SIMULATION OF VAWT AND HYDROKINETIC TURBINES WITH VARIABLE PITCH FOILS by Lindsay Damon Woods A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Boise State University May 2013 © 2013 Lindsay Damon Woods ALL RIGHTS RESERVED BOISE STATE UNIVERSITY GRADUATE COLLEGE DEFENSE COMMITTEE AND FINAL READING APPROVALS of the thesis submitted by Lindsay Damon Woods Thesis Title: Predicting Turbine Performance with a Dynamic Computer Model Date of Final Oral Examination: 12 March 2013 The following individuals read and discussed the thesis submitted by student Lindsay Damon Woods, and they evaluated his presentation and response to questions during the final oral examination. They found that the student passed the final oral examination. John F. Gardner, Ph.D. Chair, Supervisory Committee James R. Ferguson, Ph.D. Member, Supervisory Committee Jairo E. Hernandez, Ph.D. Member, Supervisory Committee The final reading approval of the thesis was granted by John F. Gardner, Ph.D., Chair of the Supervisory Committee. The thesis was approved for the Graduate College by John R. Pelton, Ph.D., Dean of the Graduate College. ACKNOWLEDGEMENTS This thesis was funded through the generous support of Idaho National Laboratory through the Center for Advanced Energy Studies. Without the support of Kurt Myers, Dr. Gardner, and the LDRD team, this thesis would not have been possible. I would like to thank Max Badesheim and Kate Huebschmann at the writing center for your patience in helping me find a way through all the words and figures to a coherent thesis and for making this report ten times better than it would have been otherwise. Thank you Dr. Hernandez for your work on the committee and providing me with your insight and comments. Thank you Dr. Ferguson for asking the hard questions that helped me refine this thesis and for informing me of more turbine research to be done in Switzerland. It was a wonderful opportunity that I would not have had were it not for you. For all the other grad students in room ME 314: Mike, Paul, Derek, Dan, and Hope, thank you for answering dozens of questions and turning a dreary lab at BSU into a friendly place where I could laugh and learn while I worked. Thank you to my parents for their unwavering support and overwhelming love and encouragement. Lastly, thank you Dr. Gardner for the amazing gift of this opportunity. Thank you for your abiding commitment to always do the right and sensible thing, even if it doesn’t win you praise and recognition. When we first met, I could have gone in any number of different directions in engineering. Thank you for your inspiration to pursue energy efficiency and to look at every solution with a fair but critical eye. You’ve changed my life. iv ABSTRACT A dynamic computer model of a turbine was developed in MATLAB in order to study the behavior of vertical axis wind and hydrokinetic turbines with articulating foils. The simulation results corroborated the findings of several empirical studies on various turbines in both wind and water currents. The model was used to analyze theories of pitch articulation and to inform the discussion on turbine design. Several new patents and proposed configurations were tested. Simulation results showed that pitch articulation allowed Darrieus-style vertical axis wind turbines to start from rest. The tip speed ratio was found to increase rapidly, carrying the turbine into very fast rotational velocities. The simulations revealed a region of high efficiency for wind turbines at high rates of rotation and demonstrated the advantages of using a dynamic generator load. The model was also used to study the behavior of hydrokinetic turbines in restricted environments like irrigation canals – a situation where the Betz analysis is not suitable. Further study showed that when a turbine is inserted in a channel, the resulting blockage causes the development of potential energy in the form of hydraulic head upstream of the turbine. The model was used to predict the efficiency of hydrokinetic turbines in this situation and the water level rise that would occur upstream. v TABLE OF CONTENTS ACKNOWLEDGEMENTS ..................................................................................................... iv ABSTRACT .............................................................................................................................. v LIST OF TABLES ................................................................................................................... ix LIST OF FIGURES .................................................................................................................. x LIST OF ABBREVIATIONS ................................................................................................ xiii CHAPTER ONE: INTRODUCTION ....................................................................................... 1 The Scope of the Study ................................................................................................. 1 CHAPTER TWO: BACKGROUND ....................................................................................... 6 Early Examination of the Topic .................................................................................... 6 The Betz Theory of Efficiency ................................................................................... 16 Articulating Foils ........................................................................................................ 19 Turbine Modeling ....................................................................................................... 25 CHAPTER THREE: METHODOLOGY ............................................................................... 27 The Structure of the Model ......................................................................................... 27 Defining the Geometry and the Initial Conditions ...................................................... 30 Calculating the Lift and Drag at Each Foil ................................................................. 33 Calculating Torque...................................................................................................... 36 CHAPTER FOUR: MATLAB MODEL ................................................................................ 43 vi Developing the Model for a VAWT ........................................................................... 43 Model Verification ...................................................................................................... 52 CHAPTER FIVE: RESULTS ................................................................................................. 62 Using the Model to Inform Design Decisions ............................................................ 62 Ideal Blade Articulation .............................................................................................. 63 Ideal Generator Loading ............................................................................................. 66 CHAPTER SIX: HYDROKINETIC TURBINES .................................................................. 70 Differences between Wind and Water Turbine Behavior ........................................... 70 Feedback Loops: Power Production, Position Changes, and Induction ..................... 76 Determining Water level Rise ..................................................................................... 77 Adapting the Model for Hydrokinetic Turbines ......................................................... 78 CHAPTER SEVEN: CONCLUSION .................................................................................... 85 Rotational Behavior .................................................................................................... 85 Articulation ................................................................................................................. 86 Generator Loading ...................................................................................................... 87 Hydroturbines ............................................................................................................. 88 Future Work ................................................................................................................ 89 Summary ..................................................................................................................... 89 REFERENCES ....................................................................................................................... 91 APPENDIX A ......................................................................................................................... 96 Geometric Transformation by Matrices ...................................................................... 96 APPENDIX B ....................................................................................................................... 100 MATLAB Code and Simulink Figures ..................................................................... 100 vii APPENDIX C ....................................................................................................................... 104 Additional Tables and Figures .................................................................................. 104 viii LIST OF TABLES Table 1 Outputs of MATLAB model simulations for sinusoidal pitch articulation and low resistance ................................................................ 109 Table 2 Outputs of MATLAB model simulations for sinusoidal