đĚ
Efficiency
đ
đ2
(
)
đť(m) đ(m)
RPM (m/s)
đ
đ 2
1.90 0.031 13.62
73.35
350 11.00
1.85 0.044 14.64
78.62
300 9.42
1.88 0.060 15.31
82.48
250 7.85
1.88 0.086 15.93
84.95
200 6.28
1.88 0.105 16.02
85.38
170 5.34
1.85 0.126 15.85
85.49
140 4.40
1.87 0.142 15.80
84.65
120 3.77
1.92 0.027 10.07
52.57
497 15.61

đ
0.53
0.62
0.75
0.94
1.10
1.34
1.56
0.38

ÎŠđ 
4.19
3.67
3.01
2.42
2.05
1.71
1.45
5.91

đ
0.11
0.16
0.25
0.40
0.56
0.82
1.11
0.04

đĚ
(kW)
14.30
15.37
16.08
16.73
16.83
16.64
16.59
10.57

Peak efficiency was observed at a specific speed of 1.7, RPM of 140. A sharp decrease in
efficiency was measured at rotational speeds above 250 RPM.

84

Table 3.3. Case 3 Speed Study CFD Results
Case 3
đ3
đĚ = 2.85

Case
3
3
3
3
3
3
3

đ 

đâ = .3 m
đđ  = .5 m

Run
f
h
j
l
n
p
CORD.

đť(m)
1.90
1.89
1.92
1.91
1.88
1.88
1.84

đ2

đ(m) đĚ ( đ 2 ) Efficiency đ RPM đ (m/s)
đ
ÎŠđ 
0.034
15.45
82.56
250
13.09 0.45 4.92
0.061
15.58
83.98
200
10.47 0.56 3.96
0.104
16.65
88.24
150
7.85 0.74 2.94
0.148
16.83
89.59
120
6.28 0.93 2.36
0.199
16.59
89.76
90
4.71 1.24 1.79
0.260
16.14
87.39
60
3.14 1.86 1.19
0.042
12.40
68.15
295
15.45 0.38 5.95

đ
0.09
0.14
0.27
0.43
0.75
1.64
0.05

đĚ (kW)
44.04
44.41
47.44
47.97
47.28
46.01
35.35

Peak efficiency was observed at a specific speed of 1.8, corresponding to an RPM of 90.
A sharp decrease in efficiency was measured at rotational speeds above 150 RPM.

Case 4
đ3
đĚ = 0.6

Case
4
4
4
4
4
4
4
4

đ 

đâ = .12 m
đđ  = .2 m

Run
đť(m)
b
3.73
d
3.72
f
3.68
h
3.70
j
3.70
l
3.70
n
3.70
CORD. 3.73

Table 3.4. Case 4 Speed Study CFD Results
Efficiency
đ2
đ(m) đĚ ( đ 2 )
RPM
đ (m/s)
đ
0.05
29.43
79.62
450
9.42
0.06
29.89
81.05
400
8.38
0.07
29.92
81.96
350
7.33
0.08
30.11
81.99
300
6.28
0.09
29.81
81.22
250
5.24
0.10
29.30
79.98
220
4.61
0.05
28.61
77.94
500
10.47
0.03
13.95
37.98
997
20.88

đĚ (kW)
đ
0.80
0.90
1.03
1.20
1.45
1.64
0.72
0.36

ÎŠđ 
2.45
2.19
1.93
1.64
1.37
1.21
2.74
5.44

đ
0.33
0.43
0.56
0.76
1.09
1.38
0.26
0.03

17.66
17.93
17.95
18.07
17.89
17.58
17.16
8.37

Peak efficiency was observed at a specific speed of 1.6, RPM of 300. Decreased
efficiency was measured at rotational speeds above 400 RPM.
Case 5
đ3
đĚ = 1.5
đ 

đâ = .2 m
đđ  = .3 m

Case
5
5
5
5
5
5
5
5
5

Run
đť(m)
b
3.73
d
3.67
f
3.71
h
3.70
j
3.72
l
3.72
n
3.70
p
3.72
CORD. 3.68

Table 3.5. Case 5 Speed Study CFD Results
Efficiency
đ
đ2
đ(m) đĚ ( đ 2 )
RPM (m/s)
đ
0.04
28.62
77.54
450 14.14
0.05
29.36
80.70
400 12.57
0.06
30.01
82.43
350 11.00
0.08
31.18
85.21
300 9.42
0.10
31.69
86.17
250 7.85
0.11
31.62
86.29
220 6.91
0.13
31.64
85.96
190 5.97
0.15
30.15
82.82
160 5.03
0.04
19.13
52.44
704 22.12

85

đĚ (kW)
đ
0.59
0.67
0.76
0.89
1.07
1.21
1.41
1.67
0.38

ÎŠđ 
3.88
3.49
3.03
2.60
2.16
1.90
1.65
1.38
6.13

đ
0.14
0.19
0.25
0.35
0.51
0.66
0.89
1.19
0.04

42.93
44.04
45.01
46.77
47.54
47.43
47.47
45.23
28.69

Peak efficiency was observed at a specific speed of 1.7, RPM of 190. A sharp decrease in
efficiency was measured at rotational speeds above 300 RPM.

Case 6
đ3
đĚ = 4.1
đ 

đâ = .3 m
đđ  = .5 m

Case
6
6
6
6
6
6
6

Table 3.6. Case 6 Speed Study CFD Results
Efficiency
đ
đ2
Run
đť(m) đ(m) đĚ ( đ 2 ) đ
RPM (m/s)
b
3.79 0.11
33.37
89.22
200 10.47
d
3.79 0.14
33.65
90.02
170 8.90
f
3.78 0.18
33.70
90.21
140 7.33
h
3.78 0.22
33.44
89.70
110 5.76
j
3.79 0.26
32.61
87.25
80 4.19
l
3.75 0.08
32.12
86.74
250 13.09
CORD. 3.69 0.06
25.08
68.84
419 21.94

đĚ (kW)
đ
0.79
0.93
1.13
1.43
1.97
0.63
0.38

ÎŠđ 
2.82
2.39
1.98
1.55
1.13
3.55
6.02

đ
0.30
0.42
0.63
1.01
1.86
0.19
0.05

Peak efficiency was observed at a specific speed of 2.0, RPM of 140. Decreased
efficiency was measured at rotational speeds above 170 RPM.
3.2. Discussion of Cordier Recommendations and Location on Diagram
The specific speed (and rotational speed) recommended by the Cordier line are greater
than the speed at which highest efficiency was measured in the CFD simulations. The efficiency
at the Cordier-recommended speed was lower for each case tested in the Speed Study; however
cases 3 and 6, which had the largest flow rate, had the smallest decrease in efficiency as speed
increased. The efficiency obtained in CFD was graphed against the specific speed for each case
in the Speed Study, shown in Figure 3.1 below.

86

136.83
137.97
138.17
137.12
133.70
131.68
131.68

Speed Study: Specific Speed ÎŠs vs. Efficiency
95.00
90.00
85.00

Efficiency

80.00

Case 1

75.00

Case 2

70.00

Case 3
Case 4

65.00

Case 5

60.00

Case 6

55.00
50.00
0.00

1.00

2.00

3.00

4.00

5.00

6.00

Specific Speed ÎŠs
Figure 3.1 Specific speed đşđ  Vs. Efficiency of Speed Study designs

Efficiency at the specific speeds recommended by the Cordier line was lower than the
efficiency at lower rotational speeds. Highest values of efficiencies were measured at specific
speeds ÎŠđ  between 1.4 and 1.9, lower than the values of specific speed which were
recommended by the Cordier line, between 5.5 and 6.1. The highest efficiency was measured for
the designs which ran at speeds between 12 and 33% of the Cordier-recommended speed. The
RPM, specific speed, and specific diameter where highest efficiency was measured in CFD for
each case was logged, and compared to efficiency at the RPM and specific speed recommended
by the Cordier line, recorded below in Table 3.7.

87

Table 3.7. Performance of Cordier Recommendations of Cases in Speed Study

Case

Units

1

2

đŻ

đ

1.85

1.88

đđ

đ

0.2

đđ

đ
đ3
đ 

đ˝Ě
đŤđ
RPM at highest đź
đđ (highest đź)
Highest đź from
Speed Study

3

4

5

6

1.91

3.70

3.70

3.70

0.3

0.5

0.2

0.3

0.5

.12

.18

.3

.12

.18

.3

đđđ
đđđ

0.46
1.218
170
1.37

1.05
1.214
140
1.69

2.85
1.232
90
1.77

0.60
1.268
300
1.65

1.50
1.202
90
0.78

4.10
1.212
140
2.01

%

80.58

85.49

89.76

81.99

85.96

90.21

6.1
704
22
52.44

6.0
419
22
68.84

Cordier Recommendations: Changing Speed
đđ (Cordier)
RPM (Cordier)
U (Cordier)
đź at Cordier point

đđđ
đđđ
đ/đ 
%

5.9
737
15
34.87

6.0
497
16
52.57

5.8
295
15
68.15

5.5
997
21
37.98

The below Figure 3.2 shows the six highest efficiency cases from the first study placed
on the Cordier diagram from Wright [18].

Figure 3.2. Location of peak efficiency points for each case on đşđ  vs đˇđ  diagram with Cordier line

88

Although the designs have specific diameter which the literature would recommend using
an axial turbine, the designs with highest efficiency measured in the CFD study were located
below the Cordier line, at rotational speeds less than half of that recommended by the Cordier
line. The peak efficiency designs operated at specific speeds which were close to the specific
speeds at which radial turbines typically operate. The specific diameter of the designs suggest the
designs fall into Wrightâs region A on the Cordier diagram, typically populated by axial turbines.
The specific speeds of the peak efficiency designs, however, fit with Wrightâs region F on the
Cordier diagram, where high pressure blowers, centrifugal compressors, and high head liquid
pumps are typically used.
3.3. Discussion of Speed Study Results
3.3.1. Overall Summary

For each of the designs simulated in the Speed Study, specific diameter was relatively
constant, and had a value of đˇđ  between 1.20 and 1.27. The Speed Study analysis used three sets
of hub and shroud radii and blade thickness, with two values of total available head. Cases 1 and
4, 2 and 5, and 3 and 6 had the similar geometry, and had similar efficiencies, although the
higher head designs (4 meters) were slightly higher efficiency than the designs for 2 meters. For
each of the cases it was observed that as specific speed ÎŠđ  was increased past 2, turbine
efficiency decreased. This is shown in Figure 3.3. As rotational speed was changed, tip speed
đ changed, and flow coefficient đ changed. The below Figure 3.4, which shows efficiency
graphed vs. flow coefficient, designs which had flow coefficients between 1.1 and 1.6 had the
highest efficiency measured in simulations. Efficiency was lowest in the designs with the lowest
flow coefficients, where rotational speed was high.

89

95.00

Speed Study: Flow Coefficient đ vs. Efficiency

90.00

Efficiency

85.00
80.00

Case 1

75.00

Case 2

70.00

Case 3
Case 4

65.00

Case 5

60.00

Case 6

55.00
50.00
0.00

0.50

1.00

1.50

2.00

2.50

Flow Coefficient đ
Figure 3.3 Flow coefficient vs. Efficiency of Speed Study designs

Designs which produced the highest efficiency had blade loading coefficients between
0.63 and 1.17. This is shown below in Figure 3.4. Efficiency was lowest for the designs with the
lowest head coefficients, the designs which ran with ÎŠs above 2.5.

Speed study: Blade Loading Coefficient đ vs.
Efficiency

95.00
90.00

Efficiency

85.00
80.00
Case 1
75.00

Case 2

70.00

Case 3

65.00

Case 4

60.00

Case 5
Case 6

55.00
50.00
0.00

0.50

1.00

Blade Loading Coefficient đ

1.50

2.00

Figure 3.4. Blade loading coefficient vs. Efficiency of Speed Study designs

90

3.3.2. Flow Visualization and Vectors

The simulations of designs with high rotational speed predicted flows with higher relative
flow speed. The higher speed designs also experienced larger flow velocity gradients at the
leading and trailing edges. Towards the trailing edge on the pressure side, the flow accelerated
and pressure decreased. This effect was more severe for higher rotational speed designs. Relative
velocity vector diagrams were generated for each design, and show the velocity gradients at the
leading and trailing edges. Figure 3.5 shows the flow of three designs in case 1, Figure 3.6 shows
the flow of three designs in case 2, Figure 3.7 shows the flow of three designs in case 3, Figure
3.8 shows the flow of three designs in case 4, Figure 3.9 shows the flow of three designs in case
5, Figure 3.10 shows the flow of three designs in case 6.

1j ÎŠđ  = 1.37
đ =78.4%

1CORD ÎŠđ  = 5.5
đ = 34.9%

1b ÎŠđ  = 2.8
đ = 77.1%

Figure 3.5. Relative velocity vectors for case 1: Span 50%
2CORD ÎŠđ  = 5.9
đ = 52.6%

2d ÎŠđ  = 3.4
đ = 78.7%

Figure 3.6. Relative velocity vectors for case 2: Span 50%

91

2l ÎŠđ  = 1.7
đ = 85.5%

3CORD ÎŠđ  = 5.9
đ = 68.2%

3n ÎŠđ  = 1.8
đ = 89.8%

3f ÎŠđ  = 4.9
đ = 82.6%

Figure 3.7. Relative velocity vectors for case 3: Span 50%

4CORD ÎŠđ  = 5.4
đ = 37.9%

4h ÎŠđ  = 1.6
đ = 81.9%

4b ÎŠđ  = 2.4
đ = 79.6%

Figure 3.8. Relative velocity vectors for case 4: Span 50%

5CORD ÎŠđ  = 6.1
đ = 52.4%

5b ÎŠđ  = 3.9
đ = 77.5%

5n ÎŠđ  = 1.6
đ = 85.9%

Figure 3.9. Relative velocity vectors for case 5: Span 50%

92

6CORD ÎŠđ  = 6.0
đ = 68.8%

6b ÎŠđ  = 2.8
đ = 89.2%

6f ÎŠđ  = 1.9
đ = 90.2%

Figure 3.10. Relative velocity vectors for case 6: Span 50%

To show additional flow details, more detailed relative flow velocity vector diagrams
were generated at the leading and trailing edges using the simulation results of each design.
Figures 3.12, 3.13, and 3.14 show relative velocity vectors at leading edge for three designs
within cases 2,1, and 3 respectively. At the leading edge, a region of flow deceleration was
measured on the pressure side, located close to a region of rapid flow acceleration towards the
suction side. The higher speed designs exhibited flows with a higher magnitude flow
deceleration on the leading edge pressure side, and higher flow acceleration near the leading
edge suction side. This same effect was observed for each of the cases simulated.
2d ÎŠđ  = 3.4
đ = 78.7%

2CORD ÎŠđ  = 5.9
đ = 52.6%

2l ÎŠđ  = 1.7
đ = 85.5%

Figure 3.11. Relative velocity vectors at leading edge for case 2: 50% span

93

1b ÎŠđ  = 2.8
đ = 77.1%

1CORD ÎŠđ  = 5.5
đ = 34.9%

1j ÎŠđ  = 1.37
đ = 80.6%

Figure 3.12. Relative velocity vectors at leading edge for case 1: Span 50%

3CORD ÎŠđ  = 5.9
đ = 68.2%

3f ÎŠđ  = 4.9
đ = 82.6%

3n ÎŠđ  = 1.8
đ = 89.8%

Figure 3.13. Relative velocity vectors at leading edge for case 3: Span 50%

Similarly, vector diagrams were produced for the trailing edge. The flow for the higher
speed designs experienced more severe velocity gradients at and after the trailing edge. This
relationship was observed for every case simulated in the Speed Study. This is shown below in
Figure 3.14 for case 2, Figure 3.15 for case 1, and Figure 3.16 for case 3.
2CORD ÎŠđ  = 5.9
đ = 52.6%

2d ÎŠđ  = 3.4
đ = 78.7%

2l ÎŠđ  = 1.7
đ = 85.5%

Figure 3.14. Relative velocity vectors at trailing edge for case 2: 50% span

94

1b ÎŠđ  = 2.8
đ = 77.1%

1CORD ÎŠđ  = 5.5
đ = 34.9%

1j ÎŠđ  = 1.37
đ = 80.6%

Figure 3.15. Relative velocity vectors at trailing edge for case 1: 50% span

3CORD ÎŠđ  = 5.9
đ = 68.2%

3f ÎŠđ  = 4.9
đ = 82.6%

3n ÎŠđ  = 1.8
đ = 89.8%

Figure 3.16. Relative velocity vectors at trailing edge for case 3: Span 50%
3.3.3 Trailing Edge Effects

The above figures illustrate how varying rotational speed effected velocity gradients at
the trailing edge. If the magnitude of the velocity gradients at the trailing edge are larger,
additional viscous shear could occur, resulting in flow turbulence and entropy generation [35].
The designs with higher speed exhibited flows with more extreme velocity gradients at the
trailing edge, which results in added viscous shear compared to the lower speed designs. The
designs recommended by the Cordier line experienced the largest velocity gradients at the
trailing edge, as well as the largest flow deceleration measured after the trailing edge. This effect
was observed for every case simulated in the Speed Study.
Flow will be more turbulent in areas where velocity gradients are more severe, and the
higher speed designs in the Speed Study experienced greater velocity gradients at the trailing
95

edge. Turbulence kinetic energy was recorded to track the effect of increasing rotational speed
on turbulence. Turbulence kinetic energy at and after the trailing edge was higher for the designs
which operated at higher rotational speeds. Figure 3.17 shows a contour of turbulence kinetic
energy for two designs 3Cord, 3n, and Figure 3.18 shows this for two designs in case 1, 1Cord
and 1j.
3CORD ÎŠđ  = 5.9
đ = 68.2%

3n ÎŠđ  = 1.8
đ = 89.8%

Figure 3.17. Turbulence kinetic energy contours for case 3: Span 50%
1CORD ÎŠđ  = 5.5
đ = 34.9%

1j ÎŠđ  = 1.37
đ = 80.6%

1l ÎŠđ  = 1.1
đ = 77.5%

Figure 3.18. Turbulence kinetic energy contours for case 1: Span 50%

Figure 3.19 below shows the relative flow velocity for simulations of designs 1b ,1j and
1Cord of case 1 logged from 80% along the blade chord to the trailing edge. These figures
illustrate the flow acceleration at the pressure side near the trailing edge, and rapid flow
deceleration near the trailing edge on the suction side. As rotational speed was increased, flow
96

velocity at the pressure side trailing edge increased and velocity at the suction side trailing edge
decreased. This was observed for every case simulated in the Speed Study.

1j ÎŠđ  = 1.37
đ = 80.6%

đž (đ/đ)

1CORD ÎŠđ  = 5.5
đ = 34.9%

1b ÎŠđ  = 2.8
đ = 77.1%

Normalized position along blade
Figure 3.19. Relative velocity of case 1 Speed Study designs near the trailing edge: Span 50%
3.3.4. Blade Loading

Simulations showed that for the higher speed designs, the adverse pressure gradient on
the suction side of the blades after the point of minimum pressure became more severe. Another
adverse pressure gradient at the trailing edge at the suction side became more severe as speed
increased. Adverse pressure gradients are correlated with boundary layer growth, entropy
generation, and increased losses [35].
For each of the designs simulated in CFD, highest pressure was measured at the leading
edge at the pressure side where flow was decelerated, the higher rotational speed designs
demonstrated a higher pressure at this point. This effect was most extreme for the designs with
rotational speeds recommended by the Cordier line. This point was located close to an area of
lower pressure at the leading edge on the suction side, where flow accelerates. Simulations of the
97

designs with higher rotational speed predicted decreased pressure at this point compared to lower
speed designs. This effect was observed for each case simulated in the Speed Study. For many
designs, at this point pressure reached below vapor pressure.
Towards the trailing edge on the pressure side, the flow accelerates and pressure
decreases. This effect was magnified for higher rotational speed designs. The designs with
Cordier speed experienced the most severe drop in pressure near the trailing edge on the pressure
side. The area of low pressure at the trailing edge pressure side takes up a larger portion of the
blade chord for the higher speed designs. Near the trailing edge on the suction side, pressure
increases as the flow decelerates. Each of the higher rotational speed designs simulated recorded
greater flow deceleration near the trailing edge on the suction side of the blades as the flow
experienced an adverse pressure gradient. This is illustrated in blade loading diagrams for two of
the cases, which chart pressure on the blades over the normalized position along the blade chord.
Figure 3.20 shows design 2d, 2l, and the Cordier-recommended design for case 2, 2Cord, and
Figure 3.21 shows design 1b, 1j, and the Cordier-recommended design for case 1, 1Cord. The
adverse pressure gradients are highlighted with thicker lines.

98

2d ÎŠđ  = 3.4
đ = 78.7%

Pressure (Pa)

2l ÎŠđ  = 1.7
đ = 85.5%

2CORD ÎŠđ  = 5.9
đ = 52.6%

Normalized position along blade
Figure 3.20. Blade loading diagrams for Speed Study case 2: Span 50%
1b ÎŠđ  = 2.8
đ = 77.1%

Pressure (Pa)

1j ÎŠđ  = 1.37
đ = 80.6%

1CORD ÎŠđ  =
5.5 đ = 34.9%

Normalized position along blade
Figure 3.21. Blade loading diagrams for Speed Study case 1: Span 50%

99

Static pressure contours were produced for each design simulated, which display areas of
increased pressure at the leading edge pressure side, decreased pressure at the leading edge on
the suction side, and decreased pressure at the trailing edge pressure side. These figures illustrate
that for the higher speed designs, the pressure gradients are of larger magnitude at each of these
areas. This effect was observed for each case simulated in the Speed Study. Static pressure
contours are shown for case 3 (Figure 3.22), case 1 (Figure 3.23), and case 3 (Figure 3.24)

2l ÎŠđ  = 1.7
đ = 85.5%

2d ÎŠđ  = 3.4
đ = 78.7%

2CORD ÎŠđ  = 5.9
đ = 52.6%

Figure 3.22. Static pressure contours for Speed Study case 2: Span 50%

1CORD ÎŠđ  = 5.5
đ = 34.9%

1b ÎŠđ  = 2.8
đ = 77.1%

1j ÎŠđ  = 1.37
đ = 80.6%

Figure 3.23. Static pressure contours for Speed Study case 1: Span 50%

100

3CORD ÎŠđ  = 5.9
đ = 68.2%

3n ÎŠđ  = 1.8
đ = 89.8%

3f ÎŠđ  = 4.9
đ = 82.6%

Figure 3.24. Static pressure contours for Speed Study case 3: Span 50%

Figure 3.25 and 3.27 below shows velocity triangles for two designs within the Speed
Study, designs 1cord and 1j respectively, superimposed onto relative velocity vector contours.
These figures illustrate how the relative velocities were higher for the designs with the Cordier
recommended speeds.

101

1CORD ÎŠđ  = 5.5
đ = 34.9%
đ = 15.4

Figure 3.25. Velocity triangles with relative velocity contours for design 1CORD of the Speed Study:
(Span 50%)

102

1j ÎŠđ  = 1.37
đ = 80.6%
đ = 3.6

Figure 3.26. Velocity triangles with relative velocity contours for design 1j of the Speed Study: (Span
50%)
3.3.5. Hub and Shroud Effects

Decreased flow velocity was recorded near the hub and shroud surfaces for each design
simulated. The higher speed designs saw relative velocity decrease the most near the hub and
shroud. This effect was observed across each case in the Speed Study. Figure 3.27 shows the
relative flow velocity đ over the blade span at the trailing edge for four designs in case 2 of the
Speed Study, and Figure 3.28 shows this for four designs within case 1.

103

Span Normalized

2d ÎŠđ  = 3.4
đ = 78.7%
đ = 9.4

2n ÎŠđ  = 1.4
đ = 84.6%
đ = 3.8

2l ÎŠđ  = 1.7
đ = 85.5%
đ = 4.4
2CORD ÎŠđ  = 5.9
đ = 52.6%
đ = 15.6

Span Normalized

đž (đ/đ)
Figure 3.27. Relative velocity of Case 2 Speed Study designs at the trailing edge along span

1j ÎŠđ  = 1.37
đ = 80.6%
đ = 3.6

1b ÎŠđ  = 2.8
đ = 77.1%
đ = 7.3

1l ÎŠđ  = 1.1
đ = 77.5%
đ = 2.9

1CORD ÎŠđ  = 5.5
đ = 34.9%
đ = 15.4

đž (đ/đ)
Figure 3.28. Relative velocity of Case 1 Speed Study designs at the trailing edge along span

Figure 3.29 and Figure 3.30 display relative flow velocity for the designs at which
highest efficiency was recorded in the Speed Study along the span of the blades, near the leading
and trailing edges respectively. Decreased flow velocity close to the hub and shroud was
recorded for each design simulated.
104

The designs with higher head (cases 4,5,6) exhibited flows with increased relative
velocity compared to those with lower head (cases 1,2,3). The designs with the smallest diameter
experienced the most flow deceleration near the hub and shroud at the trailing edge, these are
designs in cases 1 and 4. It can be observed that lower velocity was recorded over a larger
portion of the span at the trailing edge for the simulations of the designs with lower diameter,
thus the boundary layer size was larger for the smaller diameter designs. The simulations of the
designs with lower diameter predicted lower efficiency; this could be attributed in part to
additional friction losses at the hub and shroud. Losses in the boundary layer are higher for
turbine designs with increased surface area of the blades, as well as the hub and shroud surfaces

Span Normalized

relative to the volume of the fluid region.

2l ÎŠđ  = 1.7
đ = 85.5%
đ = 4.4

5n ÎŠđ  = 1.6
đ = 85.9%
đ = 5.9

3n ÎŠđ  = 1.8
đ = 89.8%
đ = 4.7

6f ÎŠđ  = 1.9
đ = 90.2%
đ = 7.3

1j ÎŠđ  = 1.37
đ = 80.6%
đ = 3.6

4h ÎŠđ  = 1.6
đ = 81.9%
đ = 6.3

đž (đ/đ)
Figure 3.29. Relative velocity of highest efficiency Speed Study designs at the leading edge along span

105

Span Normalized

4h ÎŠđ  = 1.6
đ = 81.9%
đ = 6.3
1j ÎŠđ  = 1.37
đ = 80.6%
đ = 3.6

2l ÎŠđ  = 1.7
đ = 85.5%
đ = 4.4
5n ÎŠđ  = 1.6
đ = 85.9%
đ = 5.9

3n ÎŠđ  = 1.8
đ = 89.8%
đ = 4.7

6f ÎŠđ  = 1.9
đ = 90.2%
đ = 7.3

đž (đ/đ)
Figure 3.30. Relative velocity of highest efficiency Speed Study designs at the trailing edge along span

Below Figure 3.31 shows the turbulence kinetic energy đ at the trailing edge for the
highest efficiency designs in the Speed Study. Turbulence kinetic energy of the flow was higher
near the hub and shroud for the designs with higher head, and the designs with lower diameter
(cases 1 and 4) showed the more turbulent region of the flow takes up a larger portion of the
blade span. This could be attributed to additional friction losses at the hub and shroud in the
designs with smaller diameter.

106

Span Normalized

3n ÎŠđ  = 1.8
đ = 89.8%
1j ÎŠđ  = 1.37
đ = 80.6%
4h ÎŠđ  = 1.6
đ = 81.9%

2l ÎŠđ  = 1.7
đ = 85.5%
6f ÎŠđ  = 1.9
đ = 90.2%

5n ÎŠđ  = 1.6
đ = 85.9%

Turbulence kinetic energy đ
Figure 3.31. Turbulence kinetic energy of highest efficiency Speed Study designs at the trailing edge
along span
3.3.6. Work Extraction

The simulations of each design predicted reduced work extraction near the hub and
shroud. This is illustrated in the figures below, which display absolute circumferential flow
velocity over the span of the blades at the trailing edge. Figure 3.32 shows absolute
circumferential flow velocity for three designs in case 2 of the Speed Study, Figure 3.33 shows
this for case 3. Designs which used the Cordier-recommended speed exhibited simulated flows
with the largest decrease of circumferential flow velocity near the hub and shroud. This effect
was observed for each case simulated in the Speed Study.

107

Span Normalized

2l ÎŠđ  = 1.7
đ = 85.5%
đ = 4.4

2CORD ÎŠđ  = 5.9
đ = 52.6%
đ = 15.6

2n ÎŠđ  = 1.4
đ = 84.6%
đ = 3.8

2d ÎŠđ  = 3.4
đ = 78.7%
đ = 9.4

đŞđ (đ/đ)

Span Normalized

Figure 3.32. Absolute circumferential flow velocity for designs in case 2 of Speed Study at the trailing
edge along span

3CORD ÎŠđ  = 5.9
đ = 68.2%
đ = 15.4

3p ÎŠđ  = 1.2
đ = 87.4%
đ = 3.1

3f ÎŠđ  = 4.9
đ = 82.6%
đ = 13.1

3n ÎŠđ  = 1.8
đ = 89.8%
đ = 4.7

đŞđ (đ/đ)
Figure 3.33. Absolute Circumferential flow velocity for designs in Case 3 of Speed Study at the trailing
edge along span

108

For each design, đśđ˘ reached a maximum near the trailing edge, and between the trailing
edge and the exit the flow decelerates, and đśđ˘ decreases. Figure 3.34 below shows average đśđ˘
from annulus inlet to outlet for three designs within case 1 of the Speed Study.

đŞđ (đ/đ)

1l ÎŠđ  = 1.1
đ = 77.5%

1j ÎŠđ  = 1.37
đ = 80.6%

1b ÎŠđ  = 2.8
đ = 77.1%

Streamwise Location
Figure 3.34. Average absolute circumferential flow velocity from inlet to outlet for designs case 1 of
Speed Study

Figure 3.35 below shows the absolute circumferential flow velocity for the designs with
the highest predicted efficiency in the Speed Study along the blade span at the trailing edge. đśđ˘
was higher near the hub than the shroud, this is because the relative flow angles were designed to
have an equal value of đđ đśđ˘đ across the span.

109

Span Normalized

5n ÎŠđ  = 1.6
đ = 85.9%
đ = 5.9
4h ÎŠđ  = 1.6
đ = 81.9%
đ = 6.3

1j ÎŠđ  = 1.37
đ = 80.6%
đ = 3.6
2l ÎŠđ  = 1.7
đ = 85.5%
đ = 4.4

6f ÎŠđ  = 1.9
đ = 90.2%
đ = 7.3

3n ÎŠđ  = 1.8
đ = 89.8%
đ = 4.7

đŞđ (đ/đ)
Figure 3.35. Absolute circumferential flow velocity for highest efficiency designs in Speed Study at the
trailing edge along span
3.3.7. Cavitation

It is important to track how low static pressure reaches when evaluating the performance
of a hydraulic turbine design in CFD. If there are any areas where pressure reaches below the
vapor pressure of water in simulation, cavitation could occur in a real-life application. Contour
diagrams were produced for each design, recording areas where static pressure was close to or
below the vapor pressure of water. Figure 3.36 shows the low static pressure contour for case 1,
Figure 3.37 for case 2, Figure 3.38 for case 3, Figure 3.39 for case 4, Figure 3.40 for case 5, and
Figure 3.41 for case 6. The figures show the simulations of designs at higher speeds exhibited
flows with lower pressure at the leading edge on the suction side of the blades, and have added
risk of experiencing cavitation.

110

1b ÎŠđ  = 2.8
đ = 77.1%

1j ÎŠđ  = 1.37
đ = 80.6%

Figure 3.36. Areas of low static pressure for case 1 of Speed Study: 50% Span
2l ÎŠđ  = 1.7
đ = 85.5%

2d ÎŠđ  = 3.4
đ = 78.7%

Figure 3.37. Areas of low static pressure for case 2 of Speed Study: 50% Span
3f ÎŠđ  = 4.9
đ = 82.6%

3n ÎŠđ  = 1.8
đ = 89.8%

Figure 3.38. Areas of low static pressure for case 3 of Speed Study: 50% Span

111

4h ÎŠđ  = 1.6
đ = 81.9%

4b ÎŠđ  = 2.4
đ = 79.6%

Figure 3.39. Areas of low static pressure for case 4 of Speed Study: 50% Span
5n ÎŠđ  = 1.6
đ = 85.9%

5b ÎŠđ  = 3.9
đ = 77.5%

Figure 3.40. Areas of low static pressure for case 5 of Speed Study: 50% Span
6b ÎŠđ  = 2.8
đ = 89.2%

6f ÎŠđ  = 1.9
đ = 90.2%

Figure 3.41. Areas of low static pressure for case 6 of Speed Study: 50% Span

112

3.4. Second Study: Blade Length Study
The Blade Length Study consisted of simulations of designs with varying axial blade
lengths at the rotational speeds at which highest efficiency was measured in the Speed Study.
Different axial blade lengths will result in different values of Zweifel coefficient. The below
tables show how varying Zweifel coefficient effected isentropic efficiency and trailing edge flow
deviation.
Table 3.8. Case 1 Blade Length Study Results
Case 1 - 1j
đť = 1.86 m
đ3
đĚ = 0.46
đ 

đâ = .12 m
đđ  = .2 m

đ
Case Run (đ/đ )
1j
1
3.56
1j
2
3.56
1j
3
3.56
1j
4
3.56
1j
5
3.56
1j
6
3.56
1j
7
3.56
1j
8
3.56

Efficiency
đ2
đĚ ( đ 2 ) đ (m) đ
RPM
14.45 0.162
78.40
170
14.69 0.122
79.04
170
14.51 0.090
79.72
170
14.64 0.070
79.70
170
14.34 0.050
78.21
170
14.13 0.190
77.43
170
14.83 0.080
80.06
170
14.78 0.096
79.96
170

Trailing edge
deviation đż â˛
(deg)
7.42
8.94
11.86
15.35
19.68
5.94
13.03
10.85

đĚ
(kW)
đ
0.48
0.63
0.85
1.09
1.49
0.41
0.96
0.80

6.65
6.76
6.68
6.73
6.59
6.50
6.82
6.80

Table 3.9. Case 2 Blade Length Study Results
Case 2 - 2l
đť = 1.85
m
đĚ =
đ3

1.050

đ 

đâ = .2 m
đđ  = .3 m

đ
Case Run (đ/đ )
2l
1
4.40
2l
2
4.40
2l
3
4.40
2l
4
4.40
2l
5
4.40
2l
6
4.40
2l
7
4.40
2l
8
4.40

đĚ (

đ2
)
đ 2

15.35
15.59
15.80
15.83
15.70
16.02
15.96
15.79

Efficiency
đ
(m) đ
RPM
0.24
82.86
140
0.19
83.79
140
0.15
84.65
140
0.11
85.25
140
0.08
84.88
140
0.10
85.47
140
0.13
85.14
140
0.12
85.52
140

113

Trailing
edge
deviation
đż â˛ (deg)
5.71
7.15
8.99
12.61
17.10
14.57
10.34
10.63

đĚ
(kW)
đ
0.42
0.53
0.67
0.91
1.24
1.05
0.78
0.84

16.12
16.37
16.59
16.62
16.48
16.82
16.76
16.58

Table 3.10. Case 3 Blade Length Study Results
Case 3 - 3n
đť = 1.9 m
đ3
đĚ = 2.85
đ 

đâ = .3 m
đđ  = .5 m

Case Run
3n
1
3n
2
3n
3
3n
4
3n
5
3n
6
3n
7
3n
8
3n
9

đ
(đ/đ )
4.71
4.71
4.71
4.71
4.71
4.71
4.71
4.71
4.71

đĚ (

đ2
)
đ 2

16.33
16.33
16.64
16.61
16.58
16.84
16.78
18.01
16.75

đ (m)
0.40
0.35
0.30
0.25
0.20
0.15
0.13
0.11
0.18

Efficiency
RPM
đ
86.79
90
86.76
90
88.11
90
88.57
90
89.30
90
89.77
90
89.71
90
87.58
90
89.94
90

Trailing
edge
deviation
đż â˛ (deg)
5.27
6.41
6.74
8.10
10.05
13.79
15.99
19.99
11.41

đĚ
(kW)
đ
0.40
0.46
0.53
0.64
0.79
1.05
1.20
1.42
0.90

46.54
46.55
47.44
47.35
47.26
47.99
47.81
51.33
47.73

Table 3.11. Case 4 Blade Length Study Results
Case 4 - 4h
đť = 3.69
m
đ3
đĚ = 0.6
đ 

đâ = .12 m
đđ  = .2 m

Efficiency
đ
đ2
Case Run (đ/đ ) đĚ ( đ 2 ) đ (m) đ
RPM
4h
1
6.28 29.11
0.16
79.40
300
4h
2
6.28 29.33
0.14
80.17
300
4h
3
6.28 29.06
0.18
79.02
300
4h
4
6.28 29.73
0.11
81.15
300
4h
5
6.28 30.04
0.08
81.86
300
4h
6
6.28 29.60
0.06
81.99
300
4h
7
6.28 30.14
0.09
81.69
300
4h
8
6.28 30.14
0.05
81.77
300
4h
9
6.28 30.05
0.07
82.01
300

114

Trailing
edge
deviation
đż â˛ (deg)
5.45
6.25
4.99
7.76
10.77
14.63
9.52
15.88
12.54

đĚ
(kW)
đ
0.40
0.46
0.36
0.58
0.80
1.06
0.71
1.26
0.91

17.47
17.60
17.44
17.84
18.02
17.76
18.08
18.08
18.03

Table 3.12. Case 5 Blade Length Study Results
Case 5 â 5n
đť = 3.73 m
đ3
đĚ = 1.5
đ 

đâ = .2 m
đđ  = .3 m

Case 6 6h
đť = 3.78
m
đ3
đĚ = 4.1
đ 

đâ = .3 m
đđ  = .5 m

Case Run
5n
1
5n
2
5n
3
5n
4
5n
5
5n
6
5n
7
5n
8
5n
9

đ
(đ/đ )
5.97
5.97
5.97
5.97
5.97
5.97
5.97
5.97
5.97

đĚ (

đ2
)
đ 2

30.85
30.48
31.07
31.05
31.41
31.48
31.40
31.55
31.49

đ (m)
0.24
0.27
0.20
0.16
0.12
0.10
0.08
0.11
0.11

Efficiency
RPM
đ
83.68
190
83.03
190
84.38
190
84.93
190
85.77
190
86.04
190
85.57
190
85.86
190
85.97
190

Trailing
edge
deviation
đż â˛ (deg)
6.08
5.45
7.16
8.80
11.45
13.89
17.90
12.61
10.66

Table 3.13. Case 6 Blade Length Study Results
Trailing
edge
Efficiency
deviation
đ
đ
đ2
Case Run (đ/đ ) đĚ ( đ 2 ) (m) đ
RPM đż â˛ (deg)
6h
1
5.76 32.47 0.40
87.06
110
5.97
6h
2
5.76 32.39 0.44
86.72
110
5.52
6h
3
5.76 32.22 0.47
86.36
110
5.21
6h
4
5.76 31.98 0.50
85.97
110
4.93
6h
5
5.76 32.44 0.37
87.26
110
6.40
6h
6
5.76 32.68 0.33
87.92
110
7.07
6h
7
5.76 32.97 0.29
88.21
110
8.05
6h
8
5.76 33.10 0.25
88.76
110
9.22
6h
9
5.76 33.24 0.21
89.20
110
10.78
6h
10
5.76 33.52 0.16
89.58
110
14.19
6h
11
5.76 32.98 0.13
89.02
110
17.37
6h
12
5.76 33.36 0.20
89.81
110
10.97

đĚ
(kW)
đ
0.44
0.39
0.52
0.65
0.87
1.03
1.28
0.94
0.80

46.27
45.72
46.60
46.58
47.12
47.22
47.11
47.32
47.23

đĚ
(kW)
đ
0.44
0.41
0.38
0.36
0.48
0.54
0.61
0.70
0.83
1.08
1.31
0.90

133.14
132.78
132.10
131.13
132.99
134.00
135.17
135.73
136.30
137.45
135.23
136.78

3.5. Blade Length Study Results Discussion
3.5.1. Overall Summary

For designs with long blade lengths Zweifel coefficient is small. These designs
demonstrated lower efficiency in the CFD simulations than those with shorter blade lengths.
Dixon suggested this is due to increased friction losses [4]. The below Figure 3.42 shows

115

efficiency obtained at differing values of Zweifel coefficient for the simulations conducted as a
part of the Blade Length Study. At Zwefiel coefficients between 0.8 and 1.1 efficiency was
highest in simulations, and stayed relatively constant across designs in that range. At Zwefiel
coefficients below 0.8 predicted efficiency was lower as friction losses increased. Designs with
Zweifel coefficients above 1.1 experienced decreased efficiency in simulations.

Blade Length Study: đ vs Efficiency

91.00
89.00

Efficiency

87.00
Series1

85.00

Series2

83.00

Series3
81.00

Series4

79.00

Series5
Series6

77.00
75.00
0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Figure 3.42. Zweifel coefficient vs. Efficiency for Blade Length Study

Flow does not follow the blade shape perfectly, and as blades are made shorter axially the
blades must be made with sharper turning to obtain the same change in absolute circumferential
flow speed and energy extraction. As the blade length was decreased and Zweifel coefficient
increased, the trailing edge flow deviation đżâ˛ increased. In the Blade Length Study, a correlation
between the Zweifel coefficient and trailing edge flow deviation was observed, this is shown
below in Figure 3.43.

116

Trailing edge Deviation (deg)

Blade Length Study: đ vs trailing edge deviation
đżâ˛
25.00
20.00
Series1

15.00

Series2
Series3

10.00

Series4
5.00

Series5
Series6

0.00
0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Zweifel Coefficient đ
Figure 3.43. Zweifel Coefficient vs. Flow deviation at trailing edge for Blade Length Study

For the Blade Length Study simulations of turbines with both long axial blade lengths
and short blade lengths were conducted, and it was observed that efficiency was highest for the
simulations of designs with Zweifel coefficients between .8 and 1.1. With Zweifel coefficients
below 0.8, efficiency decreased. This trend is illustrated in Figure 3.44, where the difference of
the peak efficiency measured in the Speed Study and efficiency of the design simulated is shown
vs. the Zweifel Coefficient for each case.

117

Peak efficiency-Efficiency simulated

4.50

Blade Length Study: Z vs Efficiency Decrease

4.00
3.50
3.00

Series1

2.50

Series2

2.00

Series3

1.50

Series4

1.00

Series5

0.50

Series6

0.00
0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Zweifel Coefficient đ

Figure 3.44. Difference of the peak efficiency and efficiency of the design vs. the Zweifel coefficient

The decrease in predicted efficiency resulting from extending the blades longer than
aerodynamically necessary can be considered modest, designs with Zweifel Coefficients near
.6(Âą.05) predicted a decrease in efficiency no larger than 1.7% compared to the peak efficiency
point. Designs with a Zweifel coefficient near .5(Âą.05) exhibited a maximum efficiency
decrease of 2.5% compared to the peak efficiency point. Blades for water turbines experience
high loads, and blades for water turbines often require large blade chords as longer blades can
demonstrate improved structural performance, as distributing the force over a larger area
decreases stresses on the blades [4]. Structural simulations were not done as a part of this study,
however if FEA analysis suggested the designs with higher Zweifel coefficients would fail,
longer blades could be employed to improve strength.
3.5.2. Flow Visualization and Vectors

The relative velocity vector diagrams shown in the below figures at span 50%, show
there are two areas where velocity gradients were had the highest magnitude, at the suction side
leading edge, and on the pressure side at the trailing edge. These large changes in flow velocity
118

over short distances can lead to shear stresses and losses [35]. A wake was observed after the
blade trailing edge where velocity is lower. The designs which had higher Zweifel coefficient
had greater magnitude velocity gradients at the leading and trailing edges, and a larger wake size.
Relative velocity vector contours were produced for each design in the Blade Length Study,
Figure 3.45 shows relative velocity contours with vectors for case 1, Figure 3.46 for case 2,
Figure 3.47 for case 3, Figure 3.48 for case 4, Figure 3.39 for case 5, and Figure 3.40 for case 6.
1j Z = 0.48
đ = 78.4%

1j8 Z = 0.8
1j8
đ = 79.9%

1j5 Z = 1.49
đ = 78.2%

Figure 3.45. Relative velocity vectors for case 1 of Blade Length Study: Span 50%
2l Z = 0.42
đ = 82.9%

2l8 Z = 0.84
đ = 85.5%

2l5 Z = 1.24
đ = 84.9%

Figure 3.46. Relative velocity vectors for case 2 of Blade Length Study: Span 50%

119

3n Z = 0.40
3n
đ = 86.8%

3n6 Z=1.05
đ = 89.8%

Figure 3.47. Relative velocity vectors for case 3 of Blade Length Study: Span 50%
4h Z = 0.40
đ = 79.4%

4h9 Z=0.91
đ = 82.0%

4h6 Z=1.06
đ = 81.9%

Figure 3.48. Relative velocity vectors for case 4 of Blade Length Study: Span 50%
5n Z = 0.44
đ = 83.7%

5n6 Z = 1.03
đ = 86.0%

Figure 3.39. Relative velocity vectors for case 5 of Blade Length Study: Span 50%

120

6h10 Z =1.08
đ = 89.6%

6h12 Z =0.9
đ = 89.9%

6h Z = .44
6h
đ = 87.1%

Figure 3.40. Relative velocity vectors for case 6 of Blade Length Study: Span 50%

For any turbine the flow velocity is lower on the pressure side, and the velocity is higher
on the suction side. For the designs with shorter blades, the simulated flow accelerated over a
shorter distance. For the designs with larger Zweifel coefficients simulated in the Blade Length
Study, the relative velocity was higher at the suction side, and the relative velocity at the pressure
side was lower compared to designs with lower Zweifel coefficients. To show more detail into
the flow features, vector diagrams were produced at the leading and trailing edges, vectors at the
leading edge are shown below in Figure 3.41 for two designs in case 5.
5n6 Z = 1.03
đ = 86.0%

5n Z = 0.44
5n
đ = 83.7%

Figure 3.41. Relative velocity vectors at leading edge for case 5 of Blade Length Study (Span 50%)

As Zweifel coefficient was increased, flow acceleration at the leading edge suction side
increased. This is shown in Figure 3.41 above, and is also seen below in Figure 3.42 for case 3,
and Figure 3.43 for case 1. For each design simulated in the Blade Length Study, this effect was
observed.

121

3n Z = 0.40
đ = 86.8%

3n6 Z=1.05
đ = 89.8%

Figure 3.42. Relative velocity vectors at leading edge for case 3 of Blade Length Study (Span 50%)
1j5 Z = 1.49
đ = 78.2%

1j Z = 0.48
đ = 78.4%

Figure 3.43. Relative velocity vectors at leading edge for case 1 of Blade Length Study (Span 50%)

The below figures show the velocity vectors at the trailing edge for designs in cases 1, 5
and 3, Figures 3.55, 3.56, and 3.57 respectively.
5n Z = 0.44
đ = 83.7%

5n4 Z=0.65
đ = 84.9%

5n6 Z = 1.03
đ = 86.0%

Figure 3.44. Relative velocity vectors at trailing edge for case 5 of Blade Length Study (Span 50%)

122

3n Z = 0.40
đ = 86.8%

3n6 Z=1.05
đ = 89.8%

Figure 3.45. Relative velocity vectors at trailing edge for case 3 of Blade Length Study (Span 50%)
1j8 Z = 0.8
đ = 79.9%

1j5 Z = 1.49
đ = 78.2%

1j Z = 0.48
đ = 78.4%

1j5 Z = 1.49
đ = 78.2%

Figure 3.46. Relative velocity vectors at trailing edge for case 1 of Blade Length Study (Span 50%)
3.5.3. Trailing Edge Effects

The above figures illustrate that as Zweifel coefficient was increased, flow acceleration
near the trailing edge on the pressure side of the flow increased. On the suction side as flow
travels towards the trailing edge, flow decelerates and experienced an adverse pressure gradient,
and the boundary layer along this surface became larger. The designs with higher Zweifel
coefficient experienced more flow deceleration towards the trailing edge on the suction side,
velocity gradients at the leading and trailing edges, and larger size boundary layers. As Zweifel

123

coefficient was increased the wake left by the trailing edge became larger, and shifted closer
toward the suction side. These effects can result in additional viscous shear, turbulence, entropy
production, and losses at the trailing edge. This effect was observed for each of the designs that
were simulated as a part of the Blade Length Study.
Figure 3.47 below shows the magnitude of the relative flow velocity near the blade
surface for designs 3n,3n3, and 3n6 of case 3, from 80% along the blade chord to the trailing
edge. Each design experienced flow acceleration at the pressure side near the trailing edge. The
figure shows the magnitude of this flow acceleration was highest for the designs with the shortest
blades. The designs with larger Zweifel coefficients had greater velocity gradients over a smaller
axial distance. This suggests the shear stresses and entropy production in this region are higher.
This effect was observed for each design simulated as a part of the Blade Length Study.

đž (đ/đ)

5n6 Z = 1.03
đ = 86.0%

5n4 Z=0.65
đ = 84.9%

5n Z = 0.44
đ = 83.7%

Normalized position along blade
Figure 3.47. Relative velocity near trailing edge of case 3 and 5 of Blade Length Study

Figure 3.48 below shows a contour of the turbulence kinetic energy two designs in case 1
of the Blade Length Study, Figure 3.49 shows two designs in case 3 and Figure 3.50 shows two

124

designs in case 5. These figures illustrate the increased turbulence at the trailing edge for the
shorter blade length designs.
1j5 Z = 1.49
đ = 78.2%

1j8 Z = 0.8
đ = 79.9%

Figure 3.48. Turbulence kinetic energy contour for cases 1j5 and 1j8: Span 50%
3n6 Z=1.05
3n6
đ = 89.8%

3n Z = 0.40
đ = 86.8%

Figure 3.49. Turbulence kinetic energy contour for designs 3n and 3n6: Span 50%
5n Z = 0.44
đ = 83.7%

5n6 Z = 1.03
đ = 86.0%

Figure 3.50. Turbulence kinetic energy contour for designs 5n and 5n6: Span 50%

125

Figure 3.51 below shows exit loss coefficient graphed vs Zweifel coefficient for each
design in the Blade Length Study. For Zweifel coefficients under .8, the exit loss coefficient đ2âđ
was relatively constant, however as Z was increased above 0.8, the exit loss coefficient
increased, which is likely the result of increased losses resulting from the trailing edge wake.

Blade length study: Z vs Exit Loss Coeff đđâđ
0.070

Exit Loss Coefficient

0.060
0.050
Series1
0.040

Series2
Series3

0.030

Series4
0.020

Series5
Series6

0.010
0.000
0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Zweifel Coefficient Z
Figure 3.51. Zweifel Coefficeint vs Exit Loss Coefficient for Blade Length Study

Turbines are usually designed to have a thin trailing edge. Blade profiles with high
Zweifel coefficient require careful design and iteration to reduce trailing edge losses and reduce
the risk of flow separation. Performance of the designs investigated in this work could be
improved by tapering the trailing edge, especially those with high Zweifel coefficients, to reduce
the trailing edge losses and wake size after the trailing edge.
3.5.4. Blade Loading

With shorter blades, total pressure must decrease over a shorter axial distance. This can
lead to more flow deviation and velocity gradients at the leading and trailing edges, which can
lead to additional losses. The designs with higher values of đ exhibited flows with lower
126

pressure on the suction side of the blade, and higher pressure on the pressure side. With shorter
axial length, the pressure difference from pressure to suction side needs to be larger to keep
power constant, as the area on which the pressure acts is smaller, and the same force must be
maintained to keep power constant.

Simulations showed that as Zweifel coefficient was

increased by decreasing blade length, an adverse pressure gradient developed on the suction side
of the blades after the point of minimum pressure. Another adverse pressure gradient at the
trailing edge at the suction side became more pronounced as Zweifel coefficient was increased.
Adverse pressure gradients are correlated with boundary layer growth, entropy generation, and
increased losses.
Below Figure 3.52 shows the pressure vs. blade position (blade loading) charts for case 3
at three Zweifel coefficients, at 50% span. Figure 3.53 shows static pressure contours for the
three designs within case 3 of the Blade Length Study. Figure 3.54 and 3.66 show the blade
loading chart and static pressure contours respectively for three designs within case 5. The
shorter blade length designs experienced flows with lower pressure on the suction side of the
blades, the figures illustrate this effect. For each design simulated in this work, towards the
trailing edge an area of low pressure and flow acceleration on the pressure side of the blade was
observed, located close next to an area of flow deceleration, an adverse pressure gradient, at the
suction side trailing edge. As the blade length was decreased this effect was magnified, and the
region was observed over a larger portion of the distance along the blade.

127

3n Z = .40

3n3 Z = .53

Îˇ=88.1%

3n6 Z = 1.05

Îˇ=89.77%

Pressure (Pa)

Îˇ=86.8%

Normalized position along blade
Figure 3.52. Blade loading diagrams for case 3 of Blade Length Study: Span 50%
3n Z = .40
đ = 86.8%

3n3 Z = .53
đ = 88.1%

3n6 Z = 1.05
đ = 89.77%

Figure 3.53. Static Pressure Distributions for Case 3 of Blade Length Study (Span 50%)

128

5n Z = .44

5n4 Z = .65

Îˇ=83.7%

5n6 Z = .1.03

Îˇ=86.0%

Pressure (Pa)

Îˇ=84.9%

Normalized position along blade
Figure 3.54. Blade loading diagrams for case 5 of Blade Length Study: Span 50%
5n Z = .44
đ = 83.7%

5n4 Z = ..65
5n4
đ = 84.9%

5n6 Z = .1.03

Îˇ=86.0%

Figure 3.55. Static Pressure Distributions for Case 5 of Blade Length Study (Span 50%)

Figures 3.67 and 3.68 below show velocity triangles superimposed over velocity vector
contours and blade loading diagrams produced in CFD-Post for two designs in case 2, designs 2l
and 2l8. The figure illustrates how as Zweifel coefficient increases, velocity on the suction side
of the blades increases, and velocity on the pressure side of the blades decreases.
129

Pressure (Pa)

Figure 3.56.Velocity triangles superimposed over velocity vector contours for design 2l of the Blade
Length Study at leading and trailing edge (span 50%)

130

Pressure (Pa)

Figure 3.57.Velocity triangles superimposed over velocity vector contours for design 2l8 of the Blade
Length Study at leading and trailing edges (span 50%)
3.5.5. Hub and Shroud Effects

Literature suggests that for the designs with higher Zweifel coefficient, losses via friction
from the blade surfaces, hub and shroud are reduced, however the losses resulting from diffusion
and shear stresses at the trailing edge are higher [24] [29]. The results obtained in the Blade
Length Study correlate with these findings.

131

To investigate the effects of friction resulting from the hub and shroud surfaces, Charts
were generated which tracked the relative velocity of the flow along the span of the blades.
Figure 3.58 below shows relative flow velocity graphed along the span of the blades at the
leading edge of the turbine for case 3, and Figure 3.59 shows the relative flow velocity graphed
along the span at the trailing edge. Decreased relative flow velocity was exhibited near the hub
and shroud for each design tested, indicating friction effects. The designs with higher Zweifel
coefficient experienced a larger decrease in relative velocity at the hub and shroud as the flow
traveled from leading to trailing edge. Although the velocity distribution at the leading edge was
similar for each of the designs, the designs with lower Zweifel coefficient experienced decreased
relative flow velocity at the trailing edge, especially near the hub and shroud. This suggests the
lower Z designs experienced additional friction resulting from the hub and shroud surfaces,
compared to the higher Z designs.

Span Normalized

6h12 Z =0.9
đ = 89.9%
6h10 Z =1.08
đ = 89.6%
6h Z = .44
đ = 87.1%

6h4 Z = .36
đ = 85.9%

đž(đ/đ)
Figure 3.58. Relative velocity of case 6 for Blade Length Study at the leading edge along span

132

Span Normalized

6h12 Z =0.9
đ = 89.9%
6h10 Z =1.08
đ = 89.6%
6h Z = .44
đ = 87.1%

6h4 Z = .36
đ = 85.9%

đž(đ/đ)
Figure 3.59. Relative velocity of case 6 for Blade Length Study at the trailing edge along span

Profile loss coefficient đđđđđđđđ was graphed vs the Zweifel coefficient to track how Z
effected losses from the leading to trailing edge. Figure 3.60 below shows that as Zweifel
coefficient was increased from the lowest values to .8, đđđđđđđđ decreased. However for the
Zweifel coefficients above .8, đđđđđđđđ stayed relatively constant.

133

Blade length study: Z vs Profile Loss đđđđđđđđ
0.80

Profile loss Coefficent

0.75
0.70

Series1
Series2

0.65

Series3
Series4

0.60

Series5
Series6

0.55
0.50
0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Zweifel Coefficent Z
Figure 3.60. Zweifel coefficeint vs Profile loss coefficient for Blade Length Study
3.5.6. Work Extraction

Average relative flow velocity was graphed from annulus inlet to annulus outlet. Figure
3.61 below shows average relative velocity from inlet to outlet for case 1, Figure 3.62 for case 6.
The designs with higher efficiency had higher relative velocity at the trailing edge, however each
design had equal relative velocities at the leading edge. As đđ stays constant from inlet to outlet,
the highest efficiency designs had the highest average relative circumferential velocity at the
trailing edge, đđ˘2 .

134

đž(đ/đ)

1j8 Z = 0.8
đ = 79.9%

1j Z = 0.48
đ = 78.4%

1j5 Z = 1.49
đ = 78.2%

Streamwise Location
Figure 3.61. Average relative velocity from inlet to outlet for case 1 for Blade Length Study
6h12 Z =0.9
đ = 89.9%

đž(đ/đ)

6h10 Z =1.08
đ = 89.6%

6h Z = .44
đ = 87.1%
6h4 Z = .36
đ = 85.9%

Streamwise Location
Figure 3.62. Average relative velocity from inlet to outlet for case 6 for Blade Length Study

The simulations of each design produced results which predicted reduced work extraction
near the hub and shroud for each design. This is illustrated in the figures below, which display
135

absolute circumferential flow velocity over the span at the trailing edge. Figure 3.63 below
shows absolute circumferential velocity graphed along the span of the blades for three designs in
case 6 of the Blade Length Study, at the trailing edge, and Figure 3.64 shows this for case 1. The
designs with Zweifel coefficients between 0.8 and 1.1 exhibited the highest absolute
circumferential velocity đśđ˘ near the hub at the trailing edge. These designs had the highest
change in angular momentum, and therefore work extraction, near the hub compared to those
with Zweifel coefficients outside this range.

Span Normalized

6h12 Z =0.9
đ = 89.9%
6h10 Z =1.08
đ = 89.6%

6h Z = .44
đ = 87.1%
6h4 Z = .36
đ = 85.9%

đŞđ (đ/đ)
Figure 3.63. Absolute circumferential velocity for case 6 of the Blade Length Study at the trailing edge
along span

136

Span Normalized

1j8 Z = 0.8
đ = 79.9%

1j5 Z = 1.49
đ = 78.2%

1j Z = 0.48
đ = 78.4%

đŞđ (đ/đ)
Figure 3.64. Absolute circumferential velocity for case 1 of the Blade Length Study at the trailing edge
along span

Graphs were generated which show đśđ˘ from inlet to outlet for each design, Figure 3.65
below shows this for case 6, and Figure 3.66 for case 1. The designs which had the highest
predicted efficiency had the highest circumferential velocity at the trailing edge (đśđ˘2 ). As
absolute circumferential velocity at the leading edge, đśđ˘1 = 0, this means the highest efficiency
designs exhibited flows with the largest change in absolute circumferential velocity đśđ˘2 â đśđ˘1 .
These designs had the most work extraction, they exhibited flows with the largest change in
angular momentum.

137

6h10 Z =1.08
đ = 89.6%
6h4 Z = .36
đ = 85.9%

đŞđ (đ/đ)

6h12 Z =0.9
đ = 89.9%

6h Z = .44
đ = 87.1%

Streamwise Location
Figure 3.65. Average Absolute circumferential velocity from inlet to outlet of case 6 of Blade Length
Study

đŞđ (đ/đ)

1j8 Z = 0.8
đ = 79.9%

1j Z = 0.48
đ = 78.4%

1j5 Z = 1.49
đ = 78.2%

Streamwise Location
Figure 3.66. Average Absolute circumferential velocity from inlet to outlet of case 1 of Blade Length
Study

To help observe the effect of reducing Zweifel coefficient on the simulated work
extraction, the product of absolute circumferential velocity đśđ˘ and rotational velocity đđ was

138

graphed over the span of the at the trailing edge for each design. Figure 3.67 below shows
đđ đśđ˘ graphed for three designs in case 1 of the Blade Length Study at the trailing edge of each
turbine. The highest efficiency designs had the highest value of đđ đśđ˘

at the trailing edge. As

Zweifel coefficient was increased, work extraction đđ đśđ˘ at the trailing edge increased. However
as Zweifel coefficient was increased above 1.2, đđ đśđ˘ at the trailing edge decreased. This effect
can be seen below in Figure 3.67 below, although the same effect was observed for each case in
the Blade Length Study. The designs were set up to have no pre-swirl, such that đśđ˘1 = 0.
Designs which had the highest đđ đśđ˘ at the trailing edge had the highest efficiency, as these
designs exhibited flows with the largest change in angular momentum, and thus the largest shaft

Span Normalized

power.

1j5 Z = 1.49
đ = 78.2%
1j Z = 48
đ = 78.4%
1j8 Z = 0.8
đ = 79.9%

đźđ đŞđ (đđ /đđ )
Figure 3.67. đđ đśđ˘ for case 1 of the Blade Length Study at the trailing edge across span
3.5.7. Cavitation

As blade length was decreased to lower Zweifel coefficient, areas of low static pressure
developed on the suction side of the blades, which could lead to cavitation in a real-world
application. The below figures show areas of low static pressure for different designs in the
139

Blade Length Study. Figure 3.68 shows the low static pressure contour for case 1, Figure 3.69 for
case 2, Figure 3.70 for case 3, Figure 3.71 for case 4, Figure 3.72 for case 5, and Figure 3.73 for
case 6. The results suggest that as Z is increased, the risk of cavitation became greater.
1j8 Z = 0.8
đ = 79.9%

1j Z = 48
đ = 78.4%

1j5 Z = 1.49
đ = 78.2%

Figure 3.68. Areas of low static pressure for case 1 of Blade Length Study (Span 50%)
2l Z = 0.42
đ = 82.9%

2l8 Z = 0.84
đ = 85.5%

Figure 3.69. Areas of low static pressure for case 2 of Blade Length Study (Span 50%)
3n Z = .40
đ = 86.8%

3n6 Z = 1.05
đ = 89.77%

Figure 3.70. Areas of low static pressure for case 3 of Blade Length Study (Span 50%)

140

4h8 Z=1.26
đ = 81.8%

4h9 Z=0.91
đ = 82.0%

4h Z = 0.40
đ = 79.4%

Figure 3.71. Areas of low static pressure for case 4 of Blade Length Study (Span 50%)
5n Z = .44
đ = 83.7%

5n6 Z = 1.03
đ = 86.0%

Figure 3.72. Areas of low static pressure for case 5 of Blade Length Study (Span 50%)
6h12 Z =0.9
đ = 89.9%

6h Z = .44
đ = 87.1%

6h10 Z =1.08
đ =Z89.6%
6h10
=1.08
đ = 89.6%

Figure 3.73. Areas of low static pressure for case 6 of Blade Length Study (Span 50%)

141

CHAPTER 4. CONCLUSIONS
No flow separation was observed for any of the designs simulated, and after optimizing
the rotational speed and blade length each set of specifications had a design with efficiency
predicted via CFD over 79%. This is despite the fact the highest efficiency designs simulated
were located below the Cordier Line. The designs with lower diameter and smaller crosssectional area showed significantly lower predicted efficiency, Cases 1 and 4 have a tip diameter
of .2 meters, designs done for those cases had a peak efficiency of 80%, compared to 90% for
cases 3 and 6, which had a tip diameter of .5 meters.
Both hypotheses introduced in section 1.17 were confirmed in this work. First, the
designs simulated with specific speed which place it on the Cordier line performed with lower
efficiency than designs with specific speed lower than that recommended by the line. Second,
peak efficiency in CFD was obtained in CFD for designs which had Zweifel coefficients between
.8 and 1.1.
4.1. Speed Study and Cordier Line
For the Speed Study, six sets of turbine specifications were used to produce turbine
designs, which were tested in CFD. Designs were generated and simulated at differing rotational
speeds. Highest efficiency was observed in the designs which ran at speeds at a range 12-33%
that of the Cordier-recommended speed. Highest efficiencies were recorded when the designs
were generated at specific speeds between 1.37 and 1.98, lower than the values of specific speed
recommended by the Cordier line, between 5.5 and 6.1.
Designs which were generated for higher rotational speeds experienced more severe
adverse pressure gradients on the suction side of the blades in simulation, and more severe
velocity gradients near the leading and trailing edges. Turbulence kinetic energy at the trailing

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edge was higher for the designs which operated at higher rotational speeds. The simulations
suggest that as rotational speed was increased, the risk of cavitation on the suction side
increased.
4.2. Blade Length Study and Zweifel Coefficient
For the Blade Length Study, the designs at which highest efficiency was measured in the
Speed Study were simulated with different Zweifel coefficients. Zweifel coefficient was changed
by adjusting axial blade length đ. Lower efficiency was predicted for the designs with Zweifel
coefficients less than 0.75. Simulated efficiency was highest in the range of 0.8 < đ < 1.1,
within this range efficiency was relatively constant.
With higher Zweifel coefficients, pressure on the suction side is reduced and pressure on
the pressure side is increased. Designs which had higher Zweifel coefficients experienced more
severe adverse pressure gradients on the suction side of the blades. The higher Zweifel
coefficient designs experienced more losses at the trailing edge and had a larger wake after the
trailing edge. As Zweifel coefficient was increased, the risk of cavitation on the suction side
increased. Increased Zweifel coefficients were correlated with larger flow velocity gradients at
the leading and trailing edges of the blades. Trailing edge flow deviation increased linearly as
Zweifel coefficient was increased.
Literature suggests that for the designs with higher Zweifel coefficient, losses via friction
from the blade surfaces, hub and shroud are reduced, however the losses resulting from diffusion
and shear stresses at the trailing edge are higher. The results obtained in the Blade Length Study
are in line with this prediction. Profile loss coefficient was lowest for designs within 0.8 < đ <
1.1, however as Zweifel coefficient was increased above 0.8, loss coefficient of the exit region
increased, the result of increased trailing edge losses. Turbine designs with Zweifel coefficients

143

below 0.8 were also tested, it was shown that longer blade lengths could be employed, although
these designs exhibited more friction losses and lower predicted efficiency in simulations.
4.3. Next Steps of the Project
Additional simulation work can be done to shed more light on the performance of
CTPATs. The simulations done in this work demonstrate the efficacy of the designs; however
the manufactured turbine will have real geometry different than the idealized 3-D modeled
geometry simulated. The Woven Wheel turbomachinery manufacturing technique offers
benefits, however additional work is required to mature the manufacturing process to the point
where it could be used to produce turbine rotors within acceptable tolerances. At high Reynolds
numbers the surface finish of the blades become more important to keep friction losses low [35].
Blade thickness was kept constant across each design within each set of specifications,
referred to as a case. To achieve more complete dynamic similarity between each design in the
Speed Study

đĄ
đ

could be kept constant across each simulation, but for some of the higher speed

simulations axial blade length must be decreased to keep Zweifel coefficient at .8, many of the
blade thicknesses would be under 1 millimeter, which would be more costly to manufacture and
could experience structural failure.
The designs tested in this analysis did not have inlet or outlet guide vanes. Further
simulations would show how this affects the performance of the turbine at off-design points, at
differing operating conditions. The simulations assumed a uniform laminar velocity profile at the
turbine inlet. This is an idealized scenario, in a real world application flow would be more
turbulent at the inlet and a nose cone/guide vanes would be required to guide the flow into the
turbine similar to the one simulated in this study. Transient simulations would reveal additional
details into the flow features. The speed and blade length studies could be repeated with different

144

specifications, altering volume flow rate, head, and turbine geometry. This could reveal
additional insights into how the inputs effect the specific speed at which peak efficiency is
measured.
ANSYS CFX includes cavitation modeling as a part of the simulation package, designs of
this type should be simulated with cavitation modeling to see where cavitation bubbles develop
and collapse. The design could then be improved to reduce the risk of cavitation. Finite Element
Analysis (FEA) simulations of the blades would validate the structural performance of the
turbine designs. FEA can be used to find the points with the highest stresses, and the design can
be changed to reduce stress in those areas.
Performance would be improved by using blade profiles designed for water turbines
rather than using constant thickness blade profiles with rounded leading and trailing edges.
Trailing edge thickness could be reduced to reduce flow deviation, reduce trailing edge losses
and flow deviation.

145

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146

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