EML 4905 Senior Design Project A SENIOR DESIGN PROJECT PREPARED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING
Solar Stirling Engine for Remote Power and Disaster Relief Final Report Denisse Aranda Kevin LaMott Stephen Wood
Advisor: Professor Yong Tao
April 5, 2010
This report is written in partial fulfillment of the requirements in EML 4905. The contents represent the opinion of the authors and not the Department of Mechanical and Materials Engineering.
Ethics Statement and Signatures
The work submitted in this project is solely prepared by a team consisting of Denisse Aranda, Kevin LaMott, and Stephen Wood and it is original. Excerpts from others‟ work have been clearly identified, their work acknowledged within the text and listed in the list of references. All of the engineering drawings, computer programs, formulations, design work, prototype development and testing reported in this document are also original and prepared by the same team of students.
Denisse Aranda
Kevin LaMott
Stephen Wood
Team Leader
Team Member
Team Member
Dr. Yong Tao Faculty Advisor
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Table of Contents Introduction ....................................................................................................................... 14 Problem Statement ........................................................................................................ 14 Motivation ..................................................................................................................... 14 Justification ................................................................................................................... 14 Disastrous Events and their Location Across the Globe ........................................................ 15 Ideal Locations for Solar Energy Power Generation ............................................................. 17
Literature Survey .......................................................................................................... 18 History ................................................................................................................................... 18 Stirling Engine Configurations .............................................................................................. 21 Solar Radiation....................................................................................................................... 22 Solar Concentrator ................................................................................................................. 24 Solar Stirling Engine .............................................................................................................. 25 Solar Tracking........................................................................................................................ 27
Discussion ..................................................................................................................... 29 Project Formulation .......................................................................................................... 30 Overview ....................................................................................................................... 30 Project Objectives ......................................................................................................... 30 Design Specifications.................................................................................................... 30 Constraints and Other Considerations .......................................................................... 31 Discussion ..................................................................................................................... 32 Design Alternatives ........................................................................................................... 33 Overview of Conceptual Designs Developed ............................................................... 33 Solar Stirling Trade Studies .......................................................................................... 34 Types of Solar Energy Conversion ........................................................................................ 34 Types of Stirling Engine Configurations ............................................................................... 35 Types of Solar Concentrators................................................................................................. 36
Conceptual Design ........................................................................................................ 38 Feasibility Assessment .................................................................................................. 39 Proposed Design ............................................................................................................... 40 Collector ........................................................................................................................ 40 3|Page
Tracking Mechanism ............................................................................................................. 42 Collector Parasitic Loss ......................................................................................................... 42
Initial Engine Design .................................................................................................... 43 Interim Engine Design .................................................................................................. 44 Final Engine Design ...................................................................................................... 45 Geometry of Heater Volume .................................................................................................. 46 Geometry of Expansion Volume............................................................................................ 46 Geometry of Regenerator Volume ......................................................................................... 46 Geometry of Cooler Volume ................................................................................................. 46 Geometry of Compression Volume ....................................................................................... 46 Design of Black Hole ............................................................................................................. 46 Design of Crankshaft ............................................................................................................. 47 Design of Rods....................................................................................................................... 47 Design of alternator................................................................................................................ 47 Operating pressure ................................................................................................................. 47 Working Fluid ........................................................................................................................ 47 Mass of Working fluid ........................................................................................................... 47 Operating Temperatures......................................................................................................... 47 CAD Rendering of Engine ..................................................................................................... 48 Kinematic Analysis and Animation ....................................................................................... 48 Cooling Reservoir .................................................................................................................. 49
Engineering Design and Analysis ..................................................................................... 50 Calculating Energy from Sunlight ................................................................................ 50 Analysis of Solar Collector ........................................................................................... 51 Analysis of Cooling Reservoir Size .............................................................................. 52 Calculation of Time of Local Sunrise and Sunset for Autonomous Tracking Capabilities ................................................................................................................... 53 Calculation of Time of Local Sunrise .................................................................................... 53 Calculation of Time of Local Sunset ..................................................................................... 54
Engine Adiabatic Analysis ............................................................................................ 56 Nomenclature ......................................................................................................................... 56 Background ............................................................................................................................ 57 Development of Equation Set ................................................................................................ 58
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Adiabatic Stirling Engine Model Set of Differential and Algebraic Equations ..................... 62 Solution .................................................................................................................................. 63 Implementation of Developed Model .................................................................................... 64 Calculation of Operating Frequency ...................................................................................... 68 Discussion .............................................................................................................................. 68
FVM Isothermal Analysis ............................................................................................. 69 Isothermal Transient Startup Simulation Results ......................................................... 72 Engine Geometry Optimization 1: Isothermal Analysis ............................................... 77 Initial Design:......................................................................................................................... 78 Intermediate Designs: ............................................................................................................ 78 Optimized Design: ................................................................................................................. 80
FVM Adiabatic Analysis .............................................................................................. 80 Adiabatic Transient Startup Simulation Results ........................................................... 83 Engine Geometry Optimization 2: Adiabatic Analysis ................................................ 85 Solution Dependant Motion .......................................................................................... 92 Material Selection ......................................................................................................... 96 Engine: ................................................................................................................................... 96 Collector: ............................................................................................................................... 97
Thermal Analysis .......................................................................................................... 98 Steady State Heat Transfer Model ......................................................................................... 98 Computer BasedSteady State Hot End Temperature ........................................................... 100 Discussion ............................................................................................................................ 102
Stress Analysis ............................................................................................................ 103 Hot End Stress Analysis....................................................................................................... 103 Displacer Piston Base Stress Analysis ................................................................................. 104 Engine Body Stress Analysis ............................................................................................... 105 Displacer Piston Rod Stress Analysis .................................................................................. 106 Power Piston Rod Stress Analysis ....................................................................................... 107 Engine Bolts/ Linear Shafts Stress Analysis ........................................................................ 108 Crankshaft Stress Analysis .................................................................................................. 109
Design Based on Static and Fatigue Failure Design Theories .................................... 110 Crankshaft Fatigue Life Analysis ........................................................................................ 110
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Power Piston Rod Fatigue Life Analysis ............................................................................. 111
Deflection Analysis ..................................................................................................... 112 Hot End Deflection Analysis ............................................................................................... 112 Displacer Piston Base Deflection Analysis .......................................................................... 113 Engine Body Deflection Analysis ........................................................................................ 114 Displacer Piston Rod Deflection Analysis ........................................................................... 115 Power Piston Rod Deflection Analysis ................................................................................ 116 Engine Bolts/ Linear Shafts Deflection Analysis ................................................................ 117 Crankshaft Deflection Analysis ........................................................................................... 118 Cost Analysis ....................................................................................................................... 119 Discussion ............................................................................................................................ 120
Prototype Construction ................................................................................................... 120 Description of Prototype ............................................................................................. 120 Prototype Design ......................................................................................................... 120 Parts List and Analysis................................................................................................ 121 Solar Concentrator Parts List ............................................................................................... 121 Stirling Engine Parts List ..................................................................................................... 122
Construction ................................................................................................................ 123 Testing and Evaluation ................................................................................................... 124 Introduction ................................................................................................................. 125 Steady State Concentrator Heat Input ......................................................................... 126 Overview .............................................................................................................................. 126 Experimental Set up ............................................................................................................. 128 Instrumentation .................................................................................................................... 129 Data Acquisition .................................................................................................................. 129 Results .................................................................................................................................. 129 Analysis ............................................................................................................................... 129
Hourly Concentrator Heat Input with Tracking .......................................................... 130 Overview .............................................................................................................................. 130 Experimental Set Up ............................................................................................................ 130 Instrumentation .................................................................................................................... 130 Data Acquisition .................................................................................................................. 130
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Results .................................................................................................................................. 131 Analysis ............................................................................................................................... 132
Stirling Engine Performance ....................................................................................... 133 Overview .............................................................................................................................. 133 Experimental Set Up ............................................................................................................ 133 Instrumentation .................................................................................................................... 133 Data Acquisition .................................................................................................................. 133 Results .................................................................................................................................. 133 Analysis ............................................................................................................................... 133
Conclusion .................................................................................................................. 134 Design Considerations .................................................................................................... 135 Assembly and Disassembly ........................................................................................ 135 Maintenance of the System ......................................................................................... 135 Regular Maintenance ........................................................................................................... 135 Major Maintenance .............................................................................................................. 135
Environmental Impact ................................................................................................. 135 Risk Assessment ......................................................................................................... 135 Project Management ....................................................................................................... 136 Overview ..................................................................................................................... 136 Important Milestones .................................................................................................. 136 Breakdown of Responsibilities Among Team Members ............................................ 137 Organization of Work and Timeline ........................................................................... 138 Cost Analysis .............................................................................................................. 139 Relevant Course Work ................................................................................................ 140 Patent/Copyright Application ..................................................................................... 140 Commercialization of the Final Product ..................................................................... 140 Discussion ................................................................................................................... 140 Design Considerations and Future Work .................................................................... 141 Lessons Learned.......................................................................................................... 141 Conclusion and Discussion ......................................................................................... 141 Works Cited .................................................................................................................... 142 Appendices ...................................................................................................................... 145 7|Page
Appendix A. Detailed Engineering Drawings of All Parts ......................................... 146 Appendix B. Detailed Raw Design Calculations and Analysis .................................. 162 Adiabatic Analysis ............................................................................................................... 162 Isothermal Analysis ............................................................................................................. 166 Developed Tracking Code ................................................................................................... 167
Appendix D. Stirling Geometry and Mesh Generation Codes ................................... 170 var.dat .................................................................................................................................. 170 setStirlingGeomertry.C ........................................................................................................ 170 stirlingGeometry.H .............................................................................................................. 172 designVariables.H ................................................................................................................ 173 blockMeshDict ..................................................................................................................... 174
Appendix E. Optimization Codes ............................................................................... 180 diffEvol.C ............................................................................................................................ 180 QsubPar_parents.sh.............................................................................................................. 189 OF_qsub.sh .......................................................................................................................... 211 pistonPressureOut ................................................................................................................ 211 pistonPlot2 ........................................................................................................................... 213 pistonNet .............................................................................................................................. 214
Appendix F. Solution Dependent Motion Codes ........................................................ 215 stirlingSDM.m ..................................................................................................................... 215
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List of Figures Figure 1. Earthquake Density Map of the Globe .............................................................. 15 Figure 2. Tsunami History of Location, Intensity, and Size ............................................. 16 Figure 3. Hurricane Emergence around the Globe ........................................................... 16 Figure 4. Average Annual Ground Solar Energy.............................................................. 17 Figure 5. The original Stirling Engine patent of 1816 ...................................................... 18 Figure 6. Automotive Stirling Engine ............................................................................... 19 Figure 7. Brayton Rotating Unit (BRU) ........................................................................... 19 Figure 8. Stirling based Fission Surface Power System ................................................... 20 Figure 9. Alpha Stirling Engine ........................................................................................ 21 Figure 10. Beta Stirling Engine ........................................................................................ 21 Figure 11. Gamma Stirling Engine ................................................................................... 22 Figure 12. Directional Nature of Solar Radiation outside the Earth's Atmosphere .......... 22 Figure 13.Spectral Distribution of Solar Radiation .......................................................... 23 Figure 14. Directional Distribution of solar radiation at the Earth's surface .................... 23 Figure 15. Parabolic trough in Sandia Figure 16. Fresnel Reflectors Ausra ............... 24 Figure 17. Solar Stirling Schematic .................................................................................. 25 Figure 18. Stirling Energy Systems Stirling Power Units ................................................ 26 Figure 19. Stirling Energy Systems - SunCatcher ............................................................ 26 Figure 20. Nellis Air Force-Single Axis SunPower T20 tracker ...................................... 27 Figure 21. Rotating house with tracking solar panels that operate independently ........... 28 Figure 22. Point Focus parabolic dish with Stirling Engine ............................................. 28 Figure 23. Power Generation per Square Methods for Different Methods ....................... 37 Figure 24. Conceptual Design........................................................................................... 38 Figure 25. Designed Solar Concentrator ........................................................................... 41 Figure 26. Cutaway view of Fresnel lens.......................................................................... 41 Figure 27. Initial Solar Stirling Configuration .................................................................. 43 Figure 28. Interim Design of 2.7 kWe Stirling Engine ..................................................... 44 Figure 29. Area Breakdown of Designed Stirling Engine. ............................................... 45 Figure 30. Designed Stirling Engine. ................................................................................ 48 Figure 31. Cutaway Views of Designed Stirling Engine. ................................................. 48 Figure 32. Diagram of cooling Reservoir ......................................................................... 49 Figure 33. Designed Solar Concentrator ........................................................................... 51 Figure 34. Power Flows for 2.7 kW Stirling Engine ........................................................ 51 Figure 35. Heat Flows for Cooling Reservoir ................................................................... 52 Figure 36. Temperatures of Ambient Air and Cooling Reservoir .................................... 52 Figure 37. Adiabatic Cycle (Berchowitz, 1984) .............................................................. 57 Figure 38. Stirling Engine Used in Development of Equation Set (Berchowitz, 1984) .. 58 Figure 39. Work per cycle ................................................................................................ 64 9|Page
Figure 40. Work Done by Compression Space for Single Cycle ..................................... 65 Figure 41. Work Done By Expansion Space for Single Cycle ......................................... 65 Figure 42. Compression Space Volume............................................................................ 66 Figure 43. Expansion Space Volume ................................................................................ 66 Figure 44. Pressure During a Single cycle ........................................................................ 67 Figure 45 Boundary patch names ..................................................................................... 71 Figure 46: Velocity Field from the end of the 9th cycle of the isothermal transient startup simulation.......................................................................................................................... 72 Figure 47: p/rho Field from the end of the 9th cycle of the isothermal transient simulation ........................................................................................................................................... 73 Figure 48: Prototype Isothermal Simulation 9th cycle Displacer Piston .......................... 73 Figure 49: Prototype Isothermal Simulation 9th cycle Power Piston ............................... 74 Figure 50: Prototype Isothermal Simulation 9th cycle Summary..................................... 74 Figure 51: Velocity Field from the end of the 10th cycle of the isothermal transient startup simulation .............................................................................................................. 75 Figure 52: p/rho Field from the end of the 10th cycle of the transient startup simulation 75 Figure 53: Prototype Isothermal Simulation 10th cycle Displacer Piston ........................ 76 Figure 54: Prototype Isothermal Simulation 10th cycle Power Piston ............................. 76 Figure 55: Prototype Isothermal Simulation 10th cycle Summary ................................... 77 Figure 56: Prototype Design1 Optimization Initial Design .............................................. 78 Figure 57: Prototype Design1 Optimization Generation 1 ............................................... 78 Figure 58: Prototype Design1 Optimization Generation 14 ............................................. 79 Figure 59: Prototype Design1 Optimization Generation 25 ............................................. 79 Figure 60: Prototype Design1 Optimization Generation 32 ............................................. 80 Figure 61:Fine and Coarse Mesh Comparison.................................................................. 83 Figure 62: Design 2 Transient Startup Pressure vs. Time ................................................ 84 Figure 63: Design 2 Transient Startup Temperature vs. Time.......................................... 84 Figure 64: Design 2 Transient Startup Velocity Magnitude vs. Time .............................. 85 Figure 65: Prototype Design2 Initial Design with parameters denoted ............................ 86 Figure 66: Initial Optimization Population ....................................................................... 87 Figure 67: Stirling Helium Geometry Design Space after 4 Generations ........................ 87 Figure 68: Optimization Population after 15 Generations ................................................ 88 Figure 69: Prototype Design2 Optimization Generation 15 ............................................ 88 Figure 70: Optimization Population after 30 Generations ................................................ 89 Figure 71: Prototype Design2 Optimization Generation 30 ............................................ 89 Figure 72: Optimization Population after 45 Generations ................................................ 90 Figure 73: Prototype Design2 Optimization Generation 45 ............................................. 90 Figure 74: Optimization Population after 70 Generations ................................................ 91 Figure 75: Prototype Design2 Final Design...................................................................... 91 Figure 76: Theta (Displacer Piston Crank Angle) and Phi (Power Piston Crank Angle) . 93 10 | P a g e
Figure 77: w (Crank Speed) vs. time and theta ................................................................. 94 Figure 78:Displacer piston position vs. time and theta ..................................................... 94 Figure 79: Power piston position vs. time and theta ......................................................... 95 Figure 80. Steady State Thermal Diagram of Stirling Engine .......................................... 98 Figure 81. Hot End Mesh and Imposed Conditions ........................................................ 100 Figure 82. Thermal Plot o Lower End of Hot End ......................................................... 101 Figure 83. Thermal Plot of Upper Portion of Hot End ................................................... 101 Figure 84. Expected Hot End Temperatures for the 2.7 kW Solar Stirling Engine ........ 102 Figure 85. Stress Analysis of Hot End ............................................................................ 103 Figure 86. Stress Analysis of Displacer Piston Base ...................................................... 104 Figure 87. Stress Analysis of Engine Body .................................................................... 105 Figure 88. Stress Analysis of Displacer Piston Rod ....................................................... 106 Figure 89. Stress Analysis of Power Piston Rod ............................................................ 107 Figure 90. Stress Analysis of Engine Bolts/ linear Shafts .............................................. 108 Figure 91. Stress Analysis of Crankshaft ........................................................................ 109 Figure 92. Fatigue Life Analysis of Crankshaft.............................................................. 110 Figure 93. Fatigue Life Analysis of Power Piston Rod .................................................. 111 Figure 94. Deflection Analysis of Hot End .................................................................... 112 Figure 95. Deflection Analysis of Displacer Piston Base ............................................... 113 Figure 96. Deflection Analysis of Engine Body ............................................................. 114 Figure 97. Deflection Analysis of Displacer Piston Rod ................................................ 115 Figure 98. Deflection Analysis of Power Piston Rod ..................................................... 116 Figure 99. Deflection Analysis of Engine Bolts/ linear Shafts ....................................... 117 Figure 100. Deflection Analysis of Crankshaft .............................................................. 118 Figure 101. Machining the finned interior finned Surface of the Hot End ..................... 123 Figure 102. Top and Bottom Images of the Solar Stirling Engine - showcasing the inside of the displacer piston, the linear bearings, and finned interior of the hot end ............... 123 Figure 103. Testing of Solar Concentrator ..................................................................... 124 Figure 104. Design of Experiment - Fresnel Lens .......................................................... 126 Figure 105. Relationship Between Test Article Temperature and Heat Input ................ 128 Figure 106. Experimental Set-Up ................................................................................... 128 Figure 107. Instrumentation for Testing the Hot end Temperatures .............................. 129 Figure 108. Reaching Temperatures of 260 ˚C (500˚ F) ................................................ 129 Figure 109. Experimental Test Article Temperature ...................................................... 131 Figure 110. Theoretical and Experimental Collected Energy ......................................... 131 Figure 111. Distribution of Labor based on hours .......................................................... 139 Figure 112. Distribution of Work based on Cost ............................................................ 139
List of Tables Table 1. Types of Solar Energy Conversion Ranked ........................................................ 35 11 | P a g e
Table 2. Types of Stirling Engine ..................................................................................... 36 Table 3. Types of Solar Concentrators ............................................................................. 36 Table 4. Cost Analysis of Solar Concentrator .................................................................. 40 Table 5. 2.7kW Stirling Engine Volume Allocations ....................................................... 45 Table 6. Characteristics of Cooling Reservoir .................................................................. 52 Table 7. Nomenclature Used for Adiabatic Stirling Engine Analysis .............................. 56 Table 8. Adiabatic Stirling Cycle Differential and Algebraic Equations (Berchowitz, 1984) ................................................................................................................................. 62 Table 9. Constants Used for Stirling Cycle Simulation .................................................... 64 Table 10: Mesh Statistics .................................................................................................. 82 Table 11. Initial Imposed Thermal Conditions ............................................................... 100 Table 12. Imposed Stresses for Stress Analysis of Hot End ........................................... 103 Table 13. Imposed Stresses for Stress Analysis of Displacer Piston Base ..................... 104 Table 14. Imposed Stresses for Stress Analysis of Engine Body ................................... 105 Table 15. Imposed Stresses for Stress Analysis of Displacer Piston Rod ...................... 106 Table 16. Imposed Stresses for Stress Analysis of Power Piston Rod ........................... 107 Table 17. Imposed Stresses for Stress Analysis of Power Piston Rod ........................... 108 Table 18. Imposed Stresses for Stress Analysis of Crankshaft ....................................... 109 Table 19. Imposed Stresses for Fatigue Life Analysis of Crankshaft............................. 110 Table 20. Imposed Stresses for Fatigue Life Analysis of Power Piston Rod ................. 111 Table 21. Imposed Stresses for Deflection Analysis of Hot End ................................... 112 Table 22. Imposed Stresses for Deflection Analysis of Displacer Piston Base .............. 113 Table 23. Imposed Stresses for Deflection Analysis of Engine Body ............................ 114 Table 24. Imposed Stresses for Deflection Analysis of Displacer Piston Rod ............... 115 Table 25. Imposed Stresses for Deflection Analysis of Power Piston Rod .................... 116 Table 26. Imposed Stresses for Deflection Analysis of Power Piston Rod .................... 117 Table 27. Imposed Stresses for Deflection Analysis of Crankshaft ............................... 118 Table 28. Parts List and Analysis for Prototype Solar Concentrator .............................. 121 Table 29. Part List and Analysis for Prototype Engine .................................................. 122 Table 30. Daily Heat Input (no atmospheric effects) .........Error! Bookmark not defined. Table 31. Breakdown of Deadlines ................................................................................. 136 Table 32. Breakdown of Responsibilities among Team Members ................................. 137 Table 33. Gantt Chart for Solar Stirling.......................................................................... 138 Table 34. Hours Worked on Design and Development .................................................. 139
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Abstract In order to satisfy the rising energy demands of global consumption, a new cleaner and renewable power source needs to be explored, conceptualized, and developed. Solar energy is a free and clean energy resource which can be used to generate power without damage to humans or the local ecosystems. To efficiently capture this solar energy as a feasible power source, a Stirling engine will be developed and will use sunlight as a source via a solar concentrator. This project intends to utilize methods of gathering solar energy that have not yet been commercially implemented, and modifications to traditional Stirling engines will be made in order to maximize the efficiency of solar Stirling engines. These modified solar Stirling engines can produce power for a wide variety of applications. The nature of the engine allows for both the scalability to create a solar farm as well as use for producing power in remote areas and disaster relief.
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Introduction Problem Statement The political, economical, environmental concerns over traditional fossil fuel power generation have led to an overwhelming amount of innovation and research into cleaner renewable sources. The United States of America currently gets 85% of our energy through fossil fuels and less than 2% from renewable energy (Systems, Technology, 2009). It is in the nation‟s best interest to invest heavily in renewable energy so that we could reap the benefits to the economy, environment, politics, and human health.
Motivation Of the existing sources of renewable energy, the most promising is the sun. It is the most abundant source of energy on the planet and it is a phenomenal source of light and heat. Scientific American magazine states, “The energy in sunlight striking the Earth for 40 minutes is the equivalent to global energy consumption for one year.” (Systems, Technology, 2009). Therefore, it behooves engineers to design way of capturing this incredible natural resource for use in power generation as an alternative to other methods such as fossil fuels.
Justification The United Nations has a difficult time quantifying the exact number of lives that are lost in nature disaster. Perhaps more surprising is not the amount of death that occur from natural disasters, but the deaths that occur after disaster hits. The lack of clean water, food, and electricity can sometime cause more deaths than the actual disastrous event. Creating a technology that provides power to such disastrous areas can provide much needed clean water, and desperately needed electricity for life saving operations such as medical equipment, communications, and food preparation. Remote power can provide a real survival opportunity for disaster victims who have been left without a home, food, water, or power.
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Disastrous Events and their Location Across the Globe Earthquakes An investigation was conducted into the location heaviest hit areas for natural disasters to occur. The image shown below illustrates the earthquake density map for the entire planet. The scale is based on the average number of earthquakes per year per 12,300 km^2 which are magnitude 5 of greater (Interior, 2009). We can see that they highest danger for earthquakes are for the eastern hemisphere of the world, with places that border with the Indian Ocean and the North Pacific Ocean. However, as seen by earthquakes that have hit Haiti and California, many other costal places are at danger.
Figure 1. Earthquake Density Map of the Globe
Tsunamis Tsunamis have become part of the collective conscience of current society due to the horrific tsunami that hit part of Asia in 2004. Though tsunamis have been recorded to occur in many different locations on the planet, the majority of its occurrences have taken place in Pacific Ocean. The map below illustrates the location of the tsunami as well as its magnitude and size. This map indicates over 2,000 tsunami events that date back from 1628 BC (Goverment, 2009).
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Figure 2. Tsunami History of Location, Intensity, and Size
Hurricanes University of California at Berkeley physics graduate student, Robert Rohde complied data available from several sources to generate a map of 150 years‟ worth of tracking hurricanes leading up to September 2005. This map below shows the areas which are worse hit by these deadly storms and can serve as a roadmap to future hurricanes‟ location here (Discover, 2007) .
Figure 3. Hurricane Emergence around the Globe
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Ideal Locations for Solar Energy Power Generation The irony of the tragedies experienced by the citizens of these locations that are in the path of disaster is that they are also the most ideal source for solar energy power. The world maps shown previously that illustrate the places that are heaviest hit by natural disasters such as earthquakes, tsunamis, and hurricanes. The same conditions that create a breeding ground for natural disasters also provide a unique ability to generate solar power. The world map shown below demonstrates the availability of solar power at different locations on the globe. What we have discovered is that the places that would most benefit from a solar Stirling engine system are the same places that the system would be the most efficient (Beta, 2008).
Figure 4. Average Annual Ground Solar Energy
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Literature Survey Stirling engines are external combustion engines which can function by using a wide variety of fuel sources such as a combustible gas, nuclear head, or solar energy. The heat supplied to the engine causes the working fluid to expand; thereby, moving a displacer piston. This piston then displaces the working fluid from the hot end into the cold end of the engine where the working fluid is compressed and the piston retracts. The displacer piston then moves the fluid into the hot end where it will once be expanded and then displaced into the cold end where it will compress and this cycle will continue as long the temperature difference exists. The Stirling cycle is a reversible cycle which closely follows the Carnot principal, making it a highly efficient cycle. Stirling engines are the simplest form of heat engine and are arguably the most efficient engine (Berchowitz, 1984).
History The first patent containing a Stirling engine was written in 1816 by the Rev'd Dr. Robert Stirling. He patented an „economizer‟ which is synonymous with today‟s regenerator, used to increase the efficiency of the engine. The Stirling engine did not gain wide popularity compared to the steam engine due to the limits that currently available materials offered. Stirling engines went relatively unnoticed and not improved on until the late 1930 when Philips selected Stirling engines to power radios for remote areas. The decision to use Stirling was based on its low audible and E&M noise and ability to run on any heat source from heating oil to wood (Berchowitz, 1984).
Figure 5. The original Stirling Engine patent of 1816
In 1972 Ford Motor Company teamed up with Philips to develop an automotive Stirling engine, and gage its potential for automobiles. What was produced was a four cylinder, 170 Horse Power Stirling engines which used a swash plate to transfer the power from the Stirling engines into torque that could be connected to a traditional transmission [7]. The engine ended up having little potential for use in automobiles due to the nature of external combustion engines inability to produce immediate power.
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There is however concepts to revive the automobile Stirling engine for use in hybrid electric vehicles because of its higher power to weight ratio and overall efficiency (Nightingale, 1986)
Figure 6. Automotive Stirling Engine
Beginning in the 1970‟s NASA‟s Glenn Research Center began investigations and development of high efficiency Stirling engines to be used in space applications. The decision to use Stirling engines was centered on their relative reliability compared to other mechanical engines, simplicity, low noise (audible, E&M), essentially nonexistent vibration (when convertors were paired), and most importantly high power to weight ratio. The Brayton Rotating Unit (BRU) Project aim at obtaining higher efficiency power conversion system for isotope, reactor, and solar receiver hear sources (Lee Mason, 2007).
Figure 7. Brayton Rotating Unit (BRU)
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NASA is now taking a serious interest in Stirling engines for their potential use on other planetary bodies. One of the most prominent possibilities is the use of a Stirlingbased Fission Surface Power System which can generate power of about 50kWe per unit. This form of power generation is a viable solution to the monumental problem of attempting a manned mission to the Lunar and Martian Surfaces for extended periods of time. This type of system could be used to provide power for rovers, remote science experiments, or as a utility power source for an outpost in any of our celestial orbiting bodies (Lee Mason, 2007).
Figure 8. Stirling based Fission Surface Power System
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Stirling Engine Configurations Stirling engines are commonly found in three different configurations; alpha, beta, and gamma. There is also a variation of each one named free-piston but due to its complexity and high cost, it will not be discussed in details for this project. Each of the three main configurations has unique advantages and disadvantages due their variation in geometry and arrangement. An Alpha Stirling engine is composed of two power pistons which are housed in two separate cylinders where one cylinder is exposed to heat while the second is subjected to cold and heat dissipation. Alpha Stirling engines will sometimes utilize a regenerator as part of its configuration. The regenerator function is to store heat as it moves from the hot end to the cold one and re-supplying the fluid with heat as it returns to the hot end.
Figure 9. Alpha Stirling Engine
A Beta Stirling Engine configuration uses one cylinder which houses both the power and displacement piston. The displacer piston purpose is to shuffle the air between the hot end and the cold end while not extracting any power from the expanding gas.
Figure 10. Beta Stirling Engine
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Lastly, a Gamma Stirling engine is similar to a Beta configuration expect save for the power piston which is housed in a separate cylinder but still connected to the same flywheel as the displacer piston.
Figure 11. Gamma Stirling Engine
Solar Radiation The sun can be considered a spherical radiation source that is 1.39 x m in diameter and at a distance of about 1.50 x m from the Earth (Frank P. Incropera, 2002). Due to Earth‟s Ozone Layer, the radiation felt by body outside our atmosphere would be different than the radiation felt on Earth surfaces as shown in Figure 12 .
Figure 12. Directional Nature of Solar Radiation outside the Earth's Atmosphere
In fact, the solar radiation reaching Earth can be treated as a series of parallel rays that would form an angle θ, the zenith angle, with respect to the normal surface of any horizontal surface outside our atmosphere. Therefore, the extraterrestrial solar irradiation is dependent on the global position of the object as well as the time of day and year.
Equation 1. Extraterrestrial Solar Irradiation
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The solar constant, , can be defined as the flux of solar energy incident on a surface which is oriented normal to the sun‟s rays at the point in which the Earth is at its mean distance away. The solar constant is given as
= 1353
and the correction
value for the eccentricity of Earth‟s orbit about the sun is given by 0.97 ≤
≥ 1.03.
Figure 13.Spectral Distribution of Solar Radiation
When solar radiation passes through Earth‟s atmosphere, it experiences a change in magnitude as well as spectrally and directional distributions. These changes can be attributed to the absorption and scattering of the radiation by the atmosphere. Since the ozone is strong in the UV region, it provides attenuation below 0.4 μm and complete attenuation below 0.3 μm (Frank P. Incropera, 2002).
Figure 14. Directional Distribution of solar radiation at the Earth's surface
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The atmosphere acts on the solar rays by redirecting the rays using two kinds of scattering, Rayleigh scattering of the gas molecules and Mie scattering of the dust and aerosol particles. The cumulative effect of the Earth atmosphere on solar ray‟s distribution on Earth‟s surface is shown in Figure 14. The emissive power associated with the Earth‟s surface is given by equation below where the surface emissivity is and is the Stefan – Boltzman Constant which is given by
The spectral distribution of atmospheric emission attributes to the irradiation of Earth‟s surface and can be estimated by using the equation below.
Solar Concentrator A wide variety of solar concentrators are currently commercially available in order to concentrate solar rays for the purpose of power generation. There are many forms of solar concentrators, but the most common forms are those which utilize curved, parabolic mirrors and those which use Fresnel lenses. Parabolic Troughs are the most widely used type of solar concentrator. It consists of a linear parabolic reflector which can concentrate sunlight onto a tube, commonly filled with a working fluid such as molten salt, and positioned along the focal length in order to generate heat for power generation. This type of solar concentrator can be found in Solar Energy Generating Systems (SEGS) plants in California, Acciona‟s Nevada Solar One, and Plataforma Solar de Almerias in Spain (Laboratories, 2009).
Figure 15. Parabolic trough in Sandia Figure 16. Fresnel Reflectors Ausra
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Concentrating Linear Fresnel lenses are defined as many thin mirror strips in the place of parabolic mirrors to focus sunlight and heat on a given point. The advantage to this method over parabolic mirrors is that flat mirrors are much cheaper than parabolic mirrors and that more reflectors can be used in the same amount of space which provides more sunlight energy at the focus. This type of solar concentrator shown in Figure 16 was constructed a company called Ausra (Ausra, 2009).
Solar Stirling Engine Due to Stirling engine‟s unique ability to produce power in the presence of any heat source, a wide variety of fuels can be utilized for the purpose of power generation which includes Solar. Using sunlight as a viable heat source for Stirling engines yields a method of producing power without harmful emissions and without using manufacturing methods which deplete the Earths of its precise natural resources. Solar energy has been utilized before for power production in heat engines, however, most of the previous applications were for steam turbines that would be only practical for very large scale installations. Stirling engines provide a methodology for generating power for use in a small system to drive an electrical generator. The schematic below illustrates a small scale electric power from solar thermal energy system which utilizes solar Stirling. In this system, the solar heat collector provides heat for the solar Stirling engine which in turn provides AC power. The electrical power can be transferred to a battery charger, then to DC control unit which can either go into a battery or into an inverter. Efficiencies for this type of small scale system can range from 18% to 23% (Communications).
Figure 17. Solar Stirling Schematic
Solar Stirling has made a tremendous impact on alternative energy in the certain years with companies like Stirling Energy Systems (SES) leading the way. This company in partnership with Sandia National Lab managed to break the world record for solar-togrid conversion efficiency at an amazing 31.25 % on January 31, 2008. SES Serial #3 was erected in May 2005 as part of the Solar Thermal Test Facility which produced up to
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150kW of grid ready electrical power during the hours of sunlight. Each dish consisted of 82 mirrors that can focus the light into an intense beam (Systems, 2008).
Figure 18. Stirling Energy Systems Stirling Power Units
SES solar Stirling engine, named SunCatcher, was awarded the 2008 Breakthrough Award winner by Popular Mechanics for its role as one of the top 10 world-changing innovations. The SunCatcher is a 25 kWe solar dish Stirling system which uses a solar concentrator structure which supports an array of curved glass mirror which are designed to follow the sun and collect the focused solar energy onto a power conversion unit. The diagram below illustrates the workings of SES‟s SunCatcher.
Figure 19. Stirling Energy Systems - SunCatcher
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Solar Tracking Due the fluctuations of the sun‟s position with respect to time, season, and position, a solar tracking device is often use to maximize the amount of sunlight that reaches the solar converter. For the solar Stirling engine, the feasibility, usability, and effectiveness of this technology are directly dependant on the amount of sunlight that can be focused on the hot end. For this reason, an extensive investigation in to the different types of solar energy was conducted. Tracker Mount Types Polar Polar is a type of solar tracker that uses a one axis alignment which is near parallel to the axis of the Earth‟s rotation around the north and south poles. This method of tracking sunlight is most useful in technology that is not the main source of power. An example of polar tracking is at Nellis Air Force Base in Nevada (Force, 2007), where the photovoltaic‟s are mostly utilized in peak summer sunlight to supply power to additional power needed to run the AC units. In this configuration, the polar axis faces north with the angle between the axis and the horizontal equal to the latitude of the locations at hand. The angle of declination is one that can be alter either manually or automated in order to angle the solar collection further north in the summer and further south in winter. Another option is to have the solar collector angled at zero degrees with it position being perpendicular to the polar axis which is where the mean path of the sun is found. This method can be even more improved with occasional shifts in the angle of declination to compensate for changes in season. Figure 20. Nellis Air Force-Single Axis SunPower T20 tracker
Horizontal Axle For the horizontal axle tracking device, a tube is place on the north-south place. This tube is then attached to the solar collector and it will rotate on its axis to track the sun through the day. This method is best for locations near the equator as is less effective at higher latitudes. However, the robustness of the structure and the simplicity of the mechanism makes it a popular option. When active mechanisms are used to track the sun, a single control and motor is used to actuate multiple rows of panels. 27 | P a g e
Vertical Axle This solar tracker used a single axis that pivots about the vertical axis. This method is best used for high altitudes where the sun path is not as high as equatorial places. Altitude- Azimuth This is a two-directional tracker which allows the solar collector to rotate about the horizontal (altitude) and the vertical (azimuth). This method is more complex due its need to a computer to control the movements.
Figure 21. Rotating house with tracking solar panels that operate independently
Two – Axis Mount This method uses active trackers to move the solar collector in two axes. One axis has a vertical pivot (horizontal ring mount) which let the solar collector move to a compass point. The second axis is a horizontal elevation pivot located in the vertical platform. The combination of these two axes allow the device to hone in on any upward hemispherical location. This method is computer controlled or may use sensors to control the motor that orient the solar collectors toward the sun. This method is popular for parabolic mirror and Stirling engine. Figure 22. Point Focus parabolic dish with Stirling Engine
Multi-Mirror Reflective Unit This device compiles multiple mirrors on a horizontal plane that will concentrate the sunlight upward to a high temperature device. This method is suited for use in flat surfaces as well as for lower latitudes.
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Drive Trackers Active Trackers Active trackers use motors and gear trains to move the tracker via a controller which responds to the solar directions. Two axis active trackers sometimes use heliostats which are mirrors that can move as they reflect the sunlight toward the collector. Each heliostat is controlled through a computer program in which gives the opportunity for the system to be shut down if need be. Light-sensing trackers are also commonly used in active trackers. This method uses photo sensors which can output a null when they get the identical light flux. These light sensors are oriented at 90 degrees apart such that the steepness of the cosine transfer function will be balanced and will therefore create maximum sensitivity. Passive Tracker There are two types of passive tracking that are commonly used. One type uses a compressed gas which has a low boiling point. This causes the gas fluid to move via the solar heat raising gas pressure which in turn moves the solar collector. These devices use viscous dampers in order to reduce the wind gusts and also use reflector to shine sunlight on the collector. The second type of passive tracker is the use of hologram. When sunlight passes through the transparent side of the solar collector, it is reflect back to the collector via the hologram. This allows for sunlight to shine on both sides of the collector and therefore increases efficiencies. Chronological Tracker A chronological tracker works by counteractive the sun rotation by rotating the solar collector at nearly the same rate but in opposite direction. This works best with Polar mount configuration and can utilize a gear motor that can rotate at any average of 15 degrees an hour.
Discussion After an in-depth assessment of current technologies in solar tracking, a decision was made an implemented for our solar Stirling engine. For disaster relief, the most simple and efficient configuration is preferred. For this reason, chronological tracker is the most appropriate. It would allow for maximum absorption of the sun without huge energy loses for the mechanism that is conducting the tracking and would also eliminate the necessarily for correction of errors that occur with photovoltaics.
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Project Formulation Overview The overall goal of this project is to conceptualize, design, and build a modified solar Stirling engine with a Fresnel lens as the solar concentrator.
Project Objectives This solar Stirling engine uses a beta configuration. This project will be considered a success if the following objectives are met. Firstly, a design is to be made of a beta Stirling engine which uses a cost effective means of producing the most electricity. This engine should have a large margin of positive net energy and net power to be considered a feasible application. Second, a proof-of-concept of this configuration should be demonstrated by the creation of a small scale prototype. Lastly, this design should prove itself to be flexible and scalable to fit the needs of varying applications such as use in remote areas and disaster relief.
Design Specifications In order to meet the objectives of this project, certain specifications need to be ascertained. Due to the nature of Stirling engines, the maximum efficiency is achieved when the temperature difference between the hot end and the cold end is sufficiently large. Therefore, the design specifications focused on achieving this goal. The solar concentrator used in this project is to be sufficiently powerful to concentrate sunlight on the surface of the engine without noticeable losses due to refraction, medium, and geometry. The material used for the cylinders, pistons, and flywheel should be able to withstand thermal cyclic loading at the high operating temperature without causing the material to weaken, undergo chemical changes, or fail. The extended surfaces used in the cold end of the engine to dissipate heat should be of such geometry and material that heat transfer would be maximized between the engine and the ambient fluid.
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Constraints and Other Considerations The major constraint of Stirling engines is the ability to generate enough heat on the hot end while cooling the cold end in order to produce the necessary change in temperature so that power generation in feasible. Therefore, the main constraint of this design is its ability to concentrate enough sunlight on the hot end while chilling the cold end. The amount of sunlight that can be concentrated is dependent on a few factors, some of which can be controlled by the design and some of which are outside of the engineering design scope. Such factors that are outside of our control are the position of the engine relative to the Earth and the climate of that region. However, these environmental factors can be improved by ensuring that there is no aerial coverage near the engine such as trees and buildings so that the solar concentrator can optimize the solar rays in that region. Due to the constraints of the sunlight in the operating region, the most important consideration when conceptualizing the engine is the optimization of the solar concentrator. In the event of low solar heat throughout the day, season, or location, the efficiency of the engine could be optimized by the following factors which work to counteract the loss due to the availability of the sun. The efficiency of the engine can be improved significantly by selecting effective extended finned surfaces to assist in the heat dissipation from the cold end. This will cause the cold end temperature to be significantly lower than the heat on the hot end and increase the change in temperature. Another way to increase efficiency is to select a working fluid within the cylinder which can adequately transfer heat.
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Discussion This senior design project will conceptualize, design, and modify a solar Stirling engine for power generation for remote areas and/or disaster relief. The Stirling engine will be a beta configuration with a power capacity equal to the amount the solar collector harvests at peak hours of the day. This power capacity will be achieved via the use of a solar concentrator large enough to supply the hot end with sufficient heat and by generating a cold end which can efficiently dissipate heat into the atmosphere or working fluid in order to produce the needed change in temperature to create the volume changes in the cylinder. The efficiency of the engine can be maximized by selecting appropriate fins and extended surfaces as well as accurately focusing sunlight on the hot end. Other important consideration when designing a solar Stirling engine is to take into account the locations of where the engine will be placed, since the sunlight reaching the engine is dependent on its location on the globe. Along the same lines, allotting adequate space without coverage from trees and building so that the sunlight reaching the engine is not blocked. One of the largest areas that need improvement in heat engines is the thermal losses of the engine to the surroundings. A innovative way in which this problem can be addressed is thorough the implementation of Aerogels. This light-weight material currently holds the world title for the lowest density solid in history, measuring in at 1.9 mg/cm3! Aerogels are extremely porous material and can be as much as 99.8% air. Its mesoporousity is an invaluable ally against heat loss due to convection, conduction, and radiation. The use of Aerogels as a high-temperature, low-weight alternative to traditional insulation will yield an engine that has less heat loss due to heat transfer as well as maintaining the low weight necessary needed for the solar Stirling applications. This project will be submitted to industry leader working both in government and the private sector. Due to our teams‟ affiliation with NASA during previous internship, the knowledge gained from those experiences will be integrated into this project to refine our design. NASA Glenn Research Center is the leading research team on Stirling engines for space nuclear power. Our overall general design will be assessed and critiqued by a team of Stirling engine experts. In addition, NASA Kennedy Space Center has a long history of conducting risk analysis which also includes feasibility, reliability, and maintainability. They, too, will look over our conclusion on risk and the stated factors and will provide comments on our solar Stirling project.
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Design Alternatives Overview of Conceptual Designs Developed Three trade studies performed in order to justify the decision made for the design of the solar Stirling engine. The first trade study compares the different methods of generating power through the use of solar energy which includes photovoltaics, and heat engines such as Brayton and Stirling. The second trade study compares the different types of Stirling engine, alpha, beta, and gamma, to justify the selection for use in our design configuration. Lastly, the third trade study compares the different methods of concentrating sunlight which are traditional glass lenses, glass mirrors, and Fresnel plastic lenses. Each trade studies that was conducted, was ranked based on a desirability scale. This scale consists of four criteria, Cost, Ingenuity, Ability, and Reliability. Each ranking is based on a 1 through 5 score on the desirability of the concept being implemented. A basic cost analysis was preformed for each option in which the expected cost of each design was analyzed. For the cost portion, a 1 corresponds to high cost which is not desirable, and a 5 correlates to low cost which is desirable. Each alternative was given a ranking for Ingenuity. Ingenuity is defined as the implementations relative degree of current implementation. For the Ingenuity portion, a 1 corresponds to high degree of current implementation which is not desirable, and a 5 correlates to low degree of current implementation which is desirable. Each alternative was given a ranking for Ability. Ability is defined as the particular concepts ability to perform the intended role. The expectation of the Stirling engine is 25 kW of net energy production. For the Ability portion, a 1 corresponds to low degree of concept not being able to perform intended role which is not desirable, and a 5 correlates to a high degree of concept being able to perform intended role which is desirable. Each alternative was given a ranking for Reliability. Reliability is defined as the ability of the concept to perform its intended role with the minimal amount of maintenance or failures. For the Reliability portion, a 1 corresponds to a low expected reliability which is not desirable, and a 5 correlates to high reliability which is desirable.
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Solar Stirling Trade Studies Types of Solar Energy Conversion Throughout the history, there have been many methods explored on gathering sunlight for power generator. Some of the most successful methods of using solar energy in order to produce power are Photovoltaics, Brayton Cycle Steam Engines, and Stirling Engines. Photovoltaics are an array of cells which contain a special material that can convert solar radiation into electrical current (Placeholder1). Photovoltaics ranked a 1 on our scale for cost due its current price which is about $3/W (Solarbuzz, 2009). Since solar panels have been around since the beginning of the space race in the late 1950‟s, its ingenuity was ranked a 1 even though there have been several advances in their efficiencies in the past few years. Photovoltaics ranked a 4 in ability because of their continuous ability to produce an electrical current whenever it is exposed to sunlight. Because photovoltaics have no moving parts, it makes the system extremely reliable and operates with minimal maintenance. It is also worthy to note that many current solar panels use silicon as the main material in the cells. Though there are many advantages to using photovoltaic, the depletion of silicon from soil and the use of rare earth metals lead to solar panels not being the best solution to our power generation problem (Placeholder2). For the reasons stated above, photovoltaics ranked a total of 11 out of 20 on the desirability scale. Brayton Cycle is a type of thermodynamic cycle used in heat engine that uses steam as the working fluid in order to produce power (Sandfort, 1962). It ranked a 3 on the cost scale due to its use of rare metals and it cost-benefit analysis is mostly good for very large scale applications but would not make sense for smaller engines. The Brayton cycle, or steam engine, also ranked a 3 on ingenuity since it has existed for many decades but has only recently been applied in solar systems. Brayton cycle was ranked a 5 in ability since it can effectively use a solar concentrator to heat a reservoir of water to create steam which then turns a turbine. However, since it is comprised of moving parts, its reliability cannot be a 5 since its maintenance may cause a problem with long-term applications (Sandfort, 1962). Stirling engine is a type of heat engine that generates power through the compression and expansion of the working gas in its cylinder via a hot end and cold end (Berchowitz, 1984). This engine was given a 4 on the cost scale due to its relative inexpensiveness. The materials used for the engines are neither exotic nor rare therefore making the parts list more cost effective than other means. The solar Stirling engine ranked a 5 in ingenuity because though the Stirling engine has been around for over 100 years, it adaptation to using solar for the hot end as opposed to nuclear is new and 34 | P a g e
innovative. It also ranked a 5 in ability because a Stirling engine will continue to compress and expand a gas as long the temperature difference is present therefore making it a very viable option for power generation with respect to other heat engines. However, like steam engines which use the Brayton Cycle, Stirling engines also have moving parts and though the ability to generate power is very reliable, its long term maintenance plan forces it rank as 4 for reliability (Berchowitz, 1984). Table 1. Types of Solar Energy Conversion Ranked
Cost
Ingenuity
Ability
Reliability
Total
Photovoltaic
1
1
4
5
11
Brayton Cycle
3
3
5
4
15
Stirling Engine
4
5
5
4
18
Conclusion The conclusion of the trade studies is that we will use a Stirling engine for the conversion of solar energy into electrical energy.
Types of Stirling Engine Configurations Due to the increasing of price for energy gathered from fossil fuels as well as the harmful consequences that they have on the environment, a new way of generating power that is both clean and efficient needs to be explored. A prominent candidate for power generation which uses natural resources are Stirling engines due to their unique functionality which allows for use of different types of fuels including solar heat. Below are listed the most common configurations for a Stirling Engine; Alpha, Beta, and Gamma. Alpha Stirling Engines ranked a 3 in cost due to lack of durability in the seals which always pose a technical problem. Commercially, alpha configurations require an insulating head in order to move the seals away from the high temperature exposure in the hot end. Though this fixes the seal problem, it also adds dead space so it was assigned a 3 on ability and reliability. Beta Stirling Engines do not have the seal problem that alpha configurations have and are therefore ranked a 4 in cost and ability respectively. The beta engine is also extremely reliable and was therefore given a 5 on reliability. Gamma Stirling Engines provides a lower compression ratio but it much simpler mechanically; this earns gamma a 4 in cost. Also, gamma offers a unique ability to be used in multi-cylinder Stirling engines and therefore gets two 5‟s for ability and reliability (Wheeler, 2007).
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Table 2. Types of Stirling Engine
Cost
Ingenuity Ability
Reliability Total
Alpha Stirling
3
4
3
3
13
Beta Stirling
4
4
4
5
17
Gamma Stirling
4
4
5
5
18
Conclusion The trade studies for the different Stirling Engines configuration showed that for the intended application and purpose of our project, the best type of Stirling engine to use in the Beta configuration.
Types of Solar Concentrators Choosing the right type of solar concentrator for use in our solar Stirling engines will greatly influence the efficiency of the engine and therefore is deserving of special attention. A wide variety of solar concentrators are currently commercially available in order to concentrate solar rays for the purpose of power generation. There are many forms of solar concentrators, but the most common forms are the use of curved, parabolic mirrors and the use of Fresnel lenses. Parabolic mirrors ranked a 3 on our cost scale due the expense of manufacturing curved mirrors. It is one the most common forms of solar concentration and therefore ranks a 2 in ingenuity. However, its popularity is well placed since it is extremely able to perform its task with a noticeable amount of reliability which has earned parabolic mirrors two 5‟s obtained in the reliability and ability. The Fresnel lens ranked a 5 on cost since it is significantly more cost effective than the parabolic mirror. This is due to is composition of many flat mirrors instead of curved. It also ranked a 5 on ingenuity since it is a fairly new form of concentrating sunlight. Though Fresnel lens is not as efficient at concentrating sunlight, they gather more sunlight over the same amount of area and are therefore ranked a 4 and 5 for ability and reliability respectively. Table 3. Types of Solar Concentrators
Cost
Ingenuity Ability
Reliability Total
Parabolic Mirrors
3
2
5
5
15
Fresnel Plastic lens
5
5
4
5
19
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Figure 23 illustrates the daily generated energy per unit area versus the sun daily energy per unit area for Stirling solar dish, central receiver, parabolic trough, and tracking photovoltaic (Systems, Technology, 2009). This image demonstrates that using Solar Stirling instead of photovoltaics and other heat engines yields a higher estimated annual energy and would therefore be more beneficial as a method of solar energy conversion.
Figure 23. Power Generation per Square Methods for Different Methods
Conclusion The trade studies for the Types of Solar Concentrators showed that for the intended application and purpose of our project, the best type of solar concentrator to use is the Fresnel lens.
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Conceptual Design Based on the trade studies conducted during extensive research, it was found that solar Stirling would be the best option for remote power generation. Stirling engines provide a huge advantage over other heat engines based on their power outputs and this solar convertor can be considered greener than photovoltaics due their life cycle impact on their environment. Fresnel lenses provide the highest amount of energy from sunlight per unit area and are therefore ideal for use for disaster relief, where high energy density can make a noticeable difference. Due to the relatively low expected temperature differences, the Stirling engine was chosen to be of beta configuration. In order to improve efficiencies of the engine, the temperature difference needs to be at a maximum. It is for this reason that the cold end of the engine would be submerged in water to increase the heat transfer rate and heat dissipation from the engine. For Stirling engines, friction is their biggest enemy, especially with low temperature difference engines. Due to the engines‟ submergence in water to compensate for low temperature differences, some of the components needed to be internalized for the liquid submergence to take place. For this reason, the engines flywheel was internalized and place within the displacer piston. This allows for the solar energy to get converted to thermal energy, then mechanical energy, which is finally converted to useful electrical energy. Due to the multitude of conversion in the system, any and all steps to increase efficiencies will be taken. In addition to the engines submergence in water and the internalization of the flywheel, the Stirling engine will also be design to minimize all possible dead volume. This is the biggest enemy within Stirling and it something that needs to be closely monitored. For this reason, the displacer piston and the power piston were designed to reduce as much dead volume as possible with very small tolerances.
Figure 24. Conceptual Design
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Feasibility Assessment This project is feasible because similar technologies have been produced earlier. Sandia National Lab paired with SES created a huge solar Stirling farm using parabolic mirrors. The method of generating power via solar Stirling, though still at its infancy, is very reliable and efficient. Our design differs in several ways. First, our design includes a Fresnel lens as the solar collector instead of parabolic mirrors. Perhaps most unique about this configuration is our heat dissipation system and our internalization of the components. The most famous solar Stirling application uses a water pump to cool the engines. Since we don not want to lose any power, the stream from already existing water will cool the engine. This type of cooling technology is commonly used with nuclear power plants so it has been proven successful. The most interesting feature of our Stirling engine that has never been done before is the internalizing of the components. This method will be tested and if proven successful, will have many positive applications for heat engines working in harsh environments. The Carnot efficiency for our engine is 69%; this is based on a 975K hot end temperature and a 300K cold end temperature.
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Proposed Design The proposed design will consist of a solar concentrator, a Stirling engine and a cooling reservoir. The Solar collector is the most important portion of the design as it dictates the power requirements for all other components.
Collector Based on the imposed constraints of the engine fitting within a 3‟x3‟x3‟ package, the size of the collector must be a multiple of 3‟x3‟. Table 4 demonstrates an overview of possible sizes based on 3‟X3‟ panels. Table 4. Cost Analysis of Solar Concentrator
Energy Electricity Concentrator Concentrated Produced Cost
Engine Cost
Total Cost
$/W
3‟X3‟ 1.5 kW
0.3 kW
$150
$250
$400
$1.33
6 kW
1.2 kW
$200
$550
$750
$0.63
13.5 kW
2.7 kW
$350
$600
$950
$0.35
24 kW
4.8 kW
$900
$900
(1 panel) 6‟X6‟ (4 panels) 9‟X9‟ (9 panels) 12‟X12‟ $1,800 $0.38
(16 panels)
The 9‟x9‟ solar concentrator yielded slightly better per kW cost versus the 12‟x12‟ solar concentrator. The 9‟x9‟ solar concentrator was selected for its lower per kW cost, the complexity of fabricating the outer lenses of the 12‟x12‟ Fresnel lens, as well as the lessened focal length, approximately 8‟, and higher wind tolerance. The style of solar tracking selected was an Altitude-Azimuth type with a chronological tracking drive. This was selected because of its adaptability to any situation with little modification. The drive unit would only need time of day and latitude in order to follow the sun. This is to be accomplished through the use of a worm gear for the Azimuth portion of the tracking, controlled by a microprocessor. The altitude tracking would be accomplished by a screw driver. The power to run the tracking would be supplied from the Stirling engine, and would be considered a parasitic loss of the engine.
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Figure 25. Designed Solar Concentrator
A Fresnel lens works like a normal magnifying glass In that it focus light on a single point based on the curvature of the surface. However, a Fresnel lens only has the surface of a traditional lens. The Fresnel lenses needed for the 9‟ by 9‟ solar concentrator should have a focal length of 12ft. Figure 26 shows a cut away view of the central Fresnel lens.
Figure 26. Cutaway view of Fresnel lens
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Tracking Mechanism The type of tracking selected for the design is a single axis horizontal chronological timer with a potentiometer as instrumentation. The timer integrated in a single PLC will dictate the desired angle, measure by the potentiometer, and control the motor accordingly. The Solar collector needs to rotate at a rate of 15° per hour {Citation}, which comes out to 6.94 revolutions per minute. The final mechanical connection to the solar concentrator will be a worm-gear gear-set in order to eliminate the need to continuously overcome gravity to keep the solar concentrator in position, and reduce the parasitic loss of the due to solar racking The Solar collector is expected to weigh 60 pounds, at a moment arm of 10 foot. This will result in a minimum torque of 600 foot-pounds (7200 in-lbs) to rotate the collector. However through the instillations of bungee cords at 6‟ up the concentrator to 6‟ away from the concentrator, a zero-torque situation can be achieved through the selection of bungee cords with a specific spring constant. The spring constant was found to be 10 pounds/inch. The solar tracking will be achieved though rotating the concentrator one degree every four minutes. This will lead to a maximum error of .6 inches, which will not deviate more than 15% from the center of the hot end to the rim of the hot end. The rotation of the collector will be done through the implementation of a 60:1 worm gear reducer connected to a 50 in-lbs compact DC gear mount with a PLC controller which will also see as a clock/time with a simple one button input and 3-digit LED display for the time. The gear motor selected draws 0.12 Amps at 12 Volts, which is 1.44 watts. The motor is expected to run for 17 seconds every 4 minutes. Converting the power draw into a constant time draw, we get a constant 0.104 We.
Collector Parasitic Loss The power draw of the PLC is 0.01WE, and the time constant power draw of the DC gear motor is 0.104 We. Adding the two losses gives us a total parasitic loss of 0.114 We, which is 0.004% of the maximum power output. This small of a loss can be neglected in future power studies.
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Initial Engine Design The initial design for the solar Stirling engine was a beta-gamma hybrid configuration. This was initially tried in order to reduce the amount on components which are submerged under water. This configuration is shown in the image below.
Figure 27. Initial Solar Stirling Configuration
This configuration was later abandoned because of the implementation of finned surfaced to achieve the same heat transfer characteristics as well as a multitude of benefits that the full beta configuration offered. As you can see, the new design is much more intricate and complete. The new engine is more compact which is desirable for ease of transportation.
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Interim Engine Design Below are some CAD Images of the interim Sirling Engine Design, this design had the general layout of the final design, however it did not incorperate the specefic geometry derived from the analisys. This was the design in which the prototype is based primaraly on. The interim engine design used air as a working fluid for its abundance, ease to working with, abiliy to find in remote locations (relative to other gasses), and cost-effectivness. The use of air was changed when the Stirling cycle computer analysis showed two major problems with its use. The first, the thermal capacity of air was too low, resulting in an extremely high operating frequency of the engine in order to transfer the heat from hot end. Secondly, the low gas constant resulted in operating presures below atmospheric for our intended heat inputs. Helium was subsuquently used as the working fluid, however mixtures of air and helium were tried but eventually abanoned since it would require expensive gas mixture analysis instrumentation.
Figure 28. Interim Design of 2.7 kWe Stirling Engine
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Final Engine Design The engine power output will be matched to the concentrated energy input. For our design, this calls for a 2.7kWe Stirling engine. The interior volume for each portion of the Stirling was based off the GPU-3 rhombic drive Stirling engine, a 7.4 kW design developed for automobiles by NASA. Since the original design was developed to produce 2.7 times as much power, it was used as a starting point for the optimization of the engine needed for our purposes. Table 5. 2.7kW Stirling Engine Volume Allocations
Engine Volumes Compression Clearance Volume (Vclc) Expansion Clearance Volume (Vcle)) Compression Sweep Volume (Vswc) Expansion Sweep Volume(Vswe) Cooler Volume(Vk) Regenerator Volume (Vr) Heater Volume (Vh)
31 cc 31 cc 32 cc 32 cc 15 cc 50 cc 75 cc
For reference, Figure 29 shows the volume allocation for the designed Stirling engine. In this design, the sides of the displacer piston will be thermally conductive in order to classify it as a regenerator.
Figure 29. Area Breakdown of Designed Stirling Engine.
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Geometry of Heater Volume The overall volume for the heater is prescribed to be 70 cubic centimeters, about 4.3 cubic inches. The inside diameter of the body of the engine is 4.75 inch, giving us 1/4 inch height before we surpass our volume allocation. Through the use of 33% volume ratio wire mesh we can increase the overall height of the heater, as well as increase the surface area for heat transfer. Implementing the 33% volume mesh the heater height comes to 3/8 inches.
Geometry of Expansion Volume The overall expansion volume for the engine is prescribed to be 63 cubic centimeters, about 3.9 cubic inches. The inside diameter of the body of the engine is 4.75 inch, giving us 1/4 inch height before we surpass our volume allocation.
Geometry of Regenerator Volume The overall regenerator volume for the engine is prescribed to be 51 cubic centimeters, about 3.1 cubic inches. The inside diameter of the body of the engine is 4.75 inch, and the height of the regenerator is 2 inches (based on minimum crankshaft clearances). This leaves the outside diameter of the displacer piston/ regenerator to vary. The outside diameter of the displacer piston/ regenerator can be 4.61 inches before the volume allocation is surpassed.
Geometry of Cooler Volume The overall expansion volume for the engine is prescribed to be 13 cubic centimeters, about 0.8 cubic inches. The inside diameter of the cold end of the engine is 2.25 inch, giving us 1/4 inch height before we surpass our volume allocation. In order to increase the length of the cooler, a foam cone will be inserted with varying diameter. The diameter will begin at 2.25 inch and end at 1 inch. This results in a 1 inch cooler volume.
Geometry of Compression Volume The overall compression volume for the engine is prescribed to be 63 cubic centimeters, about 3.9 cubic inches. The inside diameter of the cold end of the engine is 2.25 inch, giving us 1/2 inch height before we surpass the total length of the cold end. The rest of the volume, 2.1 cubic inches will be allocated to the bottom inch of the displacer piston.
Design of Black Hole The on top of the solar absorption plate will be a hemispherical structure constructed of plastic rod and aluminum foil to reflect back all diffused radiation. The expected view factor resulting from the structure is expected to be 0.8.
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Design of Crankshaft The crankshaft was designed to accomplish the sweep distance for both the power piston and the displacer piston. For both pistons, the sweep distance is ¼ inch. The displacer piston had two rod connections to the crankshaft, equally spaced from the central rod connection to the power piston. The diameter of the crankshaft should be capable of handling the expected loads transferred from the rods, which is expected to be 500 pounds from the power piston, and 1 pound from the displacer piston based on a zero weight assumption. The crankshaft is expected to rotate at 950 RPM, based on literature review of like engines (similar volume and power output).
Design of Rods The rods were designed to withstand the maximum loading expected in the engine. For the power piston, this is the cross sectional area multiplied the maximum pressure of the engine, which comes to approximately 1400 pounds.
Design of alternator The alternator will be a commercial off the shelf part. The power output of the alternator will be matched to the power output of the Stirling engine, 2.7 kWe.
Operating pressure The operating pressure of the Stirling engine was first assumed to match the NASA Rhombic Drive GPU-3, and then optimized to the solar Stirling engines operating temperatures and power input. The equation set used to find the pressure is contained within the engineering analysis portion of the report. The resulting pressure is 3.7 Mpa (500 psi).
Working Fluid The working fluid chosen for the Stirling engine is helium. Helium was chosen because of its cost, non-toxicity, and elimination for the need of environmental controls. The use of compresses helium makes any recharging of the engine in remote locations or disaster areas more feasible, and any implemented package would contain a recharge bottle for in-field recharging.
Mass of Working fluid The mass of the working fluid was found through applying the ideal gas law to the total engine volume at 500 psi, which yielded 1.4g.
Operating Temperatures The expected operating temperatures were derived from a thermal analysis, contained within the engineering design section, and are expected to be 675°C for the hot end, and 25 °C for the cold end. The resulting regenerator temperature is 272 °C, based on the Log-Mean Temperature. 47 | P a g e
CAD Rendering of Engine Figure 30 shows the CAD rendering of the outside of the designed Stirling engine.
Figure 30. Designed Stirling Engine.
Kinematic Analysis and Animation Figure 31 shows a cutaway view of the engine at three displacer piston positions, exposing the cold end, mid (power) stroke, and exposing the hot end. These images were taken from an animation of the engine used to verify that there were no unforeseen internal volume conflicts or collisions of components.
Figure 31. Cutaway Views of Designed Stirling Engine.
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Cooling Reservoir The primary idea of the cooling of the engine would be to locate the engine on the bank of a river, stream, or bay in which there is constantly moving water. In the event that this is not possible, a tarp for the creation of a cooling channel will be created. The size of the cooling reservoir was chosen based on the selected solar concentrator to dissipate 70% of the collected solar energy (assuming 30% engine efficiency) without going over 120°F (40°C). Based on a steady state energy balance at maximum input,
The heat transfer coefficient between the water surface and the ambient air is based on a slight breeze, which would result in a value of h of around 24W/m^2 K. A reservoir 3 meters by 10 meters should be capable of dissipating the heat assuming that there is some thermal capacitance of the reservoir to handle the period of time for maximum heat input. Figure 32 shows a diagram of the cooling reservoir with the relative placement of the solar concentrator and Stirling engine.
Figure 32. Diagram of cooling Reservoir
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Engineering Design and Analysis Calculating Energy from Sunlight In calculating the amount of sunlight that would be collected, a 12 hour period was selected. The energy that can be obtained from sunlight is dependent on several factors such as position on globe, surface area, and the Earth‟s orbit since days are dependent on rotation while seasons are on orbit. The energy from sunlight is a function of time, area, and incident sunlight (Mazza). 2
2
J = (W/m ) x (Area in m ) x (Time in sec)
The incident sunlight value that corresponds to having the sun directly overhead and at high noon would be the equivalent to the solar constant whose value is 1353 . Allowing the solar collector to have a surface area of a square meter and exposing it to sunlight for 12 hours, the energy incident from a square meter solar collector who is oriented perpendicular to the sun is given by the equation below.
However, this value assumed that the sun is directly overhead for 12 hours, which is a false assumption. The sun moves from the East to the West throughout the day and from North to South over the course of the year. It is also known that the sun moves ± 23.5° above and below the equator over the course of a typical year. The Sun‟s position north of the equator, , is found by using the following equation:
The value for is given by the number of days from the vernal equinox which is April 21. This means that will be negative for winter months. Using this correct value for the sun‟s position throughout the year, a new solar constant can be found.
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Analysis of Solar Collector The solar collector chosen for this design was based on the requirement to fit within a 3 foot square box, and to supply as much power as possible. Figure 33 shows a CAD rendering of the collector above the cooling reservoir.
Figure 33. Designed Solar Concentrator
The plot showing this relationship of power collected, dissipated and converted with respect to hour of sunlight is shown in Figure 34. In order to determine the size of the cooling reservoir, an iterative approach was taken, altering the dimensions of the cooling reservoir until a certain boundary condition, reservoir temperature, was reached. Energy Colected
12000
Energy Dissipated
Power, Watts Th
10000
Energy Converted
8000 6000 4000 2000 0 0
2
4 6 Hour of Sunlight (Hr)
8
10
Figure 34. Power Flows for 2.7 kW Stirling Engine
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Analysis of Cooling Reservoir Size The Cooling reservoir should be of sufficient size to not allow the cold end of the engine to go above 40°C. 8000
Heat Flow Rate, Wth
6000 4000 2000 0 0
2
-2000
4 Heat Input (W)
6
8
10
8
10
Heat Rejected (W)
-4000
Heat Stored (W)
-6000
Hour of Sunlight, Hr Figure 35. Heat Flows for Cooling Reservoir
45.0 40.0
Temperature, C
35.0 30.0 25.0 20.0 15.0 Ambient Temp (C)
10.0
Reisvour Temperature
5.0 0.0 0
2
4 6 Hour of Sunlight, Hr
Figure 36. Temperatures of Ambient Air and Cooling Reservoir
The resulting dimensions to achieve the boundary condition are, Table 6. Characteristics of Cooling Reservoir
Width (m) Length (m) Depth (m) Cubic m
3.0 5.0 0.333 5.49
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Calculation of Time of Local Sunrise and Sunset for Autonomous Tracking Capabilities The solar tracking system will be capable of tracking the sun anywhere in the world based on four inputs; Date, Time, Latitude, and Time Zone Offset. There will be a Date/Time chip that will feed input data to the control PLC for the tracking system. The following algorithm used to calculate the local sunrise or sunset is taken from the Almanac for Computers, United States Naval Observatory, 1990. The algorithm assumes that the calculations will be carried out in degrees; therefore a conversion factor of (π/180) should be multiplied to the argument of all trig functions.
Calculation of Time of Local Sunrise To begin, we calculate the Julian Date (N),
Then we convert the longitude to an hour value in order to approximate the time (t) in order to calculate the Sun‟s mean anomaly (M), and true longitude (L). The Suns true longitude may need to be brought back into the range of 0 to 360 b adding or subtracting 360.
The Sun's right ascension (RA) and conversion into hours is as follows; again, the RA may need to be brought back into the range of 0 to 360 b adding or subtracting 360.
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The Suns declination (sinDec and cosDec) as well as the local hour angle (cosH) , and conversion into hours (H) is calculated as follows,
Therefore the local mean time (T) of the sunrise is,
And adjust back to UTC time, The UTC time may need to be brought back into the range of 0 to 24 b adding or subtracting 24.
Finally including the local time zone offset in order to find the local sunrise time,
Calculation of Time of Local Sunset To begin, we calculate the Julian Date (N),
Then we convert the longitude to an hour value in order to approximate the time (t) in order to calculate the Sun‟s mean anomaly (M), and true longitude (L). The Suns true longitude may need to be brought back into the range of 0 to 360 b adding or subtracting 360.
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The Sun's right ascension (RA) and conversion into hours is as follows; again, the RA may need to be brought back into the range of 0 to 360 b adding or subtracting 360.
The Suns declination (sinDec and cosDec) as well as the local hour angle (cosH) , and conversion into hours (H) is calculated as follows,
Therefore the local mean time (T) of the sunset is,
And adjust back to UTC time, The UTC time may need to be brought back into the range of 0 to 24 b adding or subtracting 24.
Finally including the local time zone offset in order to find the local sunset time,
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Engine Adiabatic Analysis The model used to analyze the engine is a variable pressure, variable temperature, and variable volume model. The equation set was developed by Berchowitz in 1984 and leads to a system of six simultaneous differential equations as the solution of the engine.
Nomenclature Table 7. Nomenclature Used for Adiabatic Stirling Engine Analysis
Symbol Tc Tk Tr Th Te p Dp M mc mk mr mh me Dmc Dmk Dmr Dmh Dme gAck gAkr gArh gAhe W Qk Qr Qh DW DQk DQr DQh
Description Temperature of Working Gas within the compression space Temperature of Working Gas within the cooler Temperature of Working Gas within the regenerator Temperature of Working Gas within the heater Temperature of Working Gas within the expansion Pressure of the Working Gas Change in Pressure Total Mass of Working Gas Mass of Working Gas within the compression space Mass of Working Gas within the cooler Mass of Working Gas within the regenerator Working Gas Mass within the heating Working Gas Mass within the expansion space Change in mass of the compression space Change in mass of the cooler Change in mass of the regenerator Change in mass of the heater Change in mass of the expansion space Mass flow rate from compression space to cooler Mass flow rate from cooler to regenerator Mass flow rate from regenerator to heater Mass flow rate from heater to expansion space Work Done by the engine Energy flow rate from cooler to working Gas Energy flow rate from regenerator to working Gas Energy flow rate from heater to working Gas Change in work done by the engine Change in Energy flow rate from cooler to working Gas Change in Energy flow rate from regenerator to working Gas Change in Energy flow rate from heater to working Gas
Units Kelvin Kelvin Kelvin Kelvin Kelvin Pa Pa/s kg kg kg kg kg kg kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s J J J J J/s J/s J/s J/s
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Background In order to model the engine, an adiabatic process was assumed, in which the pressure and volume are not constant for the entire cycle. Shown in Figure 37 is a PV plot, as well as volume and pressure plot versus crank angle.
Figure 37. Adiabatic Cycle (Berchowitz, 1984)
A numerical approach was taken to solve the sets of linear ordinary differential equations in which the model was solved as an „Initial Value Problem‟ where the operating characteristics were chosen within the engines expected range. The model is actually a Boundary Condition problem; however through running the model through successive engine cycles, a steady state condition should be reached. The steady state condition will replace the initial values chosen for the operating characteristics as the boundary conditions (Berchowitz, 1984). The method chosen to solve the linear sets of ODE‟s was a 4th order Runge-Kutta. This has a step error to the fourth power and is the most commonly and widely used method for this type of analysis (Berchowitz, 1984). 57 | P a g e
Development of Equation Set Below is the equation set used for the adiabatic analysis of the Stirling cycle developed by Berchowitz, 1984. The equation set is based on the model of the Stirling cycle shown in Figure 36.
Figure 38. Stirling Engine Used in Development of Equation Set (Berchowitz, 1984)
To begin the analysis, certain assumptions must be made: 1. 2. 3. 4. 5.
The mass of the working fluid remains constant Use of Ideal Gas The speed of the engine is constant Cyclic state Kinetic and potential energy of the working fluid can be neglected
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The work done by the engine can be defined as follows, looking only at the expansion, contraction space, or regenerator as a control volume;
Becomes,
The Ideal Gas law
Keeping in mind,
Taking the log of both sides of the ideal gas law and differentiating,
(i) Keeping in mind that the total mass of working fluid in the engine never changes and is defines as, Differentiating,
(ii)
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Assuming a constant volume and temperature for the heat exchangers reduce to,
(iii) Applying principal (iii) to the constant volume terms, the cooler, regenerator, and heater,
Substituting the differential ideal gas law, (i),
Applying the control volume energy equation at the compression space, we are able to eliminate and , yielding an equation of
Since the compression space is adiabatic,
the work done is,
From the conservation of mass, the accumulation of working fluid is equal to the mass entering the control volume ( ),
Applying the ideal gas law,
Similarly for the expansion space,
Substituting both differential compression and mass equations into the differentiated mass equation (ii),
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Yielding,
Taking the generalized mass flow
we can define all the flows within the engine,
The total work done by the engine, in differential form is,
Based on the energy balance of a controlled volume within the engine,
The energy equations for the hot end, regenerator, and cold end become,
In which the temperature of the gas leaving the regenerator is,
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Adiabatic Stirling Engine Model Set of Differential and Algebraic Equations Table 8. Adiabatic Stirling Cycle Differential and Algebraic Equations (Berchowitz, 1984)
Pressure
Temperatures
Energy
Conditional Temperatures If Else,
If Else, Masses
Mass Accumulations and Flows
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Solution The resulting solution is a set of 6 simultaneous differential equations in which p, mc, W, Qk, Qr, and Qn needs to be solved for (Berchowitz, 1984).
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Implementation of Developed Model The model developed by Berchowitz in 1984 was implemented in order to validate the volumes and find the operating pressure for the engine. The desired Outcomes from computer model of Stirling Cycle are as follows: 1 – Solution of operating pressure to match desired power output. 2 – Verification of engine volumes Table 9 shows the values for the constant terms used for the analysis. Various other properties, such as compression and expansion clearance and sweep volumes can be found in Table5. Table 9. Constants Used for Stirling Cycle Simulation
Description Working Fluid Mass Individual Gas constant Volume in cooler Volume in Regenerator Volume in Heater Cold End Temperature Hot End Temperature
Constant M R Vk Vr Vh Tk Th
Value 2.8E-05 2077 14*10^-6 51*10^-6 70*10^-6 300 950
Units g J/m^2K m^3 m^3 m^3 K K
Figure 39 shows the Work done by the engine per cycle. The simple average of the work per cycle is taken in order to be used for the operating frequency analysis. These results are based on a steady state heat flow rate, and do not apply to any other operating point other than the imposed peak performance. 3
2
Watts
1
0
-1
-2
-3
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 39. Work per cycle
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Figure 40 shows the work done by the compression space for one cycle. Figure 41 shows the work done by the expansion space for one cycle. These results are based on a steady state heat flow rate, and do not apply to any other operating point other than the imposed peak performance. 4 3.5 3
Watts
2.5 2 1.5 1 0.5 0
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 40. Work Done by Compression Space for Single Cycle
1 0.5 0 -0.5 -1
Watts
-1.5 -2 -2.5 -3 -3.5 -4
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 41. Work Done By Expansion Space for Single Cycle
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Figure 42 shows the volume of the compression space for one cycle. Figure 43 shows the volume of the expansion space for one cycle. These results are not based on a steady state heat flow rate, and apply to all other operating point since the mass of working fluid in the engine does not change. -5
16
x 10
14
Cubic Meters
12
10
8
6
4
2
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 42. Compression Space Volume
-5
16
x 10
14
Cubic Meters
12
10
8
6
4
2
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 43. Expansion Space Volume
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Figure 44 shows the internal pressure within the engine for one cycle. This result is based on a steady state heat flow rate with certain resulting temperatures, and do not apply to any other operating point other than the imposed peak performance. For further analysis, the simple average of the pressure was used. 5
1.5
x 10
1.4 1.3
Pascals
1.2 1.1 1 0.9 0.8 0.7
0
100
200
300
400
500
600
700
Crank Angle (1/100 radian)
Figure 44. Pressure During a Single cycle
Results to Desired Outcomes from computer model of Stirling Cycle: 1 – Solution of operating pressure to match desired power output. Pressure =3.7 Mpa 2 – Verification of engine volumes Compression space and expansion space volumes altered to match desired power output During the analysis various working fluids were implemented in order to solve for an engine with the desired power output and crankshaft frequency. Initial variations focused on air, nitrogen, carbon dioxide, and helium. Final variations altered the mixture of nitrogen and helium, with the final iteration concluding with pure helium. Along with variations in the working fluid, the geometry of the compression and expansion space was varied in order to assist the variations of the working fluid match the desired operating conditions (~400rpm @ 1atm). The final iteration of the volumes led to a 63% reduction in heater and compression volume in order to create a net work output per cycle of the engine.
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Calculation of Operating Frequency Along with the modeling of the Stirling cycle, an equation was inputted with the equation set in order to derive the operating frequency of the engine. The derived equation calculated the amount of heat energy the mass of the working fluid can absorb per cycle, then divides that quantity into the total energy transferred per second (W=J/s).
The resulting operating frequency of the designed Stirling engine is 139.15 rpm. This was somewhat brought about through the variance of working fluid and internal volumes.
Discussion The development and implementation of the Stirling cycle engine analysis was essential in the design of the engine volumes, selection of the working fluid, and required pressure to meet the desired power output. The analysis led to the reduction of expansion and compression space, the calculating of the internal working pressure, and the selection of the working fluid necessary to achieve a desired power output.
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FVM Isothermal Analysis The analysis of the first engine design was based upon the assumptions of the Analytic Isothermal Analysis a finite volume method analysis was conducted with the PISO-SIMPLE dynamic mesh motion solver PimpleDyMFoam included with OpenFOAM 1.6 In order to present the solver, solution obtained for this case, and verification of the solver and solution what follows is an overview of the solver, a detailing of the temporal and spatial discretization schemes, the pressure, temperature, and velocity field boundary conditions. In appendix of the report, the code for the top level applications of OpenFOAM 1.6.x are included. Solver Overview PimpleDyMFoam is a transient solver for incompressible, flow of Newtonian fluids on a moving mesh using the PIMPLE (merged PISO-SIMPLE) algorithm. Equations solved
fvVectorMatrix UEqn ( fvm::ddt(U) + fvm::div(phi, U) + turbulence->divDevReff(U) ); solve(UEqn == -fvc::grad(p));
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Descritization Schemes Employed ddtSchemes
Euler
gradSchemes grad(p)
Gauss Linear divSchemes
div(phi,U)
Gauss Linear
div((nuEff*dev(grad(U).T())))
Gauss Linear laplacianSchemes
laplacian(nu,U)
Gauss linear corrected
laplacian(rAU,pcorr)
Gauss linear corrected
laplacian(rAU,p)
Gauss linear corrected
laplacian(diffusivity,cellMotionU)
Gauss linear uncorrected
laplacian(nuEff,U)
Gauss linear uncorrected interpolationSchemes
interpolate(HbyA)
linear snGradSchemes
default
corrected fluxRequired
default
no
pcorr p
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Figure 45 Boundary patch names
Velocity Boundary Conditions Power type
timeVaryingUniformFixedValue
filename
stirlingEngine/smoothVectorPower
outOfBounds
clamp Displacer
type
timeVaryingUniformFixedValue
filename
stirlingEngine/smoothDisplacerPower
outOfBounds
clamp powerWalls, displacerWalls, farfield, walls, hot, cold
type
fixedValue
value
uniform (0 0 0)
Pressure Boundary Conditions Power , Displacer, powerWalls, displacerWalls, farfield, walls, hot, cold type
zeroGradient
Isothermal Transient Startup Simulation Results The recommendation from (Berchowitz, 1984) to run transient startup simulations for 10 complete cycles to observe steady state operation was followed. Below are figures presenting the velocity and pressure fields at the end of the 9th and 10th cycles. Performance charts of the 9th and 10th cycles of a transient startup simulation are also presented. The figures show that the simulation has indeed converged to steady state operation despite the slight transient asymmetries.
Figure 46: Velocity Field from the end of the 9th cycle of the isothermal transient startup simulation
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Figure 47: p/rho Field from the end of the 9th cycle of the isothermal transient simulation
Figure 48: Prototype Isothermal Simulation 9th cycle Displacer Piston
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Figure 49: Prototype Isothermal Simulation 9th cycle Power Piston
Figure 50: Prototype Isothermal Simulation 9th cycle Summary
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Figure 51: Velocity Field from the end of the 10th cycle of the isothermal transient startup simulation
Figure 52: p/rho Field from the end of the 10th cycle of the transient startup simulation
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Figure 53: Prototype Isothermal Simulation 10th cycle Displacer Piston
Figure 54: Prototype Isothermal Simulation 10th cycle Power Piston
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Figure 55: Prototype Isothermal Simulation 10th cycle Summary
Engine Geometry Optimization 1: Isothermal Analysis Utilizing a differential evolutionary algorithm coded by Stephen Wood as part of EML 5509 Mechanical Design Optimization enabled a the optimization of two parameters controlling engine geometry with the objective of reducing the total energy lost to pressure drag during one cycle of the engine. Initial optimization was conducted on the engine cylinder‟s shoulder geometry utilizing the isothermal analysis and was begun February 16, 2010 on FIU‟s MAIDROC Laboratory‟s Tesla-128 Cluster. A population of 20 designs was evaluated through 32 generations. Each function evaluation consists of the simulation of an entire engine cycle of the isothermal simulation with an initial flow field mapped from the 10th cycle of the isothermal transient startup analysis. The OpenFOAM simulation of the engine cycle is followed by evaluation scripts which calculated the total energy lost due to pressure drag during the cycle.
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Initial Design:
Figure 56: Prototype Design1 Optimization Initial Design
The initial design looses 0.642916 J per cycle to pressure drag.
Intermediate Designs:
Figure 57: Prototype Design1 Optimization Generation 1
Gen 1 The best member of the first generations has chamfers of 0.28783 in. and 0.255793in. and looses 0.20939 J per cycle to pressure drag. 78 | P a g e
Figure 58: Prototype Design1 Optimization Generation 14
The best member of the 14th generation has chamfers of 0.190714in. and 0.215189in. and looses 0.132705 J per cycle to pressure drag. 43% of the population scores within 10% of the best value.
Figure 59: Prototype Design1 Optimization Generation 25
The best member of the 25th generation has chamfers of 0.123271in. and 0.338246in. and looses 0.059115 J per cycle to pressure drag. 50 % of the population is scores within 10% of the best value.
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Optimized Design:
Figure 60: Prototype Design1 Optimization Generation 32
The best member of the 32 generation Gen 32has chamfers of 0.309300in. and 0.24166in. and looses 0.0376356 J per cycle to pressure drag. 72% of the population scores within 10% of the best value. The optimization of this first stirling engine was halted after 32 generations when the results of the analytic adiabatic analysis showed that a new design was needed.
FVM Adiabatic Analysis Analysis of the second design is based upon the assumptions of the Analytic Adiabatic Analysis a finite volume method analysis was conducted with the merged PISO-SIMPLE dynamic mesh motion solver rhoPorousPimpleDyMFoam developed with OpenFOAM 1.6.x. The regenerator within the engine consists of a porous material surrounding the displacer piston which translates with it. The regenerator improves the efficiency of the engine by pre-heating and pre-cooling the working fluid as it is shuffled from the cold end to the hot end of the engine and back again by the displacer piston. Solver Overview rhoPorousPimpleDyMFoam is a transient solver for compressible, flow of Newtonian fluids on a moving mesh using the PIMPLE (merged PISO-SIMPLE) algorithm. This solver also has the capability of considering the impact of mobile porous regions on the flow field. Porous regions are modeled with a Darcy-Weisbach Friction factor added to the momentum equation. Equations solved 80 | P a g e
// Momentum equation
tmp
UEqn ( pZones.ddt(rho, U) + fvm::div(phi, U) + turbulence->divDevRhoReff(U) ); pZones.addResistance(UEqn()); volScalarField rUA = 1.0/UEqn().A(); solve(UEqn() == -fvc::grad(p)); //Pressure Equation
rho = thermo.rho(); volScalarField rUA = 1.0/UEqn().A(); U = rUA*UEqn().H(); fvScalarMatrix pEqn ( fvm::ddt(psi, p) + fvm::div(phid, p) - fvm::laplacian(rho*rUA, p) ); pEqn.solve();
//Enthalpy Equation
fvScalarMatrix hEqn 81 | P a g e
( fvm::ddt(rho, h) + fvm::div(phi, h) - fvm::laplacian(turbulence->alphaEff(), h) == DpDt ); hEqn.solve();
Mesh Independence The finite volume analysis of the adiabatic model was conducted on a coarse and a fine mesh to inspect the accuracy of the solutions obtained per the guidelines established in (ASME, 2009)Error! Reference source not found., below, presents the numeric attributes of both meshes. This reveals the relative difference of the pressures and temperatures on the power piston obtained from the 15th cycle of a transient startup analysis conducted on each mesh. Table 10: Mesh Statistics
Mesh Statistics
Coarse
Fine
Points
10462
29912
Internal points
0
0
Faces
20,080
58,380
Internal faces
9,620
28,470
Cells
4,950
14,475
Boundary patches
12
12
Point zones
0
0
Face zones
22
22
Cell zones
6
6
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Figure 61:Fine and Coarse Mesh Comparison
As seen in figure 61, the pressure varies less than 3.4% between the two solutions and the temperature varies less than 8% between the two solutions. The small deviation between the solutions indicated that the results are mesh independent. All further analysis and optimization was conducted using the fine mesh.
Adiabatic Transient Startup Simulation Results We followed the recommendation from (Berchowitz, 1984) to run transient startup simulations for 10 complete cycles to observe steady state operation and found that at 860 RPM 15 cycles were needed to reach clear steady state operation. Below are figures presenting the velocity and pressure fields at the end of the 14th and 15th cycles. Performance charts of the 14th and 15th cycles of a transient startup simulation are also presented. The figures and charts show that the simulation has indeed converged to steady state operation despite the slight transient asymmetries.
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Figure 62: Design 2 Transient Startup Pressure vs. Time
Figure 63: Design 2 Transient Startup Temperature vs. Time
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Figure 64: Design 2 Transient Startup Velocity Magnitude vs. Time
Engine Geometry Optimization 2: Adiabatic Analysis Utilizing a differential evolutionary algorithm coded by Stephen Wood as part of EML 5509 Mechanical Design Optimization enabled a the optimization of two parameters controlling engine geometry. The objective of the optimization is to reduce the total energy lost to pressure drag during one cycle of the engine. The first parameter controls the aspect ratio of the regenerator. The second controls the aspect ratio of the cold end. The volumes of both regions are maintained by constraint equations throughout to preserve the correlation with the analytic adiabatic analysis. Initial optimization was conducted on the engine cylinder‟s shoulder geometry utilizing the isothermal analysis and was begun March 16, 2010 on FIU‟s MAIDROC Laboratory‟s Tesla-128 Cluster. A population of 20 designs were evaluated through 70 generations. Each function evaluation consists of the simulation of an entire engine cycle of the isothermal simulation with an initial flow field mapped from the 15th cycle of the adiabatic transient startup analysis. This simulation takes between 1 hour and 1 and a half hours to run on one of Tesla‟s 1.3 GHz processors. Each generation was evaluated in parallel so that the total clock time required for the run was 87.5 hours. The OpenFOAM simulation of the engine cycle is followed by the evaluation scripts which calculated the total energy lost due to pressure drag during the cycle. The codes for the optimizer and the evaluation scripts are included in Appendix E. Optimization Codes. 85 | P a g e
The initial design was created directly from the results of the analytic adiabatic analysis. The Regenerator and Cold End volumes specified in the results are maintained throughout the optimization process by constraint equations included in the setStirlingGeometry.C file.
Figure 65: Prototype Design2 Initial Design with parameters denoted
Initial Design Cold End Parameter
0.000
Regenerator Parameter
0.000
Energy lost per cycle
2.584 J
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Z
Energy lost per cycle (J)
X
Y
2
1 -8 -6
Cold
-4
E nd P
0 -2
0 aram eter (
0.5
mm)
) r (mm e t e m ara ator P r e n e g 1
2
Re
Figure 66: Initial Optimization Population
Figure 67: Stirling Helium Geometry Design Space after 4 Generations
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Z
Energy lost per cycle (J)
X
Y
2
1 -8 -6
Cold
-4
E nd P
0 -2
0 aram eter (
0.5
mm)
m) ter (m e m a r a rator P egene 1
2
R
Figure 68: Optimization Population after 15 Generations
Figure 69: Prototype Design2 Optimization Generation 15
Best Member of Genration 15 is member 3 Cold End Parameter
1.160 mm
Regenerator Parameter
1.452 mm
Energy lost per cycle
0.739 J
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Z
Energy lost per cycle (J)
X
Y
2
1 -8 -6
Cold
-4
E nd P
0 -2
0 aram eter (
0.5
mm)
2
m) eter (m m a r a rP nerato R ege 1
Figure 70: Optimization Population after 30 Generations
Figure 71: Prototype Design2 Optimization Generation 30
Best Member of Genration 30 is member 18 Cold End Parameter
1.267 mm
Regenerator Parameter
2.431 mm
Energy lost per cycle
0.694 J
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Z
Energy lost per cycle (J)
X
Y
2
1 -8 -6
Cold
-4
E nd P
0 -2
0 aram eter (
0.5
mm)
2
m) eter (m m a r a rP nerato R ege 1
Figure 72: Optimization Population after 45 Generations
Figure 73: Prototype Design2 Optimization Generation 45
Best Member of Generation 45 is member 19 Cold End Parameter
1.269 mm
Regenerator Parameter
2.530 mm
Energy lost per cycle
0.690 J
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Z
Energy lost per cycle (J)
X
Y
2
1 -8 -6
Cold
-4
E nd P
0 -2
aram eter (
0.5
0
mm)
(mm) meter a r a P rator egene 1
2
R
Figure 74: Optimization Population after 70 Generations
Figure 75: Prototype Design2 Final Design
Final Design from Generation 70 Population is member 8 Cold End Parameter
1.270 mm
Regenerator Parameter
2.513 mm
Energy lost per cycle
0.689 J
100% of the 70th generation scored within 10% of the best member of the population. 91 | P a g e
From the tight clustering observed the in the final 10 generations about the final design point we can conclude that based upon the assumptions of the model and the tolerances applied within the optimization process that the design shown in Figure 5 is the best design.
Solution Dependant Motion Research into modeling solution dependant motion was begun following the successful start of optimization runs based on the adiabatic model from chapter 12.8 on Reciprocating Engine Dynamics in (Burton, 1979). Following the recommendations for nomenclature presented we posed the system of ordinary differential equations as:
Posed as an Initial Value Problem
where: =Pressure Force on ith piston i=1 denotes the power Piston, i=2 the displacer piston = generator resisting torque
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= moment of inertia about G The results obtained from the matlab implementation of the problem are shown below:
Figure 76: Theta (Displacer Piston Crank Angle) and Phi (Power Piston Crank Angle)
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Figure 77: w (Crank Speed) vs. time and theta
Figure 78:Displacer piston position vs. time and theta
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Figure 79: Power piston position vs. time and theta
The matlab code is included in Appendix F. Solution Dependent Motion Codes. The results indicate that the problem is well posed and suitable for modeling the response of the pistons to an input pressure force. Future work will include the coupling of the ODE model with rhoPorousPimpleDyMFoam, the CFD solver.
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Material Selection Engine: Below are the material requirements for various portions of the Stirling engine, and the selected material to meet the requirements. Hot End The hot end of the engine needs to withstand 975K, 300 psi internal pressure with a complex interior geometry, conduct heat effectively, be as absorptive as possible of thermal radiation, and be as inexpensive as possible. To meet these requirements, we chose a commercial bronze (k=420) coated in parsons black paint. The extended surface of the hot end needs to conduct heat effectively and be as inexpensive as possible. To meet these requirements, a bronze mesh/foam material was chosen. The extended surface wall needs to be thermally nonconductive and as inexpensive as possible. To meet these requirements, we chose a ceramic disc. Hot End - black Hole The on top of the solar absorption pate will be a hemispherical structure constructed of plastic rod and aluminum foil to reflect back all diffused radiation. Cold End The cold end of the engine needs to withstand 350K, 300 psi internal pressure, conduct heat effectively, and be as inexpensive as possible. To meet these requirements, we chose generic Aluminum Alloy. Crank Shaft The crank shaft of the engine needs to withstand 400K, loads of approximately 1400 pounds-force, rotate at 950 rpm, and be as inexpensive as possible. To meet these requirements, we chose 3/8” cast alloy steel. Rods The rods of the engine need to withstand 400K, loads of approximately 600 pounds-force and be as inexpensive as possible. To meet these requirements, we chose 1050 alloy steel. Body The body of the engine needs to withstand 700K, 500 psi internal pressure, be thermally non-conductive, and be as inexpensive as possible. To meet these requirements, we chose Grade G-10 Garolite.
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The bolts holding the engine together need to withstand 800K, 1500 poundsforce, be thermally non-conductive, and be as inexpensive as possible. To meet these requirements, we chose 18-8 Stainless Steel. Displacer The base plate of the displacer piston needs to withstand 400K, 600 pounds-force, be thermally non-conductive, and be as inexpensive as possible. To meet these requirements, we chose .125” Lexan. The pin connection the displacer piston to the crankshaft needs to withstand 400K, 600 pounds-force, be thermally non-conductive, and be as inexpensive as possible. To meet these requirements, we chose .25” 1050 alloy steel. The walls of the displacer piston need to withstand 700K, be thermally nonconductive, and be as inexpensive as possible. To meet these requirements, we chose Buna-N foam rubber. The surface of the walls of the displacer piston need to withstand 700K, be thermally conductive, and be as inexpensive as possible. To meet these requirements, we chose aluminum flashing. Power Piston The power piston of the engine needs to withstand 350K, 600 pounds-force, 300 psi, be thermally conductive, and be as inexpensive as possible. To meet these requirements, we chose generic Aluminum Alloy. The working fluid deflector /cover of the power piston of the engine needs to withstand 400K, be thermally non-conductive, and be as inexpensive as possible. To meet these requirements, we chose Buna-N foam rubber.
Collector: Below are the material requirements for various portions of the solar collector, and the selected material to meet the requirements. Lens The lenses of the solar collector need to withstand 45 mph winds, be transmittive of thermal radiation, and be as inexpensive as possible. To meet these requirements, we chose cast acrylic. Support The support structure of the solar concentrator needs to be capable of withstanding 3.6 pounds-force, be assembled easily, and be as inexpensive as possible. To meet these requirements, we chose 1.25” right angle steel.
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Thermal Analysis Steady State Heat Transfer Model A simple steady state heat transfer analysis was preformed on the stirling engine in order to derive the hot end temperatures. For this analysis the regenarator and the intermediate air was neglected, due to both components changing and having the same properties at the end of a cycle. Figure 80 shows the developed thermo-resistance diagram used. The cold end of the engine was treated as a radial system, the hot end of the engine was treated as a linear system.
Figure 80. Steady State Thermal Diagram of Stirling Engine
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The equivalent thermal resistance is,
Using the thermal resistance equation,
The resulting temperature difference is,
With a cold end temperature of 300K, the resulting hot end temperature at peak performance is 650°C. This will be compared to the developed computer model in order to verify the result. If the numbers are close, then the computer model will be used as it more accurately captures the expected conditions.
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Computer BasedSteady State Hot End Temperature In order to determine the temperatures of the engine, a Cosmo Works model of the engine was build and the expected conditions for the engine were inputted. The convective coefficients were derived from the CFD modeling of the interior of the engine. Convective heat transfer was split into two areas of the engine due to the loss in velocity of the working fluid after going through the hot end extended surface and subsequent lowering of the convective transfer coefficient, as well as the increase in working gas temperature. The resulting hot end temperatures were used for the isothermal MATLAB modeling of the engine. The radiative heat transfer from the unpainted surfaces were neglected due to the emissivity of commercial copper being negligible (ε = .045). Table 11 shows the expected conditions at peak performance. Figure 40 shows the solid mesh used in the analysis along with the areas of the imposed conditions. Figures 41 and 42, show the resulting temperature distributions within the hot end. Table 11. Initial Imposed Thermal Conditions
Imposed Conditions Heat Power In Convection Radiation Convection Convection
Magnitude 2700 W h =10 W/m^2 K T∞ = 300K T∞ = 300K View Factor=0.5 ε = 0.98 h = 80 W/m^2 K T∞ = 375K h = 120 W/m^2 K T∞ = 350 K
Location Center of top surface Top and side surfaces Top surface Interior surface Extended mesh surface
Figure 81. Hot End Mesh and Imposed Conditions
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Figure 82. Thermal Plot o Lower End of Hot End
Figure 83. Thermal Plot of Upper Portion of Hot End
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Discussion After the hot end of the engine was successfully modeled for peak performance, the power input was varied in order to obtain the hot end temperatures of the engine at different hours of the day. Figure 84 shows the expected hot end temperature versus the hour of the day.
Hot End Temperature Resuling Temperature (°C)
800 700 600 500 400 300 200 100 0 0
2
4 6 8 Hour of the Sunlight (hr) Figure 84. Expected Hot End Temperatures for the 2.7 kW Solar Stirling Engine
The convergence of the thermal circuit analysis and the CAD thermal analysis was extremely good. The difference between the two models was 60 °C for the maximum heat input, less than 10% difference
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Stress Analysis Hot End Stress Analysis Figure 85 shows the resulting Von Mises stress from the expected loading on the hot end of the engine. The highest level of stress is expected to be 66.8 MPa, the yield strength of commercial bronze is 275 MPa, which gives a factor of safety of 4.12.
Figure 85. Stress Analysis of Hot End
Table 12. Imposed Stresses for Stress Analysis of Hot End
Imposed Stress Restraint Internal Pressure Thermal Stress
Magnitude Fixed 500 psi Resultant
Direction Interior surfaces of bolt holes Normal to interior surface Resultant from thermal analysis
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Displacer Piston Base Stress Analysis Figure 86 shows the resulting Von Mises stress from the expected loading on the base of the displacer piston. The highest level of stress is expected to be 114 kPa, the yield strength of Acrylic is 207 kPa, which gives a factor of safety of 1.85.
Figure 86. Stress Analysis of Displacer Piston Base
Table 13. Imposed Stresses for Stress Analysis of Displacer Piston Base
Imposed Stress Restraint Internal Pressure Thermal Stress Force
Magnitude Fixed 20 psi Resultant 60 lbs-f
Direction Interior surfaces of bolt holes Normal to lower surface Resultant from thermal analysis Load from power piston
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Engine Body Stress Analysis Figure 87 shows the resulting Von Mises stress from the expected loading on the body of the engine. The highest level of stress is expected to be 161 kPa, the yield strength of Acrylic is 207 kPa, which gives a factor of safety of 1.29.
Figure 87. Stress Analysis of Engine Body
Table 14. Imposed Stresses for Stress Analysis of Engine Body
Imposed Stress Restraint Restraint Internal Pressure Thermal Stress
Magnitude Fixed Cylindrical 300 psi Resultant
Direction Top and bottom surfaces Crankshaft Holes Normal to interior surface Resultant from thermal analysis
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Displacer Piston Rod Stress Analysis Figure 88 shows the resulting Von Mises stress from the expected loading on displacer piston rod for the engine. The highest level of stress is expected to be 171 kPa, the yield strength of Alloy Steel is 241 kPa, which gives a factor of safety of 1.41. Thermal stresses were not considered within this system since it is only bound on one end and any thermal expansion would not induce any significant stresses.
Figure 88. Stress Analysis of Displacer Piston Rod
Table 15. Imposed Stresses for Stress Analysis of Displacer Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Interior surfaces of Crankshaft holes 60 lbs-f Interior surfaces of displacer shaft holes
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Power Piston Rod Stress Analysis Figure 89 shows the resulting Von Mises stress from the expected loading on power piston rod for the engine. The highest level of stress is expected to be 107 kPa, the yield strength of Alloy Steel is 241 kPa, which gives a factor of safety of 2.25. Thermal stresses were not considered within this system since it is only bound on one end and any thermal expansion would not induce any significant stresses.
Figure 89. Stress Analysis of Power Piston Rod
Table 16. Imposed Stresses for Stress Analysis of Power Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Interior surfaces of Crankshaft holes 60 lbs-f Interior surfaces of power piston shaft holes
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Engine Bolts/ Linear Shafts Stress Analysis Figure 90 shows the resulting Von Mises stress from the expected loading on engine bolts/ linear shaft. The highest level of stress is expected to be 2.34 GkPa, the yield strength of Stainless Steel is 6.2 GPa, which gives a factor of safety of 2.65.
Figure 90. Stress Analysis of Engine Bolts/ linear Shafts
Table 17. Imposed Stresses for Stress Analysis of Power Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Lower portion of bolt head 8000 N Top surface of threading
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Crankshaft Stress Analysis Figure 91 shows the resulting Von Mises stress from the expected loading on the crankshaft for the engine. The highest level of stress is expected to be 123 kPa, the yield strength of Cast Alloy Steel is 241 kPa, which gives a factor of safety of 1.96. Thermal stresses were not considered within this system since it is only bound on one end and any thermal expansion would not induce any significant stresses.
Figure 91. Stress Analysis of Crankshaft
Table 18. Imposed Stresses for Stress Analysis of Crankshaft
Imposed Stress Restraint Force Rotation
Magnitude Fixed 600 N 950 RPM
Direction End portions of shaft Central rod connection Central rod connection
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Design Based on Static and Fatigue Failure Design Theories Crankshaft Fatigue Life Analysis Figure 75 shows the resulting Failure Areas of the crankshaft. For the Fatigue analysis, the part was subject to 1*10^6 zero based loadings, the portions of the part that failed during the analysis are shown in red.
Figure 92. Fatigue Life Analysis of Crankshaft
Table 19. Imposed Stresses for Fatigue Life Analysis of Crankshaft
Imposed Stress Restraint Force Rotation
Magnitude Fixed 600 N 950 RPM
Direction End portions of shaft Central rod connection Central rod connection
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Power Piston Rod Fatigue Life Analysis Figure 76 shows the resulting Failure Areas of the power piston rod. For the Fatigue analysis, the part was subject to 1*10^6 zero based loadings, the portions of the part that failed during the analysis are shown in red.
Figure 93. Fatigue Life Analysis of Power Piston Rod
Table 20. Imposed Stresses for Fatigue Life Analysis of Power Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Interior surfaces of Crankshaft holes 60 lbs-f Interior surfaces of power piston shaft holes
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Deflection Analysis Hot End Deflection Analysis Figure 77 shows the resulting deflection from the expected loading on the hot end of the engine. The highest level of deflection is expected to be 11.7 μm.
Figure 94. Deflection Analysis of Hot End
Table 21. Imposed Stresses for Deflection Analysis of Hot End
Imposed Stress Restraint Internal Pressure Thermal Stress
Magnitude Fixed 500 psi Resultant
Direction Interior surfaces of bolt holes Normal to interior surface Resultant from thermal analysis
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Displacer Piston Base Deflection Analysis Figure 78 shows the resulting deflection from the expected loading on the base of the displacer piston. The highest level of deflection is expected to be 1.8 mm.
Figure 95. Deflection Analysis of Displacer Piston Base
Table 22. Imposed Stresses for Deflection Analysis of Displacer Piston Base
Imposed Stress Restraint Internal Pressure Thermal Stress Force
Magnitude Fixed 20 psi Resultant 60 lbs-f
Direction Interior surfaces of bolt holes Normal to lower surface Resultant from thermal analysis Load from power piston
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Engine Body Deflection Analysis Figure 79 shows the resulting deflection from the expected loading on the body of the engine. The highest level of deflection is expected to be 3.0 mm.
Figure 96. Deflection Analysis of Engine Body
Table 23. Imposed Stresses for Deflection Analysis of Engine Body
Imposed Stress Restraint Restraint Internal Pressure Thermal Stress
Magnitude Fixed Cylindrical 300 psi Resultant
Direction Top and bottom surfaces Crankshaft Holes Normal to interior surface Resultant from thermal analysis
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Displacer Piston Rod Deflection Analysis Figure 80 shows the resulting deflection from the expected loading on displacer piston rod for the engine. The highest level of deflection is expected to be 17 μm.
Figure 97. Deflection Analysis of Displacer Piston Rod
Table 24. Imposed Stresses for Deflection Analysis of Displacer Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Interior surfaces of Crankshaft holes 60 lbs-f Interior surfaces of displacer shaft holes
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Power Piston Rod Deflection Analysis Figure 81 shows the resulting deflection from the expected loading on power piston rod for the engine. The highest level of deflection is expected to be 26 μm.
Figure 98. Deflection Analysis of Power Piston Rod
Table 25. Imposed Stresses for Deflection Analysis of Power Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Interior surfaces of Crankshaft holes 60 lbs-f Interior surfaces of power piston shaft holes
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Engine Bolts/ Linear Shafts Deflection Analysis Figure 82 shows the resulting deflection from the expected loading on engine bolts/ linear shaft. The highest level of deflection is expected to be 45 μm.
Figure 99. Deflection Analysis of Engine Bolts/ linear Shafts
Table 26. Imposed Stresses for Deflection Analysis of Power Piston Rod
Imposed Stress Restraint Force
Magnitude Direction Fixed Lower portion of bolt head 8000 N Top surface of threading
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Crankshaft Deflection Analysis Figure 83 shows the resulting deflection from the expected loading on the crankshaft for the engine. The highest level of deflection is expected to be 96 μm.
Figure 100. Deflection Analysis of Crankshaft
Table 27. Imposed Stresses for Deflection Analysis of Crankshaft
Imposed Stress Restraint Force Rotation
Magnitude Fixed 600 N 950 RPM
Direction End portions of shaft Central rod connection Central rod connection
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Cost Analysis Description Flange-Mount Linear Ball Bearing, 1/4" ID SS Ball Bearing,1/4" ID, 3/8" OD Aluminum (Alloy 7075) 5" Diameter, 1/2" Long 18-8 SS Hex Head Screw 1/4"-20, 8-1/2" Length SS Serrated-Flange Hex Locknuts 1/4"-20 Acrylic Tube 5" OD X 4-3/4" ID, 1' Length 150cc Piston and Cylinder, 2.25" bore, 2" stroke Polyurethane Foam Rod 4" Diameter, 36" Length Aramid/Buna-N Gasket, 1/8" Thick, 6" X 6" Aramid/Buna-N Gasket, 1/16" Thick, 6" X 6" Arc Welder Leather Gloves 6-32X1 Steel Bolts 1/4" Steel rod 3' 4 1/8" hole saw bit 1/4" drill bit Lexan sheet 8"X10" 10" Metal Cutting Wheel 3/4 inch shaft collars 4" #6 threaded rod #6 nuts Nylon Bushings Ceramic Spacer Alternator, 960W, 12V Flywheel Assembly Digital Calipers
Each $25.22 $7.00
Unit Amount Total Total 4 $100.88 $100.88 8 $56.00 $56.00
$20.32
1
$20.32
$20.32
$5.19 $3.83 $19.20
4 1 1
$20.76 $3.83 $19.20
$20.76 $0.96 $4.80
$75.00
1
$75.00
$75.00
$10.65 $6.63 $3.57 $125.00 $5.35 $1.04 $3.50 $26.72 $2.75 $4.26 $4.46 $4.88 $10.00 $1.04 $2.03 $3.98 $60.00 $25.00 $15.00
1 1 2 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1
$10.65 $6.63 $7.14 $125.00 $5.35 $1.04 $3.50 $26.72 $2.75 $4.26 $4.46 $29.28 $10.00 $1.04 $2.03 $3.98 $60.00 $25.00 $15.00 $639.82
$0.30 $6.63 $3.57 $0.00 $0.00 $0.10 $1.75 $0.00 $0.00 $4.26 $0.00 $29.28 $1.25 $0.17 $2.03 $3.98 $60.00 $25.00 $0.00 $417.04
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Discussion The relative cost for the solar Stirling engine comes out to 31¢/We. This is compared to current photovoltaic systems costing 3$/We, or other on the market Stirling engines that cost 12.5$/We. This is an extremely low cost, however administrative, manufacturing, and distribution has not been included. The additional costs are not expected to alter the enormous cost savings.
Prototype Construction The design of the prototype and engine was based primarily on accessibility and cost of parts. An approach was taken in which strong bias was used for pre-manufactured parts and commercially available „like items‟ over fabrication of custom parts. The design of the prototype followed design methodology of the actual system in that the solar concentrator was designed first, and the Stirling engine was matched to the heat input of the concentrator.
Description of Prototype Based on a product search, the largest commercially available Fresnel lens that was within reasonable cost was a 2‟X4‟ lens, with a focal length of ~3‟. Due to the relatively small focal length, only one lens could be implemented for the solar concentrator, producing approximately 1.3kWth. The prototype engine should be capable of producing around 300 Watts. We plan on using the same exact plans for the original design, keeping shape and dimensions of the prototype identical to that of the designed engine; however removing some of the pressure of the working gas. The interior volumes will be identical to the designed engine in order to demonstrate the feasibility of fabricating the designed engine.
Prototype Design The prototype engine will be the same design as the original design; however it will not be pressurized.
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Parts List and Analysis Solar Concentrator Parts List Table 28. Parts List and Analysis for Prototype Solar Concentrator
Description Angle steel Fresnel Lenses 3/8" Bolts Hinges 5/8" Hexbar Shaft 8" X 10" Acrylic Sheet 0.6 RPM 50 In-Lbs Mini-Gearmount 20:1 Worm gear speed reducer Key Stock Bungee Chord PLC with programmer
Unit Price $4.00 $70.00 $12.00 $3.79 $6.44 $4.00 $42.00 $100.00 $0.70 $0.15 $15.00
Amount 24 1 1 4 1 1 1 1 1 24 1
Total $96.00 $70.00 $12.00 $15.16 $6.44 $4.00 $42.00 $100.00 $0.70 $3.60 $15.00 $364.90
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Stirling Engine Parts List Table 29. Part List and Analysis for Prototype Engine
Description Flange-Mount Linear Ball Bearing, 1/4" ID SS Ball Bearing,1/4" ID, 3/8" OD Aluminum (Alloy 7075) 5" Diameter, 1/2" Long 18-8 SS Hex Head Screw 1/4"-20, 8-1/2" Length SS Serrated-Flange Hex Locknuts 1/4"-20 Acrylic Tube 5" OD X 4-3/4" ID, 1' Length 150cc Piston and Cylinder, 2.25" bore, 2" stroke Polyurethane Foam Rod 4" Diameter, 36" Length Aramid/Buna-N Gasket, 1/8" Thick, 6" X 6" Aramid/Buna-N Gasket, 1/16" Thick, 6" X 6" Arc Welder Leather Gloves 6-32X1 Steel Bolts 1/4" Steel rod 3' 4 1/8" hole saw bit 1/4" drill bit Lexan sheet 8"X10" 10" Metal Cutting Wheel 3/4 inch shaft collars 4" #6 threaded rod #6 nuts Nylon Bushings Ceramic Spacer 4.6" diameter foam, 6"
Each $25.22 $7.00
Unit Amount Total Total 4 $100.88 $100.88 8 $56.00 $56.00
$20.32
1
$20.32
$20.32
$5.19 $3.83 $19.20
4 1 1
$20.76 $3.83 $19.20
$20.76 $0.96 $4.80
$75.00
1
$75.00
$75.00
$10.65 $6.63 $3.57 $125.00 $5.35 $1.04 $3.50 $26.72 $2.75 $4.26 $4.46 $4.88 $10.00 $1.04 $2.03 $3.98 $13.60
1 1 2 1 1 1 1 1 1 1 1 6 1 1 1 1 1
$10.65 $6.63 $7.14 $125.00 $5.35 $1.04 $3.50 $26.72 $2.75 $4.26 $4.46 $29.28 $10.00 $1.04 $2.03 $3.98 $13.60 $553.42
$0.30 $6.63 $3.57 $0.00 $0.00 $0.10 $1.75 $0.00 $0.00 $4.26 $0.00 $29.28 $1.25 $0.17 $2.03 $3.98 $0.57 $332.61
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Construction Below are photographs of the engine in various fabrication stages, as well as the prototype solar concentrator.
Figure 101. Machining the finned interior finned Surface of the Hot End
Figure 102. Top and Bottom Images of the Solar Stirling Engine - showcasing the inside of the displacer piston, the linear bearings, and finned interior of the hot end
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Figure 103. Construction of Prototype Solar Concentrator
Figure 104. Painting of support structure for the Tracking Fresnel Lens
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Testing and Evaluation Introduction In order to verify and quantify the actual values versus the theoretical values of this project, testing and evaluation will need to be conducted. There two main systems in our project that will need to be tested and evaluated are the Fresnel lenses and the Stirling engine. The testing will be divided into three tests, with the first two tests dealing with the solar concentrator and the last test focusing on the Stirling engine. To test the solar concentrator, the first test will be on the maximum steady state energy gathered. This will be done with the partially built prototype lens, in order to gather some data earlier in the development stage. The second test will be focused on the Fresnel lens with the tracking mechanism. This will be done to verify that the algorithm tracks the sun and rotated at the desired rate. The final test will be on the efficiency of the engine based on a known energy concentration from the previous testing.
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Steady State Concentrator Heat Input The solar thermal radiation reaching any square meter can be calculated via the equations shown in the solar radiation section of the report. For the given dimension of our Fresnel lens, it is anticipated that it the sun reaching will provide a heat output of 13,500 watts. The Fresnel lens will then translate this heat into a concentrated focal point on the hot end. This is the heat that the hot end of the engine will endure. It is expected that the engine will temperature in the range 20 – 700 ˚ C during the course of the day based on the steady state heat transfer analysis. This test will aim to obtain actual values which can then be used to compare with our theoretical as part of our testing and evaluation.
Overview This test will provide us the maximum steady state heat input for the engine. This value will be obtained from the experiment as follows. A prototype hot end will be used to for this experiment. This prototype hot end is comparable to the one used for our design both in material and shape. At the center of the hot end will be a perforated hole in which a thermocouple will be inserted. This will allow for temperature reading at the focal point of the Fresnel lens.
Figure 105. Design of Experiment - Fresnel Lens
The total surface areas of Fresnel lens and the hot end were found to be 0.7 m^2 and 0.024m^2 respectively. Due to the complexity of the geometry for the hot end, the total surface area for the hot end was as follows:
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The hot end was coated with black engine enamel paint which has an emissivity of 0.98 (Frank P. Incropera, 2002). The ambient temperature was selected to be a standard 300 Kelvin. The heat transfer coefficient for convection was selected to be 50 W/m^2*k due the breeze that was encountered in the location of the testing (on the intercostals of Deerfield Beach).
The experimental results where compared with theoretical values obtain from the computational model. The theoretical calculations for obtaining the heat input for the collector are as follows:
Where: = Solar Constant = Efficiency of the lens = Surface Area of the lens Furthermore, the efficiency of the lens is dependent on two variables:
Where: = Efficiency of 32mm Plexiglas = 0.9 = Efficiency due to lens defect = 0.9 = Surface Area Efficiency (gradient percent of beam hitting hot end) From this information, the theoretical sunlight concentration for prototype lens is obtained. This concentration is dependent on hour of sunlight along with the power collected measured in watts. The resultant peak heat input in watts was also obtained.
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Resulting Peak Heat Input (W)
1800 1600 1400 1200 1000 800 600 400 200 0 300
400
500
600
700
800
900
Surface Temperature (K) Figure 106. Relationship Between Test Article Temperature and Heat Input
Experimental Set up The Fresnel lens is supported by a stand made of angle iron. It is supported on all four sides and has support legs that join at a single point on either side. This configuration was selected because it allows for the ease of maneuvering when adjusting for the changing positions of the sun. The hot end was placed on top of a wooden block to insulate the bottom surface as well as thermal protection for the ground. The thermocouple was attached to the bottom and digital measurements was recorded and compiled.
Figure 107. Experimental Set-Up
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Instrumentation The prototype hot end‟s top surface will be coated with black engine enamel paint. This is done so improve thermal radiative absorption properties. A thermocouple will be placed at the bottom of the surface of the hot end in order to obtain the temperature readings.
Figure 108. Instrumentation for Testing the Hot end Temperatures
Data Acquisition Data acquisition was done manually by reading the temperature of the test article every 5 minutes. The highest value was recorded and used for further analysis.
Results The data obtained from this experiment gave values ranging from 25 ˚C to an ultimate high temperature of 260˚ C (500˚ F). At this point, the engine enamel started boiling off our part. From this peak temperature, we can evaluate the range of power inputs that we are receiving for this assembly. Based on the system model, a heat input of 414 watts is required to achieve the temperatures recorded during our test; this is close to the theoretical 432 Watts.
Figure 109. Reaching Temperatures of 260 ˚C (500˚ F)
Analysis Percent error calculations were conducted as it was found that our experimental values were within 8% of the theoretical values. 129 | P a g e
Hourly Concentrator Heat Input with Tracking Overview This test will provide us the time dependant heat input for the engine, with the solar tracking mechanism that was developed. This value will be obtained from the experiment as follows. A prototype hot end will be used to for this experiment. This prototype hot end is comparable to the one used for our design both in material and shape. At the center of the hot end will be a perforated hole in which a thermocouple will be inserted. This will allow for temperature reading at the focal point of the Fresnel lens. The same thermal system as before will be used in order to correlate the test article temperature with thermal energy input.
Experimental Set Up The experimental set up will be similar to the previous experiments in that it will include the Fresnel lens and stand; however it will be attached to a fabricated base and the solar tracking hardware will be installed.
Instrumentation The same instrumentation as before will be used; a thermocouple attached to the back of the prototype hot end.
Data Acquisition The sampling rate for the thermocouple will be one reading for every five minutes for every hour of sunlight which will be manually recorded. This test will be used in conjunction with computational analysis to determine the power input that the solar concentrator is supplying to the engine. A temperature will be obtained as mentioned before. This temperature will provide us with the surface temperature of the engine. This value along with the other known‟s of ambient temperature, emissivity, areas, and heat transfer coefficient will allow for a resultant heat input from the collector. Since solar energy is dependent on time, this calculation will be repeated for each interval of surface temperature readings.
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Results Figure 109 shows the experimental temperatures of the test article over the period of a day. Figure 110 shows the experimental and theoretical solar energy collected. The same correlation between test article temperature and collected energy developed for the previous experiment was used.
Experimental Temperature (F)
600 500 400 300 200 100
9:35 9:55 10:15 10:35 10:55 11:15 11:35 11:55 12:15 12:35 12:55 13:15 13:35 13:55 14:15 14:35 14:55 15:15 15:35 15:55 16:15
0
Time, (Hr:Min) Figure 110. Experimental Test Article Temperature
500 450
Power, (Watts)
400 350 300 250 200 150 Experimental Results
100
Theoretical Values
50 0 9:30
10:42
11:54 13:06 14:18 Time of Day, (Hr:Min)
15:30
Figure 111. Theoretical and Experimental Collected Energy
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Analysis Based on the results obtained from the second experiment, we can see that the performance and efficiency of the solar Stirling engine is heavily dependent on the heat input and resulting hot end temperature. The effects of weather, predominately clouds, affects every type of solar energy conversion. However, solar Stirling engines are even more susceptible in that the decreased temperature will lead to decreased efficiency as well as decreased output.
Recommendations There are several design measures that can be taken to increase the efficiency of the engine during cloud cover or intermittent sunlight. The experiments show that there is residual heat stored in the hot end during cloud cover. This latent heat is sufficient to produce small amount of power. An improvement to the design would be to increase the amount of heat storage of the hot end in order to produce power during these events. This can be achieved by selecting a different material with a higher heat storage capabilities as well as including regenerators. A major improvement to the design that would improve power production would be the inclusion of a black hole. This black hole would decrease heat loss by reducing the amount of the heat loss by convection due to the refraction of sunlight on the surface.
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Stirling Engine Performance Overview Experimental Set Up Instrumentation Data Acquisition Results Analysis
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Conclusion
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Design Considerations Assembly and Disassembly It is planned for the assembly to occur within a factory under decently clean conditions. Disassembly is not anticipated, as the product is not expected to be recovered after deployment.
Maintenance of the System Regular Maintenance o Monitoring of internal working pressure o Lubrication of bearings o Inspection for overheating damage
Major Maintenance Major maintenance is not anticipated, as the design life of the engine is only 4 months. It is not anticipated that the package will not be recovered after deployment, as the low cost of the power system inhibits the feasibility of reconditioning and redeploying.
Environmental Impact There is expected to be little environmental impact from the engine. No exotic metals, or toxic gases, or reactive components.
Risk Assessment There are always risks when handling pressurized objects; to mitigate the risk of explosion a pressure release valve will be installed on all engines.
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Project Management Overview In order to meet the milestones for this senior design project, a breakdown of work into specific tasks and responsibilities among the team members as well as the copyright applications and the commercialization of the solar Stirling engine will be cover through the project management section.
Important Milestones Report 10%
Due November 5, 2009
25%
December 3, 2009
-
December 11, 2009 January 14, 2010
50%
February 16, 2010
75%
March 10, 2010
100%
April 2, 2010
-
April 6 & 8 April 14 & 15
Items Introduction Design Alternatives Project Management Conclusion References Project Formulation Engineering Design Engineering Analysis Prototype Construction Team Poster IAB Project Feasibility Presentation Final Design (100% completed) Prototype Assembly (50% completed) Prototype Assembly (100% completed) Testing of Prototype (50% completed) All Report Power point Rehearsal Presentation to MME Final Presentation to IAB and MME
Competed November 5, 2009
December 3, 2009
December 10, 2009 January 14, 2010 February 16, 2010
Table 30. Breakdown of Deadlines
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Breakdown of Responsibilities Among Team Members Each team member is responsible for an assigned task in this project. This caters to the strength of each person as well as creates a cohesive plan to achieve all the milestones and deadline for this task. Table 31. Breakdown of Responsibilities among Team Members Abstract
DENISSE
Design Based on Static and
Introduction
DENISSE
Fatigue Failure Design Theories
STEPHEN
Problem Statement
DENISSE
Deflection Analysis
STEPHEN
Motivation
DENISSE
Component Design/Selection
STEPHEN STEPHEN
Literature Survey
KEVIN
Finite Element Analysis
Discussion
KEVIN
Design Overview
KEVIN
Project Formulation
KEVIN
Cost Analysis
KEVIN
Overview
STEPHEN
Discussion
KEVIN
Project Objectives
STEPHEN
Prototype Construction
KEVIN
Design Specifications
STEPHEN
Description of Prototype
KEVIN
Constraints and Other Considerations
STEPHEN
Prototype Design
KEVIN
Discussion
STEPHEN
Parts List
KEVIN
Design Alternatives
KEVIN
Construction
KEVIN
Overview of Conceptual Designs Developed
KEVIN
Prototype Cost Analysis
KEVIN
Variants of Solar Concentrator
KEVIN
Discussion
DENISSE
Design Alternate 2
KEVIN
Testing and Evaluation
DENISSE
Design Alternate 3
KEVIN
Overview
DENISSE
Feasibility Assessment
KEVIN
Description of Experiments
DENISSE
Proposed Design
KEVIN
Test Results and Data
DENISSE
Discussion
KEVIN
Evaluation of Experimental Results
DENISSE
Project Management
DENISSE
Improvement of the Design
STEPHEN
Overview
DENISSE
Discussion
STEPHEN
Breakdown of Work into Specific Tasks
DENISSE
Design Considerations
STEPHEN
Organization of Work and Timeline Breakdown of Responsibilities Among Team Members
DENISSE
Assembly and Disassembly
KEVIN
DENISSE
Maintenance of the System
KEVIN
Patent/Copyright Application
KEVIN
Regular Maintenance
KEVIN
Commercialization of the Final Product
KEVIN
Major Maintenance
KEVIN
Discussion
KEVIN
Environmental Impact
DENISSE DENISSE
Engineering Design and Analysis
DENISSE
Risk Assessment
Kinematic Analysis and Animation
STEPHEN
Conclusion
KEVIN
Dynamic/Vibration Analysis of the System
STEPHEN
Conclusion and Discussion
KEVIN
Structural Design
DENISSE
Patent/Copyright Application
STEPHEN
Force Analysis
DENISSE
Commercialization
STEPHEN
Deflection Analysis
DENISSE
Future Work
DENISSE
Material Selection
DENISSE
References
DENISSE
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Organization of Work and Timeline
Table 32. Gantt Chart for Solar Stirling
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Cost Analysis Below is a breakdown of the hours invested in the design and development of the Solar Stirling Engine for Remote Power and Disaster Relief.
Kevin Denisse Stephen Project Total
Survey
Prototype
Design
Modeling
Report
Presentation
Total
27 46 12
68 22 12
45 25 12
48 12 183
42 85 11
23 60 15
253 250 245
85
102
82
243
138
98
748
Table 33. Hours Worked on Design and Development
Survey Prototype Design Modeling Report Presentation
Figure 112. Distribution of Labor based on hours
Each subtask that was conducted by the team was then evaluated at the worth of each category. Categories such as survey, prototype earned $20/hour while report and presentation work earns $25/hour and the design and modeling section earns $30/hour. The breakdown by cost is shown in the pie chart below.
Survey Prototype Design Modeling Report Presentation
Figure 113. Distribution of Work based on Cost
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Relevant Course Work The following engineering courses were instrumental in achieving this solar Striling engine design to be in accordance with all engineering principles. Linear Algebra
Heat Transfer
Mechatronics
Thermodynamics
Mechanical Design I &II
Simulation Software
Programming Engineering
Transport Phenomena
Analysis of Engineering Systems
Materials in Engineering
Computational Fluid Dynamics
Mechanical Design Optimization
Differential Equations (ODE & Partials)
Design of Thermal and Fluid Systems
Patent/Copyright Application To the best of the team‟s knowledge, this configuration of solar Stirling engine has never been done before and is therefore a completely unique and innovative design. This fact allows provides the team with a unique opportunity to apply for a patent and /or copyright for this particular configuration of the engine.
Commercialization of the Final Product Solar Stirling Engines have the ability to produce a relatively large amount of power using nothing more than sunlight and other free, clean, and natural resources that prove to be of an alternative than other heat engines and solar panels. This fact provides our design with a unique opportunity to use the final product for commercial applications as well as for use in disaster relief and remote locations.
Discussion Project Management is perhaps one of the most important aspects of this project. Without it, this concept of solar Stirling engine will be just a concept and will be hard pressed to find a place in engineering applications as well as the commercial and humanitarian sector. The project management includes accomplishing objectives, meeting deadlines, and reaching milestones. The breakdown of work into specific tasks as well as the Gantt chart is shown in Table 32 and Table 33.
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Design Considerations and Future Work Future work would include patent applications for various components of the design including the internalization of the crankshaft inside of the engine, application of a Fresnel lens to power a Stirling, and utilization of a cooling channel to remove waste heat from the engine. Further work could be done in further designing the power distribution and conditioning of the engine in order to expand the concept into domestic energy production. This would include long term design analysis, as well as a more intricate cooling system. The need for a more permanent tracking system would increase the overall cost of the system, however, is still expected to be extremely competitive with current solar energy conversions. If a domestic version is to be expanded upon, then it would justify the refinement of the internal geometry to focus more on efficiency instead of cost. Optimization of internal geometry based on internal aero-dynamic flow consideration would be preformed which would be based on a CFD run with heat transfer as well as aerodynamic considerations.
Lessons Learned A wealth of knowledge was generated from this design project that could help future design teams in their endeavors. Through the design process, a methodology for analyzing and modeling of a Stirling engine was established. This analysis was extended to include the cyclic thermal loading with variable magnitude heat input due to the sun‟s position relative to the Earth.
Conclusion and Discussion A solar Stirling engine for remote power generation and disaster relief was successfully conceptualized, designed, and prototype fabricated. Engineering analysis and modeling indicates that this design can achieve a 31 % solar to electrical power conversion efficiency. Through testing and evaluation, the feasibility of this application was successfully illustrated. The prototype Fresnel lens was capable of concentrating 414 watts on the simulated hot end of the engine which is within 8% of the 452 watts which was theoretically calculated. This testing can be extrapolated to support the value of 12 kW for the full scale Fresnel lens assembly. In conclusion, the design of a solar Stirling engine could save millions of lives by its implementation as part of disaster relief effort by providing power to areas which are in dire need of electricity to power medical equipment, purify water, cook food, as well as numerous other basic necessities.
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Works Cited ASME. (2009). Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer. New York: The American Society of Mechanical Engineers. Ausra. (2009). Solar Thermal Energy. Retrieved 10 25, 2009, from Technology & Product: www.ausra.com Berchowitz, I. U. (1984). Stirling Cycle Engine Analysis. Bristol: Adam Hilger. Beta, C. (2008, JUNE 27). Goverment Study of Solar Energy on Federal Lands Suggests Nefarious Motives for US Solar Energy Freeze. Retrieved Novemeber 12, 2009, from Clean Tech Law and Business: http://cleantechlawandbusiness.com/cleanbeta/index.php/2008/06/us-stops-solar-projectsen-masse-citing-environmental-concerns/ Burton, P. (1979). Kinematics & Dynamics of Planar Machinery. Prentice Hall. Communications, W. (n.d.). Electropaedia. Retrieved october 20, 2009, from mpoweruk: www.mpoweruk.com D.M. Berchowitz, M. D. (1987). Development and Performance of a 3kW(e) Air Charged Free-Piston Stirling Engine with Linear Alternator. 22nd Intersociety Energy Conversion Engineering Conference. Philadelphia, Pennsylvania. Discover. (2007, september). Where do the Nastiest Hurricane Emerge ? Retrieved january 10, 2010, from discovermagazine.com: http://discovermagazine.com/2007/sep/map-where-do-hurricanes-do-the-most-damage Force, N. A. (2007, Decemebr 17). Nellis Air Force . Retrieved February 2010, from www.nellis.af.mil: http://www.nellis.af.mil/shared/media/document/AFD-080117043.pdf Frances Hurwitz, D. A. (2008). Aerogels in the AL2O3 - SiO2 System. American Chemistry Society. Philadelphia, Pennsylvania. Frank P. Incropera, D. P. (2002). Fundamentals of Heat and Mass Transfer. Danvers: John Wiley & sons. Goverment, A. (2009). Introduction to Tsunami. Retrieved February 02, 2010, from Tsunami Education and Awareness: http://beachsafe.org.au/tsunami/ema/pages/05_history.html
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Interior, U. D. (2009, October 21 21). Earthquake Density Maps for the World. Retrieved January 5, 2010, from Earthquake Hazards Program: http://earthquake.usgs.gov/earthquakes/world/world_density.php Laboratories, S. N. (2009). Sandia. Retrieved 10 25, 2009, from www.sandia.gov Lee Mason, J. S. (2007). A Historical Review of Brayton and Stirling Power Conversion Technologies for Space Applications. Cleveland: NASA. Lee S. Mason, J. G. (2007-214976). A Historical Review of Brayton and Stirling Power Conversion Technologies for Space Applications. NASA Glenn Research Center Technical Memorandum . Mazza, D. (n.d.). Calculating Sunlight over a 12 hour Period. Retrieved 10 06, 2009, from NASA : http://www.grc.nasa.gov/WWW/K12/Numbers/Math/Mathematical_Thinking/sun12.htm Nightingale, N. (1986). Automotive Stirling Engine. Cleveland: NASA. R. Van Giessel, F. R. (1977). Design of the 4-215 Automotive Stirling Engine. Society of Automotive Engineers . Richard K. Sheltend, W. A. (2007-214930). Advanced Stirling Technology Development at NASA Glenn Research Center. NASA Glenn Research Center Technical Memorandum . Sandfort, J. F. (1962). Heat Engines. Garden City, New York: Anchor Books. Schreiber, J. G. (2006-214429). Summary of Stirling Convertor Testing at NASA Glenn Research Center. NASA Glenn Research Center Technical Memorandum . Senft, J. R. (2007). Mechanical Effeciency of Heat Engines. Cambridge: Cambridge University Press. Solarbuzz. (2009). European PV Markets 2009 Report : 2008 Market Outcome; 20092013 Scenerio Forecast. San Francisco, United States: Solarbuzz. Systems, S. E. (2008, February 12). SES Sets World Record for Solar-to-Grid Conversion Efficiency. Systems, S. E. (2009, October 20). Technology. Scottsdale, Arizona, USA. Wheeler, R. (2007, 02 23). Stirling Engines. Reading, Berkshire, England.
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Wong, W. A. (2004). Advanced Radioisotope Power Conversion Technology Research & Development. 2nd International Energy Conversion Engineering Conference. Providence, Rhode Island.
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Appendices Below are the appendices referenced throughout the report. A. Engineering Drawings B. Documentation of the Developed Software Code C. Scanned Information on Important Document Specifications
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Appendix A. Detailed Engineering Drawings of All Parts
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Appendix B. Detailed Raw Design Calculations and Analysis Adiabatic Analysis Main Computer Program clear all clc R=8.314; Cv=450; Cp=50; gamma=Cp/Cv; Vclc=28.68; Vcle=30.52; Vswc=114.13; Vswe=120.82; Vc=28.68+114.13; Vk=13.18; Vr=50.55; Vh=70.28; Ve=30.52+120.82/2; Th=977; The=Th; Trh=Th; Tk=288; Tkr=Tk; Tck=Tk; Tr=(Tk+Th)/2; M=1.1362; dt=.0001; t=0:dt:(5/42); p=41.3*101800; mc=0.25; W=20000; Qk=0; Qr=0; Qh=0; for i=2:(size(t,2)+1) theta=sin(41.72*t(i-1)/pi); Vc(i)=Vclc+Vswc*sin(theta); Ve(i)=Vcle+Vswe*sin(theta+pi/4); DVc=Vc(i)-Vc(i-1); DVe=Ve(i)-Ve(i-1); Dp=-gamma*p(i1)*(DVc/Tck+DVe/The)/(Vc(i)/Tck+gamma*(Vk/Tk+Vr/Tr+Vh/Th)+Vc(i)/The); Dmc=(p(i-1)*DVc+Dp/gamma)/(R*Tck); mk=p(i-1)*Vk/(R*Tk);
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mr=p(i-1)*Vr/(R*Tr); mh=p(i-1)*Vh/(R*Th); me=M-(mc(i-1)+mk+mh+mr); Tc=p(i-1)*Vc(i)/(R*mc(i-1)); Te=p(i-1)*Ve(i)/(R*me); Dmk=mk*Dp/p(i-1); Dmr=mr*Dp/p(i-1); Dmh=mh*Dp/p(i-1); gAck=-Dmc; gAkr=gAck-Dmk; gArh=gAkr-Dmr; gAhe=gArh-Dmh; if gAck>0 Tck=Tc; else Tck=Tk; end if gAhe>0 The=Th; else The=Te; end DW=p(i-1)*(DVc+DVe); DQk=Vk*Dp*Cv/R-Cp*(Tck*gAck-Tkr*gAkr); DQr=Vr*Dp*Cv/R-Cp*(Tkr*gAkr-Trh*gArh); DQh=Vh*Dp*Cv/R-Cp*(Trh*gArh-The*gAhe); TC(i)=Tc; TE(i)=Te; TH(i)=Th; TK(i)=Tk; Kp(1)=Dp; Kmc(1)=Dmc; KW(1)=DW; KQk(1)=DQk; KQr(1)=DQr; KQh(1)=DQh; j=2; Kn(1)=Kp(1); Kn(2)=Kmc(1); Kn(3)=KW(1); Kn(4)=KQk(1); Kn(5)=KQr(1); Kn(6)=KQh(1); K=Analysis(t,i,j,dt,Kn,p,mc,W,Qk,Qr,Qh,R,Cv,Cp,gamma,Vclc,Vcle,Vswc,Vsw e,Vk,Vr,Vh,Ve,Vc,Th,The,Trh,Tk,Tkr,Tck,Tr,M); Kp(2)=K(1); Kmc(2)=K(2); KW(2)=K(3); KQk(2)=K(4); KQr(2)=K(5); KQh(2)=K(6); j=2;
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Kn(1)=Kp(2); Kn(2)=Kmc(2); Kn(3)=KW(2); Kn(4)=KQk(2); Kn(5)=KQr(2); Kn(6)=KQh(2); K=Analysis(t,i,j,dt,Kn,p,mc,W,Qk,Qr,Qh,R,Cv,Cp,gamma,Vclc,Vcle,Vswc,Vsw e,Vk,Vr,Vh,Ve,Vc,Th,The,Trh,Tk,Tkr,Tck,Tr,M); Kp(3)=K(1); Kmc(3)=K(2); KW(3)=K(3); KQk(3)=K(4); KQr(3)=K(5); KQh(3)=K(6); j=1; Kn(1)=Kp(3); Kn(2)=Kmc(3); Kn(3)=KW(3); Kn(4)=KQk(3); Kn(5)=KQr(3); Kn(6)=KQh(3); K=Analysis(t,i,j,dt,Kn,p,mc,W,Qk,Qr,Qh,R,Cv,Cp,gamma,Vclc,Vcle,Vswc,Vsw e,Vk,Vr,Vh,Ve,Vc,Th,The,Trh,Tk,Tkr,Tck,Tr,M); Kp(4)=K(1); Kmc(4)=K(2); KW(4)=K(3); KQk(4)=K(4); KQr(4)=K(5); KQh(4)=K(6); p(i)=p(i-1)+dt*(Kp(1)+2*Kp(2)+2*Kp(4)+Kp(4))/6; mc(i)=mc(i-1)+dt*(Kmc(1)+2*Kmc(2)+2*Kmc(4)+Kmc(4))/6; W(i)=W(i-1)+dt*(KW(1)+2*KW(2)+2*KW(4)+KW(4))/6; Qk(i)=Qk(i-1)+dt*(KQk(1)+2*KQk(2)+2*KQk(4)+KQk(4))/6; Qr(i)=Qr(i-1)+dt*(KQr(1)+2*KQr(2)+2*KQr(4)+KQr(4))/6; Qh(i)=Qh(i-1)+dt*(KQh(1)+2*KQh(2)+2*KQh(4)+KQh(4))/6; end TC(1)=TC(2); TE(1)=TE(2); TH(1)=TH(2); TK(1)=TK(2); plot(W)
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Subprogram function [K] = Analysis(t,i,j,dt,Kn,p,mc,W,Qk,Qr,Qh,R,Cv,Cp,gamma,Vclc,Vcle,Vswc,Vswe, Vk,Vr,Vh,Ve,Vc,Th,The,Trh,Tk,Tkr,Tck,Tr,M) p=p+dt*Kn(1); mc=mc+dt*Kn(2); W=W+dt*Kn(3); Qk=Qk+dt*Kn(4); Qr=Qr+dt*Kn(5); Qh=Qh+dt*Kn(5); theta=sin(41.72*t(i-1)/pi); Vc(i)=Vclc+Vswc*sin(theta); Ve(i)=Vcle+Vswe*sin(theta+pi/4); DVc=Vc(i)-Vc(i-1); DVe=Ve(i)-Ve(i-1); Dp=-gamma*p(i1)*(DVc/Tck+DVe/The)/(Vc(i)/Tck+gamma*(Vk/Tk+Vr/Tr+Vh/Th)+Vc(i)/The); Dmc=(p(i-1)*DVc+Dp/gamma)/(R*Tck); mk=p(i-1)*Vk/(R*Tk); mr=p(i-1)*Vr/(R*Tr); mh=p(i-1)*Vh/(R*Th); me=M-(mc(i-1)+mk+mh+mr); Tc=p(i-1)*Vc(i)/(R*mc(i-1)); Te=p(i-1)*Ve(i)/(R*me); Dmk=mk*Dp/p(i-1); Dmr=mr*Dp/p(i-1); Dmh=mh*Dp/p(i-1); gAck=-Dmc; gAkr=gAck-Dmk; gArh=gAkr-Dmr; gAhe=gArh-Dmh; if gAck>0 Tck=Tc; else Tck=Tk; end if gAhe>0 The=Th; else The=Te; end DW=p(i-1)*(DVc+DVe); DQk=Vk*Dp*Cv/R-Cp*(Tck*gAck-Tkr*gAkr); DQr=Vr*Dp*Cv/R-Cp*(Tkr*gAkr-Trh*gArh); DQh=Vh*Dp*Cv/R-Cp*(Trh*gArh-The*gAhe); K(1)=Dp; K(2)=Dmc; K(3)=DW; K(4)=DQk; K(5)=DQr; K(6)=DQh;
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Isothermal Analysis clc clear all %p is in pascals M=.000028; R=2077; Vk=14*10^-6; Vr=51*10^-6; Vh=70*10^-6; Tk=300; Th=950; Ve=(115+29)*10^-6; Vc=31*10^-6; Wc=-0; We=-.6; W=-0.6; k=Vk/Tk+Vr*log(Th/Tk)/(Th-Tk)+Vh/Th; JperCycle=M*21*550 omega=2700/(JperCycle*60) for i=2:628 theta=i/100; Vc(i)=(29+115*(1+cos(theta))/2)*10^-6; Ve(i)=(31+122*(1+cos(theta+pi/2))/2)*10^-6; p(i)=M*R/(Vc(i)/Tk+k+Ve(i)/Th); Wc(i)=Wc(i-1)+32*sin(theta)*p(i)/200*10^-6; We(i)=We(i-1)+32*sin(theta+pi/2)*p(i)/200*10^-6; W(i)=Wc(i)+We(i); End p(1)=p(628); W(1)=W(628); plot(Vc) w=mean(W) P=mean(p)
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Developed Tracking Code ' {$STAMP BS2} ' {$PBASIC 2.5} start VAR Word end VAR Word steps VAR Word i VAR Word N VAR Word month VAR Word year VAR Word day VAR Word t VAR Word longitude VAR Word M VAR Word L VAR Word RA VAR Word sinDec VAR Word cosDec VAR Word H VAR Word T VAR Word UT VAR Word localOffset VAR Word % to be % % % % % % %
Before start of program, the variables month, day, year, longitude, and localOffset need imported from parallel PLC in charge of date/time. Sunrise/set Source: Almanac for Computers, 1990 published by Nautical Almanac Office United States Naval Observatory Washington, DC 20392
N=floor(275*month/9)-floor((month+9)/12)*(1+floor((year-4*floor(year/4)+2)/3))+day-30 t=N+((6-longitude/15)/24) M=(0.9856*t)-3.289 L=M+(1.916*sin(M))+(0.020*sin(2*M))+282.634 If L>360;L-360 Elseif L<0; L+360 RA=atan(0.91764*tan(L)) If RA>360;L-360 Elseif RA<0; L+360 RA=(RA+((floor(L/90))*90-(floor(RA/90))*90))/15 sinDec=0.39782*sin(L) cosDec=cos(asin(sinDec))
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cosH=(cos(zenith)-(sinDec*sin(latitude)))/(cosDec*cos(latitude)) H=(360-acos(cosH))/15 T=H+RA-(0.06571*t)-6.622 UT=T-longitude/15 If UT>24;L-24 Elseif UT<0; L+24 start=UT+localOffset N=floor(275*month/9)-floor((month+9)/12)*(1+floor((year-4*floor(year/4)+2)/3))+day-30 t = N + ((18 - lngHour) / 24) M=(0.9856*t)-3.289 L=M+(1.916*sin(M))+(0.020*sin(2*M))+282.634 If L>360;L-360 Elseif L<0; L+360 RA=atan(0.91764*tan(L)) If RA>360;L-360 Elseif RA<0; L+360 RA=(RA+((floor(L/90))*90-(floor(RA/90))*90))/15 sinDec=0.39782*sin(L) cosDec=cos(asin(sinDec)) cosH=(cos(zenith)-(sinDec*sin(latitude)))/(cosDec*cos(latitude)) H = (acos(cosH))/15 T=H+RA-(0.06571*t)-6.622 UT=T-longitude/15 If UT>24;L-24 Elseif UT<0; L+24 end=UT+localOffset steps=(start-end)/4 Do SleepMode If time>=start GOTO Main Else Pause 15000 Loop Main For i=0 to steps GOSUB Rotate NEXT GOSUB Home GOTO SleepMode Rotate: PULSOUT 1,850
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PAUSE 17000 PULSOUT 1,650 PAUSE 223000 RETURN Home: PULSOUT 1,450 PAUSE 2550000 PULSOUT 1,650 RETURN END
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Appendix D. Stirling Geometry and Mesh Generation Codes var.dat 0.00000 //Cold End Parameter 0.00000 //Regenerator Parameter
setStirlingGeomertry.C #include #include #include #include #include #include #include #include #include using namespace std; int main(int argc, char *argv[]){ FILE *setStirlingGeometry; if ((setStirlingGeometry = fopen("constant/polyMesh/stirlingGeometry.H","w"))==NULL){ printf("Cannot open new setStirlingGeometry file.\n"); exit(1); }//end if fopen
//Print setStirlingGeometry profile to File float opta, optb; float hotR=1.5, coldOff=-4, coldR=1.125, coldEnd=-5.3125, dOff=-0.5, dEnd=-3.5, dopt=-2.0, sopt=-3.0, pOff=-4.3125, i5x=-3.3333, 170 | P a g e
org=0.0, h=0.1, pi=3.1459;
ifstream output("constant/polyMesh/var.dat"); output>>opta>>optb; output.close(); printf ("opta %f\n", opta); printf ("optb %f\n", optb);
//optimization float dR=1.375-opta, popt=0.7994+optb; i5x=-3+(-0.375/(hotR-dR))*(hotR-dR+opta); //volume constraints in cubic inches!!! conversion to meters takes place later in mesh generation process float Vreg=3.387, Vc=0.984, Vhe=3.534; //volume constraint enforcement dOff=(dEnd+(Vreg/(pi*(hotR*hotR-dR*dR))) ); coldEnd=pOff-((2*Vc)/(pi*(coldR*coldR-popt*popt))); //org=dOff-(Vhe/(pi*hotR*hotR)); org=dOff+0.4; h=org+0.1; fprintf(setStirlingGeometry,"org %f;\n", org); fprintf(setStirlingGeometry,"h %f;\n", h); fprintf(setStirlingGeometry,"hotR %f;\n", hotR); fprintf(setStirlingGeometry,"dR %f;\n", dR); fprintf(setStirlingGeometry,"dEnd %f;\n", dEnd); fprintf(setStirlingGeometry,"dOff %f;\n", dOff); fprintf(setStirlingGeometry,"coldOff %f;\n", coldOff); fprintf(setStirlingGeometry,"coldR %f;\n", coldR); fprintf(setStirlingGeometry,"coldEnd %f;\n", coldEnd); fprintf(setStirlingGeometry,"pOff %f;\n", pOff); fprintf(setStirlingGeometry,"dopt %f;\n", dopt); 171 | P a g e
fprintf(setStirlingGeometry,"sopt %f;\n", sopt); fprintf(setStirlingGeometry,"popt %f;\n", popt); fprintf(setStirlingGeometry,"i5x %f;\n", i5x); printf("Closed all\n"); fclose(setStirlingGeometry); //fclose(log); printf("yup\n"); printf("Closed all\n");
}//end main
stirlingGeometry.H org 0; hotR 1.500000; dR 1.375000; dEnd -3.500000; dOff -0.500000; coldOff -4.000000; coldR 1.125000; coldEnd -5.312500; pOff -4.312500; dopt -2.048311; sopt -3.048311; popt 0.79994;//0.630579; i5x -3.366666;
172 | P a g e
designVariables.H #include "stirlingGeometry.H"; //---verticies H0 (0 0 $org); H1 (0 $dR $org); w0 (0 $hotR $org); w1 (0 $hotR $dOff); w2 (0 $hotR $sopt); w3 (0 $coldR $coldOff); w4 (0 $coldR $coldEnd); d0 (0 0 $dOff); d1 (0 $dR $dOff); d2 (0 $dR $dopt); d3 (0 $popt $dEnd); p0 (0 $popt $pOff); i4 (0 $hotR $dopt); i5 (0 $dR $i5x); //---H0i (0.1 0 $org); H1i (0.1 $dR $org); w0i (0.1 $hotR $org); w1i (0.1 $hotR $dOff); w2i (0.1 $hotR $sopt); w3i (0.1 $coldR $coldOff); w4i (0.1 $coldR $coldEnd); d0i (0.1 0 $dOff); d1i (0.1 $dR $dOff); d2i (0.1 $dR $dopt); d3i (0.1 $popt $dEnd); p0i (0.1 $popt $pOff); i4i (0.1 $hotR $dopt); i5i (0.1 $dR $i5x); //---
xw3 5; xw4 6; xd0 7; xd1 8; xd2 9; xd3 10; xp0 11; //--xH0i 12; xH1i 13; xw0i 14; xw1i 15; xw2i 16; xw3i 17; xw4i 18; xd0i 19; xd1i 20; xd2i 21; xd3i 22; xp0i 23; xi4 24; xi4i 25; xi5 26; xi5i 27; //---
// verticies IDs-----------xH0 0; xH1 1; xw0 2; xw1 3; xw2 4; 173 | P a g e
blockMeshDict /*--------------------------------*- C++ -*----------------------------------*\ | ========= | | | \\ / F ield | OpenFOAM: The Open Source CFD Toolbox | | \\ / O peration | Version: 1.6 $ | | \\ / A nd | Web: http://www.OpenFOAM.org | | \\/ M anipulation | | \*---------------------------------------------------------------------------*/ FoamFile { version 2.0; format ascii; class dictionary; object blockMeshDict; } // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // convertToMeters 0.0254; #include "designVariables.H"; vertices ( $H0 $H1 $w0 $w1 $w2 $w3 $w4 $d0 $d1 $d2 $d3 $p0
174 | P a g e
//--$H0i $H1i $w0i $w1i $w2i $w3i $w4i $d0i $d1i $d2i $d3i $p0i // // // // // // // //
$i0 $i1 $i2 $i3 $i0i $i1i $i2i $i3i $i4 $i4i $i5 $i5i
); //Mesh Parameters xzero 25; xone 75; xtwo 45; xthree 30;//90; yzero 150; yone 15; 175 | P a g e
xopt 15;
blocks ( /* hex ($xH0 $xd0 $xd1 $xH1 $xH0i $xd0i $xd1i $xH1i) ($xzero $yzero 1) simpleGrading (1 1 1) //B0 hex ($xH1 $xd1 $xw1 $xw0 $xH1i $xd1i $xw1i $xw0i) ($xzero $yone 1) simpleGrading (1 1 1) //B1 hex ($xd1 $xd2 $xi4 $xw1 $xd1i $xd2i $xi4i $xw1i) ($xone $yone 1) simpleGrading (1 1 1) //B2 hex ($xd2 $xi5 $xw2 $xi4 $xd2i $xi5i $xw2i $xi4i) ($xtwo $yone 1) simpleGrading (1 1 1) //B3 hex ($xd2 $xd3 $xw3 $xi5 $xd2i $xd3i $xw3i $xi5i) ($xtwo $xtwo 1) simpleGrading (1 1 1) //B4 hex ($xd3 $xi0 $xi1 $xw3 $xd3i $xi0i $xi1i $xw3i) (15 $xtwo 1) simpleGrading (1 1 1) //B5 hex ($xi0 $xi2 $xi3 $xi1 $xi0i $xi2i $xi3i $xi1i) (1 $xtwo 1) simpleGrading (1 1 1) //B6 hex ($xi2 $xp0 $xw4 $xi3 $xi2i $xp0i $xw4i $xi3i) (15 $xtwo 1) simpleGrading (1 1 1) //B7 hex ($xH0 $xH0i $xH1i $xH1 $xd0 $xd0i $xd1i $xd1) (1 $yzero $xtwo) simpleGrading (1 1 1) //B0 hex ($xH1 $xH1i $xw0i $xw0 $xd1 $xd1i $xw1i $xw1) (1 $yone $xtwo) simpleGrading (1 1 1) //B1 hex ($xd1 $xd1i $xw1i $xw1 $xd2 $xd2i $xi4i $xi4) (1 $yone $xtwo) simpleGrading (1 1 1) //B2 hex ($xd2 $xd2i $xi4i $xi4 $xi5 $xi5i $xw2i $xw2) (1 $yone $xtwo) simpleGrading (1 1 1) //B3 hex ($xd2 $xd2i $xi5i $xi5 $xd3 $xd3i $xw3i $xw3) (1 $xtwo $yone) simpleGrading (1 1 1) //B4 hex ($xd3 $xd3i $xw3i $xw3 $xi0 $xi0i $xi1i $xi1) (1 $xtwo $yone) simpleGrading (1 1 1) //B5 hex ($xi0 $xi0i $xi1i $xi1 $xi2 $xi2i $xi3i $xi3) (1 $xtwo $yone) simpleGrading (1 1 1) //B6
176 | P a g e
hex ($xi2 $xi2i $xi3i $xi3 $xp0 $xp0i $xw4i $xw4) (1 $xtwo $yone) simpleGrading (1 1 1) //B7 */ hex ($xd0 $xd0i $xd1i $xd1 $xH0 $xH0i $xH1i $xH1) blockA (1 $yzero $xtwo) simpleGrading (1 1 1) //B0 hex ($xd1 $xd1i $xw1i $xw1 $xH1 $xH1i $xw0i $xw0) blockB (1 $yone $xtwo) simpleGrading (1 1 1) //B1 hex ($xd2 $xd2i $xi4i $xi4 $xd1 $xd1i $xw1i $xw1) blockC (1 $yone 125) simpleGrading (1 1 1) //B2 hex ($xi5 $xi5i $xw2i $xw2 $xd2 $xd2i $xi4i $xi4) blockD (1 $yone $xtwo) simpleGrading (1 1 1) //B3 hex ($xw3 $xw3i $xi5i $xi5 $xd3 $xd3i $xd2i $xd2) blockE (1 60 $xtwo) simpleGrading (1 1 1) //B4 //hex ($xi0 $xi0i $xi1i $xi1 $xd3 $xd3i $xw3i $xw3) blockF (1 $xtwo $yone) simpleGrading (1 1 1) //B5 //hex ($xi2 $xi2i $xi3i $xi3 $xi0 $xi0i $xi1i $xi1) blockG (1 $xtwo 1) simpleGrading (1 1 1) //B6 //hex ($xp0 $xp0i $xw4i $xw4 $xi2 $xi2i $xi3i $xi3) blockH (1 $xtwo $yone) simpleGrading (1 1 1) //B7 hex ($xp0 $xp0i $xw4i $xw4 $xd3 $xd3i $xw3i $xw3) blockF (1 $xtwo 40) simpleGrading (1 1 1) //B5 ); edges ( ); patches ( wall cylinderHead //0 ( ($xH0 $xH0i $xH1i $xH1) ($xH1 $xH1i $xw0i $xw0) ) wall cold //1 ( ($xw3 $xw3i $xw4i $xw4) 177 | P a g e
) wall liner //4 ( ($xw2 $xw2i $xi5i $xi5) ($xi5 $xi5i $xw3i $xw3) ) wall valveCold //5 ( //($xp0 $xp0i $xw4i $xw4) //($xd0 $xd0i $xd1i $xd1) ($xd2 $xd2i $xd3i $xd3) ) wall valveHot //6 ( ($xd0 $xd0i $xd1i $xd1) ) wall valveSides //7 ( ($xd1 $xd1i $xd2i $xd2) ) wall valveWalls //8 ( ($xw1 $xw1i $xi4i $xi4) ($xi4 $xi4i $xw2i $xw2) ) wall piston //9 ( ($xp0 $xp0i $xw4i $xw4) //($xd2 $xd2i $xd3i $xd3) //($xd0 $xd0i $xd1i $xd1) //($xd1 $xd1i $xd2i $xd2)
) 178 | P a g e
wall inner //10 ( ($xd3 $xd3i $xp0i $xp0) ) wall farfield //13 ( //diplacer ($xw0 $xw0i $xw1i $xw1) ) symmetryPlane axis //14 ( ($xH0 $xH0i $xd0i $xd0) )
); mergePatchPairs ( ); //********************************************************************* //
179 | P a g e
Appendix E. Optimization Codes diffEvol.C #include #include #include #include #include #include #include #include #include using namespace std; double funceval(int probnum, int varnum, double x, double y); void writeVar( char genFlag, int j, double x, double y); double readObj( char genFlag, int j); void writeChange( char genFlag, int j); int rmChange( char genFlag, int j); bool lim(double x, double y); double Fibbonacci(double delta[2], double x_old[2], double u[2], int probnum, int varnum); void bound(double (&x) [2], double u[2], double x_old[2]); void add(double (&c) [2][2], double a[2][2], double b[2][2]); void mult22(double (&c) [2][2], double a[2][2], double b[2][2]); void mult21(double (&c) [2][2], double a[2], double b[2]); void multscalar(double (&c) [2][2], double a); int numfunc=0;
int main(){ //bool stagnant; int gen=0, end, nochange, stagnant; const int Nvar=2, Npop=20; const int probnum=13, varnum=2; 180 | P a g e
double x[2], x_old[2], F, r, f_old, err, gradnorm, epsx=0.00005, epsy=0.00005, alpha; double xmin[2]={0,-0.375}, xmax[2]={0.05,0.1}, xrange[2]={xmax[0]-xmin[0], xmax[1]-xmin[1]}; double Ipop[Npop][2], P[Npop][2], C[Npop][2], Best[Npop][3]; double fIpop[Npop], fC[Npop]; char genFlag; int sel[3]; int complete; ofstream trackclear("conv.dat"); trackclear.close(); ofstream track("conv.dat",ios::app);
FILE *track2; if ((track2 = fopen("conv","w"))==NULL){ printf("Cannot open new convergence file.\n"); exit(1); }//end if fopen printf("\ntrack Opened\n"); track<<"iter\tnumfunc\tx\ty\tf"; fprintf(track2,"iter\tnumfunc\tx\ty\tf"); printf("@@@\titer\tnumfunc\tx\ty\tf\n"); //printf("\nwrittten track header"); //Initial Population for(int p=0; p
//Random selection of three individuals for (int i=0; i<3; i++){ sel[i]=(rand()%(Npop)); //cout<
for(int j=0; j
184 | P a g e
if (fIpop[j] > fC[j]){ Ipop[j][0]=C[j][0]; Ipop[j][1]=C[j][1]; Best[j][0]=C[j][0]; Best[j][1]=C[j][1]; Best[j][2]=fC[j]; cout<<"\nC["<>end; return 0; }//end of main() //--------------------------------------------------------------------------------------void writeChange( char genFlag, int j) { int complete; double f; char helper[80], helper2[80]; if (genFlag=='P'){ sprintf(helper,"P%i/change.flag",j); printf(helper); } 186 | P a g e
else{ sprintf(helper,"C%i/change.flag",j); printf(helper); } ofstream input(helper); input<<"noChange"; input.close(); }// end writeNoChange int rmChange( char genFlag, int j) { int complete; double f; char helper[80], helper2[80]; if (genFlag=='P'){ sprintf(helper,"P%i/change.flag",j); printf("rm "); printf(helper); printf("\n"); } else{ sprintf(helper,"C%i/change.flag",j); printf("rm "); printf(helper); printf("\n"); } if( remove( helper ) != 0 ) perror( "Error deleting changeFlag" ); else puts( "changeFlag successfully deleted" ); return 0; }// end rmChange void writeVar( char genFlag, int j, double x, double y) { 187 | P a g e
int complete; double f; char helper[80], helper2[80]; if (genFlag=='P'){ sprintf(helper,"P%i/constant/polyMesh/var.dat",j); printf(helper); printf("\n"); } else{ sprintf(helper,"C%i/constant/polyMesh/var.dat",j); printf(helper); printf("\n"); } ofstream input(helper); input<
//ifstream output("P1/obj.dat"); output>>f; output.close(); printf("f =%f",f); return f;
}// end readObj bool lim(double x, double y) { bool inside; if (x>=0.0 && x<=0.05 && y>=-0.375 && y<=0.1) { inside=true; } else { inside=false; } return inside; }//end of lim
QsubPar_parents.sh #!/bin/sh #$ -cwd#!/bin/sh #$ -cwd #$ -j y #$ -S /bin/bash #$ -q [email protected] . $HOME/OpenFOAM/OpenFOAM-1.6/etc/bashrc
echo "Start" cd P0 if [ -s change.flag ] then echo "cleaning P0" rm -rf 0.* result pistonData/obj.dat 189 | P a g e
echo "submitting P0" qsub -N P0 OF_qsub.sh & fi cd ../P1 if [ -s change.flag ] then echo "cleaning P1" rm -rf 0.* result pistonData/obj.dat echo "submitting P1" qsub -N P1 OF_qsub.sh & fi cd ../P2 echo "cleaning P2" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P2" qsub -N P2 OF_qsub.sh & fi cd ../P3 echo "cleaning P3" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P3" qsub -N P3 OF_qsub.sh & fi cd ../P4 echo "cleaning P4" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P4" qsub -N P4 OF_qsub.sh & fi
190 | P a g e
cd ../P5 echo "cleaning P5" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P5" qsub -N P5 OF_qsub.sh & fi cd ../P6 echo "cleaning P6" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P6" qsub -N P6 OF_qsub.sh & fi cd ../P7 echo "cleaning P7" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P7" qsub -N P7 OF_qsub.sh & fi cd ../P8 echo "cleaning P8" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P8" qsub -N P8 OF_qsub.sh & fi cd ../P9 echo "cleaning P9" if [ -s change.flag ] then 191 | P a g e
rm -rf 0.* result pistonData/obj.dat echo "submitting P9" qsub -N P9 OF_qsub.sh & fi cd ../P10 if [ -s change.flag ] then echo "cleaning P10" rm -rf 0.* result pistonData/obj.dat echo "submitting P10" qsub -N P10 OF_qsub.sh & fi cd ../P11 if [ -s change.flag ] then echo "cleaning P11" rm -rf 0.* result pistonData/obj.dat echo "submitting P11" qsub -N P11 OF_qsub.sh & fi cd ../P12 echo "cleaning P2" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P12" qsub -N P12 OF_qsub.sh & fi cd ../P13 echo "cleaning P13" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P13" qsub -N P13 OF_qsub.sh & fi 192 | P a g e
cd ../P14 echo "cleaning P14" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P14" qsub -N P14 OF_qsub.sh & fi cd ../P15 echo "cleaning P15" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P15" qsub -N P15 OF_qsub.sh & fi cd ../P16 echo "cleaning P16" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P16" qsub -N P16 OF_qsub.sh & fi cd ../P17 echo "cleaning P17" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P17" qsub -N P17 OF_qsub.sh & fi cd ../P18 echo "cleaning P18" if [ -s change.flag ] 193 | P a g e
then rm -rf 0.* result pistonData/obj.dat echo "submitting P18" qsub -N P18 OF_qsub.sh & fi cd ../P19 echo "cleaning P19" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P19" qsub -N P19 OF_qsub.sh & fi echo "****************Done Submitting***************" echo "waiting for results from P0" cd ../P0 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P1" cd ../P1 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done 194 | P a g e
echo "waiting for results from P2" cd ../P2 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P3" cd ../P3 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P4" cd ../P4 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P5" cd ../P5 readone=true; 195 | P a g e
while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P6" cd ../P6 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P7" cd ../P7 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P8" cd ../P8 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; 196 | P a g e
fi done echo "waiting for results from P9" cd ../P9 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P10" cd ../P10 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P1" cd ../P1 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P12" 197 | P a g e
cd ../P12 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P13" cd ../P13 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P14" cd ../P14 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P15" cd ../P15 readone=true; while [ $readone = true ]; do 198 | P a g e
if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P16" cd ../P16 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P17" cd ../P17 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P18" cd ../P18 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P19" 199 | P a g e
cd ../P19 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done
#$ -j y #$ -S /bin/bash #$ -q [email protected] . $HOME/OpenFOAM/OpenFOAM-1.6/etc/bashrc
echo "Start" cd P0 if [ -s change.flag ] then echo "cleaning P0" rm -rf 0.* result pistonData/obj.dat echo "submitting P0" qsub -N P0 OF_qsub.sh & fi cd ../P1 if [ -s change.flag ] then echo "cleaning P1" rm -rf 0.* result pistonData/obj.dat echo "submitting P1" qsub -N P1 OF_qsub.sh & fi cd ../P2 echo "cleaning P2" 200 | P a g e
if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P2" qsub -N P2 OF_qsub.sh & fi cd ../P3 echo "cleaning P3" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P3" qsub -N P3 OF_qsub.sh & fi cd ../P4 echo "cleaning P4" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P4" qsub -N P4 OF_qsub.sh & fi cd ../P5 echo "cleaning P5" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P5" qsub -N P5 OF_qsub.sh & fi cd ../P6 echo "cleaning P6" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P6" 201 | P a g e
qsub -N P6 OF_qsub.sh & fi cd ../P7 echo "cleaning P7" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P7" qsub -N P7 OF_qsub.sh & fi cd ../P8 echo "cleaning P8" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P8" qsub -N P8 OF_qsub.sh & fi cd ../P9 echo "cleaning P9" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P9" qsub -N P9 OF_qsub.sh & fi cd ../P10 if [ -s change.flag ] then echo "cleaning P10" rm -rf 0.* result pistonData/obj.dat echo "submitting P10" qsub -N P10 OF_qsub.sh & fi cd ../P11 202 | P a g e
if [ -s change.flag ] then echo "cleaning P11" rm -rf 0.* result pistonData/obj.dat echo "submitting P11" qsub -N P11 OF_qsub.sh & fi cd ../P12 echo "cleaning P2" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P12" qsub -N P12 OF_qsub.sh & fi cd ../P13 echo "cleaning P13" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P13" qsub -N P13 OF_qsub.sh & fi cd ../P14 echo "cleaning P14" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P14" qsub -N P14 OF_qsub.sh & fi cd ../P15 echo "cleaning P15" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat 203 | P a g e
echo "submitting P15" qsub -N P15 OF_qsub.sh & fi cd ../P16 echo "cleaning P16" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P16" qsub -N P16 OF_qsub.sh & fi cd ../P17 echo "cleaning P17" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P17" qsub -N P17 OF_qsub.sh & fi cd ../P18 echo "cleaning P18" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P18" qsub -N P18 OF_qsub.sh & fi cd ../P19 echo "cleaning P19" if [ -s change.flag ] then rm -rf 0.* result pistonData/obj.dat echo "submitting P19" qsub -N P19 OF_qsub.sh & fi
204 | P a g e
echo "****************Done Submitting***************" echo "waiting for results from P0" cd ../P0 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P1" cd ../P1 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P2" cd ../P2 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P3" cd ../P3 205 | P a g e
readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P4" cd ../P4 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P5" cd ../P5 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P6" cd ../P6 readone=true; while [ $readone = true ]; do if [ -s result ] then 206 | P a g e
cat result readone=false; fi done echo "waiting for results from P7" cd ../P7 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P8" cd ../P8 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P9" cd ../P9 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done
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echo "waiting for results from P10" cd ../P10 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P1" cd ../P1 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P12" cd ../P12 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P13" cd ../P13 readone=true; while [ $readone = true ]; 208 | P a g e
do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P14" cd ../P14 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P15" cd ../P15 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P16" cd ../P16 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; 209 | P a g e
fi done echo "waiting for results from P17" cd ../P17 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P18" cd ../P18 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done echo "waiting for results from P19" cd ../P19 readone=true; while [ $readone = true ]; do if [ -s result ] then cat result readone=false; fi done
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OF_qsub.sh #!/bin/sh #$ -cwd #$ -j y #$ -S /bin/bash . $HOME/OpenFOAM/OpenFOAM-1.6.x/etc/bashrc export PATH=/home/stephen/code/OpenFOAM/setStirlingGeomerty:$PATH echo "Start $PWD" rm -rf 0.* echo "setStirling Geometry" setStirlingGeometry blockMesh setSet -batch makeZones.setSet echo "rhoPorousPimpleDyMFoam" rhoPorousPimpleDyMFoam cd stirlingAnalysis echo "pistonPressureOut" ./pistonPressureOut cd .. echo "Writing objective value for $PWD" > result echo "Done $PWD"
pistonPressureOut #!/bin/sh #$ -cwd #$ -j y #$ -S /bin/bash . $HOME/OpenFOAM/OpenFOAM-1.6.x/etc/bashrc rm -rf ../pistonData mkdir ../pistonData echo "Writing U components to compU" 211 | P a g e
cd .. ls -d 0* > pistonData/time.out cd stirlingAnalysis ./USteps cd .. ################## #foamCalc components U > pistonData/compU echo "Calculating Average Ux on patch valveCold" #patchAverage Ux valveCold > pistonData/valveCold_Ux echo "Writing Time to time.out" #cat pistonData/valveCold_Ux | grep 'Time' | cut -d' ' -f3 > pistonData/time.out echo "Writing Average Ux on patch valveCold to valveCold_Ubar.out" #cat pistonData/valveCold_Ux | grep 'Average of Ux over patch' | cut -d' ' -f12 > pistonData/valveCold_Ubar.out echo "Writing net p/rho on patch valveCold to valveCold_p.out" patchIntegrate p valveCold > pistonData/valveCold cat pistonData/valveCold | grep 'Integral of p over area magnitude' | cut -d' ' -f15 > pistonData/valveCold_p.out echo "Writing net p/rho on patch valveHot to valveHot_p.out" patchIntegrate p valveHot > pistonData/valveHot cat pistonData/valveHot | grep 'Integral of p over area magnitude' | cut -d' ' -f15 > pistonData/valveHot_p.out ############### echo "Writing Average Ux on patch pistonCold to pistonCold_Ubar.out" #patchAverage Ux piston > pistonData/piston_Ux #cat pistonData/piston_Ux | grep 'Average of Ux over patch' | cut -d' ' -f12 > pistonData/piston_Ubar.out echo "Writing net p/rho on patch piston to piston_p.out" patchIntegrate p piston > pistonData/pistonCold cat pistonData/pistonCold | grep 'Integral of p over area magnitude' | cut -d' ' -f15 > pistonData/piston_p.out
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cd stirlingAnalysis ./pistonPlot2 ./pistonNet echo "End pistonPressureOut"
pistonPlot2 #!/usr/bin/ksh #$ -cwd #$ -j y . $HOME/OpenFOAM/OpenFOAM-1.6.x/etc/bashrc echo "Writing valve_Ubar_time.out" paste ../pistonData/time.out ../pistonData/valve_Ubar_time.out
../pistonData/valve_Ubar.out
>
echo "Writing valveCold_p_time.out" paste ../pistonData/time.out ../pistonData/valveCold_p_time.out
../pistonData/valveCold_p.out
>
echo "Writing valveHot_p_time.out" paste ../pistonData/time.out ../pistonData/valveHot_p_time.out
../pistonData/valveHot_p.out
>
echo "Writing piston_Ubar_time.out" paste ../pistonData/time.out ../pistonData/piston_Ubar_time.out
../pistonData/piston_Ubar.out
>
echo "Writing piston_p_time.out" paste ../pistonData/time.out ../pistonData/piston_p.out > ../pistonData/piston_p_time.out echo "Writing valveNet_time.out" paste ../pistonData/time.out ../pistonData/valveCold_p.out ../pistonData/valveHot_p.out ../pistonData/valve_Ubar.out > ../pistonData/valveNet_time.out echo "Writing pistonNet_time.out" paste ../pistonData/time.out ../pistonData/piston_p.out ../pistonData/valve_Ubar.out > ../pistonData/pistonNet_time.out echo "End pistonPlot2" 213 | P a g e
pistonNet #!/usr/bin/ksh #$ -cwd #$ -j y . $HOME/OpenFOAM/OpenFOAM-1.6/etc/bashrc awk '{print $1,$2,$3,$4,($3-$2),(1.2*1.0261*($2-$3)),((1.0261*($2-$3))*$5)}' ../pistonData/valveNet_time.out > ../pistonData/valveNet_time2.out awk '{a+=$5}END{print "----valve piston average NetPressure (Pa) " (a)/62}' ../pistonData/valveNet_time2.out >> ../pistonData/summary.out awk '{a+=$7}END{print "valve piston average piston (W) " a/62}' ../pistonData/valveNet_time2.out >> ../pistonData/summary.out awk '{print $1,$2,$3,($2),(1.2*0.3558*($2)),((0.3558*($2))*$4)}' ../pistonData/pistonNet_time.out > ../pistonData/pistonNet_time2.out awk '{a+=$5}END{print "piston piston average NetPressure (Pa) " (a)/62}' ../pistonData/pistonNet_time2.out >> ../pistonData/summary.out awk '{a+=$7}END{print "piston piston average piston (W) " a/62}' ../pistonData/pistonNet_time2.out >> ../pistonData/summary.out awk '{a+=+$7}END{print a}' ../pistonData/valveNet_time2.out ../pistonData/dTemp.dat awk '{a+=+$6}END{print a}' ../pistonData/pistonNet_time2.out ../pistonData/pTemp.dat paste ../pistonData/dTemp.dat ../pistonData/pTemp.dat > ../pistonData/nTemp.dat awk '{a+=$1+$2}END{print a}' ../pistonData/nTemp.dat > ../pistonData/obj.dat
> >
awk '{a+=$1+$2}END{print "-------obj value " a}' ../pistonData/nTemp.dat cat ../pistonData/valveNet_time2.out | cut -d' ' -f7 > ../pistonData/dNet.out cat ../pistonData/pistonNet_time2.out | cut -d' ' -f6 > ../pistonData/pNet.out paste ../pistonData/time.out ../pistonData/dNet.out ../pistonData/pNet.out ../pistonData/engineNet.out
>
echo "End pistonNet"
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Appendix F. Solution Dependent Motion Codes stirlingSDM.m % % necessary input constant % m = 1; l1 = 1; l2 = 2; l3 = 3; M = 0; I1 = 3; t_final = 10;
% mass of the pistons % length of the crank radius % length of the displacer connecting rod % length of the power connecting rod % constant moment at bar1 % inertia of bar1 % time duration for calculation % initial time is always 0
% % Intitial conditions for theta and theta_dot % theta = 0; % initial displacement of the crank theta_dot = 0; % initial angular velocity of the crank %theta_ddot = M/I1; % initial angular acceleration of the crank
% % Setup integration solver (ODE45) % %AbsTol = 1e-20; % tolerance for ODE45 solver % smaller it is, more accurate the solution % will be options = odeset('RelTol', 1e-10); tspan = [0, t_final]; x0 = [theta,theta_dot,];%theta_ddot]; 215 | P a g e
% % Perform ODE45 solver % %[t,x] = ode45('p3_175s1',0,t_final,x0);%,AbsTol); [t,x] = ode45('woodsPiston',tspan,x0,options); %[t,x] = ode15s('woodsPiston',tspan,x0,options); [n,m] = size(x); Theta = x(:,1); Phi = asin((l1/l2)*sin(Theta)); Psi = asin((l1/l3)*sin(Theta)); X = l1*cos(Theta)+l2*cos(Phi); Y = l1*cos(Theta+pi()/2)+l2*cos(Psi); w = x(:,2); % % Plot the results % figure(1); clf; orient tall; subplot(2,1,1),plot(t,Theta); title('Problem 3.175'); xlabel('Time (sec.)'); ylabel('Theta'); subplot(2,1,2),plot(t,Phi); ylabel('Phi'); xlabel('Time (sec.)'); figure(2); clf; orient tall; subplot(2,1,1),plot(t,X); title('Problem 3.175'); xlabel('Time (sec.)'); ylabel('Displacer position (mm)'); subplot(2,1,2),plot(Theta,X); xlabel('Theta (rad.)'); 216 | P a g e
ylabel('Displacer position (m)');
figure(3); clf; orient tall; subplot(2,1,1),plot(t,w); title('Problem 3.175'); xlabel('Time (sec.)'); ylabel('w (rad/s)'); subplot(2,1,2),plot(Theta,w); xlabel('Theta (rad.)'); ylabel('w (rad/s)'); figure(4); clf; orient tall; subplot(2,1,1),plot(t,Y); title('Problem 3.175'); xlabel('Time (sec.)'); ylabel('Power piston position (m)'); subplot(2,1,2),plot(Theta,Y); xlabel('Theta (rad.)'); ylabel('Power piston position (m)');
mxx = max(X); mnx = min(X); mxt = max(Theta); mnt = min(Theta); %axis([mnt,mxt,mnx,mxx]) function xdot = woodsPiston(t,x) % % State variables % l1 = 1; l2 = 2; l3 = 3; b2 = 0.5*l2; a2 = b2;
% length of the crank radius % length of the displacer connecting rod % length of the power connecting rod
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b3 = 0.5*l3; a3 = b3; m2 = 5; m3d = 20; m3p = 10;
sPhi = asin((l1/l2)*sin(x(1))); sPsi = asin((l1/l3)*sin(x(1))); sX = l1*cos(x(1))+l2*cos(sPhi); sY = l1*cos(x(1)+pi()/2)+l2*cos(sPsi); %PX = 100*cos(sPhi);%100*cos(2*x(1))+200; %PY = 100*cos(sPsi);%80*sin(2*x(1))+160; PX = abs( 100*cos(sPhi)*l1*sin(x(1)) );%100*cos(2*x(1))+200; PY = abs( 100*cos(sPsi)*l1*cos(x(1)) ); Mp = PX+PY;%(-sX/(x(2)+0.00001))*PX + (-sY/(x(2)+0.00001))*PY; Mr = 50*x(2)^2; J0 = 100; Jab2 = 150; Jab3 = 300; A = J0 + ((m2*b2)/l2)*l1^2 + 0.5*( (m3d+((m2*a2)/l2))*l1^2 + Jab2*(l1/l2)^2 ) + ((m2*b3)/l3)*l1^2 + 0.5*( (m3p+((m2*a3)/l2))*l1^2 + Jab3*(l1/l3)^2 ) ; B = 0.5*( (m3d+((m2*a2)/l2))*l1^2-Jab2*(l1/l2)^2 ) + 0.5*( (m3p+((m2*a3)/l3))*l1^2Jab3*(l1/l3)^2 ); CC = B*sin(2*x(1)); Gi = A-B*cos(2*x(1)); x_dot1 = x(2); %x_dot2 = (Mp-Mr-CC*x(2)^2)/Gi; x_dot2 = (Mp-Mr-CC*x(2)^2)/Gi; %x_dot3 = 0; %x_dot4 = 1; xdot = [x_dot1; x_dot2;];
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