stirling

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.

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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

Team Leader

Kevin LaMott

Team Member

Stephen Wood

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

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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



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Results .................................................................................................................................. 131

Analysis ............................................................................................................................... 132

Stirling Engine Performance ....................................................................................... 133

Overview .............................................................................................................................. 133

Experimental Set Up ............................................................................................................ 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

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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

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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 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 71: Prototype Design2 Optimization Generation 30 ............................................ 89




Figure 75: Prototype Design2 Final Design...................................................................... 91

Figure 76: Theta (Displacer Piston Crank Angle) and Phi (Power Piston Crank Angle) . 93

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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

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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.

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).

Figure 1. Earthquake Density Map of the Globe

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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 2. Tsunami History of Location, Intensity, and Size

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 Stirling-

based 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-to-

grid 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.

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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.

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.

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.

Figure 21. Rotating house with tracking solar

panels that operate independently

Figure 22. Point Focus parabolic dish with Stirling Engine

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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

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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


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


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

Concentrated

Electricity

Produced

Concentrator

Cost

Engine

Cost

Total

Cost $/W

3‟X3‟

(1 panel) 1.5 kW 0.3 kW $150 $250 $400 $1.33

6‟X6‟

(4 panels) 6 kW 1.2 kW $200 $550 $750 $0.63

9‟X9‟

(9 panels) 13.5 kW 2.7 kW $350 $600 $950 $0.35

12‟X12‟

(16 panels) 24 kW 4.8 kW $900 $900 $1,800 $0.38

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.

43 | P a g e

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.

44 | P a g e

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

45 | P a g e

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) 31 cc

Expansion Clearance Volume (Vcle)) 31 cc

Compression Sweep Volume (Vswc) 32 cc

Expansion Sweep Volume(Vswe) 32 cc

Cooler Volume(Vk) 15 cc

Regenerator Volume (Vr) 50 cc

Heater Volume (Vh) 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.

46 | P a g e

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.

47 | P a g e

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.

48 | 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.

49 | P a g e

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

50 | P a g e

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).

J = (W/m2) x (Area in m

2) 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.

51 | P a g e

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.

Figure 34. Power Flows for 2.7 kW Stirling Engine

0

2000

4000

6000

8000

10000

12000

0 2 4 6 8 10

Po

we

r, W

atts

Th

Hour of Sunlight (Hr)

Energy Colected

Energy Dissipated

Energy Converted

52 | P a g e

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.

Figure 35. Heat Flows for Cooling Reservoir

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) 3.0

Length (m) 5.0

Depth (m) 0.333

Cubic m 5.49

-6000

-4000

-2000

0

2000

4000

6000

8000

0 2 4 6 8 10

He

at F

low

Rat

e, W

th

Hour of Sunlight, Hr

Heat Input (W)

Heat Rejected (W)

Heat Stored (W)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0 2 4 6 8 10

Tem

pe

ratu

re, C

Hour of Sunlight, Hr

Ambient Temp (C)

Reisvour Temperature

53 | P a g e

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.

54 | P a g e

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.

55 | P a g e

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,

56 | P a g e

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 Description Units

Tc Temperature of Working Gas within the compression space Kelvin

Tk Temperature of Working Gas within the cooler Kelvin

Tr Temperature of Working Gas within the regenerator Kelvin

Th Temperature of Working Gas within the heater Kelvin

Te Temperature of Working Gas within the expansion Kelvin

p Pressure of the Working Gas Pa

Dp Change in Pressure Pa/s

M Total Mass of Working Gas kg

mc Mass of Working Gas within the compression space kg

mk Mass of Working Gas within the cooler kg

mr Mass of Working Gas within the regenerator kg

mh Working Gas Mass within the heating kg

me Working Gas Mass within the expansion space kg

Dmc Change in mass of the compression space kg/s

Dmk Change in mass of the cooler kg/s

Dmr Change in mass of the regenerator kg/s

Dmh Change in mass of the heater kg/s

Dme Change in mass of the expansion space kg/s

gAck Mass flow rate from compression space to cooler kg/s

gAkr Mass flow rate from cooler to regenerator kg/s

gArh Mass flow rate from regenerator to heater kg/s

gAhe Mass flow rate from heater to expansion space kg/s

W Work Done by the engine J

Qk Energy flow rate from cooler to working Gas J

Qr Energy flow rate from regenerator to working Gas J

Qh Energy flow rate from heater to working Gas J

DW Change in work done by the engine J/s

DQk Change in Energy flow rate from cooler to working Gas J/s

DQr Change in Energy flow rate from regenerator to working Gas J/s

DQh Change in Energy flow rate from heater to working Gas J/s

57 | P a g e

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).

58 | 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. The mass of the working fluid remains constant

2. Use of Ideal Gas

3. The speed of the engine is constant

4. Cyclic state

5. Kinetic and potential energy of the working fluid can be neglected

59 | P a g e

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)

60 | P a g e

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),

61 | P a g e

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,

62 | P a g e

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

63 | P a g e

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).

64 | P a g e

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 Constant Value Units

Working Fluid Mass M 2.8E-05 g

Individual Gas constant R 2077 J/m^2K

Volume in cooler Vk 14*10^-6 m^3

Volume in Regenerator Vr 51*10^-6 m^3

Volume in Heater Vh 70*10^-6 m^3

Cold End Temperature Tk 300 K

Hot End Temperature Th 950 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.

Figure 39. Work per cycle

0 100 200 300 400 500 600 700-3

-2

-1

0

1

2

3

Wat

ts

Crank Angle (1/100 radian)

65 | P a g e

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.

Figure 40. Work Done by Compression Space for Single Cycle

Figure 41. Work Done By Expansion Space for Single Cycle

0 100 200 300 400 500 600 700-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 100 200 300 400 500 600 700 0

0.5

1

1.5

2

2.5

3

3.5

4


Wat

ts


Wat

ts

66 | P a g e

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.

Figure 42. Compression Space Volume

Figure 43. Expansion Space Volume

0 100 200 300 400 500 600 7002

4

6

8

10

12

14

16x 10

-5

0 100 200 300 400 500 600 7002

4

6

8

10

12

14

16x 10

-5


Cub

ic M

eter

s C

ub

ic M

eter

s


67 | P a g e

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.

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.

0 100 200 300 400 500 600 7000.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

5


Pas

cals

68 | P a g e

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.

69 | P a g e

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));

70 | P a g e

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

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)

72 | P a g e

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

73 | P a g e

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

76 | P a g e

Figure 53: Prototype Isothermal Simulation 10th cycle Displacer Piston

Figure 54: Prototype Isothermal Simulation 10th cycle Power Piston

77 | P a g e

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.

78 | P a g e

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.

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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.


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.

80 | P a g e

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

81 | P a g e

// Momentum equation

tmp<fvVectorMatrix> 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

82 | 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

83 | P a g e

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.

84 | P a g e

Figure 62: Design 2 Transient Startup Pressure vs. Time

Figure 63: Design 2 Transient Startup Temperature vs. Time

85 | P a g e

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.

86 | 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

87 | P a g e

Figure 66: Initial Optimization Population

Figure 67: Stirling Helium Geometry Design Space after 4 Generations

Regenerator Parameter (mm)0.5

1

Cold End Parameter (mm)

-8-6

-4-2

02

En

er g

ylo

st

pe

rcycle

(J)

0

1

2

X Y

Z

88 | P a g e

Figure 68: Optimization Population after 15 Generations


Best Member of Genration 15 is member 3

Cold End Parameter 1.160 mm

Regenerator Parameter 1.452 mm



1


-8-6

-4-2

02

En

er g

ylo

st

pe

rcycle

(J)

0

1

2

X Y

Z

89 | P a g e



Best Member of Genration 30 is member 18





1


-8-6

-4-2

02

En

er g

ylo

st

pe

rcycle

(J)

0

1

2

X Y

Z

90 | P a g e



Best Member of Generation 45 is member 19





1


-8-6

-4-2

02

En

er g

ylo

st

pe

rcycle

(J)

0

1

2

X Y

Z

91 | P a g e


Figure 75: Prototype Design2 Final Design

Final Design from Generation 70 Population is member 8




100% of the 70th

generation scored within 10% of the best member of the population.


1


-8-6

-4-2

02

En

er g

ylo

st

pe

rcycle

(J)

0

1

2

X Y

Z

92 | 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

93 | P a g e

= 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)

94 | P a g e

Figure 77: w (Crank Speed) vs. time and theta

Figure 78:Displacer piston position vs. time and theta

95 | P a g e

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.

96 | P a g e

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.

97 | P a g e

The bolts holding the engine together need to withstand 800K, 1500 pounds-

force, 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 non-

conductive, 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.

98 | P a g e

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

99 | P a g e

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.

100 | P a g e

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 Magnitude Location Heat Power In 2700 W Center of top surface

Convection h =10 W/m^2 K T∞ = 300K

Top and side surfaces

Radiation T∞ = 300K View Factor=0.5 ε = 0.98

Top surface

Convection h = 80 W/m^2 K T∞ = 375K

Interior surface

Convection h = 120 W/m^2 K T∞ = 350 K

Extended mesh surface

Figure 81. Hot End Mesh and Imposed Conditions

101 | P a g e

Figure 82. Thermal Plot o Lower End of Hot End

Figure 83. Thermal Plot of Upper Portion of Hot End

102 | P a g e

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.

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

0

100

200

300

400

500

600

700

800

0 2 4 6 8

Re

sulin

g Te

mp

era

ture

(°C

)

Hour of the Sunlight (hr)

Hot End Temperature

103 | P a g e

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 Magnitude Direction

Restraint Fixed Interior surfaces of bolt holes

Internal Pressure 500 psi Normal to interior surface

Thermal Stress Resultant Resultant from thermal analysis

104 | P a g e

Displacer Piston Base Stress Analysis


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



Internal Pressure 20 psi Normal to lower surface


Force 60 lbs-f Load from power piston

105 | P a g e

Engine Body Stress Analysis


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


Restraint Fixed Top and bottom surfaces

Restraint Cylindrical Crankshaft Holes



106 | P a g e

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


Restraint Fixed Interior surfaces of Crankshaft holes

Force 60 lbs-f Interior surfaces of displacer shaft holes

107 | P a g e

Power Piston Rod Stress Analysis


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



Force 60 lbs-f Interior surfaces of power piston shaft holes

108 | P a g e

Engine Bolts/ Linear Shafts Stress Analysis


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


Restraint Fixed Lower portion of bolt head

Force 8000 N Top surface of threading

109 | P a g e

Crankshaft Stress Analysis


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


Restraint Fixed End portions of shaft

Force 600 N Central rod connection

Rotation 950 RPM Central rod connection

110 | P a g e

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





111 | P a g e

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




112 | P a g e

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





113 | P a g e

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



Internal Pressure 20 psi Normal to lower surface


Force 60 lbs-f Load from power piston

114 | P a g e

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


Restraint Fixed Top and bottom surfaces

Restraint Cylindrical Crankshaft Holes



115 | P a g e

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



Force 60 lbs-f Interior surfaces of displacer shaft holes

116 | P a g e

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




117 | P a g e

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


Restraint Fixed Lower portion of bolt head

Force 8000 N Top surface of threading

118 | P a g e

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





119 | P a g e

Cost Analysis

Description Each Amount Total

Unit

Total

Flange-Mount Linear Ball Bearing, 1/4" ID $25.22 4 $100.88 $100.88

SS Ball Bearing,1/4" ID, 3/8" OD $7.00 8 $56.00 $56.00

Aluminum (Alloy 7075) 5" Diameter, 1/2"

Long $20.32 1 $20.32 $20.32

18-8 SS Hex Head Screw 1/4"-20, 8-1/2"

Length $5.19 4 $20.76 $20.76

SS Serrated-Flange Hex Locknuts 1/4"-20 $3.83 1 $3.83 $0.96

Acrylic Tube 5" OD X 4-3/4" ID, 1' Length $19.20 1 $19.20 $4.80

150cc Piston and Cylinder, 2.25" bore, 2"

stroke $75.00 1 $75.00 $75.00

Polyurethane Foam Rod 4" Diameter, 36"

Length $10.65 1 $10.65 $0.30

Aramid/Buna-N Gasket, 1/8" Thick, 6" X 6" $6.63 1 $6.63 $6.63


Arc Welder $125.00 1 $125.00 $0.00

Leather Gloves $5.35 1 $5.35 $0.00

6-32X1 Steel Bolts $1.04 1 $1.04 $0.10

1/4" Steel rod 3' $3.50 1 $3.50 $1.75

4 1/8" hole saw bit $26.72 1 $26.72 $0.00

1/4" drill bit $2.75 1 $2.75 $0.00

Lexan sheet 8"X10" $4.26 1 $4.26 $4.26

10" Metal Cutting Wheel $4.46 1 $4.46 $0.00

3/4 inch shaft collars $4.88 6 $29.28 $29.28

4" #6 threaded rod $10.00 1 $10.00 $1.25

#6 nuts $1.04 1 $1.04 $0.17

Nylon Bushings $2.03 1 $2.03 $2.03

Ceramic Spacer $3.98 1 $3.98 $3.98

Alternator, 960W, 12V $60.00 1 $60.00 $60.00

Flywheel Assembly $25.00 1 $25.00 $25.00

Digital Calipers $15.00 1 $15.00 $0.00

$639.82 $417.04

120 | P a g e

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.

121 | P a g e

Parts List and Analysis

Solar Concentrator Parts List Table 28. Parts List and Analysis for Prototype Solar Concentrator

Description Unit Price Amount Total

Angle steel $4.00 24 $96.00

Fresnel Lenses $70.00 1 $70.00

3/8" Bolts $12.00 1 $12.00

Hinges $3.79 4 $15.16

5/8" Hexbar Shaft $6.44 1 $6.44

8" X 10" Acrylic Sheet $4.00 1 $4.00

0.6 RPM 50 In-Lbs Mini-Gearmount $42.00 1 $42.00

20:1 Worm gear speed reducer $100.00 1 $100.00

Key Stock $0.70 1 $0.70

Bungee Chord $0.15 24 $3.60

PLC with programmer $15.00 1 $15.00

$364.90

122 | P a g e

Stirling Engine Parts List Table 29. Part List and Analysis for Prototype Engine

Description Each Amount Total

Unit

Total

Flange-Mount Linear Ball Bearing, 1/4" ID $25.22 4 $100.88 $100.88

SS Ball Bearing,1/4" ID, 3/8" OD $7.00 8 $56.00 $56.00

Aluminum (Alloy 7075) 5" Diameter, 1/2"

Long $20.32 1 $20.32 $20.32

18-8 SS Hex Head Screw 1/4"-20, 8-1/2"

Length $5.19 4 $20.76 $20.76

SS Serrated-Flange Hex Locknuts 1/4"-20 $3.83 1 $3.83 $0.96

Acrylic Tube 5" OD X 4-3/4" ID, 1' Length $19.20 1 $19.20 $4.80

150cc Piston and Cylinder, 2.25" bore, 2"

stroke $75.00 1 $75.00 $75.00

Polyurethane Foam Rod 4" Diameter, 36"

Length $10.65 1 $10.65 $0.30



Arc Welder $125.00 1 $125.00 $0.00

Leather Gloves $5.35 1 $5.35 $0.00

6-32X1 Steel Bolts $1.04 1 $1.04 $0.10

1/4" Steel rod 3' $3.50 1 $3.50 $1.75

4 1/8" hole saw bit $26.72 1 $26.72 $0.00

1/4" drill bit $2.75 1 $2.75 $0.00

Lexan sheet 8"X10" $4.26 1 $4.26 $4.26

10" Metal Cutting Wheel $4.46 1 $4.46 $0.00

3/4 inch shaft collars $4.88 6 $29.28 $29.28

4" #6 threaded rod $10.00 1 $10.00 $1.25

#6 nuts $1.04 1 $1.04 $0.17

Nylon Bushings $2.03 1 $2.03 $2.03

Ceramic Spacer $3.98 1 $3.98 $3.98

4.6" diameter foam, 6" $13.60 1 $13.60 $0.57

$553.42 $332.61

123 | P a g e

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.

128 | P a g e

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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300 400 500 600 700 800 900

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ak H

eat

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Surface Temperature (K)

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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.

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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.

Figure 110. Experimental Test Article Temperature

Figure 111. Theoretical and Experimental Collected Energy

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600

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(F)

Time, (Hr:Min)

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500

9:30 10:42 11:54 13:06 14:18 15:30

Po

we

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Wat

ts)

Time of Day, (Hr:Min)

Experimental Results

Theoretical Values

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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 re-

deploying.

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

Table 30. Breakdown of Deadlines

Report Due Items Competed

10% November 5, 2009 Introduction

Design Alternatives

Project Management

Conclusion

References

November 5, 2009

25% December 3, 2009 Project Formulation

Engineering Design

Engineering Analysis

Prototype Construction

December 3, 2009

- December 11, 2009 Team Poster December 10, 2009

- January 14, 2010 IAB Project Feasibility

Presentation

January 14, 2010

50% February 16, 2010 Final Design (100% completed)

Prototype Assembly (50%

completed)

February 16, 2010

75% March 10, 2010 Prototype Assembly (100%

completed)

Testing of Prototype (50%

completed)

100% April 2, 2010 All Report

Power point

- April 6 & 8 Rehearsal Presentation to MME

- April 14 & 15 Final Presentation to IAB and

MME

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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

STEPHEN Introduction DENISSE

Fatigue Failure Design Theories

Problem Statement DENISSE

Deflection Analysis STEPHEN

Motivation DENISSE

Component Design/Selection STEPHEN

Literature Survey KEVIN

Finite Element Analysis STEPHEN

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 DENISSE

Assembly and Disassembly KEVIN

Breakdown of Responsibilities Among Team Members 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

Engineering Design and Analysis DENISSE

Risk Assessment DENISSE

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.

Survey Prototype Design Modeling Report Presentation Total

Kevin 27 68 45 48 42 23 253

Denisse 46 22 25 12 85 60 250

Stephen 12 12 12 183 11 15 245

Project Total

85 102 82 243 138 98 748

Table 33. Hours Worked on Design and Development

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.

Figure 113. Distribution of Work based on Cost

Survey

Prototype

Design

Modeling

Report

Presentation

Survey

Prototype

Design

Modeling

Report

Presentation

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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

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Dynamics and Heat Transfer. New York: The American Society of Mechanical

Engineers.

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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

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en-masse-citing-environmental-concerns/

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january 10, 2010, from discovermagazine.com:

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Chemistry Society. Philadelphia, Pennsylvania.

Frank P. Incropera, D. P. (2002). Fundamentals of Heat and Mass Transfer. Danvers:

John Wiley & sons.

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Interior, U. D. (2009, October 21 21). Earthquake Density Maps for the World. Retrieved

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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

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Conversion Technologies for Space Applications. NASA Glenn Research Center

Technical Memorandum .

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Nightingale, N. (1986). Automotive Stirling Engine. Cleveland: NASA.

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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(i-

1)*(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


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


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(i-

1)*(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;

166 | P a g e

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)

167 | P a g e

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 % Before start of program, the variables month, day, year, longitude, and localOffset need to be % 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))

168 | P a g e

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

169 | P a g e

PAUSE 17000 PULSOUT 1,650 PAUSE 223000 RETURN Home: PULSOUT 1,450 PAUSE 2550000 PULSOUT 1,650 RETURN END

170 | P a g e

Appendix D. Stirling Geometry and Mesh Generation Codes

var.dat

0.00000 //Cold End Parameter

0.00000 //Regenerator Parameter

setStirlingGeomertry.C

#include <iostream>

#include <stdio.h>

#include <stdlib.h>

#include <cstdio>

#include <vector>

#include <cctype>

#include <math.h>

#include <string.h>

#include <fstream>

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,

171 | 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);

172 | 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;

173 | 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);

//---

// verticies IDs------------

xH0 0;

xH1 1;

xw0 2;

xw1 3;

xw2 4;

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;

//---

174 | 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

175 | 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;

176 | 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)


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)


hex ($xH1 $xH1i $xw0i $xw0 $xd1 $xd1i $xw1i $xw1) (1 $yone $xtwo)


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

177 | 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)


hex ($xd1 $xd1i $xw1i $xw1 $xH1 $xH1i $xw0i $xw0) blockB (1 $yone $xtwo)


hex ($xd2 $xd2i $xi4i $xi4 $xd1 $xd1i $xw1i $xw1) blockC (1 $yone 125)


hex ($xi5 $xi5i $xw2i $xw2 $xd2 $xd2i $xi4i $xi4) blockD (1 $yone $xtwo)


hex ($xw3 $xw3i $xi5i $xi5 $xd3 $xd3i $xd2i $xd2) blockE (1 60 $xtwo)


//hex ($xi0 $xi0i $xi1i $xi1 $xd3 $xd3i $xw3i $xw3) blockF (1 $xtwo $yone)


//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)


hex ($xp0 $xp0i $xw4i $xw4 $xd3 $xd3i $xw3i $xw3) blockF (1 $xtwo 40)


);

edges

(

);

patches

(

wall cylinderHead //0

(

($xH0 $xH0i $xH1i $xH1)

($xH1 $xH1i $xw0i $xw0)

)

wall cold //1

(

($xw3 $xw3i $xw4i $xw4)

178 | 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

(


)

wall valveSides //7

(


)

wall valveWalls //8

(

($xw1 $xw1i $xi4i $xi4)

($xi4 $xi4i $xw2i $xw2)

)

wall piston //9

(

($xp0 $xp0i $xw4i $xw4)




)

179 | 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

(

);

//********************************************************************* //

180 | P a g e

Appendix E. Optimization Codes

diffEvol.C

#include <iostream>

#include <math.h>

#include <stdio.h>

#include <stdlib.h>

#include <cstdio>

#include <cctype>

#include <fstream>

#include <cstdlib>

#include <string.h>

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;

181 | 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<Npop; p++){

for (int i=0; i<2; i++){

x[i]=xmin[i]+ ((double)rand()/(double)RAND_MAX)*xrange[i];

Ipop[p][i]=x[i];

cout<<"Ipop["<<p<<"]["<<i<<"]= "<<Ipop[p][i]<<'\t';

}

//

// Write Ipop to var.dat files in parent case dirs

//

genFlag='P';

writeVar( genFlag, p, Ipop[p][0], Ipop[p][1]);

182 | P a g e

cout<<endl;

}

//Mutation parmeters

F=0.8;

//

//************************Main Generation loop

do{

//for (gen=0; gen<10; gen++){

cout<<"\n************************************************";

cout<<"\nGeneration "<<gen;

cout<<"\n************************************************\n";

//

//Evaluating all parents Ipop[j]

complete=system("./QsubPar_parents.sh");

//////////////////////////

//Reading all Ipop[j]

//fIpop=funceval( probnum, varnum, Ipop[j][0], Ipop[j][1]);

for(int j=0; j<Npop; j++){//1

// fIpop[j]=read obj.dat file from jth case dir

genFlag='P';

fIpop[j]=readObj( genFlag, j);

}//end for j 1

stagnant=0;

nochange=0;

//Cross over parameters

double CR=0.6;

int dij, dky;

/////////////////////////////////////////////

//populating a new generation

//Building intermediate Parents from three randomly selected individuals

for(int j=0; j<Npop; j++){

do{

183 | P a g e

//Random selection of three individuals

for (int i=0; i<3; i++){

sel[i]=(rand()%(Npop));

//cout<<Npop;

cout<<"sel["<<i<<"]= "<<sel[i]<<'\t';

}//end of i

cout<<endl;

//Mutation

double m=(double)rand()/(double)RAND_MAX;

int r;

if (m>0.5)

r=1;

else

r=-1;

for (int k=0; k<2; k++){

P[j][k]=Ipop[sel[0]][k]+(Ipop[sel[1]][k]-Ipop[sel[2]][k])*F*r;

//cout<<"P["<<j<<"]["<<k<<"]= "<<P[j][k]<<'\t';

}//end of k

//cout<<endl;

}while(!lim(P[j][0],P[j][1]));//end of while(lim)

for (int k=0; k<2; k++){

//P[j][k]=Ipop[sel[0]][k]+(Ipop[sel[1]][k]-Ipop[sel[2]][k])*F*r;

cout<<"P["<<j<<"]["<<k<<"]= "<<P[j][k]<<'\t';

}//end of k

cout<<endl;

//Cross over

double R=(double)rand()/(double)RAND_MAX;

if (R>CR){

dij=1;

dky=0;

}

else{

dij=0;

dky=1;

}

for (int k=0; k<2; k++){

184 | P a g e

C[j][k]=Ipop[j][k]*dij+P[j][k]*dky;

}//end of k

for (int k=0; k<2; k++){


cout<<"C["<<j<<"]["<<k<<"]= "<<C[j][k]<<'\t';

}//end of k

cout<<endl;

//

// Write to var.dat files in children case dirs

//

genFlag='C';

writeVar( genFlag, j, C[j][0], C[j][1]);

cout<<endl;

}//end of j

//////////////////////////

//Evaluating all C[j]

complete=system("./QsubPar_children.sh");


//fC=funceval( probnum, varnum, C[j][0], C[j][1]);

//////////////////////////

//Reading all C[j]


for(int j=0; j<Npop; j++){//1

// fIpop[j]=read obj.dat file from jth case dir

genFlag='C';

fC[j]=readObj( genFlag, j);

}//end for j 1

for(int j=0; j<Npop; j++){

185 | 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["<<j<<"] is better than its parent!= "<<fC[j]<<endl;

//write Change flagFile to parent caseDir root

writeChange( 'P', j);

}

else{

Best[j][0]=Ipop[j][0];

Best[j][1]=Ipop[j][1];

Best[j][2]=fIpop[j];

cout<<"\nIpop["<<j<<"] is better than its child!= "<<fIpop[j]<<endl;

nochange++;

//remove Change flagFile from parent caseDir root

rmChange( 'P', j);

}

track<<'\n'<<gen<<'\t'<<numfunc<<'\t'<<Best[j][0]<<'\t'<<Best[j][1]<<'\t'<<Best[j][2];

fprintf(track2,"%i\t%i\t%f\t%f\t%f\n",gen,numfunc,Best[j][0],Best[j][1],Best[j][2]);

printf("@@@\t%i\t%i\t%f\t%f\t%f\n",gen,numfunc,Best[j][0],Best[j][1],Best[j][2]);

//

// Write best to var.dat files in parent case dirs

//

genFlag='P';

writeVar( genFlag, j, Best[j][0], Best[j][1]);

cout<<endl;

}//end of j

cout<<"\nnochange= "<<nochange<<"\tgen= "<<gen;

//cin>>end;

if (nochange==19){

stagnant++;

}

else{

186 | P a g e

stagnant=0;

}

gen++;

}while(stagnant<10&&numfunc<10000);//end of for gen

*************************************

cout<<"\nnochange= "<<nochange<<"\tgen= "<<gen;

cout<<"\n!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\n";

//cout<<"\nBest\nx\ty\tfunc\n";

for (int i=0; i<Npop; i++){

for (int k=0; k<3; k++){


cout<<"Best["<<i<<"]["<<k<<"]= "<<Best[i][k]<<'\t';

}//end of k

cout<<endl;

}//end of i

cout<<"Converged!\n";

track<<"\nConverged!";

track.close();

fclose(track2);

cin>>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);

}

187 | 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;


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)

{

188 | P a g e

int complete;

double f;


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<<x<<'\n'<<y;

input.close();

}// end writeVar

double readObj( char genFlag, int j)

{

numfunc++;

int complete;

double f=1;


if (genFlag=='P'){

sprintf(helper,"P%i/pistonData/obj.dat",j);

printf(helper);

printf("\n");

}

else{

sprintf(helper,"C%i/pistonData/obj.dat",j);

printf(helper);

printf("\n");

}

//sprintf(helper2,"rm ");

//complete=system(strcat(helper2,helper));

ifstream output(helper);

189 | P a g e

//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

190 | P a g e

echo "submitting P0"

qsub -N P0 OF_qsub.sh &

fi

cd ../P1


then

echo "cleaning P1"




fi

cd ../P2

echo "cleaning P2"


then




fi

cd ../P3

echo "cleaning P3"


then




fi

cd ../P4

echo "cleaning P4"


then




fi

191 | P a g e

cd ../P5

echo "cleaning P5"


then




fi

cd ../P6

echo "cleaning P6"


then




fi

cd ../P7

echo "cleaning P7"


then




fi

cd ../P8

echo "cleaning P8"


then




fi

cd ../P9

echo "cleaning P9"


then

192 | P a g e




fi

cd ../P10


then

echo "cleaning P10"




fi

cd ../P11


then

echo "cleaning P11"




fi

cd ../P12

echo "cleaning P2"


then




fi

cd ../P13

echo "cleaning P13"


then




fi

193 | P a g e

cd ../P14

echo "cleaning P14"


then




fi

cd ../P15

echo "cleaning P15"


then




fi

cd ../P16

echo "cleaning P16"


then




fi

cd ../P17

echo "cleaning P17"


then




fi

cd ../P18

echo "cleaning P18"


194 | P a g e

then




fi

cd ../P19

echo "cleaning P19"


then




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


cd ../P1

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done

195 | P a g e


cd ../P2

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P3

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P4

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P5

readone=true;

196 | P a g e


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P6

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P7

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P8

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

197 | P a g e

fi

done


cd ../P9

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P10

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P1

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


198 | P a g e

cd ../P12

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P13

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P14

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P15

readone=true;


do

199 | P a g e

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P16

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P17

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P18

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


200 | P a g e

cd ../P19

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done

#$ -j y

#$ -S /bin/bash

#$ -q [email protected]


echo "Start"

cd P0


then

echo "cleaning P0"




fi

cd ../P1


then

echo "cleaning P1"




fi

cd ../P2

echo "cleaning P2"

201 | P a g e


then




fi

cd ../P3

echo "cleaning P3"


then




fi

cd ../P4

echo "cleaning P4"


then




fi

cd ../P5

echo "cleaning P5"


then




fi

cd ../P6

echo "cleaning P6"


then



202 | P a g e


fi

cd ../P7

echo "cleaning P7"


then




fi

cd ../P8

echo "cleaning P8"


then




fi

cd ../P9

echo "cleaning P9"


then




fi

cd ../P10


then

echo "cleaning P10"




fi

cd ../P11

203 | P a g e


then

echo "cleaning P11"




fi

cd ../P12

echo "cleaning P2"


then




fi

cd ../P13

echo "cleaning P13"


then




fi

cd ../P14

echo "cleaning P14"


then




fi

cd ../P15

echo "cleaning P15"


then


204 | P a g e



fi

cd ../P16

echo "cleaning P16"


then




fi

cd ../P17

echo "cleaning P17"


then




fi

cd ../P18

echo "cleaning P18"


then




fi

cd ../P19

echo "cleaning P19"


then




fi

205 | P a g e

echo "****************Done Submitting***************"


cd ../P0

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P1

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P2

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P3

206 | P a g e

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P4

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P5

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P6

readone=true;


do

if [ -s result ]

then

207 | P a g e

cat result

readone=false;

fi

done


cd ../P7

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P8

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P9

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done

208 | P a g e


cd ../P10

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P1

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P12

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P13

readone=true;


209 | P a g e

do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P14

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P15

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P16

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

210 | P a g e

fi

done


cd ../P17

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P18

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done


cd ../P19

readone=true;


do

if [ -s result ]

then

cat result

readone=false;

fi

done

211 | P a g e

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


rm -rf ../pistonData

mkdir ../pistonData

echo "Writing U components to compU"

212 | 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

213 | P a g e

cd stirlingAnalysis

./pistonPlot2

./pistonNet

echo "End pistonPressureOut"

pistonPlot2

#!/usr/bin/ksh

#$ -cwd

#$ -j y


echo "Writing valve_Ubar_time.out"

paste ../pistonData/time.out ../pistonData/valve_Ubar.out >

../pistonData/valve_Ubar_time.out

echo "Writing valveCold_p_time.out"

paste ../pistonData/time.out ../pistonData/valveCold_p.out >

../pistonData/valveCold_p_time.out

echo "Writing valveHot_p_time.out"

paste ../pistonData/time.out ../pistonData/valveHot_p.out >

../pistonData/valveHot_p_time.out

echo "Writing piston_Ubar_time.out"

paste ../pistonData/time.out ../pistonData/piston_Ubar.out >

../pistonData/piston_Ubar_time.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"

214 | P a g e

pistonNet

#!/usr/bin/ksh

#$ -cwd

#$ -j y


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"

215 | P a g e

Appendix F. Solution Dependent Motion Codes

stirlingSDM.m

%

% necessary input constant

%

m = 1; % mass of the pistons

l1 = 1; % length of the crank radius

l2 = 2; % length of the displacer connecting rod

l3 = 3; % length of the power connecting rod

M = 0; % constant moment at bar1

I1 = 3; % inertia of bar1

t_final = 10; % 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];

216 | 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');



subplot(2,1,1),plot(t,X);



ylabel('Displacer position (mm)');

subplot(2,1,2),plot(Theta,X);

xlabel('Theta (rad.)');

217 | P a g e

ylabel('Displacer position (m)');


subplot(2,1,1),plot(t,w);



ylabel('w (rad/s)');

subplot(2,1,2),plot(Theta,w);


ylabel('w (rad/s)');


subplot(2,1,1),plot(t,Y);



ylabel('Power piston position (m)');

subplot(2,1,2),plot(Theta,Y);


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; % length of the crank radius

l2 = 2; % length of the displacer connecting rod

l3 = 3; % length of the power connecting rod

b2 = 0.5*l2;

a2 = b2;

218 | P a g e

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))*l1^2-

Jab3*(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;];

stirling

Documents

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solar radiation

solar concentrator

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team of students

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solar energy power generation

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