DESIGN - DEVELOP - TEST A LOW HEAD HYDRAULIC TURBINE USING NEW THEORY FOR THE STANDARD MODULAR HYDROPOWER TECHNOLOGY By Jinbo Chen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering Doctor of Philosophy 20 20 ABSTRACT DESIGN - DEVELOP - TEST A LOW HEAD HYDRAULIC TURBINE USING NEW THEORY FOR THE STANDARD MODULAR HYDROPOWER TECHNOLOGY By Jinbo C hen Hydropower has been considered as a great renewable energy resource for decades and provides enormous clean and renewable energy every year. In terms of generation, hydropower is the primary source of renewable energy in the United States, delivering 4 8% of total renewable electricity sector generation in 2015, and roughly 62% of total cumulative renewable generation over the past decade (2006 - 2015) . However, recently, the large hydropower project is questioned because of the concerns of the large reservoir, dam , and the water channel on the local environment . Due to the smaller scale, short development time , and low environmental impact, the low - head small hydropower system gains increasing attention from the industrial and academic community. The l ow - head hydropower has the potential to generate a significant amount of electricity from rivers that traditionally were unsuitable for developing hydraulic power plant s and supporting the resiliency of the U.S electricity system. Based on the 2016 Hydro power Vision Report, across the U.S, approximate ly 65.5 GW of new stream - reach hydropower capac ities are available . Th ese new stream - reach resources are characterized by low - head, varying flows, and highly valued river function s , including fish preservation , sediment transport, and recreational usage. T he development of those resources could be possible only if the technologies for low - head hydropower that balance efficiency, economics, and environmental sustainability were developed. T he traditional hydropower design method was limited to the new challenges of the l ow - head application. Therefore, a new Standard Modular Hydropower Technology (SMH) was proposed by the U.S. Department of Energy (DOE) in 2017. This new concept offers a new perspective for small hydropower technology development s based on the premise that standardization, modularity, and preservation of stream functionality must become essential and fully realized features of next - generation hydropower technologies and projec t designs , and consists of three major modules: Generation Module, Passage Modules, Foundation Modules . Based on the need s for the new design method suitable for the SMH , t his research focuses on developing a new design methodology for the Generation Modu le , which is a low impact, damless Kaplan turbine system , suitable for the low - head new stream - reach sites application. With extensive numerical simulation results and flexible geometrical configuration methods , the new design methodology can balance the p erformance, economics, and environmental sustainability and provide new perspectives for the future low - head hydropower system designs and developments. Copyright by JINBO CHEN 20 20 v T o th is endless, meaningless , and joyful life that I attempt to make better. vi ACKNOWLEDGEMENTS To my supported advisor Dr. Abraham Engeda for this amazing project and opportunity. To my helpful and kind committee members, Dr. Norbert Mü e ller , Dr. Andre Benard, and Dr. Wei Liao , for being part of this journey and providing the crucial advice that necessary for my success. To all my friends and colleagues, Zhangrui Zhou, Mekuannint Messele , Huixiang Chen, Jintong Gu, Leilei Wang, Xu Tan, and others for supporting my works. To my p arents for letting me do things freely without any restraints . vii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ .......................... x LIST OF FIGURES ................................ ................................ ................................ ...................... xii KEY TO SYMBOLS ................................ ................................ ................................ ................. xxiii 1. CHAPTER I: Introduction ................................ ................................ ................................ ...... 1 1.1. History of hydropower and hydraulic turbines ................................ ................................ 1 1.2. Small hydropower and its application (A literature review) ................................ ............ 3 Classification of hydro - turbines and related low head applications ......................... 4 Impulse Turbines ................................ ................................ ................................ ...... 5 Reaction Turbines ................................ ................................ ................................ ..... 7 The Axial hydro - turbine for low head application ................................ ................... 9 1.3. Standard Modular Hydropower Technology (SMH) ................................ ..................... 14 S MH Generation Module ................................ ................................ ........................ 17 Design Specifications for SMH Generation Module ................................ .............. 19 1.4. Research Objectives ................................ ................................ ................................ ....... 19 2. CHAPTER II: Basic Mechanics and Physics for Hydraulic Turbine System ...................... 21 2.1. Main Conservation Equations ................................ ................................ ........................ 21 The Continuity Equation ................................ ................................ ......................... 21 First Law of Thermodynamics for Hydraulic Turbine ................................ ........... 22 .......................... 23 ................................ ................................ ... 24 2.2. Definitions of Efficiency and Loss Estimation ................................ .............................. 26 3. CHAPTER III: Design Methodology Developments ................................ ........................... 29 3.1. Turbine selection and current development status ................................ ......................... 29 3.2. Design methodology developments ................................ ................................ ............... 32 Additional de sign considerations ................................ ................................ ............ 32 Generation module initial size and operation condition design .............................. 33 1 - D Vortex analysis for axial turbine ................................ ................................ ...... 36 1 - D Blade geometry construction ................................ ................................ ........... 39 Summaries o f 1 - D blade design ................................ ................................ .............. 44 3.3. ................................ ........................... 45 Stator Inlet Angle ................................ ................................ ................................ .... 45 Stator - blade Number ................................ ................................ ............................... 46 Runner - blade Number ................................ ................................ ............................. 47 Stator and Runner blade Solidity ................................ ................................ ............ 48 Stator and Runner blade Thickness ................................ ................................ ......... 50 3.4. Summary ................................ ................................ ................................ ........................ 52 viii 4. CHAPTER IV: Numerical Methods ................................ ................................ ..................... 53 4.1. General governing equations ................................ ................................ .......................... 53 4.2. Numerical method de velopments ................................ ................................ ................... 54 Computational domain construction ................................ ................................ ....... 54 Turbulence model selection ................................ ................................ .................... 58 Boundary conditions setting ................................ ................................ ................... 61 4.3. Summary ................................ ................................ ................................ ........................ 63 5. CHAPTER V: Numerical Results and Discussions ................................ .............................. 64 5.1. General hub size consideration ................................ ................................ ...................... 64 5.2. Stator - blade cons ideration ................................ ................................ .............................. 67 Stator - blade stagger - angle setting constant consideration ................................ ...... 67 Stator - blade inlet angle considerations ................................ ................................ ... 72 Stator - blade configuration considerations ................................ .............................. 77 Stator - blade number consideration ................................ ................................ .. 77 Stator - blade solidity consideration ................................ ................................ .. 79 Stator - blade thickness consideration ................................ ............................... 88 5.3. Runner - blade consideration ................................ ................................ ............................ 92 Runner - blade stagger - angle setting constant consideration ................................ .... 92 Runner - blade configuration considerations ................................ ............................ 95 Runner - blade number consideration ................................ ................................ 95 Runner - blade solidity consideration ................................ ................................ 97 Runner - blade thickness consideration ................................ ........................... 105 5.4. Off - des ign consideration ................................ ................................ .............................. 118 max ) considerations ................................ ...................... 126 Off - design operating range considerations ................................ ........................... 128 6. CHAPTER VI: Conclusions and Future Works ................................ ................................ . 138 6.1. Summary of the SMH technology ................................ ................................ ................ 138 6.2. Summary of the design methodology ................................ ................................ ........... 138 6.3. Summary of the numerical results ................................ ................................ ................ 139 6.4. Future works suggestions ................................ ................................ ............................. 143 APPENDICES ................................ ................................ ................................ ............................ 145 APPENDIX A: The DO - friendly hydraulic turbine ........ 146 APPENDIX B: Initial design parameters with description and sugge sted setting values ...... 148 APPENDIX C: Additional results and plots for various hub size considerations. ................. 151 APPENDIX D: Additional results and plots for stator Cssa constant considerations. ........... 159 APPENDIX E: Additional results and plots for stator - blade inlet angle considerations. ....... 163 APPENDIX F: Additional results and plots for stator - blade solidity considerations. ............ 167 APPENDIX G: Additional results and plots for stator - blade thickness considerations. ........ 170 APPENDIX H: Additional results and plots for runner - blade Crsa constant considerations. 174 APPENDIX I: Additional results and plots f or runner - blade solidity considerations ............ 179 APPENDIX J: Additional results and plots for runner - blade thickness considerations ......... 181 ix BIBLIOGRAPHY ................................ ................................ ................................ ....................... 195 x LIST OF TABLES Table 1.1. Key Features of hydroelectric power plants [1] ................................ ............................ 2 Table 1.2. The operating ranges of the three most popular turbine for traditional hydraulic systems [1] ................................ ................................ ................................ ................................ .................... 2 Table 1.3. A general classification of hydropower turbines and their applicable head ranges. ..... 4 Table 1.4. Current configurations of axial turbines for low head applications (Adapted from [36]) ................................ ................................ ................................ ................................ ....................... 12 Table 1.5 . Current turbines on the market suitable for low head applications (Adapted from [37]) ................................ ................................ ................................ ................................ ....................... 13 Table 1.6. The basic functions for the SMH modules and their sub - modules ............................. 16 Table 2.1. Loss models for axial hydraulic turbines (Adapted from [43]). ................................ .. 27 Table 3.1. Five major types of turbines for low - head applications. (Adapted from [46]) ........... 30 Table 3.2. Seven blade configuration parameters for stator and runner blade ............................. 45 Table 4.1. Detail mesh information for the five selected mesh schemes ................................ ..... 57 Table 4.2 . Design and numerical simulation conditions ................................ .............................. 63 Table 5.1. The design conditions and geometrical configurations for four selected models ................. 119 Table 5.2. Maximum efficiency values and its location for all selected models under the design rotational ................................ ................................ ................................ ......................... 126 Table 5.3. Maximum efficiency values and its location for all selected models under the lower rotational ................................ ................................ ................................ ......................... 126 Table 5.4. Maximum efficiency values and its location for all selected models under the higher rotational ................................ ................................ ................................ ......................... 127 Table 5.5. Operating range and the corresponding performance for all selected models under the ................................ ................................ ................................ 131 xi Table 5.6. Operating range and the corresponding performance for all selected models under the lower ................................ ................................ ................................ ........... 132 Table 5.7. Operating range and the corresponding performance for all selected models under the higher rotational speed ................................ ................................ ................................ . 134 Table 6.1 . Geometrical and operational conditions comparison between a real - size model and a 2/7 scale model ................................ ................................ ................................ ........................... 144 Table Appendix.1. - friendly hydraulic turbine [50] ................................ ................................ ................................ ................................ .. 146 Table Appendix.2. Initial design parameters with description and recommend setting values 148 xii LIST OF FIGURES Figure 1.1. A low - head Turgo wheel turbine developed by Energy System & Design Ltd [23] .. 5 Figure 1.2. A crosssection of a low - head six jet Pelton turbine (Right), and a single jet Pelton turbine (Left) [24,25] ................................ ................................ ................................ ...................... 6 Figure 1.3. Low - head crossflow turbine in the vertical position (Left), horizontal position (Right) [26] ................................ ................................ ................................ ................................ .................. 6 Figure 1.4. Cross - section of a low head Francis turbine with scroll case, wicket gates and draft tube [28] ................................ ................................ ................................ ................................ .......... 7 Figure 1.5. Basic structure of a kinetic turbine with multiple unit configurations [29] ................. 8 Figure 1.6. Water flow geometry in a low - head Archimedes screw turbine [32] .......................... 9 Figure 1.7. Cross - section of a horizontal bulb turbine and generat or for low - head power station [33] ................................ ................................ ................................ ................................ ................ 10 Figure 1.8. An example layout of the Straflo turbine for low head application [34] ................... 10 Figure 1.9. A horizontal tube turbine configuration for lo w - head hydropower plant [20] .......... 11 Figure 1.10. Basic components of a traditional Kaplan turbine power station [35] ................... 11 Figure 1.11. Conceptual schematic showing the primary modules of an SMH facility design [39] ................................ ................................ ................................ ................................ ....................... 15 Figure 1.12. Daily average flow rate for a selected new Stream - reach site for the past 50 - years. (Data extrapolated from Oak Ridge National Laboratory SMH Explorer website: https://smh.ornl.gov/explorer/) ................................ ................................ ................................ ..... 18 Figure 2.1. Demonstration of an elementary mass across an area element (dA) with an absolute ................................ ................................ ................................ ... 21 Figure 2.2. The v elocity triangle for a generalized turbine system that shows each velocity component at each turbine station: Station - 1 is the Turbine inlet, Station - 2 is the Stator Outlet, S tation - 3 is the Runner Outlet ................................ ................................ ................................ ....... 25 xiii Figure 3.1. Assembly Model; Bottom: Close - up for each component) ................................ .......................... 31 Figure 3.2. A typical specification map for different turbine types [48] ................................ ..... 34 Figure 3.3. A conventional Kaplan turbine specificat ion map using for initial design [49] ....... 34 Figure 3.4. A radial fluid element in a typical axial turbomachinery system. ............................. 37 Figure 3.5. Three example runner - blades configurations with different vortex assumptions. (From left to right: Free vortex, Force Vortex, Constant Vortex) ................................ ........................... 38 Figure 3.6. Three example runner - blade configurations with different hub diameter settings. (From left to right: hub - to - tip ratios (D hub /D tip ) =0.685, 0.743, 0.8.) ................................ ............ 38 Figure 3.7. A standard unit runner - blade profile for the proposed turbine system ...................... 40 Figure 3.8. Runner - blade and Stator - blade with different stagger - angle setting constants ......... 42 Figure 3.9. Superimposition method using the thickness distribution function to construct blade profile [56] ................................ ................................ ................................ ................................ .... 44 Figure 3.10. Five stator - blades with different inlet blade angles ( 1blade ). ( From left to right: 1blade 55 , 70 , 90 , 110 ,130 ) ................................ ................................ ................................ ........... 46 Figure 3.11. Five stator models with different stator number. (From left to right: Stator Number = 20, 30, 40, 50 ,60) ................................ ................................ ................................ ......................... 47 Figure 3.12. Five runner - blades with different runner - blade number. (From left to right: Runner Number = 5, 6, 8, 10, 12) ................................ ................................ ................................ .............. 48 Figure 3.13. The run ner - blade profile loss as a function of the invert of solidity [56] ( The circle marks the optimum value) ................................ ................................ ................................ ............ 49 Figure 3.14. Various Run ner - blades with different solidity values. (Form left to right: r =0.7,0.8,0.9,1,1.1,1.2) ................................ ................................ ................................ ................ 50 Figure 3.15. Various stator - s =0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 2.0, 3.0.) ................................ ................................ ................................ . 50 Figure 3.16. The definition of the blockage at the runner - blade inlet ................................ ......... 51 xiv Figure 3.17. Five runner - blades with different relative thickness values. (From left to right: R T =5%, 7.5%, 10%, 12.5%, 15%.) ................................ ................................ ............................... 52 Figure 3.18. Five stator - blade with different relative thickness values. (From left to right: S T =5%, 7.5%, 10%, 12.5%, 15%) ................................ ................................ ................................ .............. 52 Figure 4.1. The nu merical grids for the three computational fluid domains. .............................. 56 Figure 4.2. The mesh quality and normalized performance relation for th e five selected schemes. ................................ ................................ ................................ ................................ ....................... 57 Figure 5.1. The relation between overall hydraulic efficiency and design flow rate (Q 11 ) for five selected Hub - to - Tip ratio configurations ................................ ................................ ...................... 65 Figure 5.2. The circumferential velocity difference (C u2 - C u3 ) distribution plots from hu b to tip at two Q 11 conditions for all five different hub - size configurations, (a). Low Flow Rate Condition, Q 11 =0.103. (b). High Flow Rate Condition, Q 11 =0.31 ................................ ................................ .. 66 Figure 5.3. The relation between overall hydraulic efficiency and stator - blade stagger - angle setting constant C ssa for four flow rate conditions ................................ ................................ ........ 67 Figure 5.4. The runner - 2 flow angle distribution plots for four selected models in 2 blade value ................................ ................................ .................. 69 Figure 5.5. The runner - blade inlet absolute flow angle ( 2 ) distribution plots for four selected 2 value ................................ ................................ ........ 70 Figure 5.6. The stream - line pattern for the (a) Model - 1and (b) Model - 4, at 50% span location 71 Figure 5.7. The Inflow angle ( 1flow ) distribution across all span location for three general - inclined angle configurations ................................ ................................ ................................ ...................... 73 Figure 5.8. The Relation between the overall hydraulic efficiency, Power Generation, and the Stator - 1 blade ) for thr ee general - inclined angle configurations under the same design flow rate condition (Q 11 =0.259) ................................ ................................ ........................ 73 Figure 5.9. The runner - 2 flow distribution plots for four selected 2 blade value ................................ ................................ .. 74 Figure 5.10. The runner - blade inlet absolute flow angle ( 2 ) distribution plots for four selected models in comparison with the designed 2 value ................................ ................................ ........ 76 xv Figure 5.11. The stream - line pattern for the (a) Model - 1and (b) Model - 4, at 13% span location ................................ ................................ ................................ ................................ ....................... 76 Figure 5.12. The relation between stator - blade number, overall hydraulic efficiency, and normalized shaft power for four designed flow conditions ................................ .......................... 78 Figure 5.13 . The relation between stator - blade solidity with (a) overall hydraulic efficiency and (b) normalized shaft power for four designed flow rate conditions ................................ .............. 79 Figure 5.14. The runner - 2 flow angle distribution plots for three selected models in 2 blade value ................................ ................................ .................. 82 Figure 5.15. The runner - 2 ) distribution plots for three selected 2 value ................................ ................................ ........ 83 Figure 5.16. The stream - line pattern for t he (a) Model - 1 and (b) Model - 2, at 60% span location ................................ ................................ ................................ ................................ ....................... 83 Figure 5.17. Three different velocity distribution at the stator exit (runner - blade inlet) for all three selected models: (a) Absolute velocity (C 2 C x2 ................................ ................................ ................................ ................................ ................. 85 Figure 5.18. Static pressure distribution at 60% span location for all three selected models, (a) Model - 1, s =0.8 (b) Model - 2, s =1.6, (c) Model - 3, s =2.6 ................................ .......................... 87 Figure 5.19. Velocity Triangle for Model - 1, s =0.8 (Green Arrow), Model - 3, s =2.6 (Black Arrow), and Designed value (Red Arrow) at 60% span location ................................ ................. 87 Figure 5.20. The relation between stator - blade relative thickness with (a) overall hydraulic efficiency and (b) normalized shaft power for four designed flow rate conditions ..................... 88 Figure 5.21. The relation between (a). Overall performance (b). Normalized Power, an d stator - blade solidity under three stator - blade relative thickness values for one flow rate condition (Q 11 =0.207) ................................ ................................ ................................ ................................ ... 89 Figure 5.22. Th e runner - 2 ) distribution plots for the two selected 2 value ................................ ................................ ........ 91 Figure 5.23. The runner - 2 flow angle distribution plots for the two selected models in 2 blade value ................................ ................................ .................. 91 Figure 5.24. The normalized total pressure drop across all span locations for the two selected models at the stator exit ................................ ................................ ................................ ................ 92 xvi Figure 5.25. The Relation between overall hydraulic efficiency and runner - blade stagger - angle setting constant C rsa for four flow rate conditions. ................................ ................................ ....... 93 Figure 5.26. The circumferential velocity difference (C u2 - C u3 ) distribution plots from hub to tip for four selected mo dels. ................................ ................................ ................................ ............... 94 Figure 5.27. The runner blade stream - line pattern for the (a) Model - 2, and (b) Model - 4, at 80% span location ................................ ................................ ................................ ................................ . 95 Figure 5.28. The relation between runner - blade number, overall hydraulic efficiency, and normalized shaft power at four flow conditions ................................ ................................ ........... 96 Figure 5.29. The relation between runner - blade solidity, (a) overall hydraulic efficiency, and (b) normalized shaft po wer for four flow conditions ................................ ................................ ......... 98 Figure 5.30. Two runner - blade geometries at two flow rate conditions with two blade solidity values ................................ ................................ ................................ ................................ ............ 99 Figure 5.31. The relative flow angle at (a). runner inlet ( 2 ) and (b). runner outlet ( 3 ) for all four selected models with comparison of the designed value ................................ ............................ 100 Figure 5.32. The streamline pattern comparison of (a) Model - 1 and (b) Model - 2 at the 50% span lo cation ................................ ................................ ................................ ................................ ........ 101 Figure 5.33. The runner - blade circumferential velocity difference (C u2 - C u3 ) distribution for all four models ................................ ................................ ................................ ................................ . 102 Figure 5.34. The comparison between (a). Model - - contour at the 50% span location ................................ ................................ ................................ 102 Figure 5.35 . Normalized - pressure distribution along the streamwise direction of the runner - blade for Model - 2 and Model - 4 at 50% span location ................................ ................................ ......... 103 Figure 5.36. The radial velocity (C r ) distribution at runner - blade (a). Inlet and (b). Outlet for Model - 2 and Mode l - 4 ................................ ................................ ................................ ................. 104 Figure 5.37. The runner - blade (a). Inlet and (b). Outlet velocity triangle for the four selected models and the designed val ues at 50% span location. [Green Arrow: Model - 1; Blue Arrow: Model - 2; Black Arrow: Model - 3; Go l d Arrow: Model - 4; Red Arrow: Designed Value] ......... 105 Figure 5.38. The relation between runner - blade relative thickness with (a). Overall hydraulic efficiency and (b). Normalized shaft power for four designed flow rate conditions ................. 107 xvii Figure 5.39. The relation between runner - blade relative thickness and runner - blade maximum - blockage ratio for four flow conditions ................................ ................................ ...................... 108 Figure 5.40. The relation between mean runner - blade maximum - blockage ratio with (a). Overall hydraulic efficiency and (b). Normalized shaft power for four designed flow rate co nditions. . 108 Figure 5.41. (a). Runner - blade inlet absolute flow angle ( 2 ) and (b). runner - blade inlet relative flow angle ( 2 ) distribution across all span locations for the three selected models. ................. 110 Figure 5.42. (a). The runner - blade inlet axial velocity (C x2 ), and (b).The runner - blade inlet radial velocity (C r2 ) distribution across all span locations for the three selected models ..................... 111 Fi gure 5.43. RM distribution from tip to hub for the four flow rate conditions at 10% relative thickness settings ................................ ................................ ................................ ........................ 112 Figure 5.44. The relation between runner - blade solidity with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickness conditions at one flow rate condition (Q 11= 0.207) ................................ ................................ ................................ ................................ .. 113 Figure 5.45. The relation between the mean runner - RM ) with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickne ss conditions at one flow rate condition (Q 11 =0.207) ................................ ................................ ....................... 114 Figure 5.46. Streamline pattern for the Selected Models - II (Low solidity) at three different span locations, (a) - (c): Small runner - blade thickness, R T= 0.05; (d) - (e): Large runner - blade thickness, R T =0.15 ................................ ................................ ................................ ................................ ....... 116 Figure 5.47. Pressure Distribution Pattern for the Selected Models I (High solidity) at three different span locations, (a) - (c): Small runner - blade thickness, R T =0.05 ; (d) - (e): Large runner - blade thickness, R T =0.15 ................................ ................................ ................................ ............ 117 Figure 5.48. Normalized Pressure distribution along the streamwise direction of the runner - blade for the selected Models I (High solidity) at 50% span location ................................ .................. 118 Figure 5.49. N 11 and Q 11 four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 .......................... 120 Figure 5.50. N 11 and Q 11 four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4. ......................... 122 Figure 5.51. N 11 and Q 11 four selected models, (a). M odel - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4. ......................... 124 xviii Figure 5.52. Power and head based performance map at the design rotational spe for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 .................... 128 Figure 5.53. Power and head for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 .................... 129 Figure 5.54. Power and head based for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 .................... 130 Figure 5.55. Simplified power operating range for all four models under the design rotational speed ................................ ................................ ................................ ................................ ........... 131 Figure 5.56. Simplified head operating range for all four models under the design rotational speed ................................ ................................ ................................ ................................ ..................... 132 Figure 5.57. Simplified power operating range for all four models under the lower rotational speed ................................ ................................ ................................ ................................ ..................... 133 Figure 5.58. Simplified head operating range for all four models under the lower rotational speed ................................ ................................ ................................ ................................ ..................... 134 Figure 5.59. Simplified power operating range for all four models under the higher rotational speed ................................ ................................ ................................ ................................ ........... 135 Figure 5.60. Simplified head operating range for all four models under the higher rotational speed ................................ ................................ ................................ ................................ ..................... 136 Figure.Appendix C.1 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31 ................ 151 Figure.Appendix C.2 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; f or the five different hub sizes at low flow rate condition Q 11 =0.103 ............... 152 Figure.Appendix C.3 . Runner - blade Radial velocity (C r ) distr ibution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31 ................ 153 Figure.Appendix C.4 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flow rate condition Q 11 =0.103 ............... 154 Figure.Appendix C.5 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runne r Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31, with xix comprision to the designed values [The designed value for 3 is across all span locations, which is not shown.] ................................ ................................ ................................ .............................. 155 Figure.Appendix C.6 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flo w rate condition Q 11 =0.103, with comprision to the designed values [The designed value for 3 is across all span locations, which is not shown.] ................................ ................................ ................................ .............................. 15 6 Figure.Appendix C.7 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31, with comprision to the designed values ................................ ................................ .............................. 157 Figure.Appendix C.8 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the fi ve different hub sizes at low flow rate condition Q 11 =0.103, with comprision to the designed values ................................ ................................ .............................. 158 Figure.Appendix D.9 . The rel ation between normalized power and stator - blade stagger - angle setting constant C ssa for four flow rate conditions. (The power is normlized with C ssa =0.7 models) ................................ ................................ ................................ ................................ ..................... 159 Figure.Appendix D.10 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the four selected models ................................ ................................ ................................ ........................... 159 Figure.Appendix D.11 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 160 Figure.Appendix D.12 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 161 Figure.Appendix D.13 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the four selected models ................................ ................................ ................................ ........................... 162 Figure.Appendix E.14 . The Relation between the overall hydraulic efficiency, Power Generation, and the Stator - blade inlet angle 1 blade ) for three general - inclined angle configurations under the same design flow rate condition (Q 11 =0.103) ................................ ................................ ............. 163 Figure.Appendix E.15 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the four selected models ................................ ................................ ................................ ........................... 163 Figure.Appendix E.16 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 164 xx Figure.Appen dix E.17 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 165 Figure.Appendix E.18 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the four selected models ................................ ................................ ................................ ........................... 166 Figure.Appendix F.19 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the three selected models ................................ ................................ ................................ ........................... 167 Figure.Appendix F.20 . Runner - blade Axial velocity (C x ) distribution at Runner Outlet, for the three selected models ................................ ................................ ................................ .................. 167 Figure.Appendix F.21 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the three selected models ................................ ................................ .............. 168 Figure.Appendix F.22 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the three selected models ................................ ................................ ................................ ........................... 169 Figure.Appendix G.23 . The relation between (a). overall performance, (b). Normalized Power, and stator - blade solidity under three stator - blade relative thicknes s values for one flow rate condition (Q 11 =0.207), and 20 stator configuration ................................ ................................ .... 170 Figure.Appendix G.24 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two selected models ................................ ................................ ................ 171 Figure.Appendix G.25 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two selected models ................................ ................................ ................ 172 Figure.Appen dix G.26 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the two selected models. ................................ ................................ ................................ .......................... 173 Figure.Appendix G.27 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the two selected models ................................ ................................ ................................ ........................... 173 Figure.Appendix H.28 . The relation between normalized power and runner - blade stagger - angle setting constant for four flow rate conditions. (The power is normalized with C rsa =0.3 models) ................................ ................................ ................................ ................................ ........ 174 Figure.Appendix H.29 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 175 xxi Figure.Appendix H.30 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 176 Figure.Appendix H.31 . Runner - blade absolute velocity angle ( ) distribution at (a). Runner Inlet ( 2 ); (b). Runner Outlet ( 3 ), for the four selected models ................................ ........................ 177 Figure.Appendix H.32 . Runner - blade relative velocity angle ( ) distribution at (a). Runner Inlet ( 2 ); (b). Runner Outlet ( 3 ), f or the four selected models ................................ ........................ 178 Figure.Appendix I.33 . Runner - blade absolute velocity angle ( 3 ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ........ 179 Figure.Appendix I.34 . Runner - blade Axial velocity (C x ) distribution at ( a). Runner Inlet; (b). Runner Outlet; for the four selected models ................................ ................................ ............... 180 Figure.Appendix J.35. The circumferential velocity difference (C u2 - C u3 ) distribution plots from hub to tip for the three selected models ................................ ................................ ...................... 181 Figure.Appendix J.36. The axial velocity distribution (C x3 ) at runner outlet for the three selected models ................................ ................................ ................................ ................................ ......... 181 Figure.Appendix J.37. The radial velocity distribution (C r3 ) at runner outlet for the three selected models ................................ ................................ ................................ ................................ ......... 182 Figure.Appendix J.38. The runner outlet absolute flow angle ( 3 ) distribution at runner outlet for the three selected models ................................ ................................ ................................ ............ 182 Figure.Appendix J.39. The runner outlet relative flow angle ( 3 ) distribution at runner outlet for the three selected models ................................ ................................ ................................ ............ 183 Figure.Appendix J.40 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 ................................ ................................ ................................ ................................ .......... 184 Figure.Appendix J.41 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at High Solidity Condition r =1.6 ................................ ................................ ................................ ................................ .......... 185 Figure.Appendix J.42 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 ................................ ................................ ................................ ................................ .......... 186 xxii Figure.Appendix J.43 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at High Solidity Condition r =1.6 ................................ ................................ ................................ ................................ .......... 187 Figure.Appendix J.44 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; ; for the two different runner - blade thickness models at Low Solidity Condition r =0.7, with comprision to the designed values [The designed value for is across all span locations, which is not shown.] ................................ ................................ ................................ ... 188 Figure.Appendix J.45 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; ; for the two different runner - blade thickness models at High Solidity Condition r =1.6, with comprision to the designed values [The designed value for is across all span locations, which is not shown.] ................................ ................................ ................................ ... 189 Figure.Appendix J.46 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickn ess models at Low Solidity Condition r =0.7, with comprision to the designed values ................................ ................................ .......... 190 Figure.Appendix J.47 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at High Solidity Condition r =1.6, with comprision to the designed values ................................ ................................ .......... 191 Figure.Appendix J.48. The relation between runner - blade solidity with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickness conditions at one flow rate condition (Q 11 =0.207), 6 - blades configration ................................ ................................ ...... 192 Figure.Appendix J.49. The relation between runner - blade solidity with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickness conditions at one flow rate condition (Q 11 =0.207), 10 - blades configration ................................ ........................... 193 Figure.Appendix J.50. The relation between runner - blade relative thickness with the Overall hydraulic efficiency for all three runner - blade number conditions at one solidity condition ( r =1) ................................ ................................ ................................ ................................ ..................... 194 xxiii KEY TO SYMBOLS Abbreviations CAE Computer - Aided Engineering CFD Computational Fluid Dynamic CFT Crossflow Turbine DOE Department of Energy, U.S. DNS Direct Numerical Simulation LES Large E ddy Simulation NACA National Advisory Committee for Aeronautics NS Navier - Stokes RANS Reynolds Averaging Navier Stokes SMH Standard Modular Hydropower Technology SA Spalart Allmaras SST Shear Stress Transport Symbols A Turbine Cross - sec tional Area Blade thickness distribution coefficients B(t) Bezier curve function CL Blade Chord length C Absolute Velocity [m/s] C rsa Runner Stagger - angle setting constant C ssa Stator Stagger - angle setting constant C 1 , C 2 Bezier Curve Control points xxiv D Turbine Diameter [m] g Standard Gravity Constant [m/s 2 ] H Design Head [m] K Vortex Constant Mass Flow rate [kg/s] n s Specific Rotation Speed P 1 , P 2, P 3, P 4 Bezier Curve set of points Q 11 Unite Flow Rate [m 1/2 /s] R T /S T Runner Relative Blade Thickness/Stator relative Blade Thickness r Radius location [m] S Blade Spacing W Relative Velocity [m/s] Turbine Specific Work [J/kg] U Radial rotational velocity [m/s] W T Turbine Work [J] y t Blade thickness distribution function Greek Symbols Absolute Velocity Angle [degree] Relative Velocity Angle/ Blade angle [degree] Blade Blockage ratio Stagger - angle t Turbine Overall Hydraulic Efficiency Gen eral Inclined Angle [degree] xxv r Blade Pitch Angle [degree] Blade Solidity Fluid Density [kg/m 3 ] Auxiliary angle Rotational Speed [RPM] s Specific Rotation Speed Rotational Speed [rad] Subscript s a/x Axial Direction Tip/Hub Tip Location/Hub Location r Runner/Radial Direction s Stator u Circumferential direction 1,2,3 Stator Inlet; Runner Inlet; Runner Outlet, respectively 1 1. CHAPTER I : INTRODUCTION This c hapter summaries some basic histories, categories, and application s for the hydraulic turbine. Also , the new technologies for low head micro - hydraulic applications are briefly covered too . 1.1. H istory of hydropower and hydraulic turbines Hydropower is the longest established source for the generation of electri city, which start ed in 1880 as a small DC generating plant in Wisconsin, United States . [1] Since 1880, hydropower has been utilized for more than a hundred years, and undoubtedly is the most efficient and confident source of renewable energy. [2] Hydropow er contributes to 19% of the total global output of electricity by the end of 1999, which produced 2650 - Terawatt hour (TWh) [3 ]; and later produc ed almost 3100 TWh at the beginning of 2009 and is expected to reach 3606 TWh in 2020. [4] Compared to other po wer plant applications, large hydropower plants require significantly higher initial investments; however, the overall maintenance and operating cost are lower . Raabe (1985) listed the various advantages and disadvantages of hydropower plants , and a summar y of these is given in Table 1 . 1 . The critical component of a hydraulic power system, undoubtedly, is the hydraulic turbine, which has a long period of development; the oldest version of it was the waterwheel, first used in ancient Greece and other cultures for process ing agricultural . In 1830, a French engineer, Ben oit Fourneyron , developed the first commercially successful hydraulic turbine. Later, the American engineer James B. Francis designed the first radial - inflow hydraulic turbine (Francis Turbine) that became widely used until now. In the late 19 th century, A merican inventor, Lester A. Pelton 2 invented the first impulse turbine, the Pelton wheel turbine , in which water is piped at high pressure to a nozzle where it expands completely to atmospheric pressure. Viktor Kaplan introduced the first turbine concept, t he Kaplan turbine, for the small head application in 1913. Later , he improved his concept with swiveling blades to improve the efficiency of various flow and head conditions. Table 1 . 1 . Key Features of hydroelectric p ower p lants [1] Advantages Disadvantages Technology is relatively simple and proven. The n umber of favorable sites limited and available only in some countries . High efficiency, long useful life. Problems with cavitation and water hammer. No thermal phenomena apart from those in bearings and generator. High initial cost , especially for low head power plants, compared with thermal power plants Small operating, maintenance, and replacement costs Inundation of the reservoirs and displacement of the population, loss of arable land. No air pollution. No thermal pollution of water Facilitates sedimentation upstream and erosion downstream of a barrage Pelton, Francis, and Kaplan turbines are the three most popular turbine types worldwide for the traditional hydraulic power system, and Table 1 . 2 summaries the normal operating conditions for all three type s of turbine s . Table 1 . 2 . The o perating r anges of the three most popular turbine for traditional hydraulic systems [1] Pelton Turbine Francis Turbine Kaplan Turbine Specific speed [rad] 0.005 - 0.4 0.4 - 2.2 1.8 - 5.0 Head [m] 100 - 1770 20 - 90 6 - 70 Maximum power [MW] 500 800 300 Optimum Hydraulic efficiency [%] 90 95 94 3 Table 1.2 Pelton Turbine Francis Turbine Kaplan Turbine Regulation method Needle valve and deflector plate Stagger - angle of guide vanes Stagger - angle of rotor blades ( Note: Values shown in the table may subject to change ) 1.2. Small hydropower and its application (A literature review) Generally, a large , and high - capital - cost dam equipped with a large turbine is required to produce sufficient power supply . H owever , low - head , small hydropower stations present an attractive and efficient way for electricity generation in rural, remote, and hilly areas because of the increment in the level of greenhouse gas emissions and fuel prices in these sites , and they have become increasingly popular for application at small rivers. [5] Micro - hydropower schemes can be used to generate enough electrical power for home, farm, and plantation or small village. [6] They can also be used in mechanical end - uses like agric ulture processing , textiles fabrication, ice cream production, cooling, and drying. [7] The main advantages of the low - head micro - power system are that it is predictable if enough water supply is available and possesses a positive environmental impact. [8, 9] Therefore, the system has become the leading interest for future hydro - developments in Europe, where large - scale stations have indeed been utilized but in return giv e advers e effects on the environment. [9,10] Most low - head , small hydropower plants generate power less than 100 kW , but there are also other categories with classification below 500 kW and less than the ten - meter head. [5] Recent publication s raise the importance of using the simple and existing turbines to achieve minimum cost to p roduce power. [11] Installation of large and medium hydropower plants with dams, vast reservoirs, large turbines, electrical equipment , and controllers 4 have been proven very expensive, and have an uneconomical and negative environmental impact. [5] Though intended as a clean and cheap source of energy generation, many developing countries that need rural electrification are instead exposed to economic problem s when installing this costly hydro - equipment. [12] Using micro - hydropower with new design and arra ngement of these equipment leading to, especially the turbines , can be the perfect solution to overcome the economic and operational problems and reduc e the total cost of hydropower plants. [5] Since the micro - turbine can generate very reliable power with relativity simple designs and fabrications, it has recently gained rapid growth in the power generation field, especially in rural areas, as their power is needed to feed both baseload and peak demand requirements of grid supply. [5] Table 1 . 3 . A general c lassification of hydropower turbines and their applicable head range s. Classifications of Turbine Turbine types Head Range [m] Impulse Turbine Pelton 50 - 1300 Turgo 3 - 150 Crossflow 3 - 250 Reaction Turbine Axial Flow Turbine Bulb Turbine 2 - 20 Straflo Turbine 2 - 40 Tube Turbine 5 - 30 Kaplan Turbine 2 - 40 Other Turbine s Francis Turbine 10 - 350 Kinetic Energy Turbine No head needed Archimedes Screw Turbine As low as 1 meter C lassification of hydro - turbines and related low head application s Hydropower turbines are categorized into two types : impulse and reaction turbines . Table 1 . 3 shows the general classification and the applicable head range for each type of turbine . 5 Impulse Turbines Impulse turbines have a simple design and are inexpensive. There are three major types of impulse turbine s : Turgo , Crossflow , and Pelton. Those types are commonly used for high and medium heads. [13] Recently, they have been applied for lower hea d micro - sites, and their proven effectiveness has allowed them to become an accepted alternative practice in many countries. [14] - Energy System & Design Ltd. [15] has produced a Turgo turbine , which can be used for head between 3m and 150m. The water is a pplied on one side , usually at an angle of about 20 , goes across the blades , and exits on the other side ( Fig . 1.1 ) . In the previous researches, Williamson et al. [16, 17] optimized Turgo turbine models in micro - project and pico - project, by altering the location of low heads from 3.5 m down to 1m to improve the turbine performance. For one specific model, the Turgo turbine has an experimental efficiency of 91% at 3.5m head and 87% at 1m head. [16] Figure 1 . 1 . A low - head Turgo wheel turbine developed by Energy System & Design Ltd [23 ] - The Pelton turbine has one or multi - jet s ( Fig . 1.2 ). Pelton wheels are suitable for a large head and low - flow sites ( The Turgo turbine wheel is a modified version of Pelton wheels for low head application). [5] Recently, Pelton turbines have been used for small and micro - 6 hydropower configurations, using a single jet. [18, 19] Generally, a Pelton turbine has a maximum efficie ncy between 70 - 90%. [19] Figure 1 . 2 . A crosssection of a low - head s ix jet Pelton turbine (Right), and a s ingle jet Pelton turbine (Left) [24,25] - A Crossflow turbine (CFT) is another significant impulse turbine for low - head applications . It is commonly applied in horizontal and vertical configurations [5] (Fig . 1.3 ). The crossflow turbine allows the water to flow through the blades twice. During the first pass, water flows from the outside of the blades to the inside; the second pass is from the inside back out ( Fig. 1.3 ). [5] This type of turbine is usually used at a higher flow rate and lower head than the Pelton and Turgo turbines. [20] The average efficiency of CFT turbines is usually 80% for small and micro - power outputs but can reach up to 86% in the case of medium and large units. [21] Figure 1 . 3 . Low - head c rossflow t urbine in the vertical position (Left), horizontal position (Right) [26] 7 Reaction Turbines A reaction turbine generates power from the combined action of pressure and moving water. The runner is placed directly in the water stream flowing over the blades rather than striking ea ch individually. [20] Compared to impulse turbines, reaction turbines have a better performance in low head and high flow rate conditions . [20] At lower operation speed s , the efficiency of reaction turbines is usually higher than the impulse turbines. [22] Most reaction turbines are classified as the axial flow turbine s , as shown in Table 1.3 , and t his type of turbine is indeed practical , indicating excellent efficiency, simplicity , and cost - effective ness (More detailed descriptions for axial turbine will present in the next section). [27] Other reaction turbines include the Francis turbine and the Kinetic Energy turbine. - A Francis turbine is the most common turbine at hydropower stations. The Francis turbine is typic ally used in a large - head c ondition ; however, recent research shows it could apply to the low - head site too . This turbine has a radial or mixed flow runner, which is equipped with guide buckets. Water is introduced just above the runner and all around it and then falls through, caus ing it to spin. Besides the runner, the other components are the scroll case, wicket gates, and a draft tube. [20] A cross - sectional of a Francis turbine is shown in Fig 1.4 . Figure 1 . 4 . Cross - section of a low head Francis turbine with scroll case, wicket gates and draft tube [28] 8 - The kinetic energy turbines , also called free - flow turbines, generate electricity from the kinetic energy present in flowing wa ter rather than the potential energy from the head. [20] The system may operate in rivers, manmade channels, tidal waters, or ocean currents. [20] the manmade channel, riverbeds, or pipes is not required. [ 20] These systems do not require large civil works; instead , existing structures such as bridges, tailraces , and channels are sufficient for these turbines. [ 20] The basic structure of a kinetic turbine is sh own in Fig 1.5 . Figure 1 . 5 . Basic structure of a kinetic turbine with multiple unit configurations [29] - The Archimedes screw turbine has become a more attractive option for some lower head sites, as its heads can be set as low as 1 meter and is especially suited for sites with large flows. [5] Western renewable energy [30] and Landustries [31] have developed the Archimedean screw as a relatively novel method to generate electrical energy from a low - head source. This turbine is being used as one of a few systems that are able to maintain or even improve the wildlife in and around the river, which is the key feature for it. [31] A water flow geometry in the Archimedes screw is shown in Fig 1.6. 9 Figure 1 . 6 . Water flow geometry in a low - head Archimedes screw turbine [32] The Axial hydro - turbine for low head application As shown in Table 1.3 , most of the reaction turbines are the axial type of turbine s . An axial type turbine generally has a runner with three to six blades in which water impinges continuously at a constant rate. [20] The pitch of the blades may be fixed or adjustable , and t he major components besides the runner are a scroll case, wicket gates, and a draft tube. [20] Generally speaking, the axial hydro - turbine has four major types: Bulb turbine, Straflo turbine, tube turbine, and Kaplan turbine. - The Bulb turbine has the turbine and generator sealed and placed directly in the water stream. [20] The near - straight design of the water passage provides both size and cost advantages . Bulb turbine s can also operate in a reverse flow direction. Bulb turbines are available for power output in the range of 10 - 100MW. [20] A typical bulb turbine is shown in Fig 1.7 . This type of turbine is suitable for low - heads application s , namely below 25m; for very low heads, an extra set of gears is used to increase the rpm of the generat or. [20] 10 - The Straflo turbine is a registered brand name that stands for straight - flow. The key feature of the Straflo turbine is the combination of the turbine and generator since the generator is attached directly to the perimeter of the turbine. [20] A Straflo turbine also consists of a group of axial turbines with a concentrically arranged generator outside of the flow channel. [20] Various components of a Straflo turbine are shown in Fig 1.8 . Figure 1 . 7 . Cross - section of a horizontal bulb turbine and generator for low - head power station [33] Figure 1 . 8 . A n example layout of the Straflo turbine for low head application [34] - In a Tube Turbine , the penstock bends just before or after the runner, allowing a straight - line connection to the generator. The power output from tube turbines ranges from 20 to 700 kW . Figure 1.9 shows the main components of a tube turbine. These types of turbines have a direct drive configuration where the turbine and the generator are on the same shaft having common bearings and seals. There are five main features of the tube turbine [20]: 1. Compact structure, turbine , and generator with bearings and seals in on e unit , 2. The i nstallation angle for the unit may vary from vertical to horizontal , 3. Stainless steel structure , 4. Long service intervals , 5. Heads range from 5 to 30 m which are sufficient for most applications 11 - The Kaplan turbine has adjustable blades and the wicket gates, allowing for a broa der range of operation. T he runner - blade is attached to the turbine shaft and the generator directly at various speeds to generate electricity at the optimum efficiency . Typically, t he Kaplan turbine is equi p p ed wit h the double regulation method: adjustable blades and wicket gates . However, a recent develop ment in various speed generator technology allows a third regulation method for the Kaplan turbine, which further improve s the overall performance. The basic components of a Kaplan turbine are shown in Fig 1.10. Figure 1 . 9 . A horizontal t ube turbine configuratio n for low - head hydropower plant [20] Figure 1 . 10 . Basic components of a traditional Kaplan turbine power station [ 35] With different configurations, the Bulb turbine and Kaplan turbine are the two most common ly used turbines for low - head application s ; and Table . 1 . 4 shows some current low head turbine configuration s and corresponding features. Currently , there are many turbine types suitable for low head hydropower applications on the market , and Table 1 .5 is a summary of the current low - head turbine s and their suppl iers with corresponding performance characteristic features. 12 Table 1 . 4 . Current c onfigurations of axial turbines for low head application s (Adapted from [3 6 ]) Turbine configuration Features Kaplan turbine Inclined axis, very low head Kaplan gear turbine Inclined at an angle of 15 to 45 degrees; Very low head application between 2 and 8 meters; Maximum power capacity is about 2.6MW Horizontal axis "S" type Kaplan turbine Runner sizes range from 1.0m up to 4m; Heads range from 5m to 25m; Power output about 12MW Vertical axis small Kaplan turbine with elbow draft tube Avoiding the use of draft tube gates Horizontal axis pit type propeller turbine Fixed blade angle; Peaked efficiency curve; Suited for constant flow and head condition Bulb Turbine Belt driven bulb turbine Short installation time; Compact powerhouse structure; Up to 4m head; Up to 600kW power output Bevel Gear bulb turbine High - speed generator; Up to 12m head; Up to 2600kW power output; Axial Bulb turbine Direct driven synchronous generator; Horizontal or inclined axis; Short installation time; Low noise level; Up to 6m head; Up to 5000kW power output 13 Table 1 . 5 . Current turbines on the market suitable for low head application s ( A dapted from [37]) Turbine type Supplier Flow range [m^3/s] Head range [m] Power output [ kW ] Pelton Powerspout 0.008 - 0.01 3 - 100 <1.6 Cross - flow IREM 0.01 - 1.0 5 - 60 <100 Ossberger 0.04 - 13 2.5 - 200 15 - 3000 Wasserkraft Volk 1.5 - 150 Not given <2000 Archimedean Screw Andritz <10 <10 <500 Hydro Coil <10 4 - 20 2 - 8 3 Helix Power 0.2 - 10 1 - 10 1.4 - 700 Kaplan (Axial and bulb included) Ossberger 1.5 - 60 1.5 - 20 20 - 3500 Mavel 0.3 - 150 1.5 - 35 30 - 20000 Voith Not given 3 - 95.0 100 - 400000 Energy systems and Design 0.03 - 0.06 0.5 - 3.0 0.09 - 1 Power Pal 0.04 - 0.13 1.5 0.1 - 1 Wasserkraft Volk Not given 1 - 40 Not given Gugler 0.2 - 50 1 - 100 3 - 10000 Alstom 0.3 - 150 2 - 30.0 <130000 Voith 2 - 30.0 Not given 1000 - 80000 Voith (Minihydro) 1 - 14.0 2 - 10.0 Not given Tamanini 1.0 - 15 5 - 20 50 - 2000 Hydrolink Not given 1.5 - 25 Not given Hydrokinetic Alternate Hydro >0.8m/s >0.6 1 - 4.0 New Energy Corporation 2.4 - 3m/s Not given 5 - 25.0 Alden <2.6m/s 25 Not given Hydrovolots 1.5 - 3m/s 0.15 1.5 - 12 Vortex Zotloterer 0.05 - 20 0.7 - 2 0.5 - 160 Francis Wasserkraft Volk Not given <300 <20000 Mavel 0.1 - 30 15 - 440 20 - 30000 Gilkes 0.05 - 40 <400 <20000 Voith Not given 3 - 95 5 - 10000000 Gugler 0.03 - 25 2 - 500 3 - 10000 Tamanini 0.2 - 10 15 - 300 20 - 5000 Hydrolink Not given 20 - 120 Not given Newmills Engineering. LTD Not given 10 - 350 1 - 820 Kossler 0.8 - 60 15 - 250 500 - 15000 14 1.3. Standard Modular Hydropower Technology (SMH) Across the U.S., there are more than 65.5 GW of new stream - reach hydropower capacit ies that can be used for hydropower development. [ 38 ] Those riverine resource s are very functionally and geographically diverse and characterized by low - head, varying flows, and highly valued river function. Because of those features, low - head hydropower and hydrokinetic technologies are best - suited technology for those sites for th eir versatility and low impact on the environment. However , a complex and uncertain aspect of the new low - head hydropower development that is targeted explicit ly to the new stream - reach sites is the balancing of performance, environmental impact , and proje ct cost. Moreover , small, low - head sites , in particular, are challenging to design and develop with acceptable performance and cost. Because of those hydropower potentials and associated challenges in 2017, the U.S. Department of Energy (DOE) proposed a n ew S tandard Modular Hydropower Technology (SMH) concept that can enable hydropower technology to deploy and operate in a new stream - reach site with minimal environmental impacts and greater public acceptance at a reduced cost. This new design paradigm has three basic principles for a low - cost, environmentally sustainable hydropower growth strategy and Logistics as the following [ 39 ]: - Standardization: Guidelines, rules, and specifications (i.e. , standards) to maximize compatibility, acceptance, interoperabil ity, quality, safety, and repeatability while minimizing environmental disturbance. In a hydropower context, standardization of design, review, regulation, manufacturing, operations, maintenance, and other features are intended to reduce site - specificity and project costs. - Modularity: The physical or virtual organization of a hydropower facility is divided into discrete functional units, known as modules. In SMH, the entire facility is envisioned 15 as a modular structure, with the generation, pas sage, and foundation modules assembled to deliver energy and environmental benefits at many different sites. - Environmental Compatibility: Siting and developing hydropower facilities with an understanding that streams provide valuable environmental benefits that must be preserved. SMH development must embody an understanding of how coupled stream - hydropower systems can minimize disturbances to landscape features, water quantity, connectivity, geomorphology, water quality, and biota. The first concept of modu larity refers to the use of different module types to assemble an entire SMH facility. The SMH has three primary modules with se ve ral sub - module s for the application to different sites ; Fig 1.11 shows the conceptual schematic of this Modularity concept, an d Table 1 . 6 summaries the basic function of each module [ 39 ]. All those modules can be assembled to form an SMH facility that matches the scale, environmental attributes, and watershed context of the site selected for development [ 39 ] . Figure 1 . 11 . Conceptual schematic showing the primary modules of a n SMH facility design [39] 16 The second concept of modularity refers to scalability at many sites through multiple modules of the same type. For exam ple, an upstream fish passage module may be applicable at many sites with a watershed region; a cost - optimized, compact generation module designed with broad operational flexibility could be applied at multiple sites throughout the country [ 39 ]. Table 1 . 6 . The basic function s for the SMH m odules and their sub - modules Major Modules Basic Function Generation Module (Including Turbine Rotor Module , Generator Module ) Encompasses all hydraulic and electric machines, equipment, and systems necessary for hydroelectric power generation. Passage Module ( Including Fish Passage Module , Sediment Passage Module , Recreation Passage Module , Water Passage Module ) Allow fish, sediment, water, and recreational craft to pass the facility safe l y. Foundation Module Provides structural resistance and reliably interface with the streambed to support generation and passage modules Based on the initial investigation, a successful SMH facility should ha ve six features [ 39 ]: - Predictable and somewhat regular production of electricity. - Minimal alteration of the inflow hydrograph and minimal impoundment of inflow water. - Environmental miti gation technology (functionality) inherent within and integral to the facility design (including fish passage, water quality, and sediment management design). 17 - Minimal disruption to the aesthetics of the natural stream and stream - scape , and do not occupy th e full width of the river. - Minimal fluctuations of water surface elevation. - Enabling of safe recreational passage through and activity around the project. SMH Generation Module As the most critical module for SMH technology, the Generation Module is envis ioned as an integrated water - to - wire module that contains a hydraulic water turbine, generator, controls, and electrical equipment within a single unit [ 39 ]. The overall design goal of the generation module is to balance the performance, economic, and envi ronmental sustainability, so it is appealing for developers. - Performance Considerations The traditional hydraulic system is often equip ped with a large reservoir that helps the system maintain the optimum operating head condition . H owever, the SMH system for the new stream - reach sites requires minimal water impoundment. Additionally , one major characteristic of the new stream - reach developme nt sites is the high variability in flow rate. To demonstrate this high flow rate variability and Fig. 1.1 2 shows an example of a n average daily flow rate of one selected new stream - reach development site for the past 50 years . Furthermore , at most low - hea d new stream - reach sites, the tailwater elevation generally rises twice as fast as headwater elevation when river flow increase s , leading to a significant reduction in the available gross head [ 40 ]. As a result, the low - head new stream - reach sites with hig h variability in flow and head often result in operating beyond the acceptable performance and efficiency limits for the traditional turbine. This variability requires a different deployment of regulation methods and presents a significant design challenge for the generation module design. 18 Figure 1 . 12 . Daily average flow rate for a selected new Stream - reach site for the past 50 - years . (Data extrapolated from Oak Ridge National Laboratory SMH Explorer website: https://smh.ornl.gov/explorer/) - Economics Considerations The immediate installed cost target for an SMH project is under $6,000/kW , including all necessary modules [ 39 ]. Over time, this number should be reduced as the module deployment increase s . Conventionally, there are six main cost sectors for developing a small hydro project: 1. Installation Cost , 2. Planning Cost , 3. Civil work , 4. Infrastructure and Logistic s , 5. Electrical connection/Construction , 6. Equipment Cost. A mong those s ectors, the civil work normally represents 40% - 50% of the total project cost for a hydro - power project [ 41 ]. This civil work cost is normally high because of the construction of the dam for creating a necessary head and the water channel for diverting the river to the turbine. However, for low - head , new stream - reach sites, which feature with minimal water impoundment and run - of - river type operation, the traditional dam structure is not a necessity. Therefore, a damless concept for the generation module , wh ich means the size of the generation module can be large enough for creating enough low head condition s , will significantly reduce the overall cost, but at the same time , create s new design challenges for the turbine designs. 0 40 80 120 160 200 0 60 120 180 240 300 360 Flow Rate [m 3 /s] Yearly Time [Days] Yearly Average Flow Rate Daily Average Flow Rate 19 - Environmental Sustainability C onsiderations The preservation of stream functionality is one of the most instrumental premises for SMH technology, and there are three primary environmental considerations for the new stream - reach sites: 1. Fish Preservation , 2. Sediment Preservation , 3. Recreation Activities Preservation. Among all those three, there is one primary consideration that is critical for the generation module design : fish preservation. The new stream - - valued native fish species (anadromous, catadromous, and amphidromous), and hydropower facilities can work as barriers for them. Therefore, the low - head hydro - system needs to have maximum protection for fish migrations. T here are two ways to achieve this fish protection : i ntroducing advanced fish passage design for overall facility construction and introducing the fish - friendly turbine design concept that allows fish to pass the generation module safely. Design Specifications for SMH Generation Module Because of the performance, econ omics, and environmental considerations, the DOE has five specific design specifications for the SMH generation module [ 39 ]: - - - - - 1.4. Research Objective s Because of the need for new and innovat iv e design method s for SMH technology suitable for potential new stream - reach sites, t h is dissertation use s physical, theoretical, numerical and 20 analy t i cal modeling techniques to design , develop and test a low impact , low - head turbine generation module . The main goal of this disser tation is to introduce a new design methodology for the SMH generation module that is suitable for low - head hydraulic conditions with flexible geometry configurations , optimum overall performance , and a wide operating range . The design and development process of this low - head generation module consist s of the following three steps : 1. Runner hydrodynamic design and development of the 3 - D geometric model s and interfaces; 2. Numerical design methods development ; and 3. CFD simul ations and analysis. 21 2. CHAPTER II : BASIC MECHANICS A ND PHYSICS F OR HYDRAULIC TURBINE SYSTEM This c hapter summarizes the basic physical and thermodynamics laws and develop s them into a form that suitable for the study of the hydraulic turbine . The essential laws and theories for developing hydraulic turbine covered in this chapter include: - The continuity laws of a control volume . - The first law of thermodynamics and the steady flow energy equation. - The conservation of momentum equation s and Euler equation s . - Definitions of h ydraulic turbi ne efficiency and l oss. 2.1. Main C onservation E quations The Continuity Equation In a flow system, the m ass flow rate is related to the flow density , fluid absolute velocity , and the cross - sectional area of the system. If the fluid with a constant density , through the finite area of the system during a finite time , the elementary mass is , where the is the absolute fluid velocit y, and is the angle between fluid velocity and the normal direction of the area, referring in Figure 2.1. Figure 2 . 1 . Demonstration of an elementary mass across an area element (dA) with an absolute fluid velocity (C) and an angle 22 The velocity component perpendicular to the area is , and . The elementary rate of mass flow rate is , where is axial velocity ( 2 - 1 ) Most 1 - D analyses in this chapter are limited to the incompressible steady flows , where the axial velocity and density are considered as constant across each section of the turbine. If are the cross - section area at station 1 and 2 along a turbine passage the system has no accumulation of fluid within the control volum e and has the mass flow rate as ( 2 - 2 ) First Law of Thermodynamics for Hydraulic Turbine The First Law o f Thermodynamics reveals that if a system is taken through a complete cycle during which the heat is transferred , and work is done , then [1] ( 2 - 3 ) w here represents the heat transfer to the system during the cycle , and is the work done by the system during th is cycle. During the change from state - 1 to state - 2, there is a change in the property of internal energy ( 2 - 4 ) For an infinitesimal change of state ( 2 - 5 ) Where is the change in energy per unit mass, this term includes internal, kinetic , and potential energy ( 2 - 6 ) 23 Now consider the steady flow of fluid through a control volume representing a turbomachine system ; the fluid passes at a steady mass flow rate , entering at station - 1 and leaving at station - 2. Energy is transferred from the fluid to the turbomachinery blades and positive work b eing done via the blade at the rate . In the general case , positive heat transfer take s place at the rate from the surroundings to the control volume. Thus, the steady - flow energy equation is ( 2 - 7 ) w here h is the specific enthalpy, is the kinetic energy per unit mass , and is the potential energy per unit mass. Now defining stagnation enthalpy by , Eqn ( 2 - 7 ) becomes ( 2 - 8 ) Moreover, for incompressible flow, Eqn ( 2 - 8 ) could be re writ t e n in the following form, which ( 2 - 9 ) The Momentum Equation and Second Law o f Motion One of the most fundamental and critical principles in mechanical Engineering is S econd L aw of M otion . The momentum equation relates the sum of the external forces acting on a fluid element to its acceleration, or to the rate of change of momentum in the direction of the resultant external force [1] . Considering a system of a mass of , the sum of all the body and surface forces acting on along some arbitrary direction is equal to the time rate of change of the total of the system, i.e . [1] . 24 ( 2 - 10 ) For a steady - state control volume where fluid enters at a uniform velocity and leaves with a uniform velocity , then ( 2 - 11 ) Equation ( 2 - 11 ) is the one - dimensional form of the steady - flow , momentum equation. Now , considering a rotation system of mass for the turbomachinery . T he sum of the moments of all external forces acting on the system about some arbitrary axis fixed in space is equal to the time rate of change of angular momentum of the system . I n other word, torque must be supplied through the shaft to the rotor in order to change the tang ential momentum of mass of fluid from station - 1 to station - 2 as [ 42 ] ( 2 - 12 ) w here is the distance of the mass center from the axis of rotation measured along the normal to the axis , and is the velocity perpendicular to both the axial and radius direction [ 42 ] . For one - dimensional steady flow, Eqn ( 2 - 12 ) could be rewrit ten in the following form ( 2 - 13 ) For a pump or turbine rotor running at angular velocity , the rate at which the rotor does work on the fluid is ( 2 - 14 ) w here the blade circumferential speed is , t hus the specific work done on the fluid is ( 2 - 15 ) 25 This equation is referred to as P ump E quations . And f or a turbine , the fluid does work on the rotor , and the sign for work is then reversed. Thus, the specific work is ( 2 - 16 ) This equation is referred to as T urbine E quation . For a rotating system in turbomachinery, it is essential to define the velocity component correctly . Considering a fluid passing through the runner, by definition , is the absolute velocity that tangential to the absolute path, is the relative velocity tangential to the blade , or the relative path , and U is the blade velocity. For every instant during the particle movement through the runner, the following velocity relationship remain s true: ( 2 - 17 ) A lso , Fig 2.2 shows a velocity triangle for a generalized turbine system . Figure 2 . 2 . The v elocity triangle for a generalized turbine system that shows each velocity component at each turbine station: Station - 1 is the Turbine inlet, Station - 2 is the Stator Outlet, Station - 3 is the Runner Outlet 26 2.2. Definitions o f Efficiency a nd Loss Estimation Turbines are designed to convert the available energy in a flowing fluid into useful mechanical work to the shaft . The efficiency of this process, the , is a performance factor of considerable interest to both the des igner and user of the turbine. Thus [1] ( 2 - 18 ) Mechanical energy losses occur between the turbine runner and the output shaft coupling as a result of the work done against friction at the bearings or other mechanical devices . The magnitude of this kind of loss is difficult to estimate as it varies with size and manufacture d skills. So, another efficiency , or were wi dely used as ( 2 - 19 ) From above definitions , it is easily deduced that the , which is simply the ratio of the shaft power to the rotor power, as ( 2 - 20 ) For the hydraulic turbines, the turbine hydraulic efficiency (or the turbine total to static efficiency ), is defined as the work supplied by the rotor in unit time divided by the maximum hydrodynamic energy difference of the fluid per unit time, as [1] ( 2 - 21 ) The losses for hydraulic turbine are modeled as (where is loss coefficient ) , and could be classified into several different categories which are show n in Table 2. 1 . 27 Table 2 . 1 . Loss models for axial hydraulic turbines (Adapted from [43]). Loss Mechanism Loss model Guide vane profile loss (skin friction loss at the stator ) [4 4 ] Incidence loss [4 3 ] Runner - blade profile loss [4 4 ] Mechanical Loss [4 4 ] Losses in the flow passage between the guide vanes and runner [45] 28 Now the total head required could express as following ( 2 - 22 ) The effective head could be determined from the design specification ( 2 - 23 ) Finally, the hydraulic efficiency could be express ed as ( 2 - 24 ) Another important definition of efficiency is total - to - total efficiency ( ) which could express as, ( is the inlet total pressure; is the outlet total pressure) ( 2 - 25 ) 29 3. CHAPTER III : DESIGN METHODOLOGY DEVELOPMENT S In general, the hydraulic turbine design process has three cr itical steps: 1. g eneral size and operation design , 2. b lade profile design , 3. b lade configuration design. In order to solve the challenges that associate with the new SMH technologies, unconventional design method and configurations must be developed and tested . Therefore , t his chapter presents a comprehensive design methodology with a detailed theoretical background and in - depth geometr ical considerations. 3.1. Turbine selection and current development status Currently, there are two paths to utiliz e the potential energy of the low head or new stream - reach sites: 1. Low - head Hydropower Application , and 2. River Current Application. For the low - head hydropower application, five major types of turbines are widely used: 1. Open - flume Francis turbines , 2. Kaplan turbine , 3. Tubular turbines , 4. Crossflow turbine , 5. Archimedes screw turbines. Table 3.1 summaries some advantages and disadvantages of those types of turbines. For the River - current application, the hydr o - kinetic turbine is commonly used. This type of turbine uses ultra - low or zero head conditions and is driven by the free - flow stream. This technology has two advantages: Multi - unit arrays can be deployed for maximum power production like the wind farm , an d t he structural requirements are low; thus, the civil cost will be limited. However, this technology still suffers from some significant drawbacks, including 1. r elatively low efficiency , 2 . h igh installation cost , 3. h igh maintenance difficulties. Based s mentioned in section 1.3.2 , only the reaction turbine is allowed for SMH applications. Furthermore, since the high flow rate and head variability , among all turbine types, the Kaplan turbine (or its variants) is the best option for the 30 SMH technology. However, the new design method need s to be developed for the challenges associated with the new SMH technology. Therefore, t he proposed generation modu le is an open flume, damless Kaplan turbine system , and has eight major components. Figure 3.1 shows a complete layout of this generation module. This Generation Module has a fixed stator, and an adjustable runner, the geometry of the stator and runner can be optimized for better performance. Table 3 . 1 . Five major types of turbine s for low - head applications . (Adapted from [46]) Turbine Type Advantages Disadvantages Open - flume Francis Turbine High efficiency on the design condition Narrow operation range, expensive to manufacture Kaplan Turbine High efficiency over a wide range. The regulation method can be complex Tubular Turbine High efficiency, and various configurations for a low hydraulic loss. Need straight passage through the turbine, can increase the civil cost. Crossflow Turbine Wide operation range (head and flow). Relatively low efficiency Archimedes Screw Turbine Wide operation range, high tolerance for debris, and fish friendly. Relatively l arge for transportation , and technology still immature. 31 Figure 3 . 1 . The proposed generation module configuration for SHM technology . (Top : Total Assembly Model ; Bottom : Close - up for each component ) 32 3.2. Design methodology development s A dditional design considerations In addition to the DOE SMH generation module design specification s , there are four crucial design considerations for developing a successful SMH generation module. - For the SMH generation module, s ince the head condition is ultra - low, traditional dam structure is not a necess ity . Instead, the reinforced Kaplan turbine structure (Part - B in Fig 3.1 ) is considered as a dam that provide s the low head condition for the turbine . This configuration can reduce the cost and complexity of the civil works , and also, means the overal l turbine structure size and the position are related to the head condition and performance. - S ince the SMH generation module is directly installed into the water system, it must be fish - friendly , which means larger turbine area s for fish migration and low rotational speed for avoiding unnecessary fish damage. - The low head, relative ly low flow rate , and relati ve ly large blade area nature s of this design mean no draft tube or tiny draft tube is needed to recover the end water kinetic energy since it is already minima l. The water exits the turbine system and can directly discharge to the river or atmosphere , whic h means less piping and water guiding structure costs. - Typically, t he new stream - reach sites ha ve various flow conditions from season to season; this means the proposed generation module system must at least have one regulated feature for different flow conditions. Therefore, the blades and speed regulat ion method s are the two traditional ways to regulate the system for different flow conditions economical - effectively . However, since the general turbine structure serves as a dam for this design, a rotating head - gate (Part - H in Fig 3.1 ) could be consider ed as a new regulated method for regulating the head conditions. 33 Generation module initial size and operation condition design - Generation module overall size and positioning Initial size design is the first challenge for developing a low - head hydraulic turbine. Traditionally, a specification map is used for determining the basic turbine characteristics, including head, rating power, and rota tional speed for opti mum performance. Figure 3.2 shows a typical specification map between the specific speed and specific diameter for various turbine type s . Figure 3.3 shows a similar specification map for the Kaplan turbine . Those charts come from years of surveys from the major hydro station and often provide an accurate initial size design for the traditional hydro - system. However, f or low head condition s , those charts can result in a smaller size and higher rotational speed , which are preferentia l features for high efficiency . Those features are contradicted with the concept of SMH since it requires a large turbine area and less fish damage. This proposed turbine uses its turbine structure (Part - A in Fig 3.1 ) to provide the low - head condition; th erefore, the initial turbine size is related to the designed head and the positioning of the turbine structure. For the low - head application turbine, the turbine structure is often position ed in water with a general - inclined angle (refer as in Fig 3.1 ), and this angle is gener ally from [ 4 7 ]. A l arger inclined angle requires a more extended turbine structure and is more subject to the tailwater elevation, and a smaller inclined angel can have a shorter turbine structure, which means a smaller turb ine area. Turbine structure minimum length can be calculate d as Turbine structure minimum structure length ( 3 - 1 ) This minimum length gives a guideline for the initial size of the turbine (Turbine Overall Diameter), and addition head - gate (Part - H in Fig 3.1 ) can be equipped for more headspace for the off - design condition. 34 Figure 3 . 2 . A typical specification map for different turbine types [4 8 ] Figure 3 . 3 . A conventional Kaplan turbine specification map using for initial design [4 9 ] 35 - Generation module hub diameter After determining the overall diameter, the turbine hub diameter is the next parameter that need s to be considered with care. For the SMH generation module, the hub volume must contain the generator and control components; this means the hub diameter and its impact on turbine performance should be studied thoro ughly for future generator selections and designs. Additionally, depending on which 1 - D design method is used for the turbine runner, the hub diameter has a significant impact on the turbine runner shape, which is covered in section 3.2.3 . For traditional Kaplan turbine, two empirical equations can be used for determining the hub - tip ratio for optimum performance [ 4 8 ] ( 3 - 2 ) [ 4 9 ] ( 3 - 3 ) Those equations can only be useful for the initial guess, and the different hub - to - tip ratio must be studied for low - head application s . D ifferent hub - to - tip ratios between 0.68 - 0.8 were studied, and the resul ts were shown in section 5.1 . - Generation module operation condition For the initial design stage, the rotational speed is the only operation al condition that needs to be considered. The criteria for determining the rotational speed comes from environmental considerations. The h igh rotational speed is typically required for a low head application that can have a high risk for the fish population . DOE developed ten criteria for designing a fish - friendly hydraulic turbine system in 1999 [ 50 ] (The full list of th ese criteria is in Appendix - A ) . According to those criteria, t he peripheral runner speed should have less than 12.24m/s (preferably 6.12m/s) for a fish - friendly turbine, and t his criterion limits the maximum rotational speed for the SMH generation module. 36 1 - D Vortex analysis for axial turbine All turbine design starts with a 1 - D velocity calculation, so it is essential to define the velocity c omponent , which shown in Figure 2 . 2 . For the initial 1 - D Kaplan turbine design, there are two steps to calculate the velocity at each radial span location: 1. Determining the mean velocity value and 2. Determining other span velocit ies with appropriate assumptions. Determining the mean velocity value is simplified by using three fundamental conservation equations mentioned in Chapter 2 . Then, the n ext step is determining the velocity component at eac h radial span location , which is the most critical part of the 1 - D design. Conventionally, the free vortex assumption is widely used for initial velocity calculation. The f ree vortex assumption is an inviscid and ideal case which includes four parts: 1. Flow upstream of the runner is assumed to be free of the vortex , 2. Flow has uniform axial velocity ( distribution , 3. Flow has zero radial velocity ( ) across all span locations , and 4. Flow a t each radial location has a constant value , namely ( 3 - 4 ) This equation come s directly from analyzing the r adial e quilibrium of the blade and consider ing a radial fluid element in Fig 3.4 . T he equilibrium of this fluid element in the radial , which states as ( 3 - 5 ) The net force along the direction could be writ t e n as ( 3 - 6 ) w here, ( 3 - 7 ) 37 Figure 3 . 4 . A radial fluid element in a typical axial turbomachinery system. Consider ing the stagnation enthalpy as constant for every particle and keep it constant along a plane perpendicular on the axis ( 3 - 8 ) In a differential form ( 3 - 9 ) ( 3 - 10 ) Substituting t he above equation into Eqn ( 3 - 8 ) as ( 3 - 11 ) Equation (3 - 11 ) can be written to describe blade geometry variation from mean to hub and tip as ( 3 - 12 ) For inviscid and steady flow, assume is uniform in direction, then ( 3 - 13 ) Th e above equation is known as the free vo rtex condition . 38 T wo other assumptions could also be used for initial velocity calculation: Force vortex assumption and Constant vortex. Force vortex means that : , where is vortex constant. And c onstant vortex means: . Figure 3.5 shows examples of runner - blade geometries designed with three different vortex assumptions. Previous research [ 51 ] has shown that different vortex assumptions trend to have similar hydraulic performance, and the only differenc e was the pressure distribution pattern across the blade , which may result in different deformation behavior of the blades. This paper mainly uses the free vortex assumption for initial velocity calculation. Figure 3 . 5 . Three example runner - blades configurations with different vortex assumptions. (From left to right: Free vortex, Force Vortex, Constant Vortex) Figure 3 . 6 . Three example runner - blade configurations with different hub diameter settings. (From left to right: hub - to - tip ratios (D hub /D tip ) =0.685, 0.743, 0.8.) B ecause of the relatively low rotational speed , w hen utilizing the fre e vortex assumption for designing runner - blade , small hub diameter can result in a large velocity and small velocity, and this can cause a negative value. This negative value at specific radial span 39 location can cause a larger twist angle of the blade , and the Fig . 3.6 shows three geometries of the blade with three hub diameters. A large , twist - angle blade can increase the complexity and the manufactur ing cost of the blade. Therefore, it is crucial to define a minimum hub - to - tip ratio for preventing a large blade twist angle. When using the free vortex assumption, the minimum hub - to tip ratio is defined when , and can be calculated as ( 3 - 14 ) 1 - D Blade geometry construction There are some previous research es focus on designing the hydraulic turbine blade profile. Ferro et al . [ 52 ] used the through - flow analysis approach and streamline curvature method to design a mini hydraulic bulb turbine rotor; Stuikno et al . [ 53 ] used the minimum pressure coefficient and free vortex method to design a low - head turbine; Anagnostopoulos et al . [ 54 ] used the Lagrangian approach to develop and optimize a Turgo turbine; Höfler et al . [ 55 ] used the stream curvature method to design the runner blade row of the Saxo - type turbine. This thesis focuses on a geometry - based design methodology for a low - head hydraulic turbine system. By using the calculated velocity values from the vortex assumption, blade geometry can be constructed by two steps: 1. b lade camberline construction , and 2. b lade thickness distribution. The proposed generation module has two bla de profiles : one is the stator - blade , and the other one is the runner - blade . Those two blade profiles share the same design process. Here i s a general process for designing a runner - blade geometry (for stator - blade geometry, replace the runner - blade angle values with stator angle values). Based on Eq n ( 2 - 16 ) , the power delivered by the turbine runner - blade is only depende nt on the blade inlet and outlet velocity conditions. Therefore, for general 1 - D consideration s , the blade camberline can be any shape of curve s, as long as its inlet 40 and outlet match the velocity condition (namely velocity direction). However, the blade geometry has a significant effect on the overall hydraulic performance. Therefore, the blade camberline construction method needs to meet two main requirements: 1. n eed to match inlet and outlet velocity condition s , and 2. n eed to ha ve the flexibility to control the blade general curve shape for further optimization. Based on those considerations , a new blade construction method, five - point Bezier curve method, is invented to create and optimize the blade camberline geometry. Figure 3 . 7 . A standar d unit runner - blade profile for the proposed turbine system Bezier curve is a parametric curve constructed by a set of control points ( and ) . For constructing the blade camberline, two points are fixed, one is at the beginning of the camberline, and the other one is at the endpoint of the camberline . The rest of the points are located outside the camberline to control the camberline - overall geometry . The general n th order Bezier curve state as 41 ( 3 - 15 ) So, the fourth - order Bezier curve is ( 3 - 16 ) For a standard unit runner - blade camberline shown in Fig 3. 7 , is the inlet point, is the outlet point: . Then, to match the velocity direction, drawing two lines that are both tangential to the curve at inlet and outlet. The intersection point of two lines is the , ( 3 - 17 ) w here, and are auxiliary angles, that come from the velocity triangle ( 3 - 18 ) ( 3 - 19 ) w here, is the stagger - angle of the runner - blade ( is for stator), f or a continuous curve ( 3 - 20 ) And, the can be defined as ( 3 - 21 ) w here, is the runner - blade stagger - angle setting constant, and is between 0 and 1 (For stator - blade , stator - blade stagger - angle setting constant refer as , and can be defined the same way with stator - blade angle and ). This stagger - angle, setting constant is critical for turbine blade design and has a significant influence on the turbine overall hydraulic performance. Figure 3. 8 shows a series of runners and stators blade with different stagger - angle setting constants, and detailed numerical results were shown in section s 5.2.1 and 5.3.1 . 42 (a) Runner - blade with various Stagger - angle setting Constants. From left to right: . (b) Stator - blade with various Stagger - angle setting Constants. From left to right: . Figure 3 . 8 . Runner - blade and Stator - blade with different stagger - angle setting constants Convention al ly, three points ( Bezier curve method is enough to construct a blade camberline for the right velocity condition s . However, a three - point Bezier curve (second - order Bezier curve) has one disadvantage: for a given velocity condition (which means an d are given), the blade profile is only controlled by the stagger - angle ( ) . This limitation means, for a fixed stagger - angle , the blade profile cannot be changed, which can cause less fixable control for further optimization. So instead of a three - point Bezier curve, the five - point Bezier curve is used by adding two more points to construct the blade camberline. One point ( is on the straight line ( 3 - 22 ) Another point ( ) is on the straight line 43 ( 3 - 23 ) are two Bezier curve control points. By adding those two points, t h is method not only meet s the velocity conditions but also increase the flexibility for further optimization. Therefore, the general fourth - order Bezier curve point coordinate for the runner - blade camberline is ( 3 - 24 ) ( 3 - 25 ) After carefully constructing the blade camberline using the above method, t he last step for constructing a blade profile is to have a thickness distribution function for the camberline. For this paper, the NACA - 4 series profile is used and can be state d as ( 3 - 26 ) w here , and are the runner and stator - blade chord length ; and and are the runner and stator - blade relative maximum thickness as ( 3 - 27 ) w here, and are the runner and stator - blade maximum thickness . In Eqn ( 3 - 25 ) , the to are prescribed coefficients for a closed trailing edge ( 3 - 28 ) The superposition method , which superimposes the thickness onto the camberline , is needed for this last step . Figure 3. 9 shows a n example of this sup erimpose d method. By using 44 some points loc a t ed at the camberline ( , and in Fig 3. 9 ), this superimposes method aims to calculate ng those points, the blade profile can be created. On the suction side of the blade, the blade surface point coordination ( ) can be calculated as ; ( 3 - 29 ) On the pressure side of the blade ( ) ; ( 3 - 30 ) w here is the camberline tang e ntial angle at each chosen point. Figure 3 . 9 . Superimposition method using the thickness distribution function to construct a blade profile [ 5 6 ] Summar ies of 1 - D blade design By using the above method, the blade profile for each span location can b e determined; then , by using standard CAE software , which connect s geometry can be constructed. The above method has great flexibility for designing an SMH 45 generation module ; for example, by using 11 radial - span location s , the runner - blade has a total of 33 parameters for modifying runner - blade geometry , which is good for future pressure distribution , stress and performance optimization. In a nutshell , there are four fundamental steps for constructing a blade profile. Construct the camber lines profile by the Bezier curve method from the calculated velocity . Chose a base profile, a relative thickness , and the stagger - angle setting constant for the blade. Calculate the suction surface and pressure s urface coordinate from Eqn ( 3 - 2 9 ) , ( 3 - 30 ) . Smooth the profile and make sure there is no discontinuity and waviness on the surface . 3.3. Generation Obtaining the blade profile is the first step for constructing a n excellen t turbine blade system. Seven fundamental blade configuration parameters can further affect the blade geometry and eventually influence the over all performance. Table 3.2 lists all seven parameters, and this section will comprehensively explain all those parameters. Table 3 . 2 . Seven blade configuration parameters for stator and runner blade Stator - blade Stator Inlet Angle Stator Solidity Stator - blade Number Stator Thickness Runner - blade Runner - blade Number Runner - blade Solidity Runner - blade Thickness Stator Inlet Angle The Stator - blade i nlet a ngle is referred to as in Fig. 2.2 . Since the SMH generation module is positioned in the water with a general - inclined angle ; theoretically, this stator , the inlet - flow angle is related to the general - inclined angle as 46 ( 3 - 31 ) However, the inflow water trends to have a constant inflow angle ( ) for different general inclined angle configuration (Detail results were shown in section 5.2.2 ) . Therefore, the stator - inlet blade - angl e ( ) can be set differently to the stator inlet - flow angle ( ) to achieve a better overall hydraulic performance. Figure 3. 10 shows five stator examples with a different between and . Different angles can completely alter the stator geometry and influence the overall hydraulic performance. The detailed numerical results were shown in section 5.2.2 . Figure 3 . 10 . Five s tator - blade s with different inlet blade angle s ( 1blade ). ( From left to right: 1blade 55 , 70 , 90 , 110 ,130 ) Stator - blade Number The purpose of the stator is to guide and pre - rotate the flow to the runner - blade . High stator counts can result in be tter flow guidance, but the thickness of the stator can also increase the 47 blockage loss. Also, high stator count can decrease the inlet area, which can cause additional damage to larger fish. Therefore, it is vital to find a balance between stator number a nd overall hydraulic performance. Figure 3.1 1 shows five stator examples with different stator - blade number s counts between 20 and 60. The detailed numerical results were shown in section 5.2.3.1 . Figure 3 . 11 . Five stator models with different stator number. ( From left to right: Stator Number = 20, 30, 40 , 50 ,60 ) Runner - blade Number Just like the stator - blade number, the runner - blade number is another critical parameter that can affect the overall hydraulic performance. Since the proposed SMH generation module has an adjustable runner - blade mechanism, the less runner - blade count can decrease the complexity of the control system and th e overall module cost. Therefore, it is crucial to study how the runner - blade number affect s the overall hydraulic performance. Figure 3.1 2 shows five runner examples with different runner - blade number s count between 4 and 12. Although the runner - blade s nu mber 48 also has a profound influence on vibration, the only focus here is on the effect on overall hydraulic performance. The detailed numerical results were shown in section 5.3.2.1 . Figure 3 . 12 . Five runner - blade s with different runner - blade number. ( From left to right: Runner Number = 5, 6, 8, 10, 12 ) Stator and Runner blade Solidity The solidity of a blade is defined as ( 3 - 32 ) w here is the blade chord length, is the blade spacing , can be defined as ( 3 - 33 ) w here is the diameter at each blade span location, is the blade number. The runner - blade cascade solidity has significant influence s on the flow behavior , especially on the blade profile losses . The runner - blade profile loss has two parts ( 3 - 34 ) 49 With a defin ed spacing, there is an equilibrium between the sep a ration and friction losses. Based on previous research, the corresponding solidity ratio has an optimum when , which is shown in Fig 3.1 3 . Figure 3 . 13 . The runner - blade p rofile loss as a function of the invert of solidity [5 6 ] ( The circle marks the optimum value ) For the SMH generation module, the runner - blade solidity can determine the runner - blade size , which has a profound effect on the efficiency and costs ; therefore, different runner - blade solidity value s mus t be tested . Fig ure 3.1 4 shows six runner - blade examples with various solidity number , and detailed numerical results were shown in section 5.3.2.2 . The stator - blade solidity , on the other hand, does not have an optimum value. However, the stator profile is also influential for the SMH generation module, since it helps to prevent the runner - blade being damaged by inlet trash, and also provide a correct guide for the inlet flow and fishes to the runner - blade ; t herefore, different stator - blade solidity values need to be tested. Figure 3.1 5 shows ten stator - blade examples with various solidity number, and detailed numerical results were shown in section 5.2.3.2 . 50 Figure 3 . 14 . Various Runner - blade s with different solidity values. ( Form left to right: r =0.7,0.8,0.9,1,1.1,1.2 ) Figure 3 . 15 . Various stator - blade s with different solidity values. ( From left to right: s =0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 2.0, 3.0. ) Stator and Runner blade Thickness As shown in Eqn ( 3 - 2 6 ) , the relative blad e thickness is the ratio of the maximum blade thickness and the chord length. This value has a significant impact on blade blockages, flow 51 behaviors, deformation, and stress performance. In the hydraulic turbine, the thickness of the runner - blade can block the annulus passage area over the entire run ner - blade zone. This runner blockage can be defined as the runner - blade maximum - blockage ratio ( ) or inlet blockage ratio ( ) as ( 3 - 35 ) Where is the maximum runner thickness, is the runner - blade relative thickness, is the runner - blade chord length, is the runner - blade spacing , is the blade solidity, and is the runner - blade angle. Fig . 3.1 6 shows a schematic diagram of this blockage definition. Since at each span location the is different, the is also a function of span location, and the mean maximum - blockage ration is defined at the mean span location. Figure 3 . 16 . The d efinition of the blockage at the runner - blade inlet There are some previous studies related to the effect of the blade thickness on turbomachinery performance. Mu et al . [ 5 7 ], Tao et al . [ 5 8 ], and Yang et al . [ 5 9 ] studied how blade thickness affects the centrifugal pump performance; and Kim et al . [ 60 ] studied the effect of blade thickness Generally, the turbine runner - blade has a value around 15% - 18%. [5 6 ] However, since this proposed SMH generation module has a large and 52 irregular size, different bla de thickness values have to be tested for hydraulic performance. Figure 3.1 7 shows five runner - blade examples with different rel ati ve thicknes se s, and t he numerical simulation results were shown in section 5.3.2.3 . Since the SMH generation module has to be fish - friendly, the stator - blade , relative thickness needs special consideration. Small stator thickness may have poor flow guidance; large stator thickness may induce substantial blockage loss and cause damage to large fish species; therefore, different s tator relative thickness values were tested , Fig . 3.1 8 shows five different stator - blade examples with various relative thickness . D etailed numerical results were shown in section 5.2.3.3 . Figure 3 . 17 . Five runner - blade s with different relative thickness value s . (From left to right: R T =5%, 7.5%, 10%, 12.5%, 15%. ) Figure 3 . 18 . Five stator - blade with different relative thickness values. (From left to right: S T =5%, 7.5%, 10%, 12.5%, 15% ) 3.4. Summary Appendix - B shows all initial design parameters with description s and initial recommend setting values. 53 4. CHAPTER IV : NUMERICAL METHO DS In this chapter, the basic c omputational fluid dynamic (CFD) method are covered . A successful and accurate numerical simulation depend s on a series of steps: Computational domain definition and construction Turbulence Modeling selection Boundary Condition setting. Each step w as discussed comprehensively in this section. The widely available commercial computational fluid dynamics software ANSYS FLUENT 2018 was used for this research. 4.1. General governing equat ions For more than 20 years, the computational fluid dynamic has become a powerful tool for evaluating the turbomachinery system . Significant progress has been made , which ensure s a more accurate prediction of the performance and flow behaviors in the turbomachinery. The general governing equations for computational fluid dynamics are the Navier - Stokes (N - S) equations. The N - S equations for incompressible isothermal flow through a turbomachinery system can be written as ( 4 - 1 ) w here is the velocity vector, is the pressure, is the density of the fluid , and is the kinematic viscosity of the fluid. Generally, the hydraulic system could be consider ed as an isothermal system , which mean s no heat addition or removal in this fluid system. Since there is no general solution for the non - linear equation, the N - S equ ation need s to be solved numerically with specific boundary condition s and initial conditions. With complex geometries, solving the N - S 54 equation for turbomachinery is numerically costly . When predicting the general performance of the turbomachinery, the av eraged flow values are usually the properties of interest at the beginning of the design process. Therefore, RANS ( Reynolds Averaging Navier Stokes method ) is normally used. This method is obtained by time - averaging the Navier - Stoke equation for the averaged values of the flow properties over a sufficient time period, and are written as ( 4 - 2 ) w here is the time - average d velocity vector, is the velocity fluctuation due to turbulence and is the Reynolds shear stress. The basic concept of the RANS method is to decompose the flow velocity into two parts : time - averaged value , and the fluctuating value as ( 4 - 3 ) In Eqn ( 4 - 2 ) , the Reynolds shear stress part could not be solved directly and must to be modeled with different turbulent models, which will be covered in section 4.2.2 . 4.2. Numerical method developments A successful and effective numerical simulation requires three parts: a sufficient computational domain, correct turbulence model selection, and appropriate boundary conditions setting. This section covers all those three parts. Computational domain construction Mesh represents the geometric domain for the CFD computa tion that has a significan t impact on the accuracy and cost. A suitable mesh is a premise for a successful numerical simulation . For turbomachinery applications , the structured hexahedral mesh is preferred since it is mostly aligne d to the mai n flow direction. [ 61 ] However, a structured hexahedral mesh is challenging and time - consuming for a complex t urbomachinery geometry . An 55 u nstructured hybrid mesh scheme h as been used for simulating the turbomachinery flow physics recently, which means different mesh schemes are used in different flow regions, for example, hexahedral mesh near the wall region and unstructured mesh in the other region . In this research, t he hybrid meshes scheme is used. In order t o generate the mes h, different geometry model s w ere first defined and cleaned (any geometric errors or in - continuity) in the computer - aided design (CAD) softwar e . The n this clean geometrical model was imported into ANSYS Mesh software. Different mesh criteria were utilized for smoothing and refinement to avoid any numerical errors . In order to capture the boundary conditions and flow behaviors near the blades , at least 10 layers were needed on the wall of the blades. The dimensionless wall distance is one critical indication for turbulent boundary layer accuracy. For example, for the turbulence model , the is required to capture the near - wall region flow behaviors [ 61 ]. However, the model has a wall function that enables a smo oth switch between the near - wall region and the central flow region [ 62 ]. This wall function allows the use of some coarse grids without a significant penalty in simulation [ 63 ] . As t his research proposed a new design methodology for the Kaplan turbine , which need s a number of CFD simulations to test the geometric setting parameters , only general performance , and basic properties were focused. S o, in order to balance the computational cost and accuracy, differen t mesh size s were used. There are three computational domains for this proposed turbine system: i nlet water channel, stator, and runner , and each part has a unique meshing scheme for better computational performance. Figure 4.1 (a) - (d) shows th ose three parts and their meshing scheme. 56 Figure 4 . 1 . The numerical grids for the three computational fluid domains. 57 Generally, more elements mean more accurate results, but with a higher computational cost. Therefore, it is necessary to demonstrate how different mesh sizes influence the overall prediction of the performance. Five different mesh schemes were selected for one example model . Table 4.1 shows the detail mesh information for each scheme , and Figur e 4.2 shows the mesh quality and performance relation . Table 4 . 1 . Detail mesh information for the five selected mesh schemes Mesh - 1 Mesh - 2 Mesh - 3 Mesh - 4 Mesh - 5 Mesh size [m] 0.015 0.01 0.0075 0.006 0.005 Mesh Type Tetrahedrons Mesh + 10 layer near the wall Total element n umber [million] 13.6 33.3 70.8 178.4 304.5 Figure 4 . 2 . The m esh q uality and normalized performance relation for the five selected schemes. All values in Fig 4.2 were normalized with respect to the scheme - 2 values. The results show that with a finer mesh, the efficiency increases by around 0.68%, the mass flow rate decreases around 0.57%, and the power increases by around 0.12% . Mesh - 5 would provide more accurate 0.985 0.99 0.995 1 1.005 1.01 1 2 3 4 5 Normalized Value Mesh Scheme Normalized Efficiency Normalized Mass Flow Rate Normalized Power 58 results; however, the computational cost was too substantial . B y using the same computing power, mesh - 5 required 45 times more computational time than m esh - 2 . This methodology needs hundreds of simulations to verify the geometry settings and other parameters ; therefore, Mesh - 2 was considered sufficiently reliable for predicting the overall hydraulic performance and was mainly used in this paper. Turbulence model selection Turbulence modeling is a n essential part of any turbomachinery design and has a significant impact on design outcomes. Current approaches for turbulence modeling have four main types: 1) RANS base d simulation, 2) Large - eddy simulation (LES), 3) Direct numerical simulation (DNS) , 4) Hybrid RANS - LES. - As mention before, RANS is a time - average method ; mean velocity, pressure , and other properties were computed with some appropriate modeling scheme . RANS simulations with different modeling methods have been widely used for turbomachinery design because of its cost advantage and accuracy satisfaction. - LES simulation is another promising method, Navier - Stokes equations are solved in large - eddy scale , and small - scale eddy need s to be modeled in LES. LES simulation req uires more significant computational resources and finner mesh scheme then the RANS method, so it is less practical than the RANS method due to the cost for turbomachinery applications . - The DNS method is the most accurate method which numerical ly solve s t he Navier - Stokes equations in any fluid domain. However, the enormous computational cost makes DNS an impractical method and only for elementary geometry. - The h ybrid method combine s both the RANS and LES method to achieve a more accurate prediction of flo w pattern s . Nera the wall region, the Spalart - Allmaras (SA) turbulence model 59 ( o ne - equation RANS model) is used , and outside the boundary layer, the LES model is applied. This method is good for capturing the flow pattern s around the wall s , but it is still t o o computationally expensive for this research. In this research, turbulence model is used , which has better performance for turbomachinery applications [ 61 ] . The turbulence model is a two - equation RANS model that has two equations , one for turbulence kinetic energy and one for specific dissipation rate , which consists of blending the model and model based on the proximity to the walls . Due to the over - prediction of the shear stress in the adverse pressure gradient boundary layer of the original model, in 1994, Menter first proposed the model [ 6 4 ]. In the model, to account for the transport of the turbulence shear stress, a limiter function is used and modified, which shows b etter predictions of s epa ration and streamline curvature [ 6 4 ]. Because of t h ose features , the model is considered as a better model for the turbomachinery application than the standard two - equation models . In the model, t he Bousi nessq approximation is used to calculate the Reynolds stress for the turbulence viscosity models as ( 4 - 4 ) For model, t he transport equation for the turbulent kinetic energy can be written as ( 4 - 5 ) w here , is the production of turbulence kinetic energy and can be defined as ( 4 - 6 ) Additionally , the transport equation for the turbulence dissipation rate is 60 , , , ( 4 - 7 ) A nd is given by [ 6 5 ] ( 4 - 8 ) Similarly, for the model, the transport equation for turbulent kinetic energy can be written as ( 4 - 9 ) And , the transport equation for the specific dissipation rate is , , , , ( 4 - 10 ) Away from the wall, the model transforms into the standard model due to the better performance in the free shear layers, which solves the disadvantage of the model. The suggested boundary condition is : [ 6 4 ] ( 4 - 11 ) w here is the distance from the wall to the first - cell center . Since t he model us ing the blending functions, the transport function is rewritten as ( 4 - 12 ) Where t he blending function is given by ( 4 - 13 ) 61 ( 4 - 14 ) w here is written as ( 4 - 15 ) The turbulent viscosity is expressed as ( 4 - 16 ) ( 4 - 17 ) ( 4 - 18 ) Boundary conditions setting Numerical simulations of any flow system require the specification of all boundary conditions in the whole computational domain s . - Inlet boundary condition For the Kaplan turbine, two inlet boundary condition types are normally used, either the total pressure or the velocity (which is equal as mass flow rate for incompressible flow). Those two boundary conditions represent two design parameters : tot al pressure , which represents the design head, and the velocity , which represents the total mass flow rate. In the 1 - D design phase, both design parameter s must be specified. However, in CFD , those two parameters are highly depend ent on each other and othe r geometry settings . This means only one parameter could be fixed and another one must float. If the turbine geometry is designed appropriately, the float value should be close to the designed value, but irregular geometry settings may cause huge loss and 62 influence the mass flow rate and head values. Additionally, a 5% turbulence intensity was assumed at the inlet , and the inlet flow is assumed unaffected from the downstream flow. - Outlet boundary condition There are two boundary conditions option s for the outlet boundary: pressure and velocity. Generally, two combinations of boundary condition s could be used: Inlet pressure and Outlet pressure and Inlet pressure and Outlet velocity. The first combination fixes the head value as the designed and flo at s the mass flow rate; the second combination fixes the mass flow rate and float s the outlet pressure. Both combination s work similarly , and for this research, since the head condition is the most concerned parameter, the first combination is used. - Wall b oundary Conditions T he wall of the stator, the inlet channel, and the runner - blade s are considered as the wall boundary . All stationary walls are treated as non - slip, isothermal stationary wall s , and the rotating runner - blade are treated as no - slip, isothermal rotating wall . - Interface The frozen - rotor interface method can be used for the steady RANS models. In this method , the interface connecting the stator (stationary) and runner (rotating) domains is prescribed as the froze n rotor with a mixing plan approach [ 6 6 ]. Moreover , appropriate transformation equations in the rotating frame of reference are applied , and the solution has no dependence on the relative positioning of the rotor and the casing [ 6 7 ]. The disadvantage of th is model is the interaction between stator and runner is completely ignored due to the averaging of the circumferential non - uniformities of the upstream flow. However, for steady - state simulation, the frozen - rotor model is a good choice for simulating the average performance and was primarily used in this research. 63 4.3. Summary By using the numerical methods described above, the main goal of the numerical simulation for this thesis is to verify the proposed design methodology with different geomet rical settings and configurations. During the simulations, the overall performance , general flow behavior around the blade , and operating range are three main focuses . The overall performance is defined by using the overall hydraulic efficiency as ( 4 - 19 ) w here is the torque on the runner - blade , is the rotational speed in , is the mass flow rate, is the stan dar d gravity constant, and is the available head. The operating range is defined by using unit flow rate and unit ( 4 - 20 ) w here is the volumetric flow rate, is the turbine tip diameter, is the rotational speed in . Table 4 . 2 summaries the basic operating range and numerical simulation settings for the Table 4 . 2 . Design and numerical simulation conditions Design Conditions Range Numerical Simulation Conditions Design Head (H) 2.5m Inlet/ Outlet Boundary Condition Pressure Inlet/Outlet Unit Flow Rate (Q 11 ) 0.103~0.52 m 1/2 /s Turbulence Model Overall Diameter (D tip ) 3.5m Simulation Type Steady State Design Unit Rotation Speed ( ) 88.5 Mesh Size ~33 Million 64 5. CHAPTER V : NUMERICAL RESULTS A ND DISCUSSIONS This chapter presents detailed numerical simulation results for the design methodology discussed above. The results cover all fundamental geometrical considerations, including g eneral hub size consideration, stator - blade consideration, runner - blade conside ration, and off - design considerations. 5.1. General h ub size consideration As shown in Fig. 3.6 , the hub size has a massiv e impact on the turbine blade design, which can have a significant influence on the overall hydraulic performance. Moreover, for the SMH generation module, the hub should have enough space for generator and control components. So, it is essential to thorou gh ly investigate the hub size and its impact on overall hydraulic performance. Five different hub - to - tip ratios (D hub /D tip ) were chosen and thorough ly investigated, and Fig. 5.1 shows the results. The results show that for low , flow - rate conditions (when ), the largest hub diameter (D hub /D tip =0.8) model has the best performance, and the performance decrease with decreasing hub diameter and the maximum performance difference between different hub size s is around 3.2%. When the flow rate increases, th e smallest hub model (D hub /D tip =0.685) has the best overall hydraulic performance, and the maximum performance difference for different hub size s is around 10.1%. Also, the results indicate that the smaller hub diameter has a more stable performance than l arger hub diameter, and the hub size has a larger impact on overall hydraulic performance at a high, flow - rate condition (Red Arrow in Fig.5.1 ) . According to Eq n ( 2 - 16 ) , the runner - blade power is determined by the magnitude of the runner - blade , circumferential - velocity difference. Therefore, Figure 5.2 shows the circumferential 65 velocity difference (C u2 - C u3 ) distribution plots from hub to tip at two selected Q 11 conditions for all five different hub - size configurations. Fig ure . 5 . 1 . The relation between overall hydraulic efficiency and design flow rate (Q 11 ) for five selected Hub - to - Tip ratio configurations At lower flow - rate condition , Fig. 5.2 ( a ) , on the close to hub region (less than 25% Span Location), a smaller hub configuration has a significantly larger circumferential velocity difference (Red Arrow). However, this circumferential - velo city difference drops steeply and remains at a very low value (Black Arrow) on the close to tip region (between 60% and 100% Span Location). In general, at low flow - rate condition s , the smaller hub configuration tends to have worse performance on the close to tip region, which leads to poor overall performance. At a larger flow rate condition , Fig. 5.2(b) , the smaller hub configuration has a larger circumferential - velocity difference across all span location s (Blue Arrow). This massiv e difference causes the smaller hub configuration to have a significantly better overall performance at a larger flow rate condition. 0.75 0.775 0.8 0.825 0.85 0.875 0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Overall Hydraulic Efficiency Design Flow Rate [Q 11 ] Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.800 Two selected flow rate conditions 66 Fig ure . 5 . 2 . The circumferential velocity difference (C u2 - C u3 ) distribution plots from hub to tip at two Q 11 conditions for all five different hub - size configurations, (a) . Low Flow Rate Condition , Q 11 =0.103 . (b) . High Flow Rate Conditio n, Q 11 =0.31 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.2 0.4 0.6 0.8 1.0 Circumferential Velocity Difference (C u2 - C u3 ) [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.2 0.4 0.6 0.8 1.0 Circumferential Velocity Difference (C u2 - C u3 ) [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 67 In general, choosing the right hub size involves balancing the design flow rate and the required hub volume. At a high flow rate, the smaller hub can provide a n excellent overall hydraulic performance but with limited hub volume that can have high over - heating possibilities for the generator, which can further affect the overall electrical performance. Balancing the hub size and performance is the key for the initial sizing of the turbine unite. (Some extra results and plots are in the Appendix . C ) 5.2. Stator - blade consideration Stator - blade stagger - angle setting constant consideration As shown in Fig. 3.8 (b) and Eq n ( 3 - 21 ) , the Stator - Blade , Stagger - Angle - Setting - Constant ( ) is a critical parameter for designing the stator - blade profile. Figure 5.3 shows how this parameter affects the overall performance at four flow rate conditions. Fig ure . 5 . 3 . The r elation between overall hydraulic efficiency and stator - blade stagger - angle setting constant C ssa for four flow rate conditions 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Overall Hydraulic Efficiency Stator Blade Stagger Angle Setting Constant (C ssa ) Four Selected Models 68 The results show that at small , flow - rate condition, the overall hydraulic efficiency increase s with the increase of and reaches the maximum efficiency when is around 0.7~0.8, then decrease s with the increase of . For large , flow - rate conditions, the overall hydraulic efficiency increase s with the increase of and reaches the maximum efficiency when is around 0.5~0.7, then remain s relatively constant, then decrease s w hen is larger than 0.8. For all flow rate conditions, the initial increase of efficiency wit h the increase of is universal, and the maximum efficiency occurs when is around 0.5~0.8. Four models under one flow - rate condition (Red Square in Fig. 5.3 ) were selected for further investigations. Since t he primary function of the stator is to redirect the inflow to the runner - blade with the desired angle, for the four selected models, two angles were studied: and . - - Considerations The is the runner - blade , inlet - relative - flow angle (see in Fig. 2.2 ), and under the ideal condition, the should be the same as the , and a large difference between and means increasing the runner - blade incidence loss , which can affect the overall performance. Figure 5.4 shows the comparison between the and for the four selected models. The results show that, near the hub region, because of th e boundary flow and low runner circumferential velocity (velocity in Fig. 2.2 ), the is larger than the for all four selected models , and the maximum angle - difference is around 12 . And, near the tip region, because of the bou ndary flow and high runner circumferential velocity, the is all larger than the for all four selected models and the maximum difference is around 9 . However, results are different in the center of the span region. The smaller models ( = 0.1,0.3) trends to have a larger than the , the maximum angle difference is ~+3 . 69 The medium models ( =0.5) trends to have a very similar than the , the maximum angle difference is only ~+0.4 . The larger models ( =0.8) trends to have a smaller than the , the maximum angle difference is ~ - 4 . As shown above, the difference between the and the can undoubted ly affect the overall performance, but the results in Fig. 5.4 cannot fully explain the trend shown in Fig. 5.3 . Therefore, other angles w ere studied. Fig ure . 5 . 4 . The runner - 2 flow angle distribution plots for four selected models in comparison 2 blade value 68 73 78 83 88 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Relative Flow Angle ( 2 Flow) [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Blade Angle Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 Hub - Wall Region Tip - Wall Region Center of the span region 70 - - Considerations The is the runner - blade - inlet absolute flow angle (see in Fig. 2.2 ), and this angle determines the magnitude of the runner - blade , inlet circumferential - velocity (C u2 ). Larger means larger C u2 , which results in more substantial potential work done on the runner - blade (according to the Eq n [ 2 - 16 ] ). Figure 5.5 shows the distribution plot across all span location for the four selected models with the comparison with the 1 - D designed values. The smaller ( = 0.1) model shows significant lower values in the majority center span region ( exc lude the hub and tip boundary region), the difference is around - 60 . For larger , the ( = 0.8) model also shows lower values in the majority span center region, but the difference is only around - 10 . This huge difference in the valu es explain why Model - 4 ( = 0.8) has significantly better performance than the Model - 1 ( = 0.1). Fig ure . 5 . 5 . The runner - blade inlet absolute flow angle ( 2 ) distribution plots for four selected models in 2 value 0 10 20 30 40 50 60 70 80 0 0.25 0.5 0.75 1 Runner Blade Inlet Absolute Flow Angle ( Normalized Span Location ('0' is Hub, '1' is Tip) Designed Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 Model-1 at 50% Span Location Model-4 at 50% Span Location 71 For better visualization, Figure 5.6 shows the stream - line pattern for the Model - 1( = 0.1) and Model - 4 ( = 0.8) at 50% span location. When is small, as shown in Fig. 5.6 - ( a ) , the stator - blade camber - line has a flat profile in the font region, and the diversion of the flow only happens near the trailing edge. This profile results in poor flow redirection and leads to very small values. When is large, as shown in F ig. 5.6 - ( b ) , the stator - blade camber - line has a flat profile in the rear region, and the diversion of the flow occurs near the leading edge. This profile results in good flow redirection and leads to the larger values. In general, the value has a larger impact on values than value which ha s a profound influence on stator outlet flow behaviors and can further affect the overall performance. Base d on the overall results, initially, the suggesting range for is between 0.7 to 0.8 . Some additional results and plots are shown in Appendix .D . (a). Model - 1 at 50% span location (Green dot in Fig. 5.5 ) Fig ure . 5 . 6 . The stream - line pattern for the (a) Model - 1 and (b) Model - 4 , at 50% span locatio n 72 Figure. 5.6 . ( b ). Model - 4 at 50% span location (Blue square in Fig. 5.5 ) Stator - blade inlet angle considerations As shown in Fig. 3.10 , the s tator - b lade inlet - angle is another critical parameter for determining the stator - blade shape. Figure 5.7 shows the inflow angle ( ) distribution for the stator at three different general - inclined angle configurations and the results indicate that the general - inclined angle has minimal effect on the . The inflow angle remains very close to at all span locations (Except at the shroud and hub boundary layer regions ) for all different general inclined angle configurations. According to the velocity triangle shown in Fig. 2.2 , the stator - blade inlet angle ( ) should be set equal to the ; however, different can have a n impact on overall hydraulic performance. Figure 5.8 shows how different affect the overall hydraulic performance and normalized power (the power is normalized with results) for three different general - inclined a ngle configurations under the same flow rate condition ( ). The results show that decreasing the has a positive effect on the overall hydraulic performance, the maximum efficiency difference is 7%. 73 Fig ure . 5 . 7 . The Inflow angle ( 1flow ) distribution across all span location for three general - inclined angle configurations Fig ure . 5 . 8 . The Relation between the o verall hydraulic efficiency, Power Generation, and the Stator - blade 1 blade ) for three general - inclined angle configurations under the same design flow rate condition (Q 11 =0.259) 89.0 89.2 89.3 89.5 89.6 89.8 89.9 90.1 90.2 90.4 0.0 0.2 0.4 0.6 0.8 1.0 Inflow Angle 1 Flow [Degree] Normalized Span Location ('0' is Hub, '1' is Tip) 50° General Inclined Angle 40° General Inclined Angle 30° General Inclined Angle 0.8 0.85 0.9 0.95 1 1.05 1.1 0.8 0.82 0.84 0.86 0.88 45 60 75 90 105 120 135 150 Normalized Power Overall Hydraulic Efficiency Stator Blade Inlet Angle 1 lade [Degree] 30° General Inclined Angle (Efficiency) 40° General Inclined Angle (Efficiency) 50° General Inclined Angle (Efficiency) Four Selected Models 30° General Inclined Angle (Power) 40° General Inclined Angle (Power) 50° General Inclined Angle (Power) Inflow Angle 1 flow =90 74 Furthermore , the power generation remains relatively constant when ; however, when , the power generation decrea ses significantly with the increase of the because of the efficiency drop . For further investigations, four models (Red Marker s in Fig. 5.8 ) were chosen. As mentioned above, the primary function of the stator is to redirect the inflow to the runner - blade with the desired angle ; therefore, for the four selected models, two angles were studied: and . - - Considerations: Figure 5.9 shows the comparison between the and for the four selected models. The results show that, near the hub region, because of the boundary flow and low runner circumferential velocity (velocity ), the is larger than the for all four selected models , and the maximum difference is around 20 . Near the tip region, because of the boundary flow and large runner circumferential velocity, the is larger than the for all four selected models , and the maximum difference is around 8 . Fig ure . 5 . 9 . The runner - blade 2 flow distribution plots for four selected models in comparison with the designed 2 blade value 55 60 65 70 75 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Relative Flow Angle ( flow) [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Designed Blade Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 75 In the majority center of the span region, the is relatively close to the for Model - 2, Model - 3, and Model - 4 ; and the maximum difference is only 4 . For Model - 1, compared to the other three models, the is much smaller than the especially in the 80% - 90% span reg ion (Red Arrow in Fig. 5.9 ) ; the maximum difference is 7 . As mention ed above , the angle results can no t fully explain the overall trends ; so, the angle results were also studied. - - Considerations: Figure 5.10 shows the distribution plot across all span location for the four selected models with the comparison with the designed value. For model - 1 ( =150 ), the is significantly lower than the other three models in the 0% to 50% span location, the maxi mum difference is 27 compared to the designed value. This vast difference (Blue Arrow in Fig. 5.10 ) is the main reason that model - 1 has significantly lower efficiency. Model - 2, 3 , and 4 all have a very similar distribution pattern, in the majority span region (between 10% to 75%), the Model - 4 has higher values than the other two models (Red Arrow in Fig. 5.10 ) , which explain s why the model - 4 has the best overall efficiency. For better visualization, Figure 5.11 shows the stream - line pattern for the Model - 1 ( =150 ) and Model - 4 ( =45 ) at the 13% span location. Model - 1 ( Fig. 5.11 - a ) has a counter - clockwise vortex pattern on the stator - blade suction side that is close to the trailing e dge , which distorts the downstream flow; in comparison, Model - 4 ( Fig. 5.11 - b ) has a clockwise vortex pattern on the stator - blade , pressure side that is close to the leading edge , which improves the downstream flow. Those different vortex patterns have a sig nificant impact on the value s , which can further influence the turbine works and efficiency. Some additional results and plots are shown in Appendix.E . 76 Fig ure . 5 . 10 . The runner - blade inlet absolute flow angle ( 2 ) distribution plots for four selected models in 2 value (a). Model - 1 at 13% span location (Green dot in Fig. 5.10 ) Fig ure . 5 . 11 . The stream - line pattern for the (a) Model - 1 and (b) Model - 4 , at 13 % span location 0 10 20 30 40 50 60 70 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Absolute Flow Angle ( Normalized Span Location ('0' is Hub, '1' is Tip) Designed Value Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) Model-1 at 13% Span Location Model-4 at 13% Span Location 77 Figure. 5. 11 ( b ). Model - 4 at 13% span location ( Blue dot in Fig. 5.10 ) Stator - blade configuration considerations The stator - blade number, solidity , and relative thickness are three critical geometrical parameters for configuring the stator - blade , and each parameter has a huge role in guiding inflow to the runner, protecting inflow fish and reducing the overall complexity of the system. Each parameter and how it impact s the system overall performance is covered in this section. Stator - blade number consideration As shown in Fig.3.11 ., t he stator - blade number is essential for SMH technology ; less stator can have larger spaci ng between blade, therefore reduc ing the fish impact, improv ing the fish survival rate, and reduc ing the overall complexity of the system. However, the high stator count can act as a trash rack , which prevent s large floating object s f rom entering the runner - blade section, therefore reduc ing the runner - blade damage. Because of those reasons, the configuration of the 78 stator should be thoroughly studied. Figure .5.12 shows the normalized shaft - power output (The power results w ere normalized w ith 40 - of the turbine with the variation in the number of stator - blade s at four different flow conditions. Fig ure . 5 . 12 . The relation between stator - blade number, overall hydraulic efficiency, and normalized shaft power for four designed flow conditions The results show that among all selected four flow rate conditions , the maximum performance difference is only 0.8%, the m aximum shaft power difference is only 2 .33 %. This less e ffect means the stator number has a minimal effect on overall hydraulic performance, and a different number of stator - blade s can be chosen for various river conditions. For this paper, the 40 stator - blade was chosen as the reference configuration . 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0.8 0.82 0.84 0.86 0.88 0.9 20 30 40 50 60 Normalized Shaft power Overall Hyfraulic Efficiency Stator Blade Number Model - Efficiency Model - Efficiency Model - Efficiency Model - Efficiency Model - Normalized Power Model - Normalized Power Model - Normalized Power Model - Normalized Power 79 Stator - blade solidity consideration As shown in Eqn ( 3 - 32,33 ) and Fig . 3.15 , the stator - blade solidity ( ) can dramatically change the stator profile and alter the overall pe rformance. Unlike the runner - blade solidity that has an optimum value for initial reference ( Fig . 3.13 ) , to the there is very little information regarding the stator - blade solidity. So, it is critical to investigate how stator - blade solidity affects overall performance . For a given stator number, t he stator - blade solidity can but with additiona l friction loss es . Also, since the blade relative thickness is related to the blade chord length, shorter chord length can cause a thin blade profile , which can decrease the blade blockage and flow guidance , alter the blade velocity triangle, and eventually reduce the overall performance. For four flow - r ate conditions, Fig 5.13 shows the overall hydraulic performance of the turbine and the normalized shaft power output (The power results w ere normalized with results ) with the variation in the stator - blade solidity . Fig ure . 5 . 13 . The relation between stator - blade solidity with (a) overall hydraulic efficiency and (b) normalized shaft power for four designed flow rate conditions 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Overall Hydraulic Efficiency (a). Stator - Blade Solidity Selected Models 80 Figure. 5. 13 The results show that, for all flow rate conditions, the increase of the stator - blade solidity can first improve the overall hydraulic efficiency, then after reaching an optimum value, the overall hydraulic efficiency drops , and the maximum efficiency diff erence is around 5.5%. The optimum stator solidity - value is between 1.2 and 1. 8 depends on different flow rate conditions . Moreover, the most significant difference happens in shaft power. With the increase of the stator - blade solidity , the shaft power dro ps dramatically, and the maximum difference is around 20%. Based on Eqn ( 2 - 24 ) , the turbine - work ( ) is related to three factors : the total head difference ( ), mass flow rate ( ), and overall hydraulic efficiency ( ). As shown in Fig. 5.1 3 - (a) , the maximum efficiency difference is only 5.5%; and the total head difference is fixed. Therefore, the major reason that causes the turbine - work to drop significantly is the decrease in the mass flow rate. All example models in Fig.5.13 are designed for four flow conditions ( , , , ) , and all those flow conditions are the target , designed flow rate 0.82 0.87 0.92 0.97 1.02 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized Shaft Power (b). Stator - Blade Solidity Selected Models 81 conditions. As discussed in section 4.2.3 , during the numerical simulation, only the head difference is fixed , and the flow rate must float. With the correct design and appropriate geometry setting, the final flow rate results should be close to the designed value. However, irregular geometry, in this case, the stator - blade solidity, can cause a huge flow r ate difference and leads to the efficiency and turbine work difference shown in Fig.5.13 . For better understanding, three models (Model - 1: Model - 2 ; Model - 3 ) with one design flow rate condition ( ) are selected for furt her investigations (Red Marker s in Fig .5.13 ). - Consideration As discussed before, the first step is to evaluate how the ( runner - blade inlet relative flow angle) change with different stator solidity at various span locations . Figure 5.14 shows the comparison between the and for the three selected models. The results show that, near the hub region, because of the boundary flow and low runner circumferential velocity (velocity in Fig. 2.2 ), the is larger than the for all three selected models , and the maximum difference is around 1 6 . Additionally, there are two features worth noticing: In the center span region , model - values are close to the designed values with the maximum difference only around 2.5 (Back Arrow in Fig .5.14 ). In the center span region, compared to the model - 1, the difference between model - values and the designed values are larger , with the maximum difference around 7 (Red Arrow in Fig .5.14 ). As mention ed above, the difference between and can undoubted ly affect the overall performance by increasing the runner - blade incidence loss, but the results in Fig. 5.14 cannot fully explain the trend shown in Fig. 5.13 . Therefore, an other angle was studied. 82 Fig ure . 5 . 14 . The runner - blade 2 flow angle distribution plots for three selected models in 2 blade value - Consideration Figure .5.15 shows the distribution plot across all span location for the three selected models with the comparison with the designed value. For model - 1 ( =0.8 ), the is significantly lower than the other t wo models in the center of the span region , the maximum differ ence is around compared to the designed value. This vast difference ( Red Arrow in Fig. 5.15 ) is the main reason that model - 1 has significantly lower efficiency. For the other two models, the maximum difference in the center span region is only around 5 . For better visualization, Figure .5.16 shows the stream - line pattern for the Model - 1 ) and Model - 2 ( =1.6) at 60 % span location. Model - 1 ( Fig. 5.16 - a) which has small stator - blade solidity and shorter blade chord length has relatively poor flow guidance, and this leads to the vast value difference an d low overall efficiency. Model - 2 ( Fig. 5.16 - b) which has a longer and thicker blade profile has good flow guidance, and this leads to the small value difference and high overall efficiency. 55 60 65 70 75 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Absolute Flow Angle ( Normalized Span Location ('0' is Hub, '1' is Tip) Designed Value Model - 1 ( Model - 2 ( Model - 3 ( 83 Fig ure . 5 . 15 . The runner - blade inlet absolute flow angle ( 2 ) distribution plots for three selected models 2 value (a). Model - 1 ( ) at 60 % span location (Green dot in Fig. 5.15 ) Fig ure . 5 . 16 . The stream - line pattern for the (a) Model - 1 and (b) Model - 2, at 60 % span location 0 10 20 30 40 50 60 70 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Absolute Flow Angle ( Normalized Span Location ('0' is Hub, '1' is Tip) Designed Value Model - 1 ( Model - 2 ( Model - 3 ( Model-1 at 60% Span Location Model-2 at 60% Span Location 84 Figure. 5.1 6 ( b ). Model - 2 ( ) at 60 % span location (Blue dot in Fig . 5.15 ) - Velocity Triangle Consideration Another key feature in Fig .5.13 - (b) is the significant power increase with the decrease of the stator - blade solidity . In order t o explain this, different velocity components at the stator exit must be studied. Therefore, different velocity components at stator exit (Station 2 in Fig.2.2 ) for model - 1 , model - 2, and model - 3 are shown in Fig .5.17 . Figure .5.17 - (a) shows the magnitude of the overall velocity (Velocity in Fig.2.2 ) , Figure .5.17 - (b) shows the magnitude of the circumferential velocity (Velocity in Fig.2.2 ), and Figure .5.17 - (c) shows the magnitude of the axial velocity (Velocit y in Fig.2.2 ) . The results show that model - and velocit ies are significantly lower than model - 2, , and the maximum difference is around 1 . However, model - velocity is noticeably higher than model - 2, difference is around 0.5 . This axial velocity difference is the main reason for the shaft power difference. As shown in Eqn 2 - 2 , the system mass flow rate depends on the axial vel ocity, and 85 large axial velocity means a large mass flow rate. As mentioned at the beginning of this section, the mass flow rate is one of the three factors that influence the turbine power, and a larger mass flow rate can lead to an increase in turbine sha ft power. Fig ure . 5 . 17 . Three different velocity distribution at the stator exit (runner - blade inlet) for all three selected models: (a) Absolute velocity (C 2 ), (b) C ircumferential v , (c) Axial velocity C x2 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.00 0.20 0.40 0.60 0.80 1.00 (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.00 0.20 0.40 0.60 0.80 1.00 (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 86 Figure. 5.1 7 The main function of the stator geometry is to guide the inlet velocity to a designed direction with the desired magnitude ; therefore , the nature of the stator is similar to a nuzzle . For a better understanding, the static pressure distribution plots at the 60% span location for all three selected models are shown in Fig .5.18 . The model - those geometrical features led to a poor velocity redirection. This poor velocity redirection has two significant consequences, inadequate velocity acceleration , and less velocity redirection angle. For a given head (which means fixed inlet pressure at stator inlet), based on the Bernoulli Equation, the inadequate velocity acceleration (which means low absolute velocity shown in Fig .5.17 - a ) results in high static pressure at the stator outlet, wh ich shown in Fig .5.18 . The less velocity redirection angle causes a small values showed in Fig .5.15 . Those two consequences reshape the velocity triangle at the stator outlet and inevitably affect the overall performance and power , and Fig .5.19 shows a comparison between the model - - stator outlet velocity triangle at 60% span location. Additional results and plots are shown in Appendix.F . 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 0.00 0.20 0.40 0.60 0.80 1.00 (c). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 87 (a). Model - 1 ( ) (b). Model - 2 ( ) . (c). Model - 3 ( ) Fig ure . 5 . 18 . Static pressure distributi on at 60% span location for all three selected models, (a) Model - 1, s =0.8 (b) Model - 2, s =1.6, (c) Model - 3, s =2.6 Fig ure . 5 . 19 . Velocity Triangle for Model - 1 , s =0.8 (Green Arrow), Model - 3 , s =2.6 (Black Arrow), and Designed value (Red Arrow) at 60% span location 88 Stator - blade thickness consideration As shown in Fig .3.18 , the stator - blade relative thickness is another para meter that influences the overall stator - blade geometry. Figure .5.20 shows how relative blade thickness (between 0.05 and 0.15) affects the overall performance and normalized shaft power ( The power results w ere normalized with ) for four flow rate conditions . Fig ure . 5 . 20 . The relation between stator - blade relative thickness with (a) overall hydraulic efficiency and (b) normalized shaft power for four designed flow rate conditions 0.85 0.86 0.87 0.88 0.89 0.9 0.05 0.07 0.09 0.11 0.13 0.15 Overall Hydraulic Efficiency (a). Stator Blade Relative Thickness (S T ) 0.9875 0.9925 0.9975 1.0025 1.0075 0.05 0.075 0.1 0.125 0.15 Normalized Shaft Power (b). Stator Blade Relative Thickness (S T ) 89 The results show that the increase of the stator - blade relative thickness can slightly improve the overall performance and decrease the shaft power, the maximum efficiency increase is around 1% , and the maximum shaft power decrease is around 1.5%. Unlike t he stator - blade solidity, the stator - blade relative thickness has a limited effect on overall performance; therefore, different thickness values can be applied to different flow and environmental conditions. Additionally, the small stator - blade relative th ickness can improve the overall performance when the stator - blade solidity is too large. Figure .5.21 shows the relation between overall performance , normalized power and stator - blade solidity under three stator - blade relative thickness values for one selected flow rate condition ( . The results show that when , there is a significant efficiency drop when is larger than two. This drop is caused by the increase of the blade thickness a nd chord length , which increases the stator blockage and friction loss. However, with smaller stator - blade relative thickness, the overall efficiency can be improved and maintain relatively constant when is larger (Red Arrow in Fig. 5.21 ) . For better u nderstanding, two models were selected for further investigation (Red Square s in Fig. 5.21 ). Figure. 5 . 21 . The relation between (a). Overall performance (b). Normalized Power, and stator - blade solidity und er three stator - blade relative thickness values for one flow rate condition (Q 11 =0.207) 0.86 0.87 0.88 0.89 0.9 0.91 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Overall Hydraulic Efficiency (a). Stator Blade Solidity Two Selected Models 90 Figure. 5. 21 Figure 5.22, 5.23 show the and values for the two selected models. Unlike in Fig. 5.14 and Fig.5.15 , t he two selected and values are relatively close to each other , with the maximum difference around 5 . Undoub ted ly, t his small difference can affect the overall performance ; however, it cannot result in a nearly 3% efficiency difference between the two selected models solely . Therefore, it is important to examine the stator closely. Stator geometry can affect the overall performance in two ways, first is changing the velocity and velocity triangle, which can influence the runner - blade performance ; second is increasing local stator loss to impact the overall performance. There are two main sources of loss for the stator, and one is the blade profile loss (surface friction), anther one is the blade blockage loss. By using the design method covered in section 3.2 , changing blade thickness has very little effect on the stator surface area, which means under the same solidity, the stator - blade profile loss is similar for the two selected models. Therefore, the main source for the 3% efficiency difference is from the stator blockage. For the two selected models, Fig. 5.24 shows the total press ure drop across all span location s at the stator exit (Station - 2 in Fig.2.2 ) ; and the pressure drop value is normalized by the inlet total pressure. 0.84 0.88 0.92 0.96 1 1.04 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized Power (b). Stator Blade Solidity 91 Fig ure . 5 . 22 . The runner - blade 2 ) distribution plots for the two selected models 2 value Fig ure . 5 . 23 . The runner - blade 2 flow angle distribution plots for the two selected models in 2 blade value 65 67 69 71 73 75 77 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Absolute Flow Angle ( degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 60 62 64 66 68 70 72 74 76 78 80 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Absolute Flow Angle ( Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 92 Fig ure . 5 . 24 . The normalized total pressure drop across all span locations for the two selected models at the stator exit The results show that except near the tip and hub region where the boundary layers are, the Model - 2 has a larger total pressure drop across all span locations than Model - 1, and the overall averaged stator total pressure drop ratios (normaliz ed by the inlet total pressure) are 5.34% and 2.74% for Model - 2 and Model - 1 respectively. The difference in total pressure drop ratio between the two selected models is around 2.6% , which is the main source for the 3% overall efficiency difference mentione d above. Some additional results and plots are shown in Appendix.G . 5.3. Runner - blade consideration Runner - blade stagger - angle setting constant consideration Just like the stator - blade stagger - angle setting constant, the runner - blade stagger - angle setting constant ( ) also is a critical geometrical parameter for the runner - blade profile. As shown in Fig.3.8(a) , this parameter can alter the blade shape dramatically, and Fig .5.2 5 shows how this parameter affects the overall performance under f our different flow rate conditions. For 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 Normalized Total Pressure drop Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 93 all flow rate conditions, the efficiency first increases with the increase of , then reach the maximum efficiency when is between 0.3 and 0.5, then the efficiency decreases significantly with the inc rease of the . Four models were selected (Red Square s in Fig. 5.2 5 ) for further investigation. As described in section 5.1 , the magnitude of the circumferential velocity difference (C u2 - C u3 ) represents how effective a runner - blade is converting the hydraulic power to mechanical power. So, Fi g. 5.26 shows the circumferential velocity difference distribution plots for all four selected models. Model - 2 has a higher (C u2 - C u3 ) value than the other three models across most of the span loc ation , which is reasonable for its highest efficiency. Model - 1 has a very similar trend as Model - 2, but with a small decrease in value near the tip region (Red Arrow in Fig.5.26 ) , which cause s a small efficiency drop compared to the Model - 2. Model - 3 and Mo del - 4 have significant ly lower (C u2 - C u3 ) values, especially in the 40% to 80% span location (Blue Arrow), which cause s them to have lower efficiency. Figure 5 . 25 . The Relation between overall hydraulic efficiency and runner - blade stagger - angle setting constant C rsa for four flow rate conditions. 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Overall Hydraulic Efficiency Runner Blade Stagger Angle Setting Constant (Crsa) Four Selected Models 94 Fig ure . 5 . 26 . The circumferential velocity difference (C u2 - C u3 ) distribution plots from hub to tip for four selected models. For better visualization, Fig . 5. 27 shows the stream - line pattern for the Model - 2 ( = 0.3) and Model - 4 ( = 0.8) at 80% span location. The Model - 2 ( Fig. 5.27 - a ) shows a great flow attachment, which gives it a better performance, and the Model - 4 ( Fig. 5.27 - b ) shows some flow distortion near the leading edge on the pressure side of the blade, which cause s the drawback of the efficiency. In comparison with that has a 12% maximum efficiency difference , the with a 9% maximum efficiency difference has less impact on the performance. However, this impact is still significant enough and need s to be carefully considered. Base on the overall results, initially, the sugge sting range for is between 0.2 to 0.3 . Some additional results and plots are shown in Appendix.H . 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 Circumferential Velocity Difference (C u2 - C u3 ) [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 Model-2 at 80% Span Location Model-4 at 80% Span Location 95 (a). Model - 2 at 80% span location (Blue square in Fig .5.26 ) (b). Model - 4 at 80% span location (Green dot in Fig .5.26 ) Fig ure . 5 . 27 . The runner blade stream - line pattern for the (a) Model - 2 , and (b) Model - 4 , at 80% span location Runner - blade configuration considerations Like the stator - blade configuration, the runner - blade number, solidity, and relative thickness are three critical geometrical parameters for configuring the runner - blade , and each parameter has a huge role in converting the hydraulic energy to mechanical energy effectively, which is the key of any hydraulic turbine system. Each parameter and how it impacts the overall performance are covered in this section. Runner - blade number consideration As shown in Fig.3.12 , t he runner - blade number is another critical parameter for SMH technology. As the proposed generat ion module has an adjustable blade mechanism for various flow conditions, less blade means a s mall and straightforward adjustable mechanism; this can help 96 reduce the cost and the complexity of the generation modules. Figure .5. 28 shows the normalized shaft power output (The power results w ere normalized with 8 - hydraulic performance of the turbine with the variation in the number of runner - blade s at four different flow conditions. Fig ure . 5 . 28 . The relation between runner - blade number, overall hydraulic efficiency, and normalized shaft power at four flow conditions The results show that, with the increase of runner - blade number s , the overall hydraulic efficiency can be improved, and the shaft power remains relatively constant. Generally, with more runner - blade s, the conversion between hydraulic power and shaft power is more effective, but at the same time, more runner - blade s mean reducing turbine spacing between blades which can have a negative effect on larger fish survival rate and increase blocking loss. The runner - blade number 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0.8 0.82 0.84 0.86 0.88 0.9 4 6 8 10 12 Normalized Shaft Power Overall Hyfraulic Efficiency Runner Blade Number Model - Efficiency Model - Efficiency Model - Efficiency Model - Efficiency Model - Normalized Power Model - Normalized Power Model - Normalized Power Model - Normalized Power 97 configuration needs to balance the overall performance, costs, complexity, and fish damage and f or this thesis , eight runner - blade was chosen as the reference configuration . Runner - blade solidity consideration Compared to the stator - blade solidity, the runner - blade solidity can have a more significant effect on the overall performance, design, and cost. As shown in Fig.3.14 , the runner - blade solidity can dramatically change the flow passage shape , therefore alter the overall flow behavior . Thus , Fig. 5.29 shows how the runner - blad e solidity influences the overall hydraulic performance and the shaft powers under four flow rate conditions ( The power results w ere normalized with ). The results in Fig.5.29 - ( a ) show that with an increase of the blade solidity, the overall efficiency first increases and reaches an optimum point, then decreases with the increase of the blade solidity ; the maximum efficiency difference is around 10% , and the optimum runner - blade solidity is between 0.9 and 1.2 . Additionally, b ecause of the runner - blade configuration, t he runner - blade solidity has more impact on performance at low mass flow rate condition s than the larger flow rate . Figure .5.30 shows comparisons of different runner - blade s geometry between small flow rate and larger flow rate conditions. At lower flow rate condition, the runner - blade angles ( , and in Fig. 2.2 ) are very close ; this results in a very small stagger - angle and narrower runner - blade spacing near the trail ing edge region (Red circle in Fig.5.30 - a ) . Larger blade solidity at lower flow rate can cause n arrow flow passage , which increases the blockage loss, therefore cause s a huge performance drop (Blue arrow in Fig.5.29 - a ). Moreover , at low flow rate conditio ns, larger runner - blade solidity can make the blade intersecting with each other and should be avoided. The results in Fig.5.29 - (b) show that , at low flow rate condition, when runner - blade solidity is larger than 0.8, with the increase of the runner - blade solidity, the 98 shaft power drop significantly, the maximum power difference is around 85% ; a dditionally, the power increase when runner - blade solidity is between 0.7 and 0.8 (Red arrow in F ig.5.29 - b ) . A t higher flow rate condition, the power remains relatively constant when runner - blade solidity is between 0.7 and 0.8 , and drop with a further increase of the solidity with the maximum efficiency around 20% . For better understanding, four mode ls under one flow conditions were chosen for further investigation (Red square s in Fig.5.29 ). Fig ure . 5 . 29 . The relation between runner - blade solidity , (a) overall hydraulic efficiency, and (b) normalized shaft power for four flow conditions 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Overall Hydraulic Efficiency (a). Runner Blade Solidity Four Selected Models 0.4 0.6 0.8 1 1.2 1.4 1.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Normalized Shaft Power (b). Runner Blade Solidity Four Selected Models 99 (a). Two runner - blade geometries at low flow rate condition ( ) with two blade solidity ; Left: , Right: . (b). Two runner - blade geometry at high flow rate condition ( ) with two blade solidity ; Left: , Right: . Fig ure . 5 . 30 . Two r unner - blade geometries at two flow rate condition s with two blade solidity values Figure 5.31 shows the relative flow angle at runner inlet and outlet for all four selected models ( in Fig.2.2 ). The results show , expect M odel - and all other three models have very similar and distribution - trend and the maximum angle difference is around 10 for and 5 for compared to the designed value (Except for some point s near the tip region). However, for model - 1, the and values are significantly lower than the designed value, and the maximum difference is around 35 for and 20 for . Figure 5.32 shows an example of the streamline pattern comparison of Model - 1 and Model - 2 at the 50% span location. Model - amline pattern shows good flow attachment near the runner - blade leading and trailing edge; on the contrary, the Model - runner - blade leading edge and insufficient flow turning near the trailing edge. Beca use of the lower blade solidity value, Model - 1 has a considerably shorter blade, and this causes a mismatch of the 100 inlet stagnation point and the runner leading - edge, hence cause the flow distortion. Moreover, the shorter blade configuration also causes th e misalignment between velocity direction and the blade direction at the runner - blade trailing edge which means poor flow turning performance. Fig ure . 5 . 31 . The relative flow angle at (a). runner inlet ( 2 ) and (b). runner outlet ( 3 ) for all four selected models with the comparison of the designed value 35 45 55 65 75 85 95 105 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Relative Flow Angle ( 2 ) [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Value 55 60 65 70 75 80 85 90 0 0.2 0.4 0.6 0.8 1 Runner Blade Oulet Relative Flow Angle ( 3 ) [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Value 101 (a). Model - 1 ( ) at 5 0 % span location (b). Model - 2 ( ) at 50 % span location Fig ure . 5 . 32 . The streamline pattern comparison of (a) Model - 1 and (b) Model - 2 at the 50% span location Figure 5.33 shows the runner - blade circumferential velocity difference ( ) distribution for all four models. Because of the above - mentioned reasons, the Model - 1 has the lowest values across the majority of the span location (Red Arrow in Fig. 5.33 ) ; hence it has the worst performance among all four models. And Model - 2 has the la rgest values across the majority of the span location ; hence it has the best performance. Model - 4, which has the largest blade solidity, has the second - worst performance among all can interfere with the Figure 5.34 shows the comparison between Model - 2 and Model - at the 50% span location. The red circle i ndicates the pressure interference, and this pressure interference can be shown more directly in a blade load ing plot , as in Figure 5.35 . 102 Fig ure . 5 . 33 . The runner - blade circumferential velocity difference ( C u2 - C u3 ) distribution for all four models (a). Model - 2 pressure distribution (b). Model - 4 pressure distribution Fig ure . 5 . 34 . T he comparison between (a). Model - 2 and (b). Model - the 50% span location 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Runner Blade Circumferential Velocity Difference (Cu 2 - Cu 3 ) [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Value 103 Fig ure . 5 . 35 . Normalized - p ressure distribution along the streamwise direction of the runner - blade for Model - 2 and Model - 4 at 50% span location For Model - 2, the normalized pressure distribution plot shows a smooth pressure distribution across the pressure and the suction side of the blade; however, for Model - 4, the normalized pressure distribution plot shows an irregular distribution trend, especially near the location when streamwise=0.75 ( Green Circle in Fig. 5.35 ) . This irr egular distribution leads to a velocity disturbance for Model - 4 , especially in the radial direction. As discussed in section 3.2.3 , one of the assumptions for the free vortex is zero radial velocity ( ) across all span location. Figure 5.36 shows the radial velocity distribution at runner - blade inlet and outlet for Model - 2 and Model - 4 . The results show that the Model - ly larger radial velocity at the runner inlet and outlet than the Model - velocity difference leads to a strong flow misalignment, thus results in worse p erformance for the Model - 4. Generally, under the same head condition, the runner - blade solidity can significantly influence the overall velocity triangle for the runner - blade , which affects the overall performance -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Pressure Streamwise Location (0 - 1) Model - 2 ( r=1.0) at 50% Span Location Model - 4 ( r=1.6) at 50% Span Location Irregular Pressure Distribution 104 and power production. Figure 5.37 shows t he runner - blade inlet and outlet velocity triangle for the four selected models and the designed values at 50% span location. Some additional results and plots are shown in Appendix.I . Fig ure . 5 . 36 . The radial velocity ( C r ) distribution at runner - blade (a). I nlet , and (b). O utlet for Model - 2 and Model - 4 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Radial Velocity (C r2 ) [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 2 ( r=1.0) Model - 4 ( r=1.6) -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 Runner Blade Oulet Radial Velocity (C r3 ) [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 2 ( r=1.0) Model - 4 ( r=1.6) 105 (a) . Runner - blade Inlet Velocity Triangle. (b) . Runner - blade Outlet Velocity Triangle. Fig ure . 5 . 37 . The runner - blade (a). I nlet , and (b). O utlet velocity triangle for the four selected models and the designed values at 50% span location . [ Green Arrow: Model - 1; Blue Arrow: Model - 2; Blac k Arrow: Model - 3; G ol d Arrow: Model - 4; Red Arrow: Designed Value ] Runner - blade thickness consideration As shown in Fig 3.17 , the runner - blade thickness can hugely impact the geometry of the runner - blade , and this impact is not only important for performance consideration, but also further deformation , stress analysis , and material selection . This section only stud ies how it influences the overall performance. For four flow rate conditions, F ig . 5.38 shows the overall hydraulic performance of the turbine and the normalized shaft power output (The power results w ere normalized with ) with the variation in the runner - blade thickness . For all four flow rate conditions, ini tially, the overall efficiency increases with the increase of the thickness, then reaches an optimum value before decreasing with the increase of the thickness. However, the relative thickness is more influential at a lower flow rate condition . When , the maximum efficiency difference is around 9%, with a 10% relative thickness increase; on the contrary, when , the maximum efficiency difference is only around 0.6%, with a 10% relative W 2 C 2 U 2 W 3 C 3 U 3 106 thickness increase. The logic behind this phenomes ar e : t he overall trend for the average ( r unner - blade inlet angle, in Fig.2.2 ) decreases with the increase of the flow rate based on the proposed design method , and a ccording to the Eqn 3 - 35 , a higher means lower value and a high er ( runner - blade maximum - blockage ratio) , this will increase the blockage loss by changing the overall flow behavior for lower flow rate condition s , hence cause the vast efficiency influence shown in Fig. 5.38 , and Fig . 5.39 shows how mean change wi th the runner - blade relative thickness for the four flow rate conditions. Moreover, for all four flow rate conditions, the power decrease with the increase of the runner - blade thickness. At the lowest flow rate condition ( ), the maximum difference is around 47% with a 10% relative thickness increase; at the highest flow rate condition ( ), the maximum difference is around 13% , with a 10% relative thickness in crease. F ig ure . 5.40 shows the overall hydraulic performance of the turbine and the normalized shaft power output with the variation in the mean runner - blade maximum - blockage . For a better understanding, three models were selected (Red squares in Fig . 5.38 ) for the lowest flow rate condition. Figure .5.41 shows the runner - blade inlet absolute flow angle ( ), and the runner - blade inlet relative flow angle ( ). The results show that, with the increase in the runner - blade relative thickness, there is a significant decrease, especially in the close to tip region (Red Arrow in Fig. 5.41 - a ). As discussed before, lower value can dramatically impact the overall performance. Figure .5.42 shows the axial velocity ( ), and radial velocity ( ) distribution for the three selected models at the runner - blade inlet. The results indicate that the drops significantly near the tip for larger thickness models; also, the increases significantly toward the tip region for the larger thickness model. 107 Figure. 5 . 38 . The relation between runner - blade relative thickness with (a). Overall hydraulic efficiency and (b). Normalized shaft power for four designed flow rate conditions 0.8 0.825 0.85 0.875 0.9 0.05 0.075 0.1 0.125 0.15 Overall Hydraulic Efficiency (a). Ruuner Blade Relative Thickness (R t ) Selected Data 0.7 0.8 0.9 1 1.1 1.2 0.05 0.075 0.1 0.125 0.15 Normalized Power (b). Ruuner Blade Relative Thickness (R t ) Selected Data 108 Fig ure . 5 . 39 . The relation between runner - blade relative thickness and runner - blade maximum - blockage ratio for four flow conditions Fig ure . 5 . 40 . The relation between mean runner - blade maximum - blockage ratio with (a). O verall hydraulic efficiency and (b). N ormalized shaft power for four designed flow rate conditions. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.05 0.07 0.09 0.11 0.13 0.15 Mean Runner Blade maximum blockage ratio ( RM ) Runner Blade Relative Thickness (R t ) 0.8 0.82 0.84 0.86 0.88 0.9 0 0.2 0.4 0.6 0.8 1 Overall Hydraulic Efficiency RM ) 109 Fig ure .5.40 Although the Fig .5.39 and Fig.5.40 shows the mean runner - blade maximum - blockage for all three models, and this mean value use the mean span values; however, since near tip the local value is larger, which causes a larger local blockage near the tip region. This effect is more noticeable at lower flow rate condition, and Fig. 5.43 shows the overall distribution from tip to hub for the four flow rate conditions at 10% relative thickness settings. At a lower flow rate condition, near the tip, the blockage ratio becomes noticeably larger. This large blockage can reduce the axial velocity, and increase the radial velocity which causes a lower value , and overall performance. 0.7 0.8 0.9 1 1.1 1.2 0 0.2 0.4 0.6 0.8 1 Normalized Power RM ) 110 Fig ure . 5 . 41 . (a). Runner - blade inlet absolute flow angle ( 2 ) and (b). runner - blade inlet relative flow angle ( 2 ) distribution across all span locations for the three selected models. -60 -40 -20 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Absolute Flow Angle ( 2 ) [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - Design Value 75.0 77.0 79.0 81.0 83.0 85.0 87.0 89.0 91.0 93.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Relative Flow Angle ( 2 ) [ degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - Design Value 111 Fig ure . 5 . 42 . (a). The runner - blade inlet axial velocity (C x2 ), and (b).The runner - blade inlet radial velocity (C r2 ) distribution across all span locations for the three selected models 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity (Cx 2 ) [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity (Cr 2 ) [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - 112 Fig ure . 5 . 43 . The RM distribution from tip to hub for the four flow rate conditions at 10% relative thickness settings Additionally, the small runner - blade thickness can improve the overall performance and power generation for the larger solidity condition. Figure.5.44 show s the relation between overall performance, normalized power ( The power results w ere normalized with and ), and the runner - blade solidity with three different runner - blade thickness values under one selected flow rate conditi on ( , and Fig .5.45 shows the same results but in term s of the mean runner blockage ratio. The results show that, at lower solidity value, larger blade thickness has a better overall performance and power generation, the optimum solidity value fo r all three thickness conditions is around . However, with the increase of the runner - blade solidity, the lower thickness models show a significant performance and power generation advantage (Red Arrow in Fig. 5.44 - a ); the maximum efficiency and power difference is around 20% and 50%, respectively. For better 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.2 0.4 0.6 0.8 1.0 RM ) Normalized Span Location ('0' is Hub, '1' is Tip) 113 understanding, two pair models were selected: lower runner - blade solidity models (Green Dots in Fig. 5.44 ), higher runner - blade solidity models (Red Squares in Fig. 5.44 ). Figure .5.46 shows the str eamline pattern for the s elected Models - II (Low solidity) at three different span locations (10%, 50%, and 90%). Figure. 5 . 44 . The relation between runner - blade solidity with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickness conditions at one flow rate condition (Q 11= 0.207) 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.65 0.85 1.05 1.25 1.45 1.65 Overall Hydraulic Efficiency (a). Runner Blade Solidity ( R ) Selected Models I (High Solidity) Selected Models II (Low Solidity) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.65 0.85 1.05 1.25 1.45 1.65 Normalized Power (b). Runner Blade Solidity ( R ) Selected Models I (High Solidity) Selected Models II (Low Solidity) 114 Figure. 5 . 45 . The relation between the mean runner - blade RM ) with (a). Overall hydraulic efficiency and (b). Normalized shaft power for three runner - blade thickness conditions at one flow rate condition (Q 11 =0.207) 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Overall Hydraulic Efficiency (a). RM ) Maxmimum Power Line Maxmimum Efficiency Line 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Normalized Power (b). RM ) Maxmimum Power Line Maxmimum Efficiency Line 115 A ll those plots show that, although both low and high runner - blade thickness models have flow distortion near the leading edge for all selected span locations (Red circle in Fig .5.46 ), the low runner - blade thickness models have a sever flow separation on th e suction side of the blade, especially approaching to the tip region (Black circle in Fig .5.46 ), which do not present in large thickness models. This sever separation is the main reason, at lower solidity condition, the smaller thickness models have a sig nificantly lower performance. Figure .5.47 shows the pressure distribution for the Selected Models - I (High solidity) at three different span locations (10%, 50%, and 90%). The plots show that a larger blade solidity model but with smaller thickness has an o verall better pressure distribution pattern in both suction side and pressure side of the blade; on the contrary, the one with larger thickness has severe pressure interference between adjacent blades which is similar with Fig.5.34 . Figure. 5.48 shows the n ormalized pressure distribution plot for selected Models - I at the 50% span location. The plot shows both models have a pressure interference (Red Circles in Fig. 5.47 , green circles in Fig.5.48 ); for the smaller thickness model, this interference happens around 87% streamwise location and only decrease the pressure on the pressure side surface by a small amount (Blue arrow in Fig. 5.48 ). However larger blade thickness model shows a more profound influence, because of the larger blockage ratio, there is a steep drop of pressure on the interference near 75% streamwise location which induces a significant pre ssure dropping on the pressure side surface and lead to a significant pressure rise on the suction side surface (Red arrow in Fig. 5.48 ). This severe pressure interference (or alternation) is the main reason for the lower performance and power of the larger thickness model at larger solidity conditions. Some additional results and plots are shown in Appendix.J . 116 (a). Streamline pattern at 10% span location, . (b). Streamline pattern at 50% span location, . (c). Streamline pattern at 90% span location, . (d). Streamline pattern at 10% span location, . (e). Streamline pattern at 50% span location, . (f). Streamline pattern at 90% span location, . Fig ure . 5 . 46 . S treamline pattern for the Selected Models - II (Low solidity) at three different span locations, (a) - (c): Small runner - blade thickness , R T= 0.05 ; (d) - (e): Large runner - blade thickness , R T =0.15 117 (a). Pressure Distribution at 10% span location, . (b). Pressure Distribution at 50% span location, . (c). Pressure Distribution at 90% span location, . (a). Pressure Distribution at 10% span location, . (b). Pressure Distribution at 50% span location, . (c). Pressure Distribution at 90% span location, . Fig ure . 5 . 47 . Pressure Distribution Pattern for the Selected Models I ( High solidity) at three different span locations, (a) - (c): Small runner - blade thickness, R T =0.05 ; (d) - (e): Large runner - blade thickness, R T =0.15 118 Fig ure . 5 . 48 . Normalized Pressure distribution along the streamwise direction of the runner - blade for the selected Models I (High solidity) at 50% span location 5.4. Off - design consideration As mentioned in section 1.3.1 , an essential characteristic of the new stream - reach site is its high variability in flow and head. This high variability means any turbine system proposed for the SMH resource has to equip with different regulation methods. Conventionally, t here are three major regulation methods for a Kaplan turbine system: variable stator pitch, variable runner pitch, variable rotational speed. The variable stator pitch method is normally equipped by some large Kaplan turbine system, which has less stator n umber ( see in Fig.1.10 ). Since the proposed system has m ore stator number , the stator regulation method can significantly increase the complexity of the system and is not utilized in the proposed system. Variable runner pitch and variable rotational speed are the two major methods for the proposed system. -1.5 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Normalized Pressure Streamwise Location (0 - 1) Model - I ( Model - II ( 119 This section focuses on how different runner - blade geometrical configurations affect the overall off - design performance under the variable runner - blade pitch and variable rotation speed condition. One desi gn point was chosen as the base condition, and three different geometrical parameters were studied: Runner - blade thickness, Runner - blade solidity, and Runner - blade number. Table. 5 . 1 shows the base design - configurations. For all four models, three rotational speeds were numerical tested, =30Rpm, =40Rpm (designed condition), and =50Rpm; together with seven head conditions, H=1.5m, H=2.0m, H=2.5m (designed condi tion), H=3.0m, H=3.5m, H=7.5m, H=10m; and with nine different runner - blade pitch angles between 8 and 32 . All those data were used to construct the performance map , and Fig . 5. 49 - 5. 51 shows the Q 11 and N 11 based performance map with the blade opening angl e line for all four models under three different rotational speed conditions. Table. 5 . 1 . The design condition s and geometrical configuration s for four selected models General Design Point Condition s Design ed Head ( ) 2.5m Design ed unit flow rate ( ) 0.22 Design ed rotational speed ( ) 88.543 Design ed runner - blade pitch angle at the hub location ( ) 16 Design ed Power Output ( ) 83 k w Four selected runner - blade configuration Model - 1 Model - 2 Model - 3 Model - 4 Runner - blade Number 8 8 8 6 Runner - blade Relative Thickness 0.1 0. 1 0. 075 0.1 Runner - blade Solidity 1 1.1 1 1 120 Figure. 5 . 49 . N 11 and Q 11 based performance map at the design rotational speed ( =40Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 121 Figure. 5. 49 122 Figure. 5 . 50 . N 11 and Q 11 based performance map at the lower rotational speed ( =30Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4. 123 Figure. 5. 50 124 Figure. 5 . 51 . N 11 and Q 11 based performance map at the higher rotational speed ( =50Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4. 125 Figure. 5. 51 126 Maximum performance ( max ) considerations - At design ed rotational speed ( Rpm , Fig. 5. 49 ), the results show that all four models have similar values and the maximum performance difference between the four models is around 0.54%. However , the locations of the maximum efficiency are slightly different . Table .5 . 2 summaries the maximum efficiency and its locations for all four selected models. Table. 5 . 2 . Maximum efficiency values and its location for all selected models under the design rotational speed Model - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 89.48% 89.99% 90.02% 89.83% N 11 Location 95.78 95.59 99.28 97.21 Q 11 Location 0.211 0.212 0.2 0.205 Blade pitch angle Location ( 15.25 16.4 13.9 14.75 - At lower rotational speed ( Rpm, Fig. 5. 50 ), the results show that all four models also have similar values , the maximum performance difference between the four models is around 0.4%. However, the locations of the maximum efficiency are slightly different. Table .5 . 3 summaries the maximum efficiency and its locations for all four selected models. Table. 5 . 3 . Maximum efficiency values and its location for all selected models under the lower rotational speed Model - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 88.45% 88.84% 88.44% 88.62% N 11 Location 86.01 85.22 88.45 86.83 Q 11 Location 0.215 0.217 0.209 0.21 Blade pitch angle Location ( 15.9 17 14.8 15.5 - At higher rotational speed ( Rpm, Fig. 5. 51 ), the results show that all four models also have similar values , the maximum performance difference between the four models is 127 around 0.56%. However, the locations of the maximum efficiency are slightly different. Table .5 . 4 summaries the maximum efficiency and its locations for all four selected models. Table. 5 . 4 . Maximum efficiency values and its location for all selected models under the higher rotational speed Mod el - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 88.94% 89.40% 89.50% 89.32% N 11 Location 99.6 97.07 100 100 Q 11 Location 0.211 0.213 0.202 0.209 Blade pitch angle Location ( 15 16.3 14 14.9 All those results show that various runner - blade geometrical settings (runner thickness, solidity, and blade number) has minimal effect on the maximum efficiency values; however, the locations of the maximum efficiency are different. For model - 2 which with has a longer runner - blade profile ; this leads to the more significant blockage at lower value (according to Eqn - 3 - 35 ) . Thus, for model - 2, the maximum efficiency occurs at a slightly higher location with a higher flow rate (compared to the based model, Model - 1) . In comparison, for model - 3 that with smaller thickness, the maximum efficiency occurs at a slightly lower location with a lower flow rate (compared to the based model, Model - 1). Additionally, with a lesser blad e count, the model - 4 has very similar performance compared to the Model - 1 . The general trends for all selected models are: at lower head condition, a lower rotational speed can provide a higher performance; at higher head condition, a high rotational speed can provide higher performance . A nd with 25% difference, the maximum efficiency difference is around 1%. All those data show variable rotational speed has a very limited impact on the maximum efficiency and the overall performance. 128 Off - design operating range considerations The previous section focuses on the ma ximum efficiency for all four selected models , but for off - design consideration, the operating range is the critical concern regarding the system 's overall performance. The above figures show the and based performance map ; here are the head and power - based performance map s ( Fig. 5.52 - 5.54 ) for the four selected models. Figure. 5 . 52 . Power and head based performance map at the design rotational speed ( =40Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 129 Figure. 5 . 53 . Power and head based performance map at the lower rotational speed ( =30Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 130 Figure. 5 . 54 . Power and head based performance map at the higher rotational speed ( =50Rpm) for four selected models, (a). Model - 1; (b). Model - 2; (c). Model - 3; (d). Model - 4 - Design rotational speed At the designed rotational speed ( Rpm, Fig. 5.52 ), Table.5 . 5 summaries the head and power operating range with the corresponding performance , and for better visualization , Fig.5. 56,57 shows the simplify ing operating range (Head and Power) for all four models . 131 Table. 5 . 5 . Operating range and the corresponding performance f or all selected models under the design rotational speed Model - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 89.48% 89.99% 90.02% 89.83% location >88% Performance Range > 85% Performance Range > 80%Performance Range > 75% Performance Range (a). Power Range 88% Efficiency [kw] (b). Power Range 85% Efficiency [kw] (c). Power Range 80% Efficiency [kw] (d). Power Range 75% Efficiency [kw] Figure. 5 . 55 . Simplif ied power operating range for all four models under the design rotational speed 30 50 70 90 110 130 Model 1 Model 2 Model 3 Model 4 25 65 105 145 185 225 Model 1 Model 2 Model 3 Model 4 0 100 200 300 400 Model 1 Model 2 Model 3 Model 4 0 100 200 300 400 500 600 Model 1 Model 2 Model 3 Model 4 132 (a). Head Range 88% Efficiency [m] (b). Head Range 85% Efficiency [m] (c). Head Range 80% Efficiency [m] (d). Head Range 75% Efficiency [m] Figure. 5 . 56 . Simplif ied head operating range for all four models under the design rotational speed - Lower rotational speed At the lower rotational speed ( Rpm, Fig. 5.53 ), Table.5 . 6 summaries the head and power operating range with the corresponding performance, and for better visualization , Fig.5. 57,58 shows the simplify ing operating range (Head and Power) for all four models . Table. 5 . 6 . Operating range and the corresponding performance for all selected models under the lower rotational speed Model - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 88.45% 88.84% 88.44% 88.62% >88% Performance Range 1.4 1.8 2.2 2.6 3 Model 1 Model 2 Model 3 Model 4 1.1 1.9 2.7 3.5 4.3 Model 1 Model 2 Model 3 Model 4 0.9 1.9 2.9 3.9 4.9 5.9 Model 1 Model 2 Model 3 Model 4 0.5 2.5 4.5 6.5 8.5 Model 1 Model 2 Model 3 Model 4 133 Table. 5.6 >85% Performance Range >80%Performance Range >75% Performance Range (a). Power Range 88% Efficiency [kw] (b). Power Range 85% Efficiency [kw] (c). Power Range 80% Efficiency [kw] (d). Power Range 75% Efficiency [kw] Figure. 5 . 57 . Simplif ied power operating range for all four models under the lower rotational speed 25 35 45 55 65 Model 1 Model 2 Model 3 Model 4 15 40 65 90 Model 1 Model 2 Model 3 Model 4 5 55 105 155 Model 1 Model 2 Model 3 Model 4 0 50 100 150 200 250 Model 1 Model 2 Model 3 Model 4 134 (a). Head Range 88% Efficiency [m] (b). Head Range 85% Efficiency [m] (c). Head Range 80% Efficiency [m] (d). Head Range 75% Efficiency [m] Figure. 5 . 58 . Simplif ied head operating range for all four models under the lower rotational speed - Higher rotational speed At the higher rotational speed ( Rpm, Fig. 5.54 ), Table.5 . 7 summaries the head and power operating range with the corresponding performance, and for better visualization , Fig.5. 59,60 shows the simplify ing operating range (Head and Power) for all four models . Table. 5 . 7 . Operating range and the corresponding performance for all selected models under the higher rotational speed Model - 1 Model - 2 Model - 3 Model - 4 Maximum efficiency 88.94% 89.40% 89.50% 89.32% >88% Performance Range 1.1 1.3 1.5 1.7 1.9 Model 1 Model 2 Model 3 Model 4 0.8 1.1 1.4 1.7 2 2.3 2.6 Model 1 Model 2 Model 3 Model 4 0.5 1 1.5 2 2.5 3 3.5 Model 1 Model 2 Model 3 Model 4 0 1 2 3 4 5 Model 1 Model 2 Model 3 Model 4 135 Table. 5. 7 >85% Performance Range >80%Performance Range >75% Performance Range (a). Power Range 88% Efficiency [kw] (b). Power Range 85% Efficiency [kw] (c). Power Range 80% Efficiency [kw] (d). Power Range 75% Efficiency [kw] Figure. 5 . 59 . Simplif ied power operating range for all four models under the higher rotational speed 50 100 150 200 Model 1 Model 2 Model 3 Model 4 30 130 230 330 430 Model 1 Model 2 Model 3 Model 4 0 200 400 600 800 Model 1 Model 2 Model 3 Model 4 0 400 800 1200 1600 Model 1 Model 2 Model 3 Model 4 136 (a). Head Range 88% Efficiency [m] (b). Head Range 85% Efficiency [m] (c). Head Range 80% Efficiency [m] (d). Head Range 75% Efficiency [m] Figure. 5 . 60 . Simplified head operating range for all four models under the higher rotational speed The above results show , by using the proposed design methodology, all selected models have significant large operating range s; however, each model has its unique features. Under all rotational speed condition s , due to the larger blade solidity, the m odel - 2 has the overall best o perating range. The larger blade solidity gives the model - 2 a longer blade, which enable s a better flow redirection, especially at larger blade opening angle , flow rate , and head condition s , therefore in Fig.5.55 (a) - (c), the model - 2 has the largest upper bound among all four models. However , when the flow rate is exceedingly high, longer blades can cause high blade profile loss , and this explains in Fig.5.55 (d) , model - 2 has a slightly lower upper bound than the model - 4. On the other hand, the model - 3, which has a thinner blade profile, has the worst operating range. Due to the thinner blade profile, the model - 3 cannot provide enough flow redirection at larger flow conditions ; this limits the upper bound of operating rang e for model - 3 which precisely 2 2.5 3 3.5 4 4.5 5 Model 1 Model 2 Model 3 Model 4 1.5 2.5 3.5 4.5 5.5 6.5 Model 1 Model 2 Model 3 Model 4 1 3 5 7 9 11 Model 1 Model 2 Model 3 Model 4 1 3 5 7 9 11 13 15 17 Model 1 Model 2 Model 3 Model 4 137 shown in Fig.5.55 (a) - (d) . Although model - 3 has the best lower bound of operating range, due to the low blockage ratio at the small blade pitch opening angle, this slim advantage is neglectable compared to the huge drawback at the upper bound . Model - 4, which has the second - best operating range, compared to the model - 1 has lesser blade count which cause less blade loss at higher flow rate condition ; thus the model - 4 has a significant ly better operating range than the model - 1 . Th is section demonstrate s the proposed system had a vast operating range potential . A nd based on different flow conditions and requirements, different blade settings and configurations can be applied to achieve the desired operating range. 138 6. CHAPTER VI : CONCL USIONS AND FUTURE WORKS 6.1. Summary of the SMH technology Low - head hydropower has the potential to generate a significant amount of electricity from rivers that traditionally were unsuitable for developing hydraulic power plant s and supporting the resiliency of the U.S electricity system. The development of those resources could be possible only if the technologies for low - head hydropower that balance efficiency, economics, and environmental sustainability were developed. T he traditional hydrop ower design method was limited to the new challenges of the Low - head application. Therefore, a Standard Modular Hydropower Technology (SMH) was proposed by the U.S. Department of Energy (DOE) in 2017. This new concept offers a new paradigm for small hydrop ower technology development based on the premise that standardization, modularity, and preservation of stream functionality must become essential and fully realized features of next - generation hydropower technologies and project designs [ 39 ]. This technolo gy has three primary modules: generation module, passage modules, foundation modules . This thesis developed a new design methodology for configuring a n SMH generation module, and with the affluent amount of numerical simulation results, this design method ology has high - level flexibility that can be optimized and configurated according to different river conditions. 6.2. Summary of the design methodology The first step of the design process is to determine the overall size and operating conditions. Due to the un conventional design , the pro posed generation module has a completely different approach regarding the overall size, positioning, operating condition, and blade configuration. The 139 proposed generation module is a damless Kaplan turbine system, and it uses its structure to provide the necessary low - head condition. This unconv entional concept means the overall turbine size and positioning angle are critical to the system which are extensively studied in the thesis. Additionally, in order to achieve high - level flexibility for various application s , a new blade design method, fiv e - point Bezier curve method, was created here for configuring the stator and runner - blade profile for different operating conditions. This method allows the blade profile being further optimized for different scenarios. Ten different geometrical parameters were thoroughly studied by numerical simulations , includin g general hub size, stator - blade stagger - angle setting constant, stator - blade inlet angle, stator - blade number, stator - blade solidity, stator - blade thickness, runner - blade stagger - angle setting con stant, runner - blade number, runner - blade solidity, runner - blade thickness . 6.3. Summary of the numerical results Detailed numerical simulations were conducted for studying how each proposed geometrical parameter affects the overall performance. - General hub size At low flow rate condition s , the smaller hub configuration tends to have worse performance on the close to tip region, which leads to poor overall performance. At a larger flow rate condition , the smaller hub configuration ha s a significantly better o verall performance at a larger flow rate condition. Moreover , choosing the right hub size is about balancing the design flow rate and the required hub volume. At a high flow rate, the smaller hub can provide a n excellent overall hydraulic performance but with limited hub volume that can have high over - heating possibi lities for the generator, which can further affect the overall electrical performance. Balancing the hub size and performance is the key to the initial turbine unite sizing. 140 - Stator - blade stagger - angle setting constant In general, the stator - blade stagger - a ngle setting constant ( ) value has a larger impact on runner - blade inlet absolute flow angle ( ) values than runner - blade inlet relative flow angle ( ) values which ha ve a profound influence on stator outlet flow behaviors, and can further affe ct the overall performance. Base d on the overall results, initially, the suggesting range for is between 0.7 to 0.8 . - Stator - blade inlet angle In general, larger stator - blade inlet angle ( ) can cause a counter - clockwise vortex pattern on the stator - blade suction side , where close to the trailing edge which distorts the downstream flow; small stator - blade inlet angle ( ) can cause a clockwise vortex pattern on the stator - blade pressure side t hat close to the leading edge , which improves the downstream flow. The suggested stator inlet angle is between to . - Stator - blade number The results show that, among all selected four flow rate conditions, the maximum performance difference is only 0 .8%, the maximum shaft power difference is only 2 .33 %. This relatively small effect means the stator number has a minimal effect on overall hydraulic performance, and a different number of stator - blade s can be chosen for various river conditions. - Stator - bl ade solidity A small solidity stator - blade has a shorter and thinner blade profile , which led to a poor velocity redirection and acceleration with less velocity redirection angle. Those two consequences reshape the velocity triangle at the stator outlet and affect the overall performance and power of the turbine. Based on the results, the optimum stator - blade solidity value is between 1.2 - 1.8 depend s on flow rate conditions. 141 - Stator - blade thickness The results show that the increase of the stator - blade relative thickness can slightly improve the overall performance and decrease the shaft power, the maximum efficiency increase is around 1%, and the max imum shaft power decrease is around 1.5%. Additionally, by decreasing the stator local blockage loss, the small stator - blade relative thickness can improve the overall performance when the stator - blade solidity is too large. - Runner - blade stagger - angle sett ing constant The results show that larger runner - blade stagger - angle setting constant ( ) causes some flow distortion near the leading edge on the pressure side of the blade, which lead to low er performance; and small has an excellent overa ll flow attachment , which leads to better performance . The maximum efficiency differences between different models are around 9% and based on the results the optimum value for is between 0.2 to 0.3. - Runner - blade number The results show th at, with the increase of runner - blade number s , the overall hydraulic efficiency can be improved, and the shaft power remains relatively constant. Generally, with more runner - blade s, the conversion between hydraulic power and shaft power is more effective, but at the same time, more runner - blade s mean reducing turbine spacing between blades , which can ha rm larger fish survival rate and increase blocking loss. The runner - blade number configuration needs to balance the overall performance, costs, complexity, a nd fish damage. - Runner - blade solidity The lower blade solidity value means a considerably shorter blade and causes a mismatch of the inlet stagnation point and the runner leading - edge, hence the flow distortion. Moreover, the shorter blade configuration also causes the misalignment between velocity direction and the blade 142 direction at the runner - blade trailing edge , which means poor flow turning performance. The large blade solidity means a lo nger blade which . T his interference changes the pressure distribution of the blade and further affects the overall performance. Generally, under the same head condition , the runner - blade solidity can significantly influence the overall velocity triangle for the runner - blade , which affects the overall performance and power production . The optimum runner - blade solidity is between 0.9 and 1.2. - Runner - blade thickness. The ru nner - blade thickness has more influence at low flow rate condition than high flow rate condition, at , the maximum efficiency difference is around 9%, with a 10% relative thickness increase . O n the contrary, when , the maximum efficienc y difference is only around 0.6%, with a 10% relative thickness increase. Additionally, the power decrease with the increase of the runner - blade thickness. At the lowest flow rate condition ( ), the maximum power difference is around 47% with a 10 % relative thickness increase; at the highest flow rate condition ( ), the maximum difference is around 13% , with a 10% relative thickness increase. Moreover, the small runner - blade thickness can improve the overall performance and power generatio n at larger solidity condition s . With the increase of the runner - blade solidity, the lower thickness models show significant ly better performance and power generation, the maximum efficiency and power difference is around 20% and 50%, respectively. - Off des ign considerations Since the new stream - reach sites have high variability in flow and head, the off - design and the operating range is critical to study. For the proposed system, variable blade pitch opening and 143 variable rotational speed are the two main methods for regulating the off - design condition s . In order to demonstrate the off - design potential for the proposed system, one design condition was chosen , and four models with different runner - blade configurations were studied. With numerous simulation results, the operating ranges for all four models were calculated . The results show that the larger blade solidity model has the best overall operating rang e ; the thinner blade model has the worst overall operating range. However, all four models have a vast operating range, and the maximum operating range is between 23% and 1220% with over 75% efficiency. Those results demonstrate that the proposed system ha d a vast operating range potential . A nd based on different flow conditions and requirements, different blade settings and configurations can be applied to achieve the desired operating range. 6.4. Future works suggestions This thesis focusses on the relation be tween the selected geometrical configuration parameters and the overall performance . F uture works can emphasize structure deformation and blade material selection. Also, the stator and runner - blade interaction during the off - design condition are also neede d to study . Environmental studies, such as fish strike simulation, sediment transport simulation, are also in need to complete the overall system design. Finally, a scaled experimental model is required for validating all numerical and theoretical assumpt ions and results. To b uil d a lab - scale testing unit, two major similarity laws must be used to e n sure the scaling up performance: Kinematic Similitude and Reynolds Similitude . - Kinematic Similitude: The kinematic similarity between scaled model and full - sca le prototype turbine can be achieved by matching the Froude number: ( 6 - 1 ) w here, = Average Inflow velocity; =Designed Head; =Turbine Diameter. 144 This number will ensure the free surface flow condition is the same between the prototype turbine and the scaled model . - Reynolds Similitude: The Reynolds Similitude between scaled model and full - scale prototype turbine can be achieved by matching the normalized flow r ate and speed : ( 6 - 2 ) Those two number s will ensure the prototype performance testing results can be applied to the scaled model. The challenge for experimental testing the proposed SMH generation module is to balance the size and flow rate. By following the two similari t y laws, Table. 6.1 shows a geometrical and operational conditions comparison between a real - size model and a 2/7 scale model. As shown, the experimental challenge is to balance the scale model size and water usage, which is the key to future experimental development s . Table 6 . 1 . Geometrical and operational conditions comparison between a real - size model and a 2/7 scale model Real Size Model 2/7 Scale Model D tip [m] 5.00 1.43 D hub [m] 3.50 1.00 Head [m] 3.50 1.00 Volumetric Flow Rate [m 3 /s] 14.93 0.65 State Inlet S peed [m/s] 1.49 0.80 Power [kW] 450.20 5.61 RPM [Rpm] 40.00 74.83 Fr number 0.065 0.065 N 11 106.90 106.90 Q 11 0.32 0.32 145 APPENDI CES 146 APPENDIX A: - friendly hydraulic turbine Table Appendix . 1 . criteria for designing a fish - friendly hydraulic turbine [50] Criteria Description Value Chosen Reasoning Fish - friendly turbine runner A new runner design Hydraulic design parameter Flow=1,000 ft 3 /s (28.3m 3 /s) Head=75 ft to 100 ft (23 - 30m) A r epresentative of most hydroelectric turbines installed in the U.S., including Kaplan and Francis Tube turbines. Turbine operating efficiency 85% minimum (3 - D calculations included scroll case and draft tube) Efficiency for most peaks at 90% to 93%. 85% was chosen , so the new runner can be competitive with existing designs. Peripheral runner speed Less than 40 ft/sec (preferably 20 ft/sec) Reduces strikes injury and minimizes shear stresses and vortices between moving and statio nary parts Minimum pressure 10 P sia (68.8 kPa) Downstream migrating fish are typically found within the top 34 ft, i.e., at 30 P sia (206 kPa), and mortality occurs when the pressure drop is more than 30% of acclimation pressure. Rate of change of pressure Less than 80 psi/sec (550.3 kPa/sec) Assuming fish injury occurs at a pressure rate of 160 psia/sec in Kaplan turbines. 147 Table Appendix.1 Shear stress indicator (Rate of Strain, du/dy) Less than 15 ft/sec/in (180ft/sec/ft or 180 m/sec/m) Tests of alewives, a fragile fish, at ARL with 15 ft/sec/in did not cause injury The n umber and the total length of leading blade edges Minimize Fewer blades and shorter leading edges reduce the probability of the strike Clearance between runner and fixed turbine housing components 2 mm or less Small clearances reduce the possibility of mechanical injury. 2 mm is less than the 3 mm gap chosen by t he USACE for testing in a Kaplan turbine. Flow passage Sizes Maximize Large amounts of water between blades should reduce abrasion injury by keeping fish away from the blades Flow control and plant configuration (Not tested for during this phase of the AHTS project) Maximize distance between runner and wicket gates and minimize travel time from intake to runner Kaplan turbines are more fish - friendly than Francis turbines. A small distance between wicket gates and the runner in Francis turbines may increa se the chance for abrasion and grinding injury. 148 APPENDIX B: Initial design parameters with description and suggested setting values Table Appendix . 2 . Initial design parameters with description and recommend setting values Design Parameters Symbol Descriptions Recommend values General geometry parameters General Inclined angle The angle between the turbine structure and the inlet flow. This would affect turbine structure length and stator geometry. Between 30 50 Turbine Tip diameter D tip The turbine general diameter. Tip diameter set as equal to turbine minimum structure length Turbine Hub diameter D hub The turbine hub diameter. No Recommended value, depend ing on flow rate and generator design. General operation parameter Rotational speed The turbine design rotational speed. Must keep turbine peripheral speed lower than 12.24m/s. (preferably 6.12m/s) Initial guess design efficiency The turbine assumed design efficiency for initial velocity calculation. Kaplan turbine has a general efficiency of 86%~90%. General Blade Geometry setting Runner and stator - blade thickness distribution equation y(t) Determine the thickness distribution along the blade camber line. NACA 4 - series distribution function. 149 Table Appendix.2 Design Parameters Symbol Descriptions Recommend values Runner - blade geometry Parameters Runner - blade number N R The turbine runner - blade number. Between 4~12. 8 - blade is the reference setting value Runner - blade Stagger - angle setting constant C rsa Determine the stagger - angle of the runner - blade at each span location. Recommend C rsa =0.2~0.5 Runner - blade control point coefficient C r1 ; C r2 Control the runner - blade camber line shape Recommend Value=0.8 Runner - blade relative maximum thickness R T / R t Runner - blade relative maximum thickness. R T =0.1~0.18 Runner - blade Solidity The ratio between runner - blade chord length and spacing. Recommend Value Stator - blade geometry parameters Stator - blade number N S The turbine stator number Between 20~60, 40 - blade is the reference setting value Stator - blade Stagger - angle setting constant C ssa Determine the stagger - angle of the stator - blade at each span location Recommend C ssa =0.7~0.8 Stator - blade control point coefficient C s1 ; C s2 Control the stator - blade camber line shape Recommend Value=0.8 150 Table Appendix.2 Design Parameters Symbol Descriptions Recommend values Stator - blade geometry parameters Stator - blade relative maximum thickness S T /S t Stator - blade relative maximum thickness S T =0.1~0.18 Stator - blade Solidity The ratio between stator - blade chord length and spacing Recommend Value Stator - blade inlet angle The stator - blade inlet setting angle Recommend Value 151 APPENDIX C: Additional results and plots for various hub size consideration s. Fig ure. Appendix C . 1 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 152 Fig ure . Appendix C . 2 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flow rate condition Q 11 =0. 103 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 153 Fig ure . Appendix C . 3 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0.31 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 -0.29 -0.24 -0.19 -0.14 -0.09 -0.04 0.01 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 154 Fig ure . Appendix C . 4 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flow rate condition Q 11 =0. 103 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3[m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 155 Fig ure . Appendix C . 5 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0. 31, with comp aris on to the designed values [The designed value for 3 is across all span locations , which is not shown.] 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Absolute Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 -50 -40 -30 -20 -10 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 156 Fig ure . Appendix C . 6 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flow rate condition Q 11 =0. 103, with comp aris on to the designed values [The designed value for 3 is across all span locations, which is not shown.] 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Absolute Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 -80 -60 -40 -20 0 20 40 60 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 157 Fig ure . Appendix C . 7 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at high flow rate condition Q 11 =0. 31, with comp aris on to the designed values 35 45 55 65 75 85 95 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 65.00 70.00 75.00 80.00 85.00 90.00 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 158 Fig ure . Appendix C . 8 . Runne r - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the five different hub sizes at low flow rate condition Q 11 =0. 103, with comp aris on to the designed values 65.00 70.00 75.00 80.00 85.00 90.00 95.00 100.00 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 75.00 80.00 85.00 90.00 95.00 100.00 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Hub-to-Tip Ratio=0.685 Hub-to-Tip Ratio=0.714 Hub-to-Tip Ratio=0.743 Hub-to-Tip Ratio=0.771 Hub-to-Tip Ratio=0.8 Designed Value Hub-to-Tip Ratio=0.685 Designed Value Hub-to-Tip Ratio=0.714 Designed Value Hub-to-Tip Ratio=0.743 Designed Value Hub-to-Tip Ratio=0.771 Designed Value Hub-to-Tip Ratio=0.8 159 APPENDIX D: Additional results and plots for stator C ssa constant considerations . Fig ure. Appendix D . 9 . The r elation between normalized power and stator - blade stagger - angle setting constant C ssa for four flow rate conditions. (The power is norm a lized with C ssa = 0.7 models) Fig ure . Appendix D . 10 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the four selected models 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Power Stator Blade Stagger Angle Setting Constant (C ssa ) -100 -80 -60 -40 -20 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 160 Fig ure . Appendix D . 11 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 161 Fig ure . Appendix D . 12 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 -0.25 -0.15 -0.05 0.05 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 162 Fig ure . Appendix D . 13 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the four selected model s 76 78 80 82 84 86 88 90 92 94 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Relative Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Cssa=0.1 Model-2 Cssa=0.3 Model-3 Cssa=0.5 Model-4 Cssa=0.8 163 APPENDIX E: Additional results and plots for stator - blade inlet angle considerations . Fig ure . Appendix E . 14 . The Relation between the overall hydraulic efficiency, Power Generation, and the Stator - blade 1 blade ) for three general - inclined angle configurations under the same design flow rate condition (Q 11 =0. 103 ) Fig ure . Appendix E . 15 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the four selected models 0.9 0.92 0.94 0.96 0.98 1 1.02 0.86 0.87 0.88 0.89 0.9 40 60 80 100 120 140 Normalized Power Overall Hydraulic Efficiency Stator Blade Inlet Angle 1 blade [Degree] 30° General Inclined Angle (Efficiency) 40° General Inclined Angle (Efficiency) 50° General Inclined Angle (Efficiency) 30° General Inclined Angle (Power) 40° General Inclined Angle (Power) 50° General Inclined Angle (Power) -60 -40 -20 0 20 40 60 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle 3 [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 164 Fig ure . Appendix E . 16 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 165 Fig ure . Appendix E . 17 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 166 Fig ure . Appendix E . 18 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the four selected models 70 72 74 76 78 80 82 84 86 88 90 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Relative Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( - Blade=150 ) Model - 2 ( - Blade=120 ) Model - 3 ( - Blade=80 ) Model - 4 ( - Blade=45 ) 167 APPENDIX F: Additional results and plots for stator - blade solidity considerations. Fig ure . Appendix F . 19 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the three selected models Fig ure . Appendix F . 20 . Runner - blade Axial velocity (C x ) distribution at Runner Outlet , for the three selected models -60 -40 -20 0 20 40 60 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 168 Fig ure . Appendix F . 21 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet ; for the three selected models -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 169 Fig ure . Appendix F . 22 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the three selected models 74 76 78 80 82 84 86 88 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Relative Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( Model - 2 ( Model - 3 ( 170 APPENDIX G : Additional results and plot s for stator - blade thickness considerations. Fig ure . Appendix G . 23 . The relation between (a). overall performance, (b). Normalized Power, and stator - blade solidity under three stator - blade re lative thickness values for one flow rate condition (Q 11 =0.207), and 20 stator configuration 0.86 0.87 0.88 0.89 0.9 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Overall Hydraulic Efficiency (a). Stator Blade Solidity ( S ) 0.85 0.89 0.93 0.97 1.01 1.05 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized Power (b). Stator Blade Solidity ( S ) 171 Fig ure . Appendix G . 24 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two selected models 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 172 Fig ure . Appendix G . 25 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Out let; for the two selected models -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - -0.25 -0.15 -0.05 0.05 0.15 0.25 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 173 Fig ure . Appendix G . 26 . Runner - blade Outlet absolute velocity angle ( 3 ) distribution for the two selected models. Fig ure . Appendix G . 27 . Runner - blade Outlet relative velocity angle ( 3 ) distribution for the two selected model s -60 -50 -40 -30 -20 -10 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 70 75 80 85 90 95 100 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Relative Velocity Angle [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 174 APPENDIX H: Additional results and plots for runner - blade C rsa constant considerations. Fig ure. Appendix H . 28 . The r elation between normalized power and runner - blade stagger - angle setting constant for four flow rate conditions. (The power is normalized with C rsa =0.3 models) 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized Shaft Power Runner Blade Stagger Angle Setting Constant (Crsa) 175 Fig ure . Appendix H . 29 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 176 Fig ure . Appendix H . 30 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 177 Fig ure . Appendix H . 31 . Runner - blade absolute velocity angle ( ) distribution at (a). Runner Inlet ( 2 ); (b). Runner Outlet ( 3 ), for the four selected models 0 10 20 30 40 50 60 70 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Absolute Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 -80 -60 -40 -20 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 178 Fig ure . Appendix H . 32 . Runner - blade relative velocity angle ( ) distribution at (a). Runner Inlet ( 2 ); (b). Runner Outlet ( 3 ), for the four selected models 50 55 60 65 70 75 80 85 90 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 70 75 80 85 90 95 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Relative Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model-1 Crsa=0.1 Model-2 Crsa=0.3 Model-3 Crsa=0.7 Model-4 Crsa=0.8 179 APPENDIX I : Additional results and plots for runner - blade solidity considerations Fig ure . Appendix I . 33 . Runner - blade absolute velocity angle ( 3 ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Absolute Flow Angle ( 2 ) [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Values -90 -70 -50 -30 -10 10 30 50 0 0.2 0.4 0.6 0.8 1 Runner Blade Oulet Absolute Flow Angle ( 3 ) [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) 180 Fig ure . Appendix I . 34 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the four selected models -0.2 0.3 0.8 1.3 1.8 0 0.2 0.4 0.6 0.8 1 Runner Blade Inlet Axial Velocity (Cx 2 ) [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Value 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Runner Blade Oulet Axial Velocity (Cx 3 ) [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - 1 ( r=0.7) Model - 2 ( r=1.0) Model - 3 ( r=1.2) Model - 4 ( r=1.6) Designed Value 181 APPENDIX J : Additional results and plots for runner - blade thicknes s considerations Fig ure . Appendix J. 35 . The circumferential velocity difference (C u2 - C u3 ) distribution plots from hub to tip for the three selected models Fig ure . Appendix J. 36 . The axial velocity distribution (C x3 ) at runner outlet for the three selected models 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Circumferential Velocity Difference (Cu2 - Cu3) [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity (Cx 3 ) [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - 182 Fig ure . Appendix J. 37 . The radial velocity distribution (C r3 ) at runner outlet for the three selected models Fig ure . Appendix J. 38 . The runner outlet absolute flow angle ( 3 ) distribution at runner outlet for the three selected model s -0.2 -0.2 -0.1 -0.1 0.0 0.1 0.1 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity (Cr 3 ) [m/s] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - -80 -60 -40 -20 0 20 40 60 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Absolute Flow Angle ( 3 ) [degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - 183 Fig ure . Appendix J. 39 . The runner outlet relative flow angle ( 3 ) distribution at runner outlet for the three selected models 81 82 83 84 85 86 87 88 89 90 91 0.0 0.2 0.4 0.6 0.8 1.0 Runner Blade Inlet Relative Flow Angle ( 3 ) [ degree] Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Model - Design Value 184 Fig ure. Appendix J . 40 . Runner - blade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Axial Velocity Cx 2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Runner Outlet Axial Velocity Cx 3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 185 Fig ure . Appendix J . 41 . Runner - bl ade Axial velocity (C x ) distribution at (a). Runner Inlet; (b). Runner Outlet ; for the two different runner - blade thickness models at High Solidity Condition r =1.6 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Runner Inlet Axial Velocity Cx 2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Axial Velocity Cx 3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 186 Fig ure . Appendix J . 42 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet ; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr 2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Runner Outlet Radial Velocity Cr 3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 187 Fig ure . Appendix J . 43 . Runner - blade Radial velocity (C r ) distribution at (a). Runner Inlet; (b). Runner Outlet ; for the two different runner - blade thickness models at High Solidity Condition r =1.6 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Radial Velocity Cr 2 [m/s] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Radial Velocity Cr 3 [m/s] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 188 Fig ure . Appendix J . 44 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 , w ith comp aris on to the designed values [The designed value for is across all span locations, which is not shown.] 0 10 20 30 40 50 60 70 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Absolute Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value -60 -50 -40 -30 -20 -10 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 189 Fig ure . Appendix J . 45 . Runner - blade Absolute velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at High Solidity Condition r =1.6 , with comp aris on to the designed values [The designed value for is across all span locations, which is not shown.] -110 -90 -70 -50 -30 -10 10 30 50 70 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Absolute Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value -75 -55 -35 -15 5 25 0.0 0.2 0.4 0.6 0.8 1.0 Runner Outlet Absolute Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - 190 Fig ure . Appendix J . 46 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at Low Solidity Condition r =0.7 , with comp aris on to the designed values 30 35 40 45 50 55 60 65 70 75 80 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value 45 50 55 60 65 70 75 80 85 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value 191 Fig ure . Appendix J . 47 . Runner - blade Relative velocity angle ( ) distribution at (a). Runner Inlet; (b). Runner Outlet; for the two different runner - blade thickness models at High Solidity Condition r =1.6 , with comp aris on to the designed values 60 70 80 90 100 110 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (a). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value 77 79 81 83 85 87 89 91 93 0.0 0.2 0.4 0.6 0.8 1.0 Runner Inlet Relative Velocity Angle [degree] (b). Normalized Span Location ('0' is Hub, '1' is Tip) Model - Model - Designed Value 192 Fig ure . Appendix J. 48 . The relation between runner - blade solidity with (a). O verall hydraulic efficiency and (b). N ormalized shaft power for three runner - blade thickness conditions at one flow rate condition ( Q 11 =0.207 ), 6 - blades config u ration 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.7 0.9 1.1 1.3 1.5 Overall Hydraulic Efficiency (a). Runner Blade Solidity ( R ) 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0.7 0.9 1.1 1.3 1.5 Normalized Power (b). Runner Blade Solidity ( R ) 193 Fig ure . Appendix J. 49 . The relation between runner - blade solidity with (a). O verall hydraulic efficiency and (b). N ormalized shaft power for three runner - blade thickness conditions at one flow rate condition ( Q 11 =0.207 ), 10 - blades config u ration 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.7 0.9 1.1 1.3 1.5 Overall Hydraulic Efficiency (a). Runner Blade Solidity ( R ) 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0.7 0.9 1.1 1.3 1.5 Normalized Power (b). Runner Blade Solidity ( R ) 194 Fig ure . Appendix J . 50 . The relation between runner - blade relative thickness with the O verall hydraulic efficiency for all three runner - blade number conditions at one solidity condition ( r =1 ) 0.86 0.865 0.87 0.875 0.88 0.885 0.89 0.895 0.05 0.075 0.1 0.125 0.15 Overall Hydraulic Efficiency Ruuner Blade Relative Thickness (R T ) 6-Blade Configuration 8-Blade Configuration 10-Blade Configuration 195 BIBLIOGRAPHY 196 BIBLIOGRAPHY [1] Dixon, S. Larry, and Cesare Hall. Fluid mechanics and thermodynamics of turbomachinery. Butterworth - Heinemann, 2013. [2] Williamson, S. J., B. H. Stark, and J. D. Booker. "Low head pico hydro turbine selection using a multi - criteria analysis." Renewable Energy 61 (2014): 43 - 50. [3] World Energy Council. Survey of Energy Resources, 19th Ed. London, UK: World Energy Council, Regen cy House; 2001. [4] Cleveland, Cutler J., and Christopher G. Morris. Handbook of energy: chronologies, top ten lists, and word clouds. Elsevier ( 2013 ): 79 102. [5] Elbatran, A. H., et al. "Operation, performance and economic analysis of low head micro - hydropower t urbines for rural and remote areas: A review." Renewable and Sustainable Energy Reviews 43 (2015): 40 - 50. [6] Mohibullah, Mohd Amran, Mohd Radzi, Mohd Iqbal, Abdul Hakim. Basic design aspects of micro - hydro power plant and its potential development in Malaysia . In: Proceedings of the national power and energy conference 2004, PECON, IEEE 0 - 7803 - 8724 - 4; 29 30 November 2004. [7] Dilip Singh. Micro hydropower resource assessment handbook. Prepared for APCTT Asian and Pacific Centre for Transfer of Technology of the Un ited Nations Economic and Social Commission for Asia and the Pacific (ESCAP), September 2009. [8] Teuteberg BH. Design of a pump - as - turbine for an abalone farm, final report for mechanical project 878. Department of Mechanical and Mechatronic Engineering, St ellenbosch University; March 2010. [9] Yaakob, O. B., et al. "A review on micro hydro gravitational vortex power and turbine systems." Jurnal Teknologi 69.7 (2014): 1 - 7. [10] Paish, Oliver. "Micro - hydropower: status and prospects." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 216.1 (2002): 31 - 40. [11] Derakhshan, Shahram, and Ahmad Nourbakhsh. "Experimental study of characteristic curves of centrifugal pumps working as turbines in different specific speeds." Experimental the rmal and fluid science 32.3 (2008): 800 - 807. 197 [12] Laghari, J. A., et al. "A comprehensive overview of new designs in the hydraulic, electrical equipments and controllers of mini hydro power plants making it cost effective technology." Renewable and S ustainable Energy reviews 20 (2013): 279 - 293. [13] Paish, Oliver. "Small hydropower: technology and current status." Renewable and sustainable energy reviews 6.6 (2002): 537 - 556. [14] Wallace, A. R., and H. W. Whittington. "Performance prediction of standardized impulse turbin es for micro - hydro." Sutton., Int. Water Power & Dam Construction (2008). [15] Energy systems and design Ltd. http://www.microhydropower.com/our - pro ducts/stream - engine/ . [ A ccessed 08/21/ 13]. [16] Williamson, S. J., B. H. Stark, and J. D. Booker. "Performance of a low - head pico - hydro Turgo turbine." Applied Energy 102 (2013): 1114 - 1126. [17] Williamson, S. J., B. H. Stark, and J. D. Booker. "Experimental optimization of a low - head pico hydro Turgo turbine." 2012 IEEE Third International Conference on Sustainable Energy Technologies (ICSET). IEEE, 2012. [18] Voith hydro Pelton turbines http://voith.com/en/products - services/hydropower/turbines/ P elton - turbines - 563.html . [ A ccessed 08/10/ 13]. [19] U.S. Department of Energy, Energy efficiency and renewable energy, Small hydropower systems, FS217; July 2001. [20] Ghosh, Tushar K., and Mark A. Prelas. Energy resources and systems: volume 2: renewable resources. Vol. 2. Springer Science & Business Media, 2011. [21] Os sberger GmbH Co. The Ossberger turbine. Bayern, Germany. http://www. ossberger.de/cms/en/hydro/the - ossberger - turbine - for - asynchronous - andsynchronous - water - plants/ ; 2011. [ A ccessed 01/20/ 17]. [22] Jiménez, Edy E. "Final study report, achievable renewable energ y targets for Puerto Rico's renewable energy portfolio standard. Puerto Rico: Puerto Rico's energy affairs administration; 2009." [23] The encyclopedia of alternative energy and sustainable living. . [ Accessed 1/24/17 ] [24] Voith Hydro Pelton turbines. < https://www.voith.com/ca - en/Voith_Pelton_turbines.pdf > . [Accessed 1/22/20] 198 [25] Small - hydropower - systems DOE/GO - 102001 - 1173, FS217, 2001 . < https://www.nrel.gov/docs/fy01osti/29065.pdf>. [Accessed 1/22/20] [26] Ossberger GmbH Co. (2011) The Ossberger turbine. Bayern, Germany. . [ Accessed 1/24 /17 ] [27] Miller, Gabriel, et al. "A study of an axial - flow turbine for kinetic hydropower generation." Energy 12.2 (1987): 155 - 162. [28] Voith - Siemens (2011) Francis turbines, Hydropower generation, Voith - Siemens, Heidenheim. < http://www.voithhydro.com/media/t3339e Francis72dpi.pdf >. [ Accessed 1/25/17 ] [29] Free Flow Power Corporation (2008) Hydrokinetics. < http://www.free - flow - power.com/index.php?id=10 > . [ Accessed 1/25/17 ]. [30] Western renewable energy. < http://www.westernrenew.co.uk/wre/hydro_ basics/machines/archimedes_screw_turbines > [ A ccessed 01 / 25 / 17]. [31] Landustries Landy hydropower screw . < http://www.landustrie.nl/fileadmin/ files/Folders/Landy%20hydropower%20screws.pdf > . [ A ccessed 01 / 25 / 17]. [32] Saroinson g, Tineke, et al. "The effect of head inflow and turbine axis angle towards the three - row bladed screw turbine efficiency." International Journal of Applied Engineering Research 10.7 (2015): 16977 - 16984. [33] Bulb/pit/S - turbines, Voith - Siemens hydropower genera tion . < http://voith.com/corp - en/BulbPitS - Turbines_Generators.pdf>. [ A ccessed 01/22/20]. [34] Kydd P (2009) Severn tidal power, Overview and implications of tidal power technologies, Severn tidal power feasibility study [35] Hydroelectric design center, Portland dist rict, US Army Corps of Engineers. https://www. nwp.usace.army.mil/HDC/edu genexcit.asp. [ A ccessed 1/26/17 ] [36] Gordon, J. L., and P. Eng. "Turbine selection for small low - head hydro developments." Proc. Waterpower XII, Buffalo (2003). [37] Loots, I., et al. "A review of low head hydropower technologies and applications in a South African context." Renewable and Sustainable Energy Reviews 50 (2015): 1254 - 1268. 199 [38] < https: //www.energy.gov /eere /water/downloads/hydropower - vision - r eport - full - report >. [Accessed 1/22/20] [39] Witt, Adam, et al. "Exemplary Design Envelope Specification for Standard Modular Hydropower Technology: Draft for Public Comment." , 2016. [40] Brown Kinloch, David. Demonstration of variable speed permanent magnet generator at small, low - head hydro site. No. Final Report: DOE - WEISENBERGER - EE0005429 - 1. Weisenberger Mills, Inc., Midway, KY (United States), 2015. [41] Johnson, Megan, Rocio Uria - Martinez, and Patrick O'Connor. 2016 Hydropower Market Report Update Metadata. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States), 2017 . [42] Lewis, R. I. Turbomachinery performance analysis. Butterworth - Heinemann, 1996. [43] Albuquerque, R. B. F., N. Manzanares - Filho, and W. Oliveira. "Conceptual optimization of axial - flow hydraulic turbines with non - free vortex design." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 221.5 (2007): 713 - 725. [44] Horlock, John Haro ld. Axial flow turbines: fluid mechanics and thermodynamics. Krieger Pub Co, 1973. [45] vanes and runner of the Kaplan turbine." Journal of Hydraulic Research 55.3 (2017): 349 - 361. [46] Zhou, Daqing, and Zhiqun Daniel Deng. "Ultra - low - head hydroelectric technology: A review." Renewable and Sustainable Energy Reviews 78 (2017): 23 - 30. [47] Gordon, J. L., and P. Eng. "Turbine selection for small low - head hydro developments." Proc. Waterpow er XII, Buffalo(2003). [48] Korpela, Seppo A. Principles of turbomachinery. Hoboken, New Jersey: Wiley, 2011. [49] De Siervo, F., and F. De Leva. "Modern trends in selecting and designing Kaplan turbines." Water power and dam construction 30.1 (1978): 28 - 35. [50] Odeh, Mufeed. "A summary of environmentally friendly turbine design concepts." US Department of Energy Idaho Operations Office (1999). 200 [51] Muis, Abdul, and Priyono Sutikno. "Design and simulation of very low head axial hydraulic turbine with variation of swirl velocity criterion." International Journal of Fluid Machinery and Systems7.2 (2014): 68 - 79. [52] Ferro, L. M. C., L. M. C. Gato, and A. F. O. Falcão. "Design of the rotor blades of a mini hydraulic bulb - turbine." Renewable Energy 36.9 (2011): 2395 - 2403. [53] Sutikn o, Priyono, and Ibrahim Khalil Adam. "Design, simulation and experimental of the very low head turbine with minimum pressure and free vortex criterions." International Journal of Mechanical and Mechatronics Engineering 11.1 (2011): 9 - 16. [54] Anagnostopoulos, J ohn S., and Dimitrios E. Papantonis. "Flow modeling and runner design optimization in Turgo water turbines." World Academy of Science, Engineering and Technology 28 (2007): 206 - 211. [55] Höfler, Edvard, Janez Gale, and Anton Bergant. "Hydraulic design and analy sis of the Saxo - type vertical axial turbine." Transactions of the Canadian Society for Mechanical Engineering 35.1 (2011): 119 - 143. [56] Schobeiri, Meinhard. Turbomachinery flow physics and dynamic performance. Heidelberg: Springer, 2005. [57] Mu, Jie Gang, et al. "Response of Blade Thickness to Hydraulic Performance of Stamping and Welding Multistage Centrifugal Pump." Applied Mechanics and Materials. Vol. 229. Trans Tech Publications Ltd, 2012. [58] Shigemitsu, Toru, Junichiro Fukutomi, and Kensuke Kaji. "Influence of blade outlet angle and blade thickness on performance and internal flow conditions of mini centrifugal pump." International Journal of Fluid Machinery and Systems 4.3 (2011): 317 - 323. [59] Tao, Yi, et al. "Influence of blade thickness on transient flow characteristics of centrifugal slurry pump with semi - open impeller." Chinese Journal of Mechanical Engineering 29.6 (2016): 1209 - 1217. [60] Kim, Seung - Jun, et al. "Effect of blade thickness on the hydraulic performance of a Francis hydro turbine model." R enewable energy 134 (2019): 807 - 817. [61] Tucker, P. G. "Computation of unsteady turbomachinery flows: Part 1 Progress and challenges." Progress in Aerospace Sciences 47.7 (2011): 522 - 545. [62] Ansys Inc ( 2016). 201 [63] Kalitzin, Georgi, et al. "Near - wall behavior of RANS turbulence models and implications for wall functions." Journal of Computational Physics 204.1 (2005): 265 - 291. [64] Menter , Florian R. "Two - equation eddy - viscosity turbulence models for engineering applications." AIAA J ournal 32.8 (1994): 1598 - 1605. [65] Jones, W. P., and BEi Launder. "The prediction of laminarization with a two - equation model of turbulence." International journal of heat and mass transfer 15.2 (1972): 301 - 314. [66] ANSYS, Release. "Documentation for ANSYS, SAS IP." (18) [67] Adhikari, Ram. Design improvement of crossflow hydro turbine. Diss. University of Calgary, 2016.